Physics of Solid-State Laser Materials

y

0,0,0 a,O,O O,a,O -a,O,O 0,-a,O O,O,a 0,0,-a

0,0,0 a,n/2,0 a, 7!12,7t/2 a,7t/2,7t a,Tt/2,37!12 a, O,O a,1t,O

FIGURE 2.4. Octahedral symmetry.

is Oh , which describes a regular octahedron possessing a center of inversion as shown in Fig. 2.4. This is made up of 4 8, symmetry operations divided into ten classes: E Identity 3C2 Rotation by 1 80° about the x, y, or z axis 6C4 Rotation by ± 90° about the x, y, or z axis 6C� Rotation by 1 80° about the axes parallel to the face diagonals 8C3 Rotation by ± 120° about the body diagonals Inversion z 1s combined with each of the C2 ( ah) , C3 ( S6 ) , C4 ( S4 ), and q ( ad) operations. Note that the only means of obtaining 48 by adding various squares is 4 (3) 2 + 2(2) 2 + 4( 1 ) 2 48. Therefore by Eq. (2.2. 1 0), the Oh group has four three-dimensional irreducible representations, two two-dimensional irredu­ cible representations, and four one-dimensional irreducible representations. The character table for the irreducible representations of the Oh group, including the double-valued representations, is shown in Table 2.4. TABLE 2.4. Character table for the oh point group. oh

A 19 A 29 Eg T1 9 T29 EI /2g E3 f2g Gg

E

i

1 1 2

1 1 2

3 3

3 3

2, -2 2, -2 4, -4

2, -2 2, -2 4 , -4

8 C3

1 1 -I 0 0 1, -1 1, -1 -1, I

8 C3 i

1 1 -I 0 0 1, -1 I, -1 -1, I

3 C2

1 1 2 -1 -1 0, 0 0, 0 0, 0

3 C2 i

1 1 2 -1 -1 0, 0 0, 0 0, 0

6C4

1 -1 0 1 -1 v'2, - v'2 - v'2, v'2 0, 0

6 C4i

1 -1 0 I -1 v'2, - v'2 - v'2, v'2 0, 0

6C�

1

1

0 -I 1 0, 0 0, 0 0, 0

6C;i

1 -1 0 -1 I 0, 0 0, 0 0, 0

2.2. Elements of Group Theory

55

The characters are determined by finding the traces of the transformation matrices for one element in each class using Eq. (2.2.7). The character for the identity element will be the dimension of the irreducible representation, which is a direct consequence of the representation of E being an identity matrix of certain dimension. In This example, only the even-parity repre­ sentations are shown. There is an equal number of odd-parity representa­ tions that have the same characters for each class of elements not involving the inversion operation and 1 times the characters of all classes of ele­ ments involving i. These representations have the same designations as those shown except with a u subscript instead of g. Here u stands for ungerade and g for gerade to designate uneven or even, respectively. The ungerade repre­ sentations are used to define basis functions possessing an odd parity with respect to an inversion operation. The last three representations listed in the character Oh table are double­ valued representations. These are included when the system is characterized by half-integer angular momentum quantum numbers. These are determined by introducing a representation for a spin-! state, £1 ;2 , and deducing the other two representations by reducing the product representations of E1 ; 2 with each of the other single-valued irreducible representations of the group. As a result the identity operator represents a rotation of 4n rather than 2n, so an operator R is introduced to represent a 2n rotation in the group with double-valued representations. This is discussed in greater detail below. One important type of procedure that is done with group character tables is decomposing a product representation into irreducible representations. As an example of doing this using Eq. (2.2.8) and the character table of the 0 group (the 24 operations of the oh group without the inversion operations), consider the direct-product representation resulting from E x T2 : 0

E

Eg T1g T2g Eg x T2g

2 3 3 6

8 C3

3 C2

6C4

6 C�

1 0 0 0

2 -1 -1 -2

0 1 -1 0

0 -1 1 0

= T1g + T2g

This shows that the direct-product representation formed by Eg x T2g is a reducible representation that can be reduced as the sum of two irreducible representations Ti g and T2g· Another important procedure utilizing character tables is reducing the representations of a group in terms of those of a subgroup. A subgroup is made up of a set of operations that are part of a total group and that among themselves obey all of the criteria that define a group. An example of a subgroup of the Oh group is the D3d group, which contains the identity element, two threefold rotation operations, three q elements, and the

56

2. Electronic Energy Levels

inversion operation times each of these. The character table for this group is given below. D3d A 19 A 29 Eg

E

i

2C3

I I

I I

I I I

2

2

2C3 i I I I

3q I I

0

3qi I I

0

The correlation between the irreducible representations of the Oh group and its D3d subgroup is oh

D3d

A 19 A 29 Eg T19 T29

A 19 A 29 Eg A 29 + E9 A 1 9 + E9

Thus the three-dimensional irreducible representations of Oh decompose into the sum of one- and two-dimensional irreducible representation of the D3d subgroup. The character tables of the most important point groups have been com­ piled in numerous books. One very useful source of this information is pro­ vided by a Department of Commerce monograph. 8 One important application of group theory to quantum mechanics is the evaluation of matrix elements. For example, the coupling of angular momentum given in Eq. (2. 1 .2 1 ) can be expressed in terms of irreducible representations as I r , y) =

L l r ! , YJ , rz, Yz) ( r ! , rz, Y i , Yz l r, y ) ,

Y1 , Y2

(2.2.14)

where the final factor in this expression represents the Clebsch-Gordan coefficients. The matrix element of an operator T(ry ) , which transforms as the y row of the r irreducible representation, taken between two state func­ tions that transform as the y ' row of the r ' irreducible representation and the y " row of the r " irreducible representation is expressed as ( rx" , r" , r" I T(r, r ) l rx' , r ' , y ' ) =

:L C( T) ( rx" , r ", r" l a, r , r) ( r, rl r, r' , y , r' ) , f , jl

2.3. Crystal-Field Splitting of Energy Levels

57

where the product function representation T (r , y) J a ' , r' , y') has been expressed in terms of a linear combination of functions J a , r, y) belonging to irreducible representations. C ( T) is a normalization factor for this ex­ pansion. The orthogonality condition for irreducible representations requires that r" , y" f, y for this expression to be nonzero. For this con­ dition, the matrix element depends on a and a ' but not on y . This allows the sum to be evaluated and the matrix element rewritten as

( a ", r", y" J T ( r, y) J a ' , r' , y') 1 - ( a ", r" JJ ( T (r) JJ a ' , r') ( r", y" J r , r' , y , y') , (2.2. 1 5) dr' where dr " is the dimension of the r" irreducible representation and the double bars denote what is referred to as a reduced matrix element. This expression is called the Wigner-Eckart theorem. The final factor in this

expression is the Clebsch-Gordan coefficient. This expression will be useful in evaluating matrix elements in later chapters. 2.3

Crystal-Field Splitting o f Energy Levels

When a free ion is put into a crystal or glass host, it is no longer in an envi­ ronment of completely spherical symmetry. Instead it is surrounded by a set of nearest-neighbor ines (or ligands) in some geometric configuration. In general the active dopant ion goes into the host substitutionally for a specific type of host cation and finds itself surrounded by oxide or fluoride anions. In a crystalline host the anions and cations are arranged on an ordered lattice with a well-defined symmetry. In an amorphous host material such as a glass there is no long-range order, but there is still a short-range order in the neighborhood of a dopant ion. The host ligands and the dopant ions in­ teract with each other through an electrostatic Coulomb interaction. The mathematical treatment . of this interaction is generally called crystal-field theory or ligand-field theory. Using the fundamental concepts of group theory, it is possible to predict qualitatively how free-ion energy levels will split due to a specific type of crystal-field symmetry environment. However, it has proven to be very difficult to predict accurately from first-principle calculations the quantitative magnitude of the splitting. This is due to the very complex nature of the crystal-field interaction problem. In order to un­ derstand the importance of crystal-field effects in solid-state laser materials, it is adequate to treat the problem using a simplified, empirical point-charge model. The dopant ion and the ligands are considered to be point charges arranged in a geometric configuration, and thus the problem becomes one of the Stark effect with a specific symmetry for the physical system. The results are expressed in terms of crystal-field parameters whose magni­ tude can be found by fitting the theoretical expressions to experimental data.

58

2. Electronic Energy Levels

Thus in this approach, any effects of covalency, ionic size, the shape of the electron orbits, or higher-order interactions are taken into account only through adjusting the magnitude of the parameters. The Hamiltonian of an ion in a crystal field is given by4 • 9 H

( !!_2m Li vf - Li r,

Ho + He + Hso + He .

+ L �;L s + L e Ve( r;, B;, (/);) i J> i ry i (2.3. 1 )

+L

0

The first term describing the interaction of each electron with the nucleus is the unperturbed Hamiltonian. The remaining three terms are treated as suc­ cessive perturbations describing, respectively, the Coulomb interaction of the electrons with each other, the spin-orbit interactions of the electrons, and the interaction of the electrons with the crystal field of the ligands. As is the case with all perturbation theory problems, the order in which these pertur­ bations are considered is not critical, but the best results are obtained most easily if the perturbations are treated in order of descending magnitude. There are three different cases of interest to understanding laser materials: 1 . Weak crystal field (He < Hso, He) · The magnitude of the crystal field is small compared to both the Coulomb interaction of the electrons with each other and the spin-orbit coupling interaction. This is generally true for rare-earth ions since the outer-shell electrons shield the optically active 4f electrons from the effects of the crystal field. In this case the eigenfunctions of the unperturbed system are taken to be the free-ion multiplets described by the quantum numbers J and MJ. In the crystal, the complete set of com­ muting operators becomes He , P, r, and rz, where the latter two designate the crystal-field symmetry representations as discussed below. The crystal field is a perturbation that causes a Stark splitting of the free-ion multiplets. 2. Medium crystal field (Hso < He < He) · The magnitude of the crystal field is small compared to the electrostatic interaction of the electrons with each other but large in comparison with the spin-orbit interaction. This is generally true for first-row transition-metal ions where the optically active electrons are unshielded 3d electrons. In this case the eigenfunctions of the unperturbed system are the LS terms of the free ion described by the quan­ tum numbers L , S, ML, and Ms. In the crystal, the LS coupling breaks down and the complete set of commuting operators is He, P, r, rz, S2 , and Sz. The crystal field is a perturbation that splits the free-ion terms. After this splitting has been determined, the spin-orbit interaction is treated as an additional perturbation to determine further splitting of the energy levels resulting in a system having a complete set of commuting operators Hso, P, r, rz' ·

3. Strong crystal field (Hso, He < He) · The magnitude of the crystal field is large compared to both the Coulomb interaction of the electrons with each other and the spin-orbit coupling interaction. This may be true for

2.3. Crystal-Field Splitting of Energy Levels

59

second- and third-row transition-metal ions with unshielded 4d or 5d opti­ cally active electrons or with transition-metal chemical complexes. In this case the eigenfunctions of the unperturbed system are the single-electron wave functions of the optically active ions described by the quantum numbers I , m1 , s, and ms . The crystal-field perturbation splits these single-electron energy levels. In the crystal the complete set of commuting operators is He, P, y, and Yz , where the crystal-field symmetry representations refer to single-electron states. After the crystal-field splitting has been determined, multielectron crystal-field terms are constructed by taking into account the electrostatic interaction between the electrons. Finally, the spin-orbit inter­ action is taken as the final perturbation to determine the crystal-field mul­ tiplets, resulting in a system with a complete set of commuting operators Hso, P, r, and rz. It is more difficult to calculate energy levels using the intermediate crystal­ field scheme than using either the strong- or weak-field scheme. Therefore, it is common for first-row transition-metal ions to use either the weak-field or the strong-field approach. Also it is difficult to work with crystal-field oper­ ators of low symmetry. Therefore it is common to treat the first-order crystal-field perturbation as if the system had octahedral symmetry, and then treat lower-symmetry cases as higher-order contributions to the crystal-field perturbation. These approximations have been very successful in treating solid-state laser materials. Since the electronic wave functions have been expressed in terms of spherical harmonic functions, it is convenient to expand Vc in terms of spherical harmonics,

Vc L L L A lm Yt(Bj , rpj )Ri(rj ) , j I m

(2.3.2)

where the subscript j designates the optically active electrons and the A 1m are expansion coefficients that depend on the specific lattice structure giving rise to the crystal field. Expressions for the spherical harmonic functions in both spherical and Cartesian coordinates are given in Table 2.5. The crystal-field operator must transform as the totally symmetric representation of the ion­ ligand cluster in order for the total Hamiltonian to remain invariant under all symmetry operations of the system. The term of greatest magnitude in the expansion is the spherically symmetric term with I 0. From Table 2.5, this can be written as

Vo

L j

1 Ro(r ) · j

(2.3.3)

This term adds the same constant energy to all the unperturbed eigenvalues, and thus shifts all the energy levels uniformly without lifting any degeneracy. Next consider all odd-parity terms in the expansion, I 1 , 3, 5 , . . . . In the perturbation calculations for the effects of the crystal field, these terms

60

2. Electronic Energy Levels TABLE 2.5. Expressions for selected spherical harmonic functions.

ff4n � z = �410 cos B Yjl = �8n r iy = �(8n sm. B) + i� y1+ 1 = + y'£ 3z2 - ? {£1 610 (3 cos2B - 1 ) Yf = vg8n z(x r2 iy) = + vg(8n sm B) (cos B) + i� y2+ 1 = + )2 y2± 2 = � ( x 2 iy = �( sm2 B) ± 2i� r 32n 32n [;f z(5z2 -3 3?) = [;f1 610 (5 cos3 B - 3 cos B) Yf = , !if (x iy) ( 5z2 - ?) - !if( sm B S cos 2B ± i� 1 ± y3 ) )( + 6410 6410 ,3 z ) 2 �( 2 B sm ) (cos B) ± 2i� y3± 2 � (x 3 iy r 32n 32n J ( x iy ) 3 sm3B) e ± 3i� y3± +� + �( 64n r3 64n � 35z4 - 30z2? + 3r4 = � ( 35 cos4 B 30 cos2 B + 3) Yf = 25610 -/if ( x iy) ( 7z3 - 3z?) - /if ( sm B 7 cos3 B cos B ± i� Y± 1 + ) )( + 6410 4 6410 ) 2 ( 7z2 - ?) = IJf 2 B 7 2B ( sm ( cos - 1 ) ± 2i� y4± 2 = IJf x iy 128n 128n z( )3 . 3B) (cos B) ± 3i� sm y4± 3 + � x iy + �( 6410 6410 )4 ( y ± 4 = IJ!f x iy = !J!f( sm4 B) ± 4i� 4 5 12n 512n Y8

=

�

+

X±

±

-

e

-

--

-

±

_

e

.

+

e

.

±

±

±

-

e

.

r4

±

r4

.

r4

±

±

e

.

±

e

.

e

.

r4

e

r4

.

e

will be used in the matrix elements ( 1/1 1 1 Vodd l t/1; ) , where the 'I'z are the unperturbed electronic wave functions. For the transitions most important in solid-state laser applications, the initial and final wave functions of the transition matrix elements have the same parity (d-d transitions for tran­ sition metal ions or f-f transitions for rare earth ions). Therefore the odd­ parity terms in the crystal field expansion will result in matrix element in­ tegrals with odd-parity integrands. These integrals taken over symmetric limits are identically zero. Thus only the even-parity terms need to be re­ tained in the crystal-field expansion. ( Note that there are exceptions to this

2.3. Crystal-Field Splitting of Energy Levels

61

rule involving d-to-f transitions that will be discussed separately.) In addi­ tion, the orthogonality conditions of the spherical harmonic functions can be used to further limit the number of terms that need to be retained in the expansion of the crystal-field operator. For d electrons with l = 2, the prod­ uct of two wave functions that appears in the perturbation matrix element will result in spherical harmonic functions with I :S 4. Thus the orthogon­ ality conditions dictates that the integral appearing in the matrix element for the crystal-field perturbation will be zero for terms in the crystal-field expansion involving spherical harmonic functions with I > 4. For f elec­ trons with I = 3, the orthogonality condition limits the relevant terms in the crystal-field expansion to those with I :S 6. To further determine the form of the crystal-field Hamiltonian, it is useful to see how symmetry operations affect spherical harmonic functions. Re­ calling that Yt (O; , rp; ) oc Pi (cos O) eim rp , for a system that is quantized in the direction of the major symmetry axis, it is clear that a rotation of rx. about this symmetry axis changes the factor exp(im rp) to exp[im(rp + rx.)] . Thus for a basis function consisting of a linear combination of wave functions having the 21 + 1 different values of the m quantum number,

eil ei (l l ) rp

(

�

r 1

(2.3.4)

'

e zlrp

the matrix element for a rotation operating on this function is given by R=

eila

ei(l ! ) a

(2.3.5)

0 The trace of this matrix is the character of the rx. rotation operator, Tr(R) = x(rx.) = eila + ei(l ! ) a + . . . + e ila

!) rx]

(2.3.6)

sm(! rx) So for the most important symmetry rotations, the characters of the matrix representations are

.

for rx. = n ,

I = 0, 1 , 4 , 5 , . . for rx. = ln I = 2, 3, 6 , 7 , . . . I = 1 , 4, . . . I = 0, 3, 6 , . . . I = 2, 5 , . . .

2n

for rx. = 3 .

62

2. Electronic Energy Levels

If the spin-orbit interaction has been taken into account before treating the effects of the crystal field, the basis function set in Eq. (2.3.4) and the matrix elements of the rotation operator representation in Eq. (2.3.5), have the same form with orbital angular momentum quantum number I replaced by the total angular momentum quantum number j. The expression for the character of the representation matrix becomes

(2.3.7)

X ( a)

One important difference between I and j is that the former has only integer values while the latter can have half-integer values. For integer values of j, x(a + 2n) x(a), so a rotation by 2n is the identity operation as before. However, for half-integer values of j, x(a + 2n) x(a), so a rotation by 2n is a symmetry operation designated by R, but it is not the identity. For this case x(E) 2j + 1 and x(R) - (2j + 1 ) . The number of operations in the group are expanded by taking the product of the operator R with all of the other group operations. The resulting representations are called double­ valued representations. Half-integer values for j occur when an ion has an odd number of electrons. The resulting double-valued representations are a manifestation of Kramer's degeneracy (time-reversal degeneracy). To carry out crystal-field calculations in detail, a specific symmetry must be assumed. Consider the case of octahedral coordination of six ligands of charge Ze around a central ion as shown in Fig. 2.4. Treating all ions as point charges, the electrostatic field at the central ion due to the ligarids is given by 6 Ze Vo =

(2.3.8) i

1 1 1 1=0 m= 1 21 + l a +

;,

1

J'

'

where the standard multipole expansion in Eq. (2. 1 .28) has been used for I r1 - r; 1 1 . Here r1 designates a point in space near the central ion and r; is the position of the ith ligand. The B; , ({J; coordinates of the ligands are given in the figure, and I r; I a > I r1 I · A one-to-one correspondence of the terms in Eq. (2.3.8) with the general form of the crystal-field expansion in Eq. (2.3.2) requires that

Rl (r)

rl 1 a +1 '

(2.3.9)

2.3. Crystal-Field Splitting of Energy Levels

63

As an example, consider the case of a d electron where it is necessary to consider only crystal-field functions with I = 2 and I = 4 as discussed above. To begin with, group theory can be used to determine qualitatively what functions must be included in the crystal-field expansion without considering the radial and expansion factors associated with the exact positions of the ligands. For even-parity d-electron wave functions, only the five classes of operations of the 0 symmetry group the inversion operator need to be con­ sidered instead of the complete Oh group with ten classes. Using the expres­ sions in Eq. (2.3.6) gives th character table

1

0

1 1

= E + Tz = A 1 + E + T1 + Tz

where the r2 and r4 are the reducible representations for I = 2 and I = 4, respectively. The final column of the table gives the reduction for these rep­ resentations in terms of the irreducible representations of the octahedral symmetry group. As an example of this reduction, Eq. (2.2.8) was used with the r4 representation to give n(AI ) = -d4 (9 X 1 X 1 + 0 X 8 X 1 + 1 X 3 X 1 + 1 X 6 X I + I X 6 X 1 ) = I , n(A 2 ) = -d4 (9 x 1 x I + 0 x 8 x 1 + I x 3 x I I x 6 x I I x 6 x 1 ) = 0 ,

n(E) = -i4 (9 X 1 X 2 0 X 8 X 1 + 1 X 3 X 2 + I X 6 X 0 + 1 X 6 X 0) = 1 , n ( TI ) = -d4 ( 9 x 1 x 3 + 0 x 8 x 0 I x 3 x 1 + I x 6 x 1 I x 6 x I ) = 1 , n(T2 ) = -d4 (9 x 1 x 3 + 0 x 8 x 0 1 x 3 x 1 I x 6 x I + I x 6 x I ) = 1 .

This analysis shows that a linear combination of the Y4m functions trans­ forms as the totally symmetric A 1 representation in octahedral symmetry as required for the total Hamiltonian to remain invariant under symmetry transformations of the system. There is no linear combination of the Yf functions that satisfies this requirement for this symmetry group. Thus Vo is represented only by Y4m functions. The exact combination of the Y4' func­ tions to be used depends on the choice of the quantization axis. If the z axis is chosen, then the fourfold rotation symmetry operation gives

Using the Cartesian coordinate form of the Y4m spherical harmonic func-

64

2. Electronic Energy Levels

tions given in Table 2.4, the result of a n/2 rotation about the z axis is y44 y43 y42 y4! y4o y4- 1 y4-2 y4- 3 y4-4

y44 · l y43 - Y42

- i YJ y4o

iY4 t

- Y4-2 - l· y - 3 4 y4-4

Since functions that are part of the crystal-field expansion must remain unchanged by a symmetry operation, the expansion coefficients must be zero except for m 0, + 4, and 4. Thus from symmetry considerations alone, (2.3.10) The final two expansion coefficients d and e can be obtained in a similar way by applying q and C3 symmetry operations to the crystal-field operator in Eq. (2.3.9) and requiring the resulting function to be identical with the initial function. This gives the final form for the expression of the operator describing the electrostatic interaction of a d electron in an octahedral crys­ tal field, Vo

Y40 + 14( Y44 + y4 4 ) 4 4 = x + / + z - � r4 . __

(2.3. 1 1 )

The appropriate factors of R1 and A1m from Eq. (2.3.9) must be included in the complete expression for the crystal field. An example of this is given below. When the symmetry of the crystal field is known, the qualitative splitting of the free-ion energy levels can be determined from group theory without using quantum-mechanical perturbation calculations. This is done by form­ ing the reducible representation of the angular momentum operator of the free-ion eigenfunction and reducing this opet:ator in terms of the irreducible representations of the crystal-field group. The effect of the crystal-field envi­ ronment on the electrons of the active ion is to inhibit their abilit to move freely with the spatial orientations and shapes of their free-ion spherically symmetric orbits. This effect is referred to as quenching of orbital angular momentum. The result of this spatial restriction is that orbital angular momentum (or total angular momentum if the spin-orbit interaction has already been taken into account) is no longer a good quantum number. Since the crystal field determines the spatial orbits of the electrons, I or j is replaced as a quantum number by the designation of the irreducible repre-

2.3. Crystal-Field Splitting of Energy Levels

65

sentation of the symmetry group of the crystal field. The spin multiplicity remains unchanged since the crystal-field operator depends only on spatial coordinates and not on spin coordinates. As an example of using group theory to determine the splitting of free-ion energy levels in a crystal field, consider the case of a weak field with octahe­ dral symmetry acting on a free ion with an angular momentum quantum nubmer of 5. This applies to either I = 5 for a Russell-Saunders term such as 3 H, or j = 5 for a multiplet such as 3 G5 . For a free-ion eigenfunction, the reducible representation in 0 symmetry is given in the following table. 0

3 rs

E

8C3

3 Cz

6C4

6q

11

-1

-1

1

-1

= 3£ + z3 Tl + 3 Tz

Here Eq. (2.3.6) has been used to determine the characters of the reducible representation and Eq. (2 2 . 8 ) has been used to determine the number of times the irreducible representations of the 0 symmetry group appear in 3 r5 . Note that the electron spin state and thus the multiplicity (2S + 1 = 3, therefore S = 1) is not changed in going from free-ion states to crystal field states. This example shows that, under the influence of an octahedral crystal field, and 1 1 -fold spatially degenerate free-ion energy level of the types con­ sidered here will split into one doubly degenerate level, two triply degenerate levels with T1 symmetry, and one triply degenerate level with T2 symmetry. The total number of levels is the same, but the degeneracy has been partially lifted. The simplicity of this technique is very powerful when only · the num­ ber and symmetry types of energy levels are required and not the magni­ tudes of the splittings. Specific examples of the application of this technique to laser materials are given in Chaps. 6 and 8. If the actual crystal field of interest has a lower symmetry such as D3 d, it is still best to approach the problem by considering the splitting of the free­ ion energy levels in an octahedral field and then determine the further split­ ting of these levels in a field of D3d symmetry. For the above example, this requires determining how the 3 E, 3 T1 , and 3 T2 representations of the 0 group reduce in terms of the irreducible representations of the D3 d group. The results of this reduction are found from the group correlation table given in the preceding section: .

3 E --+ 3£,

3 TI --+ �2 + 3£,

3 T2 --+ �� + 3£ .

The results of this procedure given one nondegenerate � 1 level, two non­ degenerate � 2 levels, and four doubly degenerate 3£ levels. Again, the spin states and total number of levels are unchanged by the crystal field, but the amount of degeneracy of some of the levels is decreased. Thus the lower­ symmetry environment serves to split (or "lift") the degeneracy of some of the free-ion levels.

66

2. Electronic Energy Levels

When it is important to calculate the magnitude of the splittings, group theory is helpful but it is not enough. The quantum-mechanical perturbation matrix elements must be evaluated. For the case of transition-metal ions, it is best to attack this problem using the strong-field approach. The initial question then is, "What is the splitting of a free-ion d-electron energy level in an octahedral crystal field?" The first step is to apply the group-theory technique to the case of a single-electron orbital with angular momentum quantum number I = 2. This is done in the following table. 0

2 rz

E

8C3

5

I

3 Cz

6q

6C4 I

I

I

zE + zTz

Here Eqs. (2.3.6) and (2.2.8) have been used as usual. The results show that a fivefold orbitally degenerate free-ion level splits into one doubly degener­ ate eg and one triply degenerate t2g crystal-field level. This splitting is shown schematically in Fig. 2.5 where lower-case letters e and t are used for single­ electron levels to distinguish them from multielectron levels designated by capital letters. The spin multiplicity has been dropped since it is not altered by the effects of the crystal field. The g subscript can be added to specifically designate that the d-electron states have even parity. The energy ordering of the levels and the magnitude of the splitting designated as 1 ODq are both discussed below. The tzg crystal-field level can hold six electrons distributed in three degen­ erate orbitals with spin up or down, while the eg crystal field can hold four electrons distributed in two degenerate orbitals with spin up or down. Note that this accounts for all of the possible ten electrons in a filled d shell. The


e

t

� 6

llG H

HELD

e

t

e t

��

� LOW FIFLD

7

LOW FIFLD

e t

5

HIGH FIFLD

e t

e t

FIGURE 2.5. Ground-state configurations for d electrons in an octahedral crystal field.

2.3. Crystal-Field Splitting of Energy Levels

67

ground-state arrangement of the electrons will always reflect the minimum energy configuration as determined by the crystal-field and spin-orbit inter­ actions. The spin-orbit interaction energy is minimized for the maximum number of aligned spins ( Hund's rule) while the crystal-field interaction energy is minimized for the maximum number of electrons in the lower­ energy crystal-field level. Of course the Pauli exclusion principle must al­ ways be obeyed. Thus it is seen in Fig. 2.5 that a single-electron ion in the ground state will have one electron in one of the orbitals of the t29 level with the spin in a specific direction, giving a doublet term. As additional electrons are added for multielectron ions, these minimum energy considerations dic­ tate that the next two electrons will occupy the other two orbitals of the tz9 level with the spins of all electrons aligned in the same direction. The ground state of an ion with a d 2 configuration will be a triplet (2S + 1 3) and the ground state of an ion with a d 3 configuration will be a quartet. As a fourth electron is added, there is a choice as to whether it goes into a tz9 orbital al­ ready occupied by one electron with opposite spin, or into an unoccupied e9 orbital with spin aligned with the other three electrons that are present. The first situation will occur if the crystal-field interaction energy is larger than that of the spin-orbit interaction. In this case the ground-state configuration will be ti9 , resulting in a triplet term. For the opposite case the ground-state configuration is ti9 e9, resulting in a quintet term. For an ion with five elec­ trons, the choice of ground state will be a sextet term from a ti9 e� config­ uration with strong spin-orbit coupling, or a doublet term from a t5.9 con­ figuration with a strong crystal field. Adding a sixth electron will completely fill the tz9 orbitals for a strong crystal-field case. Thus configurations d 7 through d 1 0 have only one choice of ground state as shown in Fig. 2.5. Electronic transitions involve a change in orbitals of one of the single electrons in the configurations. As shown in Fig. 2.6, this can involve a spin flip or a change in the spatial orbital. The properties of these two types of transitions are discussed below. The nature of the terms of a multielectron ion in a crystal field can be determined from the group-theory analysis of the direct product of the single-electron crystal-field orbital representations. For example, for two

..

(A) S P I N FLIP TRANSITION

'

'

-

, "'

(B) CON FIGU RATION TRANSITION

FIGURE 2.6. Types of transitions for d electrons in an octahedral crystal field.

68

2. Electronic Energy Levels

electrons in t2g orbitals of an octahedral crystal field, the reduction of the direct-product representation is as follows. 0

r9

=

t2g

x

t2g

E

8C3

J C2

6C4

6q

9

0

I

I

I

=

A 1 g + Eg + T1 g + T2g

Equation (2.2.8) was used to determine the reduction of the direct product representation r9 in terms of the irreducible representations of the 0 point group. If a third electron is added to the ion, the terms are found by taking the direct product of the additional single-electron orbital representation (either t2g or eg) with the representations found above (A 1g, Eg, T1g, and T2g), and then reducing the direct-product representation in terms of the irreducible representations of the 0 group. In order to determine all possible terms, this procedure must be followed for all combinations of single-electron configurations. For the two-electron example, t2g eg T1g + T2g, and e� A1g + A 2g + Eg. Not all of these terms will be available for all possible spin states since the Pauli exclusion princi­ ple must be obeyed and the wave functions must be antisymmetric. The complete method for determining the allowed terms is to couple the single­ electron wave functions including spin and evaluate the Clebsch-Gordon coefficients. This is a complicated process and many times it is possible to ascertain the results through a statistical analysis of the degeneracies of the states, and in addition, the correlation of these results with the results of a weak-field analysis. An example of the weak-field analysis of the nd2 elec­ tron configuration is given below. The total degeneracy for distributing n electrons in m possible ways in a given configuration is given by

m Cn = m! [n!(m - n) !r 1 •

(2.3.12)

For the t�g configuration, there are six ways to put an electron in the triply spatially degenerate orbit with spin up or down. The total number of ways to arrange two electrons in this configuration is 6 C2 = 6 !(2!4!) 1 1 5. This total degeneracy can be obtained by several combinations of the four possi­ ble crystal-field representations distributed as singlets or triplets. One exam­ ple that is consistent with the terms available from a weak-field analysis of a two-electron ion is J,t l g, 1 Eg, 3 T1g, and 1 T2g· For the e� configuration, there are four ways to put an electron in the doubly spatially degenerate orbit with spin up or down. The total number of ways to arrange two electrons in this configuration is 4 C2 4!(2!2!)- 1 6. This total degeneracy can be obtained by several combinations of the three possible crystal-field repre­ sentations distributed as singlets or triplets. One example consistent with the terms available from a weak-field analysis is l,41g + � 2g + 1Eg. The final =

2.3. Crystal-Field Splitting of Energy Levels

FREE ION TERMS

WEAK FIELD TERMS

STRONG FIELD TERMS

69

SINGLE ELECTRON CONFIGURATIONS

FIGURE 2.7. Correlation between strong-field and weak-field states of an ion with an

nd 2 electron configuration.

possible configuration is t29e9, which involves the independent placement of one electron in each type of spatial orbital. The total number of ways of doing this is 6 C1 4 C1 = 6! ( 1 !5! ) - 1 x 4! ( 1 !3 ! ) - 1 = 24. One example consistent with the terms available from a weak-field analysis is 1 T1 9 + 1 T29 + 3 T19 + 3 T29 • These are shown as strong field terms in Fig. 2.7. In order to demonstrate how the correct choice of multiplets is made, consider the case of an nd2 electron configuration in a weak octabhedral crystal field. The free-ion terms are found from considering all possible combinations of the (mn msi ) (ml2msz) quantum numbers of the two elec­ trons consistent with the Pauli exclusion principle. Since I = 2 for each d electron, the values of mn and m12 are ± 2 , ± 1 , 0. Thus the ML values of the total orbital angular momentum run from 4 to 4 in integer steps. Sim­ ilarly, the values of ms for each electron are ± ! so the Ms values of the total spin angular momentum are ± 1 , 0. Since the largest value of ML equals the total orbital angular momentum quantum number L, and the largest value of Ms equals the total spin angular momentum quantum number S, the terms with greatest orbital and spin angular momentum can be identified. The term with greatest orbital angular momentum is 1 G. Note that the Pauli exclusion principle forbids a triplet state with L = 4. There­ fore the triplet state with greatest orbital angular momentum is 3 F. When the (mn ms ! ; m tzmsz) substates associated with these terms are eliminated

70

2. Electronic Energy Levels

from the total, the substate that is left having the greatest orbital angular momentum has L = 2, S = 0. This is associ ted with a 1 D term. Subtracting these substates from the remaining total leaves an L = 1 , S 1 substate that is part of a 3P term. The final remaining substate has L = 0 and S 0, which implies a 1 S term. These are shown as free-ion terms in Fig. 2.7. Next the splitting of these terms in a field with octahedral symmetry must be determined. The group-theory representations for each of these orbital angular momentum terms is found from charcters given by Eq. (2.3.7). Note that this is independent of spin multiplicity in LS coupling. Using Eq. (2.2.8) and Table 2.4, these reducible representations can be reduced in terms of the irreducible representations of the octahedral point group as shown in the last column of the table below. These give the terms for weak­ crystal-field splitting as shown in Fig. 2. 7. 0

E

8 C3

3 C2

6C4

6q

ro C s) r 1 eP) r2 C D) r3 eF) r4 C G)

1 3 5 7 9

1 0 1 I 0

1 1 1 1 1

1 I I I 1

1 1 1 1 1

I,.t g 3 T1l g lEg + l T2g "A 2g + 3 T1 g + 3 T2g l,.t l g + 1Eg + 1 T1 g + 1 T2g

Thus there is an exact one-to-one correspondence with the terms obtained by the strong-field analysis of an ion with two d electrons and those obtained by the weak-field analysis as shown in Fig. 2.7. This allows for the choice of the correct combination of terms in the strong-field analysis. There are two ways in which a multielectron ion can make a transition to an excited state. The first is by flipping its spin without changing its crystal­ field orbital. The second is by changing from one of the t29 crystal field orbitals to one of the e9 orbitals. It is obvious from Fig. 2. 7 that spin-flip transitions are independent of the strength of the crystal field, while chang­ ing from a t29 to an e9 orbital is highly sensitive to the magnitude of 10Dq . The different spectral characteristics of these two types of transitions play a very important role in laser materials and are discussed in detail in the fol­ lowing chapters. An example of the exact treatment of coupling the single-electron orbitals to form multielectron terms can be demonstrated by using the 3d wave functions having the form given in Eq. (2. 1 .27), with five values of m1 having I m1 l � 2. It is convenient to work with the linear combinations of these functions that yield real functions. This is

2.3. Crystal-Field Splitting of Energy Levels

71

achieved through a unitary transformation using 1 v'2 v'2

1

U

v'2 v'2 0 0

0 0

i

With the radial parts factored out, the single-electron wave functions are now given by

t/13, 2,m1 =

1

[t/1 3, 2,m1 + ( - 1 ) ml t/J3, 2, m/ R 2 ' 3, , (r) 2 ] r 3, 2, (r) ·

These can be written in Cartesian coordinates as

t/1 3 ' 2' 2 t/1 3 ' 2' 0 -

t/1 3, 2, 2

�(x2 1 If 2 (3 z 3

v'2

3

6 n

y2 )

dx2 -y2

v,

4 n

"fur.xy

�XZ 3 � yz

3 t/J 3 ' 2' I =

6

t/1 3 , 2' 1

6

n

n

dxy ( , dzx = 'I , dyz �-

(2.3 . 1 3)

Now functions v and u serve as basis functions for the E representation of the octahedral symmetry group while (, "' and ¢ serve as basis functions for the T2 representation. The symmetry operations of Oh transform these func­ tions into linear combinations of themselves within the two sets. As an example, consider two electrons both in t2g orbitals. As discussed above, all possible spin and orbital combinations lead to a total degeneracy of 15. This gives the equation aA 1 g + bEg + c T1 g + dT2g 1 5 with a , b , c, and d equal to 1 or 3. This equation has three solutions result­ ing in the following sets of terms:

� l g + lEg + I Ti g + 3 T2g, 1,4 1 g + lEg + 3 Ti g + I T2g, � l g + 3Eg + I Ti g + I T2g ·

72

2. Electronic Energy Levels

Since A,9 is symmetric for the orbital part of the wave function, the spin part must be antisymmetric to preserve the overall antisymmetry of the electron wave function. This eliminates the third combination because the spin multiplicity is 2S + 1 = 3, which indicates that the two electrons have heir spins aligned to give S = 1 . This is a symmetric spin function. To check the first two combinations, the crystal-field term wave functions can be written in antisymmetrized form. The ZS+ l r term wave function is desig­ nated by the irreducible representation r, one of the basis functions of r designated as y , the total spin quantum number S, and the quantum number representing the z component of the spin angular momentum M. Matrix elements between states with different S , r are zero. The restrictions on the term wave functions are that they must be eigenfunctions of both S2 and Sz, they must be antisymmetric with respect to electron exchanges, and an operation of the oh group on I SM ry ) will transform this function into a linear combination of functions I SMry ' ) , where y ' represents other basis functions for r belonging to the same set as y . Thus

I SMry) = L I IP (tzm! y 1 ) tp (tzmz y2 ) I (! m d mz i SM ) ( tz y 1 tz y2 j ry ) , m1 , m2 Yt , r2

where I tp 1 tp2 I is a Slater determinant with the one-electron wave functions given by

tp (tzm; y; )

tp (tz y; )x (! m ;) ,

with y being one of the basis functions of the irreducible representation r . There are 1 5 combinations of (, 17, � orbitals with spin up or spin down for the Slater determinants of one-electron orbitals. The factor (! m , ! mz l SM) in the expression for the term wave function is a spin angular momentum coupling coefficient. Values for this factor can be found in tables such as Table 2. 1 above. The final factor in the expression for the term wave function represents orbital angular momentum coupling in a cubic symmetry environment. This term is equivalent to the Clebsch­ Gordan coefficients for the free ion discussed in Sec. 2. 1 . The general ex­ pression for this coupling coefficient is ( r 1 y 1 r2 y2 j ry ) . These satisfy the fol­ lowing relationships:

L (ry j r , Y ! rzyz) ( r , Y ! rzYz l r' y' ) = J(rr')J(yy' ) , L( r , y 1 rz y2 j ry) (ry l r , y; rzy� ) = J( y 1 y; )J( y2 y� ) , r ( ry j r , y; rz y� ) = ( r , y, rzyz j ry) * , y

and

L Dy; rz (r i ) (R) n �;; (R) ( r , y; rzy� j ry' ) = L( r , y, rz y2 l ry) D}�l (R) , r; rz y

73

2.3. Crystal-Field Splitting of Energy Levels

where D��) (R) is a matrix operator for the R symmetry element in Oh . By applying the Oh matrix operators to the single-electron and term wave functions, and applying the restrictions shown above, the cubic Clebsch­ Gordan coefficients can be calculated. This procedure has been done for all d-electron configurations1 0 and the results tabulated. Table 2.6 lists the results important for these considerations. The phase conventions are

(r1 y 1 r1y2 j ry) = ± (r 1y2 r 1 y 1 j ry) , r1 = r2 where the minus sign is for the conditions r = A 2 r1 = E, r

=

T1 r 1

=

T1 ,

TABLE 2.6. Clebsch Gordan coefficients with cubic basis (Reprinted with permission from Ref. 10.)

YI

Az x Az r: A I Yz y : ei

ez

ez

Az x E r:

-1

A z X TI r: YI

Yz

ez

()(

Yz

u

u

v

u

u

v

y:

ez 1 / V'i.

rt

p rt

p y

1 / V'i.

y:

0 0

!

v'3/2 0 0

'

YI

0 0

ez

0

0 0

I 2

u

0

- 1 / V'i.

- v!J/2 0

0 0 I 0 0 0

v

0

-1 0

Yz

TI

()(

y:

.;

-1

p

y

0

0 0

-1

0 0

0

-1

E v

0 0

1 / V'i.

0

p

u

A z x Tz r:

ez

E x T1 T1

rt

v

'

1 / V'i. - 1 / V'i.

0 0

r:

u

y:

'1

r:

v

ez

0

ExE AI Az

v

Yl Y2

1

0 0

y

YI

'1 0

p

Yz

Tz

.;

y:

YI

E

c;

v'3/ 2 0 0 ! 0 0

0

1 / V'i. 1 /V'i. 0

T2

'7

0

v!J/2 0 0 I 2

0

'

0 0 0 0 0 I

Yl Y2

u

v

c;

'7 ' c;

'7 '

r: y:

rt

v!J/2 0 0

0 0

I 2

E X T2 T1 p

0

v'3/2 0 0

0

I 2

0 0 0 0 0

c;

0 0

T2 I 2

v!J/2 0 0

'7 0

I -2

0 0

v!J/2 0

'

0 0 I 0 0 0

74

2. Electronic Energy Levels

Table 2.6 (Cont.)

l'l a

p )'

l'l a

p )'

Yz

y:

el

a

)'

p )'

a

p )'

Yz

[':

A1

y:

e1

,;

( ,;

11 ( ,;

11 (

11

,;

( ,;

11 ( ,;

11 (

v

u

X

Tl

a

T1

E u

v

- I /J3 - 1 /v'2 - 1 /v'6 0 0 0 0 0 0 0 0 0 - I /J3 1 /v'2 - 1 /v'6 0 0 0 0 0 0 0 0 0 - I /J3 2/ v'6 0

11

,;

Tl

E

T1 p

Tz )'

,;

11

(

- I /J3 I v'6 - 1 /v'2 0 0 0 0 0 0 0 0 0 0 0 0 0 - 1/v'2 - 1 /v'2 0 0 0 0 1 /v'2 0 0 0 - 1/v'2 0 0 0 0 0 0 0 - 1 /v'2 1 /v'2 - I /J3 1 /v'6 1 /v'2 0 0 0 0 0 0 - 1 /v'2 0 0 0 0 0 0 0 - 1 /v'2 0 0 0 0 0 0 0 - 1 /v'2 - 1 /v'2 0 0 0 0 0 0 0 1 /v'2 - 1 /v'2 0 - I /J3 - 2 /v'6 0 0 0 0 0 0

a

Yz

(

A1

p

l'l

11

1:

1:

A1

y:

el I /J3 0 0 0 I /J3 0 0 0 I /J3

u

v

Tz

T1

a

p

0 0 0 0 0 1 /v'2 0 1 /v'2 0

0 0 1 /v'2 0 0 0 1 /v'2 0 0

Tz

E

x

x a

Tz

Tz )'

0 1 /v'2 0 1 /v'2 0 0 0 0 0

,;

(

0 0 0 0 0 - 1 /v'2 0 0 1 /v'2 0 0 1 /v'2 0 0 0 0 0 - 1 /v'2 0 0 - 1 /v'2 0 0 1 /v'2 0 0 0

T1 p

11

Tz )'

0 0 0 - 1 /v'6 1 /v'2 0 0 0 0 1 /v'2 0 0 0 0 - 1 /v'2 0 0 0 0 - 1 /v'2 - 1 /v'6 - 1 /v'2 0 0 0 0 0 0 0 1 /v'2 0 0 0 0 1 /v'2 0 0 0 0 - 1 /v'2 0 0 0 0 2 /v'6

,;

0 0 0 0 0 1 /v'2 0 1 /v'2 0

11 0 0 1 /v'2 0 0 0 1 /v'2 0 0

(

0 1 /v'2 0 1 /v'2 0 0 0 0 0

2.3. Crystal-Field Splitting of Energy Levels

75

T2 and the positive sign is for all others, and (r2 y2 r , y, l ry) (r , y, r2 y2 J ry) , r , =f. r2 . For the case of two electrons in t2 orbitals, the concepts described above can be applied with S = 0, 1 and r A 1 , E, T1 , T2 . To check the first set of possible terms, take as an example singlet T1 g term (S 0) and its specific component y . There are six states for the elec­ trons, � + ' c ' Yf + ' Yf ' c+' and c ' where the and superscripts denote and r = T, r ,

=

=

=

+

=

spin up and spin down, respectively. The Slater determinant is

I

tJit ( r , a , ) tJit ( r2 a2 ) '

+

with m, m2 term is then

=

- 2 . The expression for the crystal-field I �+ C I ( ! cx ! PI OO ) ( t2�+t2 C I T, y) + 14 other possible combinations.

M = 0, so

I OOT, y)

=

m,

=

m

The second factor on the right-hand side of this equation involves the spin angular momentum coupling and can be found from the Clebsch-Gordon coefficients in Table 2. 1 . The third factor on the right involves the Clebsch­ Gordon coefficients in the cubic basis and can be found in Table 2.6. For this case, the Clebsch-Gordon coefficients found from the tables are all zero except for the terms

I OOT, y) = I �+ rt 1

C rt+

+

+ l rt �+

= 0.

Therefore there is no singlet T,g term in the t�g configuration. This means that the second set of terms listed above is the correct choice. Using this procedure, all of the crystal-field terms for the t�g configuration can be found in terms of linear combinations of single-electron orbitals. Examples of some of these are listed in Table 2. 7. TABLE 2.7. Example wave functions for the Ref. 1 0)

I � t.4t > = I � 3TtrM = I > = I �9 3Tt M = ia> =

1

( 1 .;�

+ I 'I'l l + I (C I l

1 .;'1 1 1'1(1

t292

configuration (reprinted with permission from

I - I 'I'l l ) I I� 1Ev> = I � 3Tt rM = - i> = I t� 1T2(> = +

( 1 .;�

101 1'1� 1)

76

2. Electronic Energy Levels

Now that the qualitative nature of the energy-level splitting has been determined, the quantitative amount of the splitting can be determined from calculating the crystal field matrix elements. Consider the effects of oper­ ators of the c2 class as typical symmetry operations of the 0 point group acting on Cartesian coordinates,

C2x : x -- x, y -- -y, z -- -z; C2y : x --- -x, y -- y, z -- -z; C2z : x -- -x, y -- -y, z -- z. Applying these transformations to the wave functions in Eq. (2.3. 1 3) gives Function

1/13, 2, 2 ![13, 2, 2 1/13, 2, I ![13,2 , I 1/13, 2, 0

C2x

C2y

C2z

1/13, 2, 2 -![13 , 2, 2 - 1/13, 2, 1 ![13, 2, I 1/13, 2, 0

1/13, 2, 2 -![13, 2, 2 1/13, 2, I - ![13, 2, 1 1/13, 2, 0

1/13, 2, 2 ![13, 2, 2 - 1/13, 2, 1 - ![13, 2, I 1/13, 2, 0

Thus for this class of operations the five d-electron wave functions just transform into ± themselves. For operations belonging to the C4 class,

C4x : x -- x, y -- z, z -- -y; C4y : X -- -z, y -- y, z -- x; C4z : X -- y, y -- -X, Z -- Z . Applying these transformation to the d-electron wave functions gives:

1/13, 2, 2 ![13,2, 2 1/13,2 , 1 ![13,2 , 1 1/13,2, 0

C4y

C4z

1 3 1/2 "' 3, 2, 0 + 21 1/1 3,2,2 - ![13, 2, 1 -1/1 3, 2, 1 ![13,2 ,2 1 3 1/2 2 "' 3, 2, 0 + ( 2 ) "' 3, 2, 2

- 1/13, 2, 2 - ![13, 2, 2 1/13,2 , 1 - ![13, 2, 1 1/13, 2, 0

C4x

Function

1 3 1/2 "' 3,2, 0 + 21 1/13,2,2 1/13,2 , 1 -![1 3, 2, 2 - ![13, 2, 1 1/2 1 ( 23 ) "' 3,2,2 2 "' 3, 2, 0 2 (2 )

2 (2)

This shows that the two functions l/1 3 2 ' 2 and l/1 3 2 0 transform into linear combinations of each other, while the three fu�ctions lfr 3 2 2 , lfr 3 2 1 , and l/1 3 2 ' 1 transform into linear combinations of each other. Applying this anal­ ysis with the operations belonging to the other three classes of the 0 group (E , C3 , and q) give the same results. Since it was shown above that the five degenerate free-ion d-electron orbitals become one doubly degenerate set of crystal-field orbitals transforming as the e9 irreducible representation and

2.3. Crystal-Field Splitting of Energy Levels

77

FIGURE 2.8 Plots of charge dis­ tributions for d-electron orbitals.

(A) Orbitals transfonning as eg (higher energy because lobes are along ligand directions).

dxy

dzx

d yz

(B) Orbitals transfonning as t2g (lower energy because

lobes are between ligand directions).

one triply degenerate set of crystal-field orbitals transforming as the t2g irre­ ducible representation in an octahedral crystal field, the functions lj; 3 2 2 and lj; 3 2 0 act as basis functions for the representation and the f�n'ctions 1/1 3 : 2 : 2 , 1/1 3 , 2 , " and 1/1 3 , 2 , 1 act as basis functions for the t2g representation. Figure 2.8 shows two-dimensional plots of the shapes of the five symme­ trized d-electron wave functions. Note that the primary lobes of the charge distributions of the two wave functions transforming as are along the major cubic symmetry axes, while the charge distribution lobes for the three wave functions transforming as t2g are along directions between the major symmetry axes. Since the ligands in octahedral symmetry lie along the di­ rections of the major axes as shown in Fig. 2.4, there is a stronger Coulomb interaction for the former set of orbitals than for the latter. For this reason, the level is shown as the higher energy level in Fig. 2.5. For other types of symmetry configurations, the ligands are in different positions, and thus the ordering of the energy levels may be the opposite. This is true for tetrahedral symmetry. To calculate the magnitude of the crystal-field splitting, the matrix ele­ ments of the crystal-field operator between the states of the system must be evaluated. The form of the azimuthal parts of the matrix elements is

eg

eg

eg

M

(lj; i f Vo i i/IJ ) oc =

J: (eim;\l ) * eim,\leimJ\ldtp J: ei(m, +mrm;) \ldtp

O(unless m v + m1 - m ;

0) .

Since mv 0, ± 4, this shows that l'lmiJ 0, ± 4. For d-electron wave func­ tions, m ; 0, ± 1 , ± 2, so all diagonal matrix elements ( l'lmiJ 0) are non­ zero and the matrix element with m ± 2, m1 + 2 are nonzero. The

i

78

2. Electronic Energy Levels

expression for the crystal field in spherical coordinates is found from Eq. (2.3. 1 1 ) to be

=

{f � (35 cobJ +

·

30 cos2 8 + 3) C(R) 4 8( eierp + e - i4rp ) C(R)

'

where the C(R) contains the expansion coefficients and the �(r) factor. The wave functions expressed in spherical coordinates are

{f � (3 cos28 1 ) , d ± l = + R3 , d {f /¥ (cos 8) (sin 8)e ± irp ) , do = R3 , d

d ± 2 - R3 , d

. ( sm2 8) e ± i2rp .

Collecting the R functions and the factors of (2n) 1/2 into a parameter K, the angular parts of the matrix elements are (do I Vo I do )

5 = K 8 Us

(d ± l l Vo i d± , ) =K

0

(3 cos2 8 1 ) (35 cos4 8 30 cos2 8 + 3) (3 cos2 8 1 ) sin 8 d8,

� J: (cos 8 sin 8) (35 cos4 8

30 cos2 8 + 3) ( cos 8 sin 8) sin 8 d8 ,

(d ± 2 l Vo l d± 2 ) =K

� J: (sin2 8) (35 cos48

(d ± 2 1 Vo l d n ) =K

30 cos2 8 + 3) (sin2 8) sin 8 d8 ,

{-£ f/J J: (sin2 8) (sin48) (sin28) sin 8 d8.

2.3. Crystal-Field Splitting of Energy Levels

79

These can be evaluated using

I: (sin2n+ I B) (cosm B) dB = I: (cos2n+ I B) (sinm B)dB 2n+ 1 n!

(m + l ) (m + 3)

·

·

·

(m + 2n + 1 )

to give (do i Vo l do ) = 6Dq , 4Dq , (d ± I I Vo l d ± I ) (d ± 2 1 Vo l d ± 2 ) Dq , (d ± 2 1 Vo l d :n) = 5Dq ,

where D and q are constants multiplied by the parameter K and will be evaluated below. The secular determinant for the d-electron orbitals in an octahedral crys­ tal field is d! do d2 d2 Dq E 0 0 d! 0 0 4D q E do 6D q E 0 0 d_ J 0 0 0 0 0 5D q d2 which can be box-diagonalized as

(

Dq E 5Dq

5D q Dq E

0 and expanded to give

)

d2 d_ J 0 5Dq 0 0 0 0 0 4Dq E 0 Dq E

0 4Dq - E

=0 6Dq E

4Dq E

( 4D q E) 2 (6Dq E) [ (D q E) 2 (5Dq) 2 ] = 0. The solutions give five values for E, three are 4Dq and two are + 6Dq . This results in the energy-level splitting shown in Fig. 2.6. Note that if the energies of the levels are weighted by their degeneracies, the center of grav­ ity of the free-ion energy level remains unchanged by the crystal-field split­ ting [2 x 6Dq + 3 x ( 4Dq) = 0] . In order to determine the definitions for D and q explicitly, consider the

80

2. Electronic Energy Levels

matrix element

M = (d+2 i e Vo l d- 2 ) = 5Dq .

The exact form of the crystal field operator can be found from Eq. (2.3. 1 1 ) with the expansion coefficients and radial factor included from Eq. (2.3 .9), Vo = A 4 , o� Y2 + A 4 , 4 � Y44 + A 4, -4 � Yi4 4n r4 4 0; 30 cos2 0; + 3) = ez 9 as

(

+

/fJ t (sin40;) (e 41PJ + e41P,i) ) ,

where a is the distance from the central ion to the positions of the ligands. Using the angular coordinates of the six ligands shown in Fig. 2.4, these sums can be evaluated to give

The matrix element then becomes

M-

·

x

2 O) (ei21P )

* 1 7 ..fic3 Ze2 ar4s

I {f JH sin20) (e-i21P ) ) ,

(sin4 0) ( ei41P ) R3d(r)

where only Yt connects the mt = ± 2 wave functions. This can be factored into a constant times the product of three matrix elements, ze 2 M = _!_ 2n 3 1 6 a5

7

14

be

256

2 2 x (R3d (r) I r4 I R3d(r)) (sin 0l sin4 0 I sin 0) (e- i21P l e i41P l e - i21P ) 2Ze 35e = 5 l4 5D q . 105 4a5 (R3d(r) r I R3d(r) ) The definitions of D and q are 35e D= 5' (2.3. 14) 4a

_

=

(2.3. 1 5) Thus the strength of the crystal field 1 ODq varies as the ligand distance a- 5 and as the ligand charge Ze.

2.3. Crystal-Field Splitting of Energy Levels

81

In order to determine the energy of a specific term in the strong-field approach, the contributions of Coulomb and exchange interactions must be added to the crystal-field energy. For the example of the two-electron configuration t�9 , the crystal field energy is Ec (t�9 ) = 2( 4Dq) = 8Dq . Assuming the two electrons are in the dxy and dyz orbitals, the Coulomb matrix element that must be evaluated is

J

1 = 4 (d, di ) � (d, + d_ , ) ; _2_ (d, d_, ) 1 (d, + d 1 h dr '1 2 1 = l [J( 1 , 1 ) + 1(1 , 1) K( 1 , - 1 )] = F0 - 2F2 - 4F4 , where J(a , b) = (ab l 1 /r 1 2 l ab ) , K( a , b) ( ab l l /r 12 l ba) , rna + mb = me+ md, and the Slater parameters have been defined previously. In a similar way, the exchange integral is

J

K(xz, yz) = (xz) � (yz) ; _2_ (yz) 1 (xzh dr = -1 K(1 , - 1 ) = 3Fz + 20F4 . rl 2 2 Thus the interaction between the two electrons contributes an energy to the 3 T,9 term of Ee e T,9) = J(xz, yz) - K(xz, yz) = F0 - 5F2 - 24F4 . The total term energy is Ee T,9) = Ec + Ee = 8Dq + F0 - 5F2 - 24F4 , or in terms of Racah parameters, Ee T1 9) = 8Dq + A 5B. This type of calculation can be done for each term of each electron configuration. Table 2.8 lists some of the important two-electron integrals for d electrons in an octahedral crystal field in terms of the Racah parameters. The splitting of crystal-field term energies due to the spin-orbit inter­ action can be treated as an additional perturbation. The expression for the spin-orbit coupling Hamiltonian given in Eq. (2. 1 .6) is strictly valid only for spherical symmetry. This form can be used as an approximate expression for Hso in a crystal field of lower symmetry. In this case, the orbital angular momentum quantum number is replaced by the irreducible representation

TABLE 2.8. Two-electron integrals for d electrons in an octahedral crystal field in terms of Racah parameters [J(uv) (uv J J uv) , K(uv) (uv J J vu)] . =

J((() = A + 4B + 3C J(?;q) = A 2B + C K(?;q) = 3B + C J((u) = A 4B + C K((u) = 4B + C

=

J(uv) = A 4B + C K(uv) = 4B + C J((v) = A + 4B + C K((v) = C (?;ull(u) = ../3B

82

2. Electronic Energy Levels

designation for the orbital motion in the crystal. For Oh symmetry, Hso = L

�i ti si ,

(2.3.16)

·

where the tx , ty , tz spatial orbitals must transform as the T1 irreducible rep­ resentation since the set of spin functions sx , sy , Sz transform as TJ . The decomposition of the direct product T1 x T1 contains A 1 as required for the total Hamiltonian to be invariant. Note that for pure d orbitals, ti = li , which does transform as T1 as required. Since the crystal field is generally small compared to the Coulomb interactions, it is a common approximation to use the form of Hso for the free ion and treat the magnitude of �i as an adjustable parameter. This is generally a satisfactory approach for under­ standing solid-state laser materials. More rigorous methods for calculating the magnitudes of spin-orbit splittings of crystal-field terms are described in Refs. 4 and 9. Thus for a crystal-field term transforming as r and a spin-orbit coupled vector multiplet,

(r i Ti g l r)

=

A1g +

·

·

·

.

By using the Oh character table and direct-product analysis, it is found that this relationship is only true if r = T1g or T2g · For example, a 4 T1g term is split by the spin-orbit interaction into two types of two-dimensional terms and two four-dimensional terms: Gg x T1g = E1;2g + E3 ;2g + 2 Gg. The quan­ titative splitting is found by treating the spin-orbit coupling parameter as an adjustable parameter and fitting these predictions to the experimentally observed optical spectra. Other examples such as a 2 E term will not have any splitting due to the spin-orbit interaction. The energy levels discussed in this chapter are those involved with optical pumping and laser emission shown in the diagram in Fig. 1 .4. The tran­ sitions between these levels are discussed in Chap. 3. When treating rare­ earth ions with /-electron configurations, it is useful to employ a tensor op­ erator formalism. This is best understood through examples and is discussed further in Chap. 7. References I . E.U. Condon and G.H. Shortley, The Theory of Atomic Spectra (Cambridge,

University Press London, 1 935). 2. E. Merzbacher, Quantum Mechanics ( Wiley, New York, 1 96 1 ); A.R. Edmonds, Angular Momentum in Quantum Mechanics ( Princeton University Press, Prince­ ton, 1 968); and Chaps. 2 and 10 in A tomic, Molectular and Optical Physics Handbook, edited by G.W.F. Drake (AlP, New York, 1 996) . 3. B. Di Bartolo, Optical Interactions in Solids ( Wiley, New York, 1 968) . 4. J.S. Griffith, The Theory of Transition Metal Ions (Cambridge University Press, London, 1 96 1 ) .

References

83

5. M. Tinkham, Group Theory and Quantum Mechanics (McGraw-Hill, New York, 1 964) . 6. F.A. Cotton, Chemical Applications of Group Theory (Wiley, New York, 1 963) . 7. B. Di Bartolo and R.C. Powell, Phonons and Resonances in Solids ( Wiley, New York, 1 976). 8. J.L. Prather, Atomic Energy Levels in Crystals ( Department of Commerce, Washington, 1 96 1 ) . 9 . C.J. Ballhausen, Introduction to Ligand Field Theory (McGraw-Hill, New York, 1 962). 10. S. Sugano, Y. Tanabe, and H. Kamimura, Multiplets of Transition Metal Ions In Crystals (Academic, New York, 1 970).

3

R adi ative Transitions

The fundamental physical process underlying laser operation is the radiative absorption and emission of light by the optically active centers of the laser material. Typically this involves laser ions undergoing transitions between electronic energy levels through the absorption and emission of photons. This can be treated in the semiclassical theory of radiation in which the elec­ tromagnetic field of the radiation is treated classically while the energy levels of the ions are treated quantum mechanically (as discussed in the preceding chapter). However, it is also useful to apply the techniques of second quan­ tization and treat the system as an ensemble of ions in a photon field. The interaction between the ions and the radiation is treated as a perturbation and standard perturbation theory techniques are employed. The radiative transition strengths or rates ( probabilities per unit time) for absorption or emission involving specific energy levels are the main parameters of interest. Qualitative estimates of transition strengths can be determined from group­ theory considerations. This can be used to classify absorption or emission of radiation between specific energy levels as allowed or forbidden transitions. The quantitative magnitudes of the transition strengths are important for calculating laser gain and threshold conditions. This chapter outlines the basic concepts of the quantum theory of radia­ tion that is relevant to understanding solid-state laser materials. A detailed treatment is given in Heider' book. 1 The ion-photon interaction is treated by perturbation theory and the spectral line shapes resulting from different types of transitions are discussed. The general expressions for transition strengths and selection rules are derived, and the use of group theory for determining allowed and forbidden transitions is outlined. The application of these concepts to specific types of laser materials is discussed in Chaps. 6-1 0. 3. 1

The Photon Field

If each ion in the laser material interacts independently with the radia­ tion field, the physical system of interest can be represented by the 84

3. 1 . The Photon Field

85

Hamiltonian H

(3. 1 . 1 )

H; + H, + H;_,,

where H; represents the electronic energy levels of the ion in the electro­ magnetic field, H, represents the electromagnetic field, and H;_, represents the interaction between the radiation field and the ion. The first step in de­ riving radiative transition rates is to determine appropriate expressions for each of these contributions to the Hamiltonian. The classical radiation field is defined by integrating Maxwell's equations and choosing the Coulomb gauge (V · A 0) to give2

2 V2 A(r, t) - 21 8 A(r, t)

0,

c

-1 oA

- ot '

E

c

B

v

X

( 3. 1 . 2)

A,

plus appropriate boundary conditions. Here E, B, and A represent the elec­ tric field, the magnetic field, and the vector potential of the electromagnetic radiation. For the case of interest here, box normalization with periodic boundary conditions in a rectangle of sides Lx, Ly , and Lz can be used. This requires that A(r + Lxux) A(r), where Ux is a unit vector in the x direc­ tion. There are similar expressions for the y and z directions. Assuming product solutions to the field equations, Ao: (r, t) qo:(t)A(r), the technique of separation of variables can be used to give

w2"' A "' 0 , q.. + w2 q"' 0 "' + 2 "' "' c

with the solutions (3. 1 .3) and

( 4 1to:

) 1 /2

_!2

(3. 1 .4) eik.·r , V where V LxLy Lz, ito: is the unit polarization vector for the rx mode, and ko: wo:/ c ( or ko: 2n/ A.o:) is the magnitude of the wave vector of the radiation. Using the solutions derived above, the Coulomb gauge condition gives

A"' (r)

_

�

Therefore ito: · ko: 0 so the polarization and propagation vectors are per­ pendicular and the field has transverse waves.

3. Radiative Transitions

86

This can be generalized to include all frequency and polarization compo­ nents (..1.), (3. 1 .5) A(r, t) 2: l: [q�(t)A � (r) + q� (t) * A � (r) * ] .

rx

A

The boundary conditions can then be used to define the modes designated by rx. The imaginary exponents imply k

k

2nnrxy 2nn rxy _ Ly ' krxz _ Lzrxz ' (nrxx, nrxy , nrxz = 0 , 1 , 2, 3, . . . ) . Therefore rx is defined by (nrxx , nrxy , nrxz) . 2nn rxx _ Lxrxx '

The electric field is now given by

_ ! oA c at

E

iwrx A A A A . (qrx A rx qrx A rx ) A c _

rx

•

•

(3. 1 .6)

The magnetic field has the same magnitude as the electric field. Therefore, defining the Hamiltonian in terms of the energy in the radiation field gives

J (£2 + B2 ) dr

H

J E2dr

However, since

J A rxA rx' dr = 4nc2Jrx, rx' , J ArxA rxdr = J A:A: dr = 0 , J A:Arxdr 4nc2 , the Hamiltonian can be expressed as H

2: 2: w� ( q�q�* + q�* q� ) rx

A

2 2: 2: w�q� q�* . rx A

(3. 1 .7)

This expression can be put in the form of the Hamiltonian for a harmonic oscillator through the canonical transformation

where

rx

and ). have been combined into one mode designation f3 for sim-

3. 1 . The Photon Field

87

plicity in notation. Therefore

qp

!2

so

1 ( Qp � lWp Pp) ' qp* 2 ( Qp + lWp Pp )

(3. 1 .8)

Q P H = 'L) p ! w� j + ! j ) Lp Hp,

(3. 1 .9)

1

where (3. 1 . 1 0)

Hp is the Hamiltonian for a harmonic oscillator. Therefore the dynamics

of electromagnetic radiation have a mechanical analog in a simple har­ monic oscillator. Thus, the radiation field can be treated as an ensemble of harmonic oscillators, one for each mode having a specific frequency and polarization. To transform this treatment into quantum mechanics, and are treated as operators. The commutation relations for these operators are

Q, P, q,

q*

[Qp, Pp'] = iMpp' , [Qp, Qp'] = [Pp, Pp'] = 0, [qp, qP, J = Jpp' , [qp, qp'] = [qp- , qp, J = 0. Making use of the harmonic oscillator analogy, creation and annihilation operators are defined in the usual way as

ap =

(3 . 1 . 1 1 )

Here a�t creates and a� annihilates a photon of frequency Wrt. and polar­ ization it�. In terms of ap and ap- the vector potential and Hamiltonian are A=

LL rt. A

and

H,

it� (a�eik.·r + a�t e - ik. ·r ) L L liwrt. (a�t a� + !) . rt. A

(3. 1 . 1 2)

(3. 1 . 1 3)

This is the Hamiltonian for the radiation field to be used in Eq. (3. 1 . 1 ) . In this expression the number operator is defined as

Nrt.A art.At aArt. =

3. Radiative Transitions

88

because In this second-quantized formalism, the eigenstates of the radiation field are written in terms of the occupation numbers of the oscillators,

l t/l' (nti ' n�Z ,

. . . )) = IT I n� ) , ct,l

(3. 1 . 14)

where n� is the number of r:x, A oscillators excited. The energy levels are the same as those of an ensemble of harmonic oscillators,

E(nt � , n�Z ,

. . . ) = (ljl(nt� , n�2 . . . ) I Hr l t/J (nt� , n�2 . . . ) ) = L L hw" (n� + !). "

;.

(3. 1 . 1 5)

Next consider the Hamiltonian for an electron in a radiation field, 1 e 2

H 2m

Squaring the first term gives

(p + - A) - erjJ.

+

c

p2 e2 A 2 e (p · A 2m + 2mc2 2mc but

+ A · p) '

- ihV · ( At/1) = -ihA · V ljl - iht/J V · A = A · p t/J . Here the Coulomb gauge has been used. The p2 j2m and erjJ terms are taken to be part of the Hi in Eq. (3. 1 . 1 ). The latter term gives a small correction (p · A)t/1

to the energy levels derived using the Hamiltonian for the ion derived in the preceding chapter. Thus the Hi is essentially the Hamiltonian found pre­ viously that includes the effects of Coulomb, spin-orbit, and crystal-field interactions. The remaining terms become part of the ion-radiation inter­ action Hamiltonian. The nonlinear term involving A 2 represents second­ order effects such as two-photon absorption, and will not be considered here. ( The formal elimination of this term is accomplished through a unitary transformation of the Hamiltonian. 3 ) Thus the interaction Hamiltonian in Eq. (3.1 . 1 ) is now written as

e Hi r m e A · p .

(3. 1 . 1 6)

Expressing the vector potential in terms of photon creation and annihila­ tion operators through Eq. (3. 1 . 1 2 ) allows Hi r to be written as

Hi-r = m !!. L k,A

( it1 · p) (a1 eik·r + azt e - ik·r ) .

(3. 1 . 1 7)

3. 1 . The Photon Field

89

Here the wave vector k is used to designate a specific mode to simplify the notation. In the case of atoms with more than one optically active electron, there is an additional sum over all active electrons. The interaction between a radiation field and an electron bound to an atom causing a radiative transition between electronic states is treated as a perturbation on the system whose eigenstates are the product functions I I/I� ;

n, , n2 , . . . )

I I/I�)

IT I n� ) . k,A

(3. 1 . 1 8)

Here the first factor represents the eigenfunction of the bound electron states discussed in Chap. 2 and the remaining factors are the states of the radiation field given by Eq. (3. 1 . 14) . The interaction Hamiltonian in Eq. (3. 1 . 1 7) causes one-photon transitions to take place between these eigenstates. The matrix element describing this interaction is

(3. 1 . 19) Here the upper sign and top occupation numbers in the { } brackets repre­ sent the creation of a photon in an emission transition, while the lower sign and bottom occupation number representation the annihilation of a photon in an absorption transition. The sum over j is the sum over optically active electrons. In evaluating the electronic part of the matrix element, it is convenient to expand the exponential in a Taylor series as

e+ ik r

=

1 ± ik · r · · · .

Since l k l 2nj A, this is an expansion in powers of rj A, where r is the posi­ tion of the electron with respect to the nucleus, which falls off exponentially from the nucleus. It will be of the order of a few angstroms. A is the radia­ tion wavelength and for light in the uv-visible-ir spectral range it is :;: 1000 A. Thus, A » r, so the terms in the expansion rapidly get smaller and only the first one or two need to be considered. With the above approximation the interaction Hamiltonian can now be written as

90

3. Radiative Transitions

The first two terms can be simplified by using the commutation relationships

[H, x] in x. ih pmx ' � 1m (t/lj iPx l t/ID (t/lj [H, x] lt/1�) (t/lj i Hx xH it/1�) (E} Ef)(t/lj lxl t/1�) nwk (t/lj lxlt/JD, =

=

=

=

=

=

so

(3. 1 .20) Therefore, the first two terms in H;_, are

where the sum over optically active electrons has been included explicitly. Here the ED superscript is used to denote electric dipole interactions. As stated before, this is valid when ion-ion interactions are negligible. The next two terms can be simplified by using the vector identity

(k · r)p ! [(k · r)p + (k · p)r] + ! [(k · r)p (k p)r] [ (k · r)r + (k r)r] ! [k (r p)] iwk [(k r)r] (k L) =

·

X

·

X

x

·

where the relationship

dF � [H F dt = n ' ] was used as above. The resulting expressions are

(3.1 .22)

HMD

-i L ,).

k

+

i

� a�t it� B

[k ·

x

[k

e (L 2S) 2m e + x

]

]

(L + 2S) .

(3 . 1 .23)

The superscripts EQ and MD stand for electric quadrupole and magnetic

3 . 1 . The Photon Field

91

dipole, respectively. Here the spin magnetic moment was simply added to

the orbital magnetic moment because it is already known as part of the total magnetic moment. For a more rigorous treatment, the spin term in the inter­ action Hamiltonian can be written as

e e H;_, oc 2mc 2S · B = m e S(V x A) . -

Then V x A can be evaluated and the exponents expanded as before to give the spin terms as written above. The interaction Hamiltonian can now be written as

(3 . 1 .24) where the individual interaction Hamiltonians are given in Eqs. (3. 1 .21 )­ (3. 1 .23).

Now that an expression has been derived for the interaction Hamiltonian, the next step is to determine the probability per unit time for the ion-radiation interaction to cause the system to undergo a radiative transition. This requires the use of time-dependent perturbation theory. The general expression for the transition rate is derived from the time-dependent Schrodinger equation,

H I 'P (r, t))

ih

t) )

where the stationary states of the system are given by

I 'P(r, t) )

l t/l(r)) e iEt/h .

If the Hamiltonian can be expressed as the sum of Ho describing the unper­ turbed system, and a time-dependent perturbation H'(t), then the new wave functions of the system are linear combinations of the unberturbed wave functions

I 'P) = L an (t) l t/l� ) eiEn t/ h , n where the nonstationary time dependence is incorporated in the expansion coefficients an (t) . To evaluate the expansion coefficients, this expression for the new wave functions can be substituted into the Schrodinger equation to give

so

I 'P�) . L an (t)H' I 'P� ) = ih L n n Specific expressions for ak (t) can be found by projecting ( t/12 1 on this rela-

92

3. Radiative Transitions

tion to yield

L an ('P� I H' I 'P�) n

i

h L tin ('P� I 'P�) ihak , n where the orthogonality of the unperturbed states I 'P� ) has been utilized. Because I 'P� (r, t) ) I '¥� ) exp( - En t/h), ak L an ( !J;� I H' I !J;�) ei(Ek En) tjh n or

ak -'1h 'L an e'.wk. t H'kn ' l n

(3 . 1 .25)

where

(3.1 .26) The most important type of perturbation to consider here is the case where H' is a step function turned on at time t 0 and otherwise is con­ stant, because this perturbation most closely models optical pumping. If the system is initially in a specific state I !J;n ) so an =f. 0 and all other a; 0, the equation for dk can be integrated to give

Jt e'wkn t H£ndt'

H' . - hcokn ( e'wkn t - 1 ) . kn The probability for the system being in state k at time t is 4 H 2 sin2 (! cokn t) I ak 1 2 _ I 2£n l 2 h (Okn 1 ak -:th

.

I

__

o

The total transition probability involves integrating over the density of final states g ( co ) dco . This can be treated as a continuum of closely spaced states in energy since the spectral lines of interest are always broadened in frequency as discussed below. Thus,

J

J

4 I H�k 1 2 sin2 (! conk l) g(conk ) dconk · h2 (Okn2 If H�k and g (conk ) are slowly varying functions of k, this expression becomes sin2 (! conk l) 4H 2 I H�k 1 2 (3.1 .27) Pnk I 2�k 1 g(conk ) 2 conk h h Pnk

oo

oo

l ak (t) l 2 g(conk ) dconk

J

oo

oo

oo

oo

The transition rate ( probability per unit times) is then

(3.1 .28)

3 . 1 . The Photon Field

93

This expression is referred to as Fermi's golden rule. It is important to keep in mind the various assumptions used in deriving Eq. (3. 1 .28) since there are some specific cases where these assumptions are not valid and thus this ex­ pression can not be used. The interaction matrix element appearing in Eq. (3. 1 .28) involves one of the H; , terms derived above. Since the transition rate involves the square of this matrix element, the electronic part of the expression is the same for absorption and emission. For an electric dipole transition

{

nk + l for emission for absorption. m Wk V X n k The factor for the photon occupation numbers is ni + 1 for emission and ni I M'nk l2 =

f p,. . 1tk

z

for absorption. The first part of the expression for the emission transition represents stimulated emission and is the same factor appearing in the ex­ pression for absorption. The second part of the expression for the emission transition represents spontaneous emission and is independent of the photon population density. The density offinal states is found from the box normalization conditions used above in the derivation of the Hamiltonian of the radiation field. The number of final states with wave vector k between k and k + dk is

where (3 . 1 .30) and spherical coordinates in k space have been used with dQk a solid angle in the k direction. Note that if the final state is degenerate, an additional factor to account for this degeneracy must also be included in Eq. (3. 1 . 30). Substituting the expressions for the transition matrix element given in Eq. (3. 1 . 1 9) and the density of final states given in Eq. (3. 1 .30) into Eq. (3. 1 .28), it is possible to write the following expressions for transition rates involving photon absorption or emission in solid angles represented by dQk ·

Spontaneous emission:

(3. 1 .3 1 )

Induced (stimulated) emission or absorption: (3. 1 .32)

94

3. Radiative Transitions

The total rate of spontaneous emission of a photon of frequency wk is given by integrating over all solid angle directions and summing over all polarization directions,

A

J

L PZ (spontaneous) dQk ). e2 Wk • ,\ · P; t/1 ; eik-r1 1tk t/11 hc3 m2

I ) 1 dQk·

�J I\ I � e

e

2

(3. 1 .33)

The rate of induced (or stimulated) absorption or emission of a photon of frequency Wk is given by (3 . 1 .34) where Pv is the energy density of the radiation field. This is related to the photon occupation number nk by

E -v

J Pvdv

k, ).

hvk (nZ + !) V

J

2 -v dk g(k)hvk (nk,1. + 21 ) .

But 4n V 2 dk 2 g(k) dk 4nk 3 c3 v dv. 8n V

Thus,

J

Pv dv

J

Therefore,

Pv

(3. 1 .35)

7 hv(n v + !) dv.

8nv2

8nhv3 c nv .

(3. 1 .36)

Note that the integral over the ! term in parentheses has been dropped since this just gives a constant that shifts the position of the zero-point potential energy. Combining Eqs. (3. 1 .34) and (3. 1 .36) gives

A Pv B

g I Babsorption

g2Bstimulated emission ·

(3. 1 .37)

These expressions are called the Einstein relations and A and B are the Ein­ stein coefficients for spontaneous and induced transitions, respectively. 3 .2

Selection Rules

For an electric dipole transition, Eqs. (3. 1 .20) and (3. 1 .2 1 ) can be used in Eqs. (3. 1 .3 1 ) and (3. 1 .32) to give the transition rates for absorption or emis-

3.2. Selection Rules

95

FIGURE 3 . 1 . Vector components of magnetic dipole radiation.

sion of a photon of frequency wk in any direction with either polarization direction, Pkb Pkm =

� J dQk I ( l/1/1 1 M · it� l l/1�1) I , (nk + 1 ) � J dQk I ( l/1/1 1 M · it� 11/1�1) I , nk

(3.2. 1 ) (3.2.2)

where the dipole moment operator is given by MEn

L er;.

(3.2.3)

The vector components of MEn in a specific coordinate system can be con­ sidered separately. For example, for the geometry shown in Fig. 3 . 1 the fol­ lowing relations hold: it!k . it2k 0 ' it!k . k it2k . k 0 ' (3.2.4) ( M · iti)z M sin e, M · itl 0. Also dQk sin e dO dr/J. (3.2.5) Thus the total rate for photon emission is given by (3.2.6) The power radiated is expressed as (3.2.7) For magnetic dipole and electric quadrupole transitions the expressions

96

3. Radiative Transitions

given above are the same except that MEn is replaced by /1 ·

'

-

2mc (L + 2S)

(3.2.8 )

or

(3.2.9 ) Instead of doing a detailed evaluation of the matrix elements, it is some­ times enough just to determine whether they are zero (forbidden transition) or nonzero (allowed transition). This can be determined by considering the parity of the mathematical expressions for the operators and wave func­ tions. A function is even parity if it does not change sign with inversion of the coordinates and is odd parity if there is a sign change with respect to coordinate inversion. Any multipole function 2' has associated with it a spherical harmonic function of order /. Thus it can be considered to have an angular momentum of hi. Equation (3.2.3) shows the electric dipole moment operator to be a vector with components x, y, z that change sign under in­ version. Thus it has odd parity. Equation (3.2.8) shows that the magnetic dipole moment operator is determined by the angular momentum with components that are products of the components of the position and mo­ mentum coordinates, such as YPz · These products will not change sign under inversion and therefore MMD is an even operator. Similarly, Eq. (3.2.9) shows that the electric quadrupole operator involves the products of coordi­ nate components, such as xy which will not change sign under inversion. Thus MEQ is an even parity operator. As discussed in Chap. 2, the wave functions of the electronic energy levels have their parity given by the parity of the spherical harmonic function ( 1 ) 1 , where I is the single electron an­ gular momentum quantum number. Thus the matrix element for an elec­ tronic transition involving the absorption or emission of radiation will be determined by the integral over all space of Y} Mt Yf. This integrand must be an even function for the integral to be nonzero. This can also be looked on as the conservation of angular momentum since the angular momentum of the electronic state of the atom is changed to balance the angular mo­ mentum of the photon that is added to or subtracted from the system. Since photons are bosons, this momentum change is ± 1 . For electric dipole selection rules, the Laport rule requires that the parity of the initial and final states must change for an allowed transition. Since the parity of the electronic states of interest here is determined by their single­ electron configuration, only transitions between energy levels of different single-electron configurations are electric dipole-allowed; all transitions be­ tween energy levels having the same single-electron configuration are "Laport forbidden." An additional requirement for an electric dipole­ allowed transition, if the energy levels of interest are Russell-Saunders terms, is that the spin angular momentum quantum number of the final

3.2. Selection Rules

97

state must be the same as that of the initial state since the operator causing the transition has no spin dependence. Also for LS coupling, the change in the orbital angular momentum quantum number between the initial and final states of the transition must be + 1, - 1, or 0 with the stipulation that the value of L cannot be 0 for both states. This is a requirement for angular momentum conservation as discussed above. The selection rules for transitions taking place through magnetic dipole interactions are the same as those stated in the preceding paragraph except that the parity of the initial and final states of the transition must be the same. For electric quadrupole transitions the selection rules are the same as for magnetic dipole transitions except for the change in L for Russell­ Saunders states. Since MEQ has an equivalent angular momentum quantum number of 2, the change in the orbital angular momentum quantum num­ bers of the states involved in the transition can be ± 2, ± 1, or 0 if neither the initial nor final states have L = 0, and ± 2 if either state has L = 0. Again, having a transition between states that both have L = 0 is forbidden. If multiplets derived from spin-orbit splitting are used instead of LS cou­ pling terms, the relevant angular momentum is the total angular momentum represented by the quantum number J. For allowed transitions, the value of J for the initial and final states must change according the rules given above for L. Transitions meeting the requirements for electric dipole transitions are referred to as allowed transitions. All other transitions are referred to as forbidden transitions. Forbidden transitions still occur through higher-order interaction operators such as magnetic dipole or electric quadrupole, or through perturbations such as spin-orbit interaction and crystal-field effects. The strength of allowed transitions is generally several orders of magnitude greater than that of forbidden transitions, but they are still important in solid-state laser applications. Order of magnitude estimates for the strengths of a magnetic dipole and electric quadrupole transitions compared to an electric dipole transition can be found from

(0.927 I Q -20 ) 2 f1.2 PMD pED ( er) 2 (4 .8 I Q - 1 0 I Q - 8 ) 2 PEQ ,. ( k . r) 2 ,. (2nao ) 2 ,. 10 _ 7 PEo A2 � �

�

�

�

X

X

�

X

�

3 10 _ 6 , X

.

Here typical values have been used for the magnetic dipole given in Eq (3.2.8), the radius of an electron orbit, and the wavelength of light in the visible region of the spectrum. When an ion is placed in a crystal field, the local symmetry determines the selection rules. The parity requirements stated above remained unchanged but the angular momentum requirements are now associated with the crystal-field orbitals. The requirement for an allowed transition is still that the matrix element for the transition be nonzero. Stated in terms of the

98

3. Radiative Transitions

symmetry group of the crystal-field environment,

(3.2. 10) (l/11 1 M, I l/l i ) -=1- o =} r1 x r, x ri :=1 r 1 , where C and r/ are the irreducible representations according to which the initial and final states of the transition transform in the point symme­ try group at the site of the ion in the host material. r, is the irreducible rep­ resentation according to which the electron-photon interaction operator causing the transition transforms, and r 1 is the totally symmetric repre­ sentation. As seen from Eqs. (3.2.3), (3.2.8), and (3.2.9), the electric dipole operator transforms as the components of a vector, the magnetic dipole op­ erator transforms as components of as pseudovector, and the electric quad­ rupole operator transforms as the product of vector components (dyadic). The concepts of group theory developed in Sec. 2.2 can be used to eval­ uate Eq. (3.2. 10). Character multiplication techniques are used to form the direct-product representation, and Eq. (2.2.8) is then used to reduce the direct­ product representation in terms of irreducible representations of the group. Instead of determining whether or not the triple direct product contains r 1 , it is equivalent to determine if the reduction of the direct product of the representations of the initial and final states contains the representation of the interaction operator r, . As an example, consider the octahedral point group Oh with the character table given in Sec. 2.2. The vector components x, y, z form a basis for the T1 u irreducible representation, and thus r, for an electric dipole transition is the T1 u representation. The magnetic dipole interaction operator transforms as T1 g in this point group, and the various components of the electric quad­ rupole interaction operator transform as the A J g , Eg, and Tzg irreducible representations. The allowed transitions for electric dipole interaction are given in Table 3. 1 . Similar tables of selection rules can be formed for tran­ sitions caused by the magnetic dipole and electric quadrupole operators. As a specific case, consider an initial state that transforms as the Eg rep­ resentation and a final state that transforms as Tzu · The relevant characters are given in Table 3.2. The bottom line of the table gives the characters of the reducible representation constructed from the product of the irreducible representations of the initial and final states of the transition. Since the reduction of this product representation contains T1 u (the irreducible repre­ sentation according to which the electric dipole moment interaction oper­ ator transforms), the transition matrix element is nonzero and therefore the transition is an allowed transition. 3.3

Properties of Spectral Lines

The properties of radiative transitions are manifested in absorption and emission spectroscopy experiments. The relevant parameters and definitions

3.3. Properties of Spectral Lines

99

TABLE 3. 1 . Electric dipole selection rules.

A 19 A 1u A z9 A zu Eg Eu T19 T1u Tz9 Tzu ElfZ g EljZu £3/Z g £3/Zu Gg Gu

T1u T19 Tzu Tz9 T1u, Tzu T19 , Tz9 A 1u, Eu, T1u, Tzu A 1 9 , £9 , T19 , Tz9 A zu, Eu, T1u, Tzu A z9 , £9 , T19 , Tz9 ElfZ u, Gu El/Z g , G9 £3/Z u> Gu E3Jz 9 , G9 EljZu> £3/Z u, Gu Elfz 9 , £3/Zg , G9

used in optical spectroscopy such as oscillator strengths, transition cross sections, and line shapes are described here. In the preceding section, the strength of a radiative transition was dis­ cussed. It is common practice to characterize the strength of a spectral line by a dimensionless parameter called the oscilator strength or f number. The name is derived from the analogy between the quantum-mechanical expres­ sion for an harmonic oscillator transition strength and the classical expres­ sion for the strength of a radiating electric dipole. The classical expression for the energy emitted by a three-dimensional radiating dipole treated as a simple harmonic oscillator is E = 3mw6 x6 , where wo is the resonant oscil­ lation frequency and x0 is the equilibrium position. 4 For a system with f oscillators, the power radiated is directly proportional to the energy per oscillator and the number of oscillators, P = yfE, where y is a proportion­ ality constant given by y = ( e2 w6 ) / ( 6nmc3 ) . From the preceding section, the TABLE 3.2. Selection rule for the Eg to Tzu transition

(r1 r; => rEo). x

oh

E

8C3

3Cz

6C4

6q

Eg T2u T1u £9 x T2u

2 3

0 0 0

2 I I

0 I I

0 I I

0

2

0

0

3 6

x, y, z (ED) T1u + T2u =

100

3 . Radiative Transitions

energy of a quantum mechanical oscillator is E 3hw and the power radi­ ated is P hwA 2 I · Using the classical expression to equate power and en­ ergy, hwA 2 I yf3hw, orf A 2 1 /(3y) . Using the expression for the Einstein coefficient given in Eq. (3.2.6) divided by 4n to account for the angular dis­ tribution of the emission, the expression for the oscillator strength of a pho­ ton emission transition between states 2 and I is given by !zi

2mw M I 2 . 3he2 I 2 I

(3.3. 1 )

Here m and e are the mass and charge of an electron, w is the frequency of the transition, and M2 1 is the transition matrix element. If there is more than one emission transition from level 2 to different terminal levels, it is necessary to sum over all such transitions. If the initial level is degenerate, the degeneracy factor g2 must be included in the denominator of the oscil­ lator strength. The sum of the oscillator strengths from a given initial state to all possible final states equals the number of electrons in the system that can take part in these transitions. The direct relationship between h i and the Einstein's spontaneous emis­ sion coefficient from Eq. (3.2.6) is

A21

8nw3 M I 2 2e2 w2 3h c3 I 2 I m e3 h i ·

(3.3.2) Since the emission rate is the inverse of the radiative lifetime, r2} A 2 I ,

there is a useful expression for the product of the radiative lifetime and the oscillator strength of a transition,

hi r2 I

mc3 2 e2 w 2

-

1 .5 U22 I .

(3.3.3)

Note that the wavelength is for the light in the host material and is ex­ pressed in centimeters in this expression. [Equation ( 1 . 1 . 1 1 ) gives this ex­ pression for f with the factor for the refractive index included.] Since the Einstein coefficient for stimulated emission Bz 1 is related to the A 2 I coefficient by Eq. (3. 1 .36), the oscillator strength of the transition can be expressed in terms of the B2 I coefficient as

2n2 e2 mw

Bz i

(3.3.4)

The absorption and stimulated emission coefficients are related by the ratio of the degeneracies of the two levels of the transition, B 1 2 / B2 I gzl g i . Thus the oscillator strengths for absorption and emission transitions between the same levels are related by !zi

gi /!2 gz

(3.3.5)

In addition to the strength of a transition, the shape of the spectral line is important. The shape is determined by the sum of the energy (or frequency)

3 . 3 . Properties of Spectral Lines cr(v)

101

FIGURE 3 . 2 . Gaussian (inhomogeneous) and Lorentzian (homogeneous) line shapes.

11v single ion c �s vo (A) Gaussian

v

v

(B) Lorentzian

widths of the initial and final energy levels for the transition. There are three major types of line shapes: Lorentzian, Gaussian, and Voigt. Different types of physical processes give rise to Gaussian or Lorentzian line shapes as dis­ cussed below. Any physical process that has the same probability of occur­ rence for all atoms of the system produces a Lorentzian line shape, while a physical process that has a random distribution of occurrence for each atom produces a Gaussian line shape. The former is known as homogeneous broadening and the latter inhomogeneous broadening. Figure 3.2 shows a schematic representation of the difference of these two types of line shapes. If both types of broadening processes are present, the line shape is the con­ volution of Lorentzian and Gaussian contributions, and this is called a Voigt profile. Lorentzian broadening is sometimes referred to as lifetime broadening be­ cause the physical processes that produce Lorentzian line shapes are gen­ erally ones that shorten the lifetimes of the energy levels involved in the transition. The most basic process of this type is associated with the Heisen­ berg uncertainty relationship relating time and energy. This contribution is referred to as the natural line width for a transition. The derivation of the expression for the spectral shape of a radiative transition can be found in most quantum-mechanics textbooks5 and the procedure is summarized here. Consider an atomic system with a time-dependent perturbation H', causing a transition from a specific state n to a continuum of energy states m. The

102

3. Radiative Transitions

expressions describing this system are

ihcm (t)

=

ihcn (t)

=

cn H'mn eiwmnl ' cm H'nm e- iwm•1 dm

J

'

where the c; are the probability amplitudes for finding the system in state i. At time t = 0, cn (O) = 1 and cm (O) = 0. The probability of finding the sys­ tem in state n decreases exponentially with time as

Cn (t)

=

e y l/ 2 ,

where y is a constant. This is introduced as a phenomenalogical damping term. Integrating the first expression over time gives h

i H e Uwm. r/2)1 �n y lWmn - 2

Cm =

_

.

1

Substituting these two expressions for en and em into the second expression above and solving for y gives

2i h

J

I Y = 2 I Hmn I 2P (Wnm )

1

e Uwnm +Y/2)1 · dwnm , zy Wnm - 2

where the density of final states is derived from

dm

=

dm dw ) dw Wnm nm p(Wnm nm · =

Since y is the indeterminacy of the initial state, it is much smaller than the transition frequency Wnm · Thus,

2i

J

2 Y�h I H�n I P ( Wnm )

1

- eiOJnml dw nm · Wnm

The exponent can be expressed in terms of trigonometric functions. For large times the cos (wt) function gives no contribution to the integral except when w = 0 at which point it is necessary to use the principal value of the integral. The definition of the J, function can be used to evaluate the integral of the second term at long times. The results give y = R e ( y ) + i lm (y) , where Re ( y )

=

lm ( y )

=

I H l 2p(wn = Wm ) , h �n (w ) P I H�n 1 2 p n Wm dwnm · Wnm

Wnm h

=

J

=

The expression for Wnm in Eq. (3.3.6) is the transition rate.

(3.3.6) (3.3.7)

3 . 3 . Properties of Spectral Lines

103

Substituting these expressions into the em expression and taking the limit of long times gives

l cm ( oo ) I 2

�h 1 H�n l 2

W2 [Wm + 21 Im ( Y ) - Wn ] 2 + nm

•

(3.3.8)

The intensity of a spectral line is directly proportional to the steady state probability of finding the atom in state m, which is given by I em ( oo) 1 2 . As seen in Eq. (3.3.8), this has the form of a Lorentzian function

I(w ) �

11w (w - wo ) 2 +

(

2,

(3.3.9)

where wo is the frequency of the line peak and 11w is the full width at half maximum of the line. Comparing Eqs. (3.3.8) and (3.3.9) shows that the linewidth is equal to the transition rate. w 1 is the lifetime r of the state and related to the width of the energy level through the uncertainty principal r/1£ h . Im(y) is a self-energy shift of the energy levels of the atom in the photon field, called the Lamb shift. It is due to the continual absorption and emission of virtual photons. In general, the observed spectral linewidth is significantly greater than the natural linewidth due to the presence of additional broadening mechanisms that shorten the lifetime of the initial state of the transition. An example of this type of mechanism in gas systems is collision broadening. In solids, radiationless relaxation processes produce Lorentzian broadening. These are discussed in the following chapter. Gaussian line shapes are produced when different subsets of atoms have different peak frequencies. Each subset produces a spectral line with a Lor­ entzian shape, but the envelope produced by the addition of a random dis­ tribution of the lines from these subsets has a Gaussian shape. The Gaussian function describing the superposition of random events is

I (w)

--4n ( n ) (-4 11w

In 2 ' /2 - exp

In 2(w - w0) 2

(11w ) 2

)

(3.3 . 1 0)

where the linewidth and peak position are the same as defined previously. In gas systems, Gaussian broadening is associated with the Doppler shifts in the frequencies due to a Maxwellian distribution of velocities. In solids Gaussian shapes arise due to random distributions of local crystal fields at the sites of ions due to microscopic strains. The differences between Gaussian and Lorentzian line shapes can be im­ portant in determining laser characteristics. For transitions with Lorentzian shapes, all of the ions can participate in laser emission at a specific fre­ quency and single longitudinal mode operation can be obtained. For tran­ sitions with Gaussian line shapes, several subsets of ions may lase simulta­ neously if the Gaussian linewidth covers several free spectral ranges of the

1 04

3 . Radiative Transitions

cavity, and multimode laser operation will ensue. Inhomogeneously broad­ ened lines exhibit spectral hole burning while homogeneously broadened lines exhibit only spatial hole burning. The magnitude of a line-broadening contribution generally depends on temperature and concentration of atoms. In Sec. 1 . 1 the Beer-Lambert Law was derived, demonstrating the ex­ ponential decay in the intensity of a beam of light traveling through an absorbing medium. The dynamics of the energy density of photons in the beam Pv , is linked to the transition probabilities and the absorption spec­ trum of a sample by the equation

- dp

n 1 W12 hvS(v) - nz Wz 1 hvS(v) + AnzhvS(v), where S(v) i s the normalized line-shape function and n; represents the den­ sity of ions in level i. The induced transition rates are related to the Einstein coefficients by W1 2 B 1 2pv , and the spontaneous emission coefficient is related to the radiative lifetime of the level by A c 1 . For absorption of

Tt

a directional beam of photons, the spontaneous emission term can be neglected and dt replaced by dxj c. Then the absorption expression can be integrated to give (3.3. 1 1 ) Pv (x) Pv (O) e a ( v)x where the absorption coefficient is given [as in Eq. ( 1 . 1 .3)] by

a(v)

(

)

hv 92 B n 9 1 nz c 9 1 21 1 - 92 S(v) 92 n S(v) . � 8nr2 9 1 n 1 - � 92 2

(

)

(3.3 . 1 2)

Here Eqs. (3. 1 . 8 ) and (3. 1 .20) have been used to relate the Einstein B2 1 co­ efficient to the radiative lifetime of the excited state. The integrated absorp­ tion coefficient is found from (3.3 . 1 3 )

where the normalized nature of the line-shape function has been used and I is the average wavelength (centroid) of the spectral line in the material. Under normal conditions of weak pumping n2 /n 1 « 1 , so

2 n A. 92 1 . (3.3. 14) a(v) dv � 8n 9 1 r2 This expression is known as the fundamental formula for absorption. Another useful parameter is the absorption cross section defined by [see

I

Eq. ( 1 . 1 .7)]

(3.3 . 1 5)

3 . 3 . Properties of Spectral Lines

105

For a Lorentzian line shape, the peak absorption cross section is given by

;. 92 1 a(vp) 4n2; v g; � r2 '

(3.3. 1 6)

while for a Gaussian line shape the peak absorption cross section is given by

;.; 92 _!_ _ a(vp) �v 8n 9 1 r2 � The integrated absorption cross section is given by 2 1 ). 92 , a(v) dv 8n 9 1 r2

(3.3. 1 7)

I

for both types of line shapes. Thus, the radiative lifetime and the sponta­ neous transition rate can be expressed in terms of the integrated absorption cross section as

-1 A 2 1 '2

8n 9 1 a(v) dv. =/ 92

I

(3.3. 1 8)

The Einstein B coefficients can also be expressed in terms of the integrated absorption cross section,

e f!.!. h v 92 a(v) dv B1 2 a(v) dv. B2 1

I

(3.3. 19)

Finally, the oscillator strength and the integrated absorption cross section are related by

2 I (v) dv _ neme (]"

+

9n

1 2·

(3.3.20 )

The intensity of a spectral line associated with a fluorescence transition of an ensemble of atoms with concentration n0 in length l of sample and ab­ sorption transition cross section a(va ) excited by a beam of photons with intensity Io is described by

Ij(VJ) Pj (va , VJ )hvt dva dvf nolo(va ) la(va ) w2 1 ( vJ )r2 dva dvj . (3.3.21 ) p1 (va, VJ ) is the number of photons emitted per second at frequencies be­ tween VJ and VJ + dv1 after absorption of photons with frequencies between Va and Va + dva . Multipling the photon emission rate by hv1 gives the energy emitted per second. The total integrated intensity in the spectrum is found by integrating over dva and dv1 . Rate equations are very useful in understanding the intensity and time evolution of a fluorescence transition. The expressions describing the con-

106

3 . Radiative Transitions

centration of ions in each level of a three-level system such as the one shown in Fig. 1 .4 (A) are

n 1 - w1 3 n 1 + r2 11 n2 , n2 r321 n 3 - r2 11 n 2 , n3 w1 3 n 1 - r321 n3 , n n 1 + n2 + n3 . 0

0

0

(3.3.22)

The transitions characterized by the rates used in these equations are shown in Fig. 1 . 4 (A) . For steady-state conditions (continuous pumping and con­ stant loss, that is, no Q switching), the time derivatives can be set equal to zero and the set of equations solved for the population of the metastable state n2 ,

'!'321 -<2 1 _ n. _1 1 1 + '!'32 + '!'321 W1 3 '!'2 1

n2

The fluorescence intensity is a product of the concentration of ions in the fluorescing level, the rate of radiative emission 1 / r2 1 , and the photon energy /z

hv172 n 1 1 1 . + + -w -<32 1 3 <2 1

hvn2 172 r2 11

(3.3.23)

Here the quantum efficiency is defined as the ratio of the radiative and the fluorescence decay rates, 172 (r2 1 ) 1 / ri} . For typical laser materials, the decay rate from the pump band to the metastable state is much faster than the metastable state decay rate or the pump rate. This simplifies Eq. (3.3.23) somewhat. Then two limiting cases can be considered, one for a pump rate much smaller than the fluorescence decay rate, and one for a pump rate much larger than the fluorescence decay rate, l

2

� �

W1 3 h V n Wl3 1'/2 1 + -'!'2 1

� �

{ <W2 11 3hhv17V1722n,n, 1

(3.3.24)

For the first case the fluorescence intensity increases linearly with the pump rate. For the second case, saturation occurs and the fluorescence intensity is independent of the pump rate. For fast pulse excitation, the rate equations can be solved for the time evolution of the population of level 2 after a J-function pulse. The result is

(3.3.25)

3.4. Nonlinear Optical Properties

1 07

This describes a population for level 2 that is zero at the time the pulse ends, increases to a maximum at a time given by

( ),

' tm ' _ 1 -1 r _ 1 In r32 21 21 32

(3.3.26)

and then decays exponentially with a decay rate equal to the slower of the two rates involved, r32 or r21 . For laser materials, the radiationless relaxa­ tion to the metastable state is very fast compared to the metastable-state lifetime, r32 « r21 · This essentially means that all ions excited into level 3 immediately decay to level 2, which is the same as pumping level 2 directly. With this assumption, Eq. (3.3.25) reduces to a simple exponential decay

(3.3.27) The fluorescence intensity associated with the radiative emission from level 2 is then given by

(3.3.28) The absorption and fluorescence intensities derived above are important in describing the dynamics of optical pumping and emission in laser systems as discussed in Chaps. 6-10.

3 .4

Nonlinear Optical Properties

For high-power laser operation, the intensities of both the pump and laser optical beams are quite high and can cause new types of physical effects to occur. These are referred to as nonlinear optical processes. These effects are associated with light-induced changes in the optical constants of the mate­ rial, either the absorption coefficient, the refractive index, or both. They are best treated by considering the interaction of the of the light beams with the atoms of the material as a driving force acting on an ensemble of oscillators with a natural resonance frequency. The term nonlinear derives from the ex­ pression for the polarization of the material P induced by the electric fields E associated with the optical beams,

(3.4. 1 )

where the susceptibility x(n) is a tensor of rank n + 1 . The optical constants of the material are contained in the real and imaginary parts of the complex susceptibility. For low optical beam intensities, the first term dominates and the linear susceptibility describes the normal dispersion and absorption of light beams in the material. For the classical oscillator model discussed in

1 08

3. Radiative Transitions

the previous section, x(l )

1 IJLI 2 (wo - w) - iy 4nN h (wo - w) 2 + y2 h (wo - w) + iy . (I) _ . NA (I) - Xre + ZXim - 4nNr:t.p l 2n 2 4nN IJLI

_

CJ.

(3.4.2)

Here N is the number of atomic oscillators with a resonant transition at frequency wo , y and JL are the dephasing rate and the dipole moment of the atomic transition, respectively, and w is the frequency of the laser beam. The real and imaginary parts of the susceptibility are related to the atomic polarizability r:t.p and the transition cross section through the expressions

(1 Xre ) where

rxp

(!) Xim

4nNrxp ,

CJ

NA - 2n

CJ ,

IJLI 2

wo - w h (wo - w) 2 + y2 '

(3.4.3) (3.4.4)

and A is the wavelength of light in the material. In deriving the expression for the transition cross section, Eqs. (3.3.9), (3.3.12) and (3.3. 15) were used. The Clausius-Mossotti relationship can be used to relate the atomic polar­ izability to the index of refraction n,

4n N rxp. n2 - 1 3 2 n +2

(3.4.5)

Defining the Lorentz local field factor !L as fL

n2 + 2

(3.4.6)

the refractive index is related to the polarizability and the real part of the susceptibility by

n2 - 1

4nN/Lrxp,

(I) Xre

/L

n2 - 1

(3.4. 7)

If some physical process changes the optical constants of the material, these changes are reflected as changes in the real and imaginary components of the susceptibility through Eqs. (3.4.3), (3.4.6), and (3.4.7) as

(3.4.8) Some of the important processes that can change the optical constants of the material are discussed below. As the light intensity increases, the higher order terms become important. The second-order nonlinear term gives rise to second-harmonic generation

3.4. Nonlinear Optical Properties

1 09

(SHG) . This is a very important effect for frequency-doubling the output of solid-state lasers. However, the normal procedure for frequency-doubling involves the use of a nonlinear optical crystal external to the laser itself and thus is not relevant to studying the properties of solid-state laser materials. There are some exceptions to this that occur when a doped nonlinear optical material is used for the lasing medium to produce a self-doubling laser. This situation is discussed further in Chap. 9, but thus far this class of lasers has not been able to compete favorably with external cavity doubled lasers. The nonlinear optical term of greatest interest to the optical pumping dy­ namics of solid-state laser materials is the one associated with the third­ order susceptibility. For off-resonance excitation, the real part of the normal x( 3 ) term results in an intensity-dependent contribution to the index of re­ fraction6

n no + An n2 ( E2 )

yl,

(3.4.9)

where

(3.4. 10) Here the optical electric field is the value inside the material and the con­ version between the nonlinear refractive coefficient y and nonlinear refractive index n2 is

cno n2 (em3 /erg) = 40n y (m2 /W) ,

(3.4. 1 1 )

dn An dT AT.

(3.4. 12)

where c is the speed of light in vacuum. The nonlinear constants contain contributions from the Kerr effect, electrostriction, and thenhal changes. For the case of optical pumping of solid-state lasers, the contribution from electrostriction is generally negligible. For high-intensity laser beams the Kerr effect can create a local change in the refractive index. If the transverse profile of the laser beam is not uniform (for example, a Gaussian shape), it is possible for An to have a radial distribution that causes the laser material to act as a lens. This can result in "self-focusing" of laser beams. This intensity-dependent beam shape can be used with an aperature for another element that defines the geometry of the beam in the cavity to produce an intensity-dependent gain that favors a specific cavity mode. This is referred to as Kerr lens mode locking. Table 3.3 lists the nonlinear refractive co­ efficients for some typical host materials for solid state lasers. The thermal contribution to the An in Eq. (3.4.9) was discussed in Sec. 1 .2. If thermal expansion effects can be neglected, Eq. ( 1 .2. 1) shows that this can be expressed as

The temperature rise in the material can occur from radiationless relaxation

1 10

3 . Radiative Transitions TABLE 3 . 3 . Nonlinear refractive coefficients for typical solid-state laser materials (data from Ref. 6). Sample Al2 03 BeAh04 MgAh04 YAI03 MgO Y3Als0 12 Gd3 Sc2 Ah0 12 Gd3 Sc2 Ga30 12 Gd3 Gas0 12 Y3 Gas0 12 La3 Lu2 Ga3 0 12 LiF YLiF4 MgF2 5038 ( phosphate glass) ED2 (silicate glass)

n2 ( I 0- 1 3 esu)

y ( 1 0- 16 cm2 /W)

1 .25 1 .46 1 . 50 3.37 1 .6 1 2.7 4.0 5.5 5.8 5.2 5.8 0.26 0.6 0.25 1 .7 1 1 .58

2.98 3.50 3.66 7.32 3.93 6.22 8.88 1 1 .89 12.52 1 1 .42 12.62 0.787 1 . 54 0.76 6.30 4.71

processes that occur during the optical pumping and lasing cycle, or might be associated with multiphoton absorption of the laser beam (as discussed further below). This depends on the intensity of the light beam I, the ab­ sorption coefficient a, the fraction of absorbed energy converted to heat <1>, and the thermal conductivity of the material K. The most important effects again occur when the heat generated has a nonuniform distribution in the radial direction resulting in a radial gradient for 11n that produces lensing. For a Gaussian distribution the thermal change in the refractive index is _ aIr2 dn

11n,

4K dT .

(3.4. 1 3)

The relevant thermal properties of some important solid state laser materials are listed in Table 1 .2. There are additional terms in 11n involving the stress induced by the heat generated. These can lead to optical damage of the material and thus be the limiting factor in determining how much optical power the material can withstand. If the optical beams (either pump or lasing) are on resonance with energy levels having the appropriate splitting, the imaginary part of the normal x ( 3 ) term results in two-photon absorption transitions. 7 The normal equation describing the decrease in the irradiance I of a beam of photons traveling through a distance z in a material must be modified to dl 2 (3.4. 14) dz = al PI ' where a is the linear absorption coefficient discussed previously and p is the

3.4. Nonlinear Optical Properties VTEP

STEP

ESA

(A)

(B)

(C)

111

FIGURE 3.3. Different types of two-photon absorption processes. (A) Virtual two­ photon excitation process, ( B) sequential two-photon excitation process, and (C) ex­ cited state absorption process.

two-photon absorption coefficient. Note that the second term on the right depends on the square of the irradiance and thus the experimental obser­ vation of a quadratic dependence of the transmitted light intensity on the intensity of the incident light beam can be used to identify the presence of two-photon transitions. There are three different types of processes that fall in the general classification of two-photon processes. These are depicted schematically in Fig. 3.3. The process that is normally referred to as two­ photon absorption is the one shown in part (A) of the figure. The major dif­ ference between this process and the other two that are shown is that there is a virtual intermediate state instead of a real intermediate state for the tran­ sition. The processes shown in parts ( B) and (C) of the figure are actually cascaded one-photon absorption processes and these are discussed further below. The two-photon absorption coefficient from Eq. (3.4. 14) that is asso­ ciated with process (A) in the figure is given by the imaginary part of the third-order susceptibility as 7

(3.4. 1 5) The magnitude of the two-photon absorption coefficient has been measured for some crystals and glasses of interest as solid-state lasers. In general it is found that x�; » x� and thus this type of two-photon process is not im­ portant for the normal optical pumping dynamics of solid-state lasers. However, the other two types of two-photon absorption processes shown in Fig. 3.3 can be important, as discussed below. In addition to the normal x ( 3 ) term discussed above, there is an effective 3 ( x ) term associated with the difference in x (ll for ions in the metastable state

1 12

3 . Radiative Transitions

';? �m m � � llga

/�:{

� mg

lg>

l m> I I

�

I

�

FIGURE 3.4. Schematic diagram for a beam of photons interacting with a four-level atomic system (after Ref. 8).

versus the ground state, 8

(xm( l )

Xe(3rr)

_

Nm Xg( 1 ) ) N '

(3.4. 1 6)

where Nm represents the number of ions in the metastable state. Using a rate-equation approach to describe the transitions and energy-level pop­ ulations as described in Sec. 1 . 1 , the fraction of ions in the metastable state under steady-state pumping conditions is

Nm N

W W+r 1 '

(3.4. 17)

where W and r 1 are the pump and decay rates for populating the meta­ stable state, respectively. Below saturation conditions W < , 1 so the frac­ tion of excited ions is proportional to the pump rate. Since W depends on the intensity of the light beam and thus the square of the optical electric field, this has the same optical electric field dependence as the normal x ( 3 ) term. For optically pumped laser materials, this term can produce intensity­ dependent absorptive, dispersive, and thermal changes in the complex re­ fractive index. The effective x ( 3 ) term can be understood by considering the four-level atomic system shown schematically in Fig. 3.4. l g ) and l m ) represent the ground and metastable states, respectively, of an ensemble of ions such as the active ions of a laser material. These are split by an energy difference hA. I a) and l b ) represent all of the other excited states of the system. Optical pumping distributes some of the ions in the metastable state and leaves some in the ground state with Pam and Pmu representing the relaxation rates from the pump band to the metastable state and from the metastable state to the ground state, respectively. A beam of photons of frequency w interacts with the ions in the ground state through a dipole moment operator flua and with

3.4. Nonlinear Optical Properties

1 13

the ions in the excited state through a dipole moment operator flmb · The components of the density matrix p for the system can be found from the solution of the Liouville-Schrodinger equation [H , p] + ihPdecay·

ihp

(3.4. 1 8 )

The Hamiltonian for the system is H

Ho + Hint ,

where

� �} c 0 0 a 0 0 m 0 0

f.

Hint

e

e

b

�

V:,g

Vga 0 0 0 (3.4. 19 ) 0 0 0 Vbm -pij Eo , where Eo is the mag­

�}

and the interaction operator is given by Vij nitude of the electric field of the optical beam e; represents the energy of the ith state. The susceptibility for this system is calculated by evaluating the trace of the product of the density matrix operator and the dipole moment operator, 8

x

4nN Tr

E

(pp ) .

(3.4.20 )

In most cases of interest the relaxation rates between excited states are large compared to the rate of decay of the metastable state, and the metastable­ state decay rate is large compared to the pump rate below saturation con­ ditions. Under these conditions the expressions given above can be solved to give the real and imaginary components of the susceptibility,

( (

)

2n 2Wmb lflmb 1 2 _ 2Wga lflga 1 2 2 - h wm2 b w2 wga2 - w2 - L lflga I 2 T�1 lflmd Tu,1 � I) 2n WN , �Xlm gm h (wmb - w ) 2 + T'f.b (wga - w ) 2 + T�2 n2Nm �(J/4n , ( I)

_

)

(3.4.21 )

where the inverse of the metastable-state decay rate has been replaced by the lifetime of the state Tm , the T2; parameters are the dephasing times of the ith level, and � r:xp and � (J are the differences in the polarizability and absorption cross section of the ion in the excited state versus the ground state. For most of the materials of interest for solid-state lasers, the real part of this effective susceptibility is greater than the imaginary part. Using the ex­ pression in Eq. (3.4.8), the polarizability difference given in Eq. (3.4.21) can be converted into a refractive-index difference. If the population distribution is not uniform throughout the material this can lead to a spatial variation in

1 14

3. Radiative Transitions

!J.n that alters the optical path of a laser beam propagating in the material. A radial distribution of !J.n results in a population lens. The importance of this effect is discussed further for specific laser materials in Chaps. 7 and 9.

It is important to note that the radiationless relaxation processes occurring in the optical pumping dynamics of the active laser ions is one major source for generating heat that leads to a thermal change in the refractive index as described by Eq. (3.4.8). Since the same pumping dynamics are associated with producing this heat and producing the metastable-state population, the shapes of the spatial distribution of the heat generated and the metastable­ state population are the same. Thus the spatial patterns of !J.n for the ther­ mal lens and the population lens are the same. If these have opposite signs, one effect can be offset by the other. The processes shown schematically in Figs. 3.3( B) and 3.3(C) both in­ volve absorption of light by ions that are already in an excited electronic state. Thus both of these types of processes are actually excited-state ab­ sorption processes. It is usual to distinguish between these two types of pro­ cesses because of the different ways they are involved in optical pumping dynamics. The distinguishing feature between these processes is that in one case the terminal state of the first photon absorption transition is the initial state for the second absorption transition, while in the other case this is not true. In general, the radiationless relaxation processes between excited states of optically active ions in solids is very fast. Thus under normal conditions for optically pumped laser materials, the sequential two-photon excitation process (STEP) is negligible. However, for pumping with ultrafast laser pulses ( picoseconds or faster), STEP transitions can be important. The cross section for the second absorption transition for a STEP mechanism can be determined experimentally from the fluorescence spectrum by using the ex­ pression9

p hv P2 !J.t CT23 p�; hh hv2 0.375/p ' 3

(3.4.22)

where h and h are the fluorescence intensities of the emission from the in­ termediate and terminal states of the STEP transition immediately after a Gaussian pump pulse of intensity lp and temporal width !J.t. hvi represents the energy of the photons emitted from the ith level, p; represents the radi­ ative emission rate for the ith level, and pi is the total decay rate of the ith level. Thus the cross section for a STEP transition depends on the ratio of the excitation pulse width to the intermediate-state lifetime. The STEP mechanism has been found to be important when high-power, short-pulse lasers are used to pump ions at a wavelength in resonance with a two­ photon transition having a real intermediate state. This can be very useful as a spectroscopic tool to elucidate the radiative and radiationless relaxation properties of excited states of ions in laser materials. Examples of this are discussed in Chaps. 8 and 9.

References

1 15

The third type of two-photon absorption processes is excited state ab­ sorption ( ESA), which generally occurs from the metastable state in lasing ions. Since a significant number of ions can be in the metastable state during the optical pumping of laser materials, this can be a significant process in the pumping dynamics. Two types of ESA processes are important, one in­ volving the absorption of laser photons and the other involving the absorp­ tion of pump photons. The first of these can act as a significant loss mecha­ nism that degrades laser performance. It is a major factor in limiting the tuning range of vibronic laser materials and can prevent some materials from lasing at all. These effects are discussed for specific laser materials in Chaps. 6-10. The second type of ESA process can be especially effective for monochromatic pump beams if they are at a wavelength in resonance with an ESA transition. This can be very important for diode laser-pumped solid­ state lasers. The ESA of pump photons results in decreased pumping effi­ ciency for the laser. However, in some cases there are higher-energy meta­ stable states of the ion that can laser. In this case ESA processes are termed up-conversion pumping for the upper laser level. The effects of these pro­ cesses on specific solid-state lasers are discussed in Chaps. 8 and 9. As discussed above, the nonlinear optical effects associated with the re­ fractive index can be important in determining the beam quality of solid­ state laser operation, while the effects associated with multiphoton absorp­ tion provide both loss mechanisms and alternate pumping schemes for laser operation. Both types of effects will be discussed further in Chaps. 6-10 where specific types of lasers are considered. References

1 . W. Heider, Quantum Theory of Radiation (Oxford University Press, London, 1 944) . 2. J.D. Jackson, Classical Electrodynamics, ( Wiley, New York, 1 975) . 3. C. Coen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Photons and A toms ( Wiley, New York, 1 989). 4. H.A. Bethe and E.E. Salpeter, Quantum Mechanics of One- and Two-Electron A toms (Springer-Verlag, Berlin, 1 957) . 5. E. Merzbacher, Quantum Mechanics ( Wiley, New York, 1 96 1 ) . 6. R. Adair, L.L. Chase, and S.A. Payne, J. Opt. Soc. Am. B 4 , 875 ( 1 987); Phys. Rev. B 39, 337 ( 1 989) . 7. M.J. Weber, D. Milam, and W.L. Smith, Opt. Eng. 17, 463 ( 1 978) . 8. R.C. Powell, S.A. Payne, L.L. Chase, and G.D. Wilke, Phys. Rev. B 4 1 , 8593 ( 1 990) . 9. G.J. Quarles, G.E. Venikouas, and R.C. Powell, Phys. Rev. B 31, 6935 ( 1 985).

4

Electron-Phonon Interactions

The ions in a crystalline or glass solid are never completely at rest. The thermal vibrations of these ions modulate the local crystal field at the site of an optically active point defect. This modulation can have several types of effects on the optical properties of the defect. For example, it can modulate the position of the electronic energy levels, thus leading to a broadening and shifting in peak position of the spectral transition. Also it can cause tran­ sitions to occur between electronic energy levels accompanied by the ab­ sorption or emission of thermal energy but with or without the emission or absorption of photons. Since the motion of the host ions is due to thermal vibrations, all of these effects depend on the temperature and the thermal conductivity of the material. In addition, they depend on the strength of the coupling between the electrons of the optically active ion and the local crys­ tal field. Ions with unshielded optically active electrons exhibit stronger thermal effects than those with their optically active electrons in inner-shell orbitals shielded by outer-shell electrons. Finally, these effects depend on the frequency distribution and density of states of the host vibrational modes. For example, hosts with high-frequency vibrational modes can be more effective in causing radiationless transitions between electronic energy levels with a large energy separation. When the optically active defect differs in mass or charge from the host ion for which it substitutes, local modes of vibration can be introduced that have frequencies that are different from the lattice modes. These local modes can be effective in contributing to the thermal properties of the optical spectrum before dissipating into the host lattice modes. Thermal effects can improve laser performance through processes such as enhanced optical pumping, through efficient depopulation of the terminal level of the laser transition, and through provision of broad emission bands for tunable laser emission. However, thermal effects can also be detrimental to laser performance through processes such as decreasing the quantum efficiency of the laser transition, decreasing the gain cross section through spectral line broadening, and thermally populating the terminal level of the laser transition. In the following sections thermal effects on the optical 1 16

4. 1 . The Phonon Field

1 17

spectra of ions in solids are described in terms of the weak coupling and strong coupling cases. Thermal effects on specific laser systems are discussed in Chaps. 6-10. 4. 1

The Phonon Field

The thermal vibrations of the atoms or ions in a solid are best described as quantized harmonic oscillators. This is similar to the treatment of the radia­ tion field outlined in Sec. 3. 1 . The quantized thermal vibrations provide a field of phonons instead of the field of photons associated with electro­ magnetic radiation. The atoms in the solid vibrate about their equilibrium positions, and since the atoms are bound to each other, their vibrational characteristics are coupled to each other. If there are N atoms in the solid, there are 3N modes of vibration grouped into two branches, acoustic pho­ nons and optic phonons. The problem is thus to find the solutions of the coupled harmonic oscillator equations in terms of the normal modes of vibration and normal coordinates of the solid. For molecular solids and glasses having only short-range order, local modes of vibration are impor­ tant, whereas for crystals with long-range order the extended lattice vibra­ tional modes provide the critical contribution to the thermal properties of interest. In general, the term phonon will be taken to include both types of vibrational modes, even though it is usually reserved for vibrational modes where the wave vector is a good quantum number. Placing dopant ions in the solid can alter the vibrational modes of the host as discussed below. The mathematical approach to any of these cases begins with the fundamental procedure for describing the vibrations of ions in a simple crystal lattice. This can be found in any solid-state physics textbook1 and is outlined below. The first step in the problem is to determine the Hamiltonian for the sys­ tem. This can be expressed as H

Hion + H,

+ H? ,

where the three terms on the right-hand side of the equation are the Hamil­ tonians for the ion, the lattice, and the electron-phonon interaction. To de­ rive the lattice Hamiltonian, consider a crystal lattice with basis vectors a , , a2 , and a . The position of an ion in the lattice is given by the primitive 3 lattice vector (4. 1 . 1 ) where the n; are integers. The crystal is assumed to have one atom per unit cell and contains N N1N2 N3 cells, where N; is the number of cells in the a; direction. Next, consider the vibration of atoms on this crystal lattice. This is done by expanding the expressions for the kinetic and potential energies of the vibrating atoms about their equilibrium positions. In the harmonic approx-

118

4. Electron Phonon Interactions

imation, only the quadratic terms of the expansions are retained. Assuming that the interatomic forces are proportional to the relative displacements of the atoms, the kinetic energy T and potential energy V of the atomic motion are expressed as

T = 2I m V=4

N 3

i=l !i.=l

U' ;2a '

( 4. 1 .2)

N N 3 3 L L L L A ;a,jp U;a �p ,

( 4. 1 .3)

i= l j=l a= ! P=l

where U;a is the displacement of the ith atom in the et. direction at time t, and the A ;a,jft are constants describing the interatomic forces. The Hamil­ tonian for the lattice is then given by H=

N N 3 3 1 N 3 T+ V=l ! f + L L L L L A;a,jp U;a �p , L P a m

(4. 1 .4)

where P ia = m U;a . Because the Hamiltonian is intrinsically harmonic, the complex normal coordinates can be introduced with the following Fourier transform pairs, i= l a=!

Ur.a QqA =

i=l j=l a=! P=l

q,A l, !i.

A eiq · r; QqA eqa ,

(4. 1 .5)

iq r; Ul·!i. i• ' qa e

( 4. 1 .6)

where q ranges from 1 to N and A ranges over the different acoustic and optic branches. Since Q� q = Q;* , the coordinates associated with q and - q are not independent. In order to deal with only the 3N-independent real co­ ordinates, Eq. (4. 1 .5) can be rewritten as U!!1.·

=

1

__

N/2 q> O A

q qa e

iq · r,

+ QqM iqa• e

iq · r, )

·

( 4. 1 . 7)

Using Eqs. (4. 1 .5)-(4. 1 .7) in Eqs. (4. 1 .2) and (4. 1 .3), the kinetic and po­ tential energies can be expressed in terms of the normal coordinates. The kinetic energy becomes

=_

1 2N

z,rx

q,A q',A'

)

. . A eA' . QqA QqA'' ei(q+q' ) · r, eqa q'a

1 19

4. 1 . The Phonon Field

The sums over i and IJ. can be evaluated using the orthogonality conditions, 1

which reduces this expression to

T=2

(4. 1 . 8)

q,.l.

Similarly, the potential energy becomes V = l2

i, rx j,p

1 A· · rrx , JP m

N

q,.l.

Qq.l. e.l.qrx e iq·r,

)

q ' , .l.'

Qq.l.'' eq.l.''P eiq' ·r1

)

This can be simplified by defining a function Grxp (q' ) through /

1

L. A trx,JP erq -r1 m

_

L. A trx,. J.P e - r"q' (r,-r1 ) erq ·r; m 1

•

·

J

J

I

. Grxp (q' ) erq r, · •

I

Therefore the potential energy can be written as

=2

q, .l.

w2q.l. Qq.l. Q .l. q ·

(4. 1 . 9)

In deriving the above expression, the following relationship was used 1 :

3

p

' GrxP ( - q ) e.l._qp

w2q.l.' e.l.' qrx

_

1 m

p

j

A

i ·(r, r1 ) e.l.' irx,jp e q qp ·

(4 . 1 . 1 0)

The Lagrangian for the system is (4 . 1 . 1 1 )

1 20

4. Electron Phonon Interactions

Thus the momentum conjugate to Q; is ).

aL

Pq = a . ). = Qq

=

PA*q ·

(4. 1 . 1 2)

The Hamiltonian is then given by

H = L P;Q; - L = ! L p� q p; + ! L W�;. Q� q Q: q, ). q, ). q, ). 2 ). ). M ! =2 q pq + Wq). Qq Qq ) . q, ).

(4. 1 . 1 3 )

For a quantum-mechanical system the form o f the Hamiltonian i s the same as Eq. (4. 1 . 1 3) except that the normal coordinates and conjugate momenta are operators,

H = ! L (P;t p; + W�;_�t Q: ) . q, ).

(4. 1 . 14)

This has the form of the Hamiltonian of an ensemble of harmonic oscil­ lators. The usual creation and annihilation operators can be formed (see Sec. 3 . 1 ), ( 4. 1 . 1 5)

These expression can be inverted to express the coordinates and momenta in terms of the creation and annihilation operators: (4. 1 . 1 6)

These operators create and annihilate phonons in the normal way for har­ monic oscillator models

b; I n; ) = b;t 1 n; ) =

;; I n; - 1 ) ,

+ l i n; + 1 ) ,

(4. 1 . 1 7) (4. 1 . 1 8 )

where the n� are the occupation numbers for phonons with wave vector q and branch A.. The Hamiltonian for the phonon field can now be written in the form

3N H1 = L hwq;. (b;tb; + !) . q, ).

(4. 1 . 19)

The electron-phonon interaction Hamiltonian can be determined from the expressions derived above. Substituting the expression for Q; from Eq. (4. 1 . 1 6) into Eq. (4. 1 . 5) gives an expression for the displacement of the

4. 1 . The Phonon Field

121

atom from its equilibrium position in terms of the creation and annihilation operators UIIX

=

=

2,; m

�" eiq · r• (bAq + bAtq ) -

q, A

;:; q ,.<

2mqA q"

- q"

q

e- iq · r; bAtq ) -

·

(4 . 1 .20)

The electron-phonon interaction takes place through the change in the crystal field due to the change in the relative positions of the active ions and their surrounding ligands. This is proportional to the local strain defined by the tensor 8"p

= 21

(

a u"

+

)

a up ax" '

(4. 1 .2 1 )

where r:x and fJ represent the three possible directions of motion. Approx­ imating this expression by the average strain e�

au ax '

I

(4. 1 .22)

b� q ) , =iL q where M = mN and the phonon velocity is given by v mq /q. In Eq. (4. 1 .22) the phonon polarization and branch designations e�" have been suppressed and it is implied that the index q refers to a phonon with a spe­ cific wave vector, polarization, and branch. The expansion of the crystal field in terms of the strain is e�

X=0

=

(4. 1 . 23)

The first term Vo is the static crystal field term that was discussed in detail in Chap. 2. The remaining terms represent the electron-phonon interaction H? = V1 e + V2 e2 +

·

·

·

= H�P + H�P , where the linear interaction term is Hiep

- l VJ ·

·

l

q and the quadratic interaction term is H�p =

v2

LiL q

q'

bq

bt q ) .

b � q ) (bq'

( 4. 1 . 24)

b � q' ) .

(4. 1 .25)

122

4. Electron Phonon Interactions

When specific normal modes of vibration dominate the electron-phonon interaction, it is useful to express the electron-phonon interaction Hamil­ tonian in terms of these modes instead of the general ion displacement ex­ pressions used above. In this case, the crystal-field expansion in terms of the strain field given by Eq. (4. 1 .23) is replaced by an expansion of the crystal field in terms of the normal displacement coordinates of the complex, Qq , a2 v av 1 Vc = Vo + L a Qq + 2! L a a QqQq' + . . . q Qq 0 qq' Qq Qq' 0 (4. 1 .26) = Vo + L Vq Qq + L Vqq' Qq Qq' + q qq' Each normal mode Qq is a linear combination of the displacements of the ions as given by Eq. (4. 1 .6) and can be expressed in terms of creation and annihilation operators as given by Eq. (4. 1 . 1 6) . The coupling parameters Vq are functions of the coordinates of the electrons on the central ion. The electron-phonon interaction Hamiltonian is now given by

I

·

·

·

I

.

H? = L VqQq + L Vqq' Qq Qq' + . . . q

where

= HfP + H;P ,

qq'

(4. 1 .27)

(4. 1 .28) (4. 1 .29)

The interaction Hamiltonians expressed in terms of phonon creation and annihilation operators in Eqs. (4. 1 .24), (4. 1 .25) and (4. 1 .28) , (4. 1 .29) along with the eigenfunctions of the lattice vibrations or normal modes expressed in terms of the phonon occupation numbers lnq ), can be used to treat phys­ ical processes involving electron-phonon interactions. The choice between using the strain Hamiltonian or the normal-mode Hamiltonian depends on the properties of the electron-phonon coupling interaction for the specific physical process being treated. If the interaction is dominated by local modes of the active ion and its ligands the normal mode approach is appropriate, whereas if the interaction involves lattice modes the strain approach is appropriate. Several of these types of processes that are impor­ tant for understanding laser materials are discussed in the following sec­ tions. When only a few phonons are involved, simplifying approximations can be made that lead to the weak-coup ling limit. For multiphonon inter­ actions, the strong-coupling limit must be used. These two extreme cases are treated separately below.

4. 1 . The Phonon Field

1 23

Group theory is an important tool for evaluating selection rules for radi­ ationless transitions just as it was used in Chap. 2 for radiative transitions. The phonons present in the host material can be expressed in terms of the normal modes of vibration of the lattice. These transform as irreducible rep­ resentations of the symmetry group describing the geometry of the lattice. For an undoped material, the symmetry operations include translational symmetry and the full space group of the lattice must be used in determining all possible phonon modes. However, when the material is doped with an impurity ion, translational symmetry is destroyed and the representations of the vibrational modes can be reduced in terms of the irreducible representa­ tions of the point symmetry group at the site of the impurity ion. It also may happen that the presence of the impurity ion may introduce local modes of vibration that are not present in the undoped material. Thus the most important procedure for doped materials is to find the normal modes of vibration of the local molecular complex consisting of the dopant ion and its ligands, determine how they transform in the local site symmetry group, and evaluate the selection rules for nonradiative transitions using these irreducible representations. If it is of interest to determine the contribution to radiation­ less transitions of a pure lattice phonon with specific translational symmetry as determined from lattice dynamics studies such as neutron-scattering measurements, it is possible to use group theory to do this. The reader is referred to Ref. 1 for the details. For a complex of N ions in a specific geometrical configuration described by a point site group, there will be a representation of the group that de­ scribes the motion of the ions. The irreducible representations contained in the reduction of this representation are those that describe the translation, rotation, and normal modes of vibration of the complex. In order to deter­ mine the characters of the total motion representation 1m , a set of coordi­ nate vectors are attached to each ion to describe their displacement from their equilibrium positions, and the transformation properties of these vec­ tors under the operations of the group are determined. These 3N vectors form the basis set for the motion representation with dimension 3N. All of the matrices describing operations of the group are 3N x 3N matrices con­ sisting of 3 x 3 submatrices for each ion of the complex. If the matrix oper­ ation on the complete set of basis functions is written in complete form, it is obvious that the submatrices associated with ions that do not change posi­ tion under that symmetry operation appear on the diagonal of the matrix, while the submatrices associated with ions that change positions appear off­ diagonal. Therefore, only those ions that do not change position under a specific symmetry operation will contribute to the character of the motion representation for that symmetry operation. The character is then given by the number of ions left in the same position multiplied by the character of the 3 x 3 matrix of the operator operating on one of these ions. As discussed in Sec. 2.3, the general matrix operator for a symmetry operation on a posi-

1 24

4. Electron Phonon Interactions

tion vector is

=

with a character given by

Sil ()(

0 0 ±1

COS ()(

Xi

0

=

± 1 + 2 cos a.

) YZii ,

(4. 1 .30)

(4. 1 .3 1 )

Here a is the angle of rotation for the operation and + 1 and - 1 refer to proper and improper rotations, respectively. As an example, consider the octahedral complex shown in Fig. 2.4. This has Oh symmetry with the character table given in Table. 2.4. Equation ( 4. 1 .3 1 ) can be applied to a symmetry operation of each of the ten classes of the group to determine the characters for the ion motion representation rm of this complex. For example, all seven ions remain unchanged under the E operation and each of these contributes three to the total character giving rm ( E ) = 2 1 . A C4 operation around any of the major axes leaves only the three ions located on the rotation axis unchanged and each of these con­ tributes + 1 to the total character giving rm ( C4 ) = 3. The inversion oper­ ation leaves only the central ion of the complex unchanged in position and according to Eq. (4. 1 . 3 1 ) this gives rm ( i) = - 3. A similar procedure is carried out for elements of the rest of the classes resulting in the characters given in Table 4. 1 . Using the full character table for the Oh group and Eq. (2.2.8), the motion representation can be reduced in terms of the irreducible representations of the group. As shown in Table 4. 1 , the results of doing this give one one-dimensional, one two-dimensional, and six three-dimensional representations for the rm motion representation reduction. The total degeneracy of the motion representation is associated with the 3N degrees of freedom of the complex. Three of these are associated with translational motion and three with rotation. Since these degrees of freedom are not of interest for an ion in a solid, the irreducible representations in the reduction of rm associated with translation and rotation need to be elimi­ nated to leave only those associated with the vibrational modes of the complex. The three translational degrees of freedom transform as a position vector while the three rotational degrees of freedom transform as an angular momentum vector. TABLE 4. 1 . Character table for the motion representations of an octahedral complex. oh E

rm rT rR

2 1 -3 3 -3 3 3

6C4 6iC4 3Cz 3iCz 6q 6iq 8C3 8iC3 3

1 1 1

-3 1 1

5 1 1

1 1 1

3 1 1

0 0 0

0 0 0

A t 9 + E9 + 3Ttu + Tt 9 + Tz9 + Tzu Ttu Tt 9 A t 9 + E9 + 2Ttu + Tz9 + Tzu fv =

4. 1 . The Phonon Field

1 25

For the example of the octahedral complex discussed above, the total number of degrees of freedom for the motion is 21 . By considering the effects of an operator from each class of the Oh symmetry group on the (x, y, z) components of a vector, it is seen from Table 2.3 that the a vector transforms as the three-dimensional T1 u irreducible representation in this group. Similar considerations show that the ( Lx, Ly, Lz) components of the angular momentum vector transform as the T1 9 irreducible representation of Oh. If the T1 9 and one of the T1 u representations are subtracted from the rm representation, it is seen in Table 4. 1 that there are six normal modes of vibration of the octahedral complex transforming as A 1 9, E9, Tz9, T1 u, T1 u, and Tzu · By performing the usual vibrational analysis of mechanics, it is possible to determine the direction of motion of the ions associated with each of the normal modes of vibration. The results are shown in Fig. 4. 1 . The relation­ ships between the symmetry designations of the normal modes and the di­ rections of ion motions can be found from group theory. The relationship between the normal-mode symmetry coordinates and the coordinates giving

T2u

FIGURE 4. 1 . Fifteen normal modes of vibration of an octahedral complex.

126

4. Electron Phonon Interactions

the relative positions of the ions is given by

S(ry ) = N L Xy( R ) RSi, R

(4. 1 .32)

where S(ry) is the symmetry coordinate transforming as the rr representa­ tion, R is one of the symmetry operators, Si is a specific coordinate of the position of an ion, and N is a normalization factor. For the octahedral complex shown in Fig. 2.4, there are six radial coor­ dinates ri and twelve angular coordinates ()iJ representing the positions of the ligands around the central ion. Let i = 1 and 6 represent the ions along the +z and -z axes in the figure, i = 2 and 4 the ions along the +x and -x axes, and i = 3 and 5 the ions along the +y and y axes, respectively. To begin the analysis, r1 is chosen as the ion coordinate. To apply Eq. (4. 1 .32) it is necessary first to find the coordinates into which r, transforms under all R symmetry operations. The results are easily found by inspection using Fig. 2.4. They are listed in the following table: R7; R E 6C4 3Cz 6q 8C3 i 6S4 3 ah 6a� 8S6

71 2 0 0 0 0 2 2 0

7z

73

74

7s

76

0

0

0

0

0 I 2 0 I 0

0 I 2 0 I 0

0 I 2 0 I 0

0 I 2 0 I 0

0 0 2 2 0 I 2

2

2

2

2

0 0

The results given in this table can be used for the RSi factor in Eq. (4. 1 .32) . Then S(ry) can be found using the characters for the symmetry operators in the irreducible representations describing the normal vibrational mode given in Tables 2.4 and 4. 1 . The result for the A ig representation is

where the normalization factor has also been calculated. Similar procedures for the E9 and T1 u vibrational modes give

4.2. Weak-Coupling: Radiationless Transitions

1 27

and

This procedure can be carried out for each of the r; and ()iJ coordinates until the expressions for the ion coordinate motion is determined for all fifteen of the symmetry vibrational modes shown in Fig. 4. 1 . The results of doing this are listed in Table 4.2 in both radial and Cartesian coordinates. When nearest-neighbor interactions are dominant, it is useful to express the electron-phonon interaction Hamiltonian in terms of the symmetry modes of collective ion motion instead of the lattice phonons. The local symmetry vibrational modes are linear combinations of the normal phonon modes. In this case, the crystal-field expansion in Eq. (4. 1 .26) in terms of the Qq normal modes is used. Since H? must be invariant under all symmetry operations of the point group of the complex and therefore transforms as A 1 9, the reduction of the direct product of the representations according to which Vq and Qq transform must contain A 1 9 • The only way for this to occur is for Vq to transform according to the irreducible representation of the q normal vibrational mode r(Qq) · This fact will be used below in deter­ mining selection rules for transitions involving electron-phonon interactions. 4.2

Weak-Coupling: Radiationless Transitions

The transitions between electronic states of an atom accompanied by the emission or absorption of phonons without photons being involved are called radiationless transitions. If the energy levels are close enough together that the energy difference involved in the transition is less than the highest­ energy phonons allowed in the material, then direct transitions involving single phonons are possible. These processes will be considered first and then the treatment will be extended to processes involving more than one phonon. The quantum-mechanical interaction diagrams ( Feynman diagrams) for direct processes involving phonon creation and annihilation are shown in Fig. 4.2( B). The system under consideration consists of the electronic states of the atom and the lattice vibrations. This can be represented in the weak­ coupling limit by product-state functions including one part describing the atom and another part describing the phonon field,

(4.2. 1 ) Since no interaction with the radiation field is involved, it may be possible for phonons in any region of the Brillouin zone and in any branch to par­ ticipate in these processes. Therefore the interaction causing the transition is described by the linear electron-phonon interaction Hamiltonian in Eq.

QI Qz Q3 Q4 ·Q5 Q6 Q? Qg Qg Qw Qu Qu Ql 3 Q I4 QI5

Normal mode

Cartesian coordinates

(xi - x4 + Y2 - Y5 + z3 - z6)/ V6 (xi - X4 - Y2 + Y5)/2 ( -x 1 + X4 - Yz + Y s + 2z3 - 2z6)/2../3 (z2 - z5 + Y3 - Y6)/2 (x3 - X6 + ZI - z4)j2 (yi - Y4 + x2 - x5) j 2 (x2 + X3 + x5 + x6)/2 (Y I + Y3 + Y4 + Y6)/2 (z 1 + z2 + Z4 + z5)/2 (x1 + x4)/V2 (Y2 + Y 5 ) j,fi (z3 + z6)/V2 (x2 + X5 - X3 - X6)/2 (Yl + Y6 - Yl - Y4)/2 (zi + Z4 - z2 - z5)/2

Radial coordinates

(r 1 + r4 + rz + r5 + r3 + r6)/V6 (r4 - rz - r3 - r5)j2 (2ri + 2r6 - r4 - rz - r5 - r3 )/2 .f3 (BI5 + 863 - 813 - 865 )/2 ( Bu - BI4 + 864 - B6z) /2 (Bz3 + 845 - 834 - B5z)/2 (2B6z - 2B5z + B6I + 864 - Bn - 834 + Biz + Bz4 - B5I - 854) /4 (2Bu - 2845 + B6z + B1z - 865 - 835 + 86 1 + Bn - 864 - 834)/4 (2Bn - 2864 + Bu + B5I - 845 - Bz4 + 83z + 835 - 862 - 865 )/4 (r2 - r4)j,fi (r3 - r5) j,fi (ri - r6)/V2 (2862 - 2823 - 2853 + Bn + 834 - 86 1 - 864 + 85 1 + 854 - BI 2 - B24)/2v'5 (2Bu - 2815 - 2845 + 835 + 856 - 862 - 83 2 + 864 + 843 - B6I - 83I ) j 2.j5 (2Bn - 2834 - 2864 + 824 + 845 - 85 1 - Bu + B6z + 865 - 83 2 - B35)/2v'5

Symmetry

A 19 E9 (v) E9 ( u) Tz9 ( ¢ ) Tz9 (1'!) Tz9 ( () T[u (x) T[u (Y) T[u (z) Ttu (x) Ttu (Y ) Ttu (z) Tzu ( ¢ ) Tzu ( l'!) Tzu(()

TABLE 4.2. Symmetry and ion motion coordinates for an octahedral complex.

N 00

"'

0 ::l

::l 0 ::l

g

7 d1 0

t!l 0

:I'"

-

4.2. Weak-Coupling: Radiationless Transitions

1 29

(B)

(A)

FIGURE 4.2. (A) Energy-level diagram and ( B) diagrams quantum-mechanical inter­ action for single-phonon absorption and emission processes.

(4. 1 .24). The transition rate for weak interactions is given as usual [see Eq. (3. 1 .27)] by the golden rule of time-dependent perturbation theory (4.2.2) where Mnr is the matrix element for the transition and p1 is the density of final states. The matrix element involves the interaction Hamiltonian and the wave functions given in Eq. (4.2. 1 ), Mnr = ( \('J I VI e l \(1;) . The transition rates for direct phonon absorption and emission processes involving specific phonons are given by

(I f l i V1 � bq I ) 1 2p1 (EJ

w�� =

\(I

l

\(I ;

=

E;)

l ( t/Jj1 1 VI It/1�1 ) 1 2 (nq - l l bq l nq) i 2pf (EJ = E;) nq I ( t/IY I VI It/1�1 ) 1 2PJ (EJ = E;)

=

=

(4.2.3)

and (4.2.4) The density of final states is expressed as the product of the densities of states of the electronic and vibrational parts of the system,

p1 (EJ = E;) = J (EJ E; )p(EJ ) dEJ 1 1 = J[(EY + Ej ) - (E; + Ef )]p(Ej )p(Ej ) dEj =

�

J[(w� wq)] g(w� wo)p(wq ) dw�dwq.

(4.2.5)

1 30

4. Electron Phonon Interactions

Here the electronic transition is defined by a frequency wt = ( Ef1 - E;t)/h and a normalized line-shape function g(wt - wo) � J(wt - wo) , where wo is the central frequency of the transition. This involves the assumption that the electronic part of the transition is very narrow compared to the spread in vibrational energies. In order to develop this expression further, a model must be chosen to describe the vibrational density of states of the solid. There is a variety of ways to determine this density of states ranging from experimental measurements of vibronic sidebands in optical spectra to fun­ damental theoretical lattice dynamics calculations. The most common pro­ cedure is to assume a Debye model. 2 The phonon density of states in the Debye approximation is given by

p (w) = �n2 v3

3 Vw2

for w :: wn

(4.2.6)

for w > wn where wn is the Debye cutofffrequency. In addition, in this model for lattice vibrations, the phonon occupation numbers are given by the Bose-Einstein distribution function (4.2.7)

where kn is Boltzmann's constant and T is the temperature. The total transition rates for direct phonon absorption and emission pro­ cesses are found by substituting Eqs. (4.2.5)-(4.2.7) into Eqs. (4.2.3) and ( 4.2.4) and integrating over Wif and Wq . Because of the J, functions involved, the results of the integrations can be immediately written as

Wnrab =

npv npv

3w�

2

-

w�� =

p

I

1 2 no , 2 ;t Vt lt/l�l ) (no

I (t/l

Vl

l

1

( 4.2.8) +

1),

(4.2.9)

where the density is M/ V . These expressions can be rewritten in terms of Eq. (4.2.7) in order to show explicitly the temperature dependences of the transition rates. Considering radiationless transitions from an initial elec­ tronic state to all possible final states, the absorption and emission rates are

) hwofk8T e Pif ( � ehwo /ks T 1 ) ' f
w�; = � Pif

1 '

f>l

w�� where

-

(4 . 2. 10 ) (4.2. 1 1 )

( 4.2. 1 2)

4.2. Weak-Coupling: Radiationless Transitions

131

theoretical calculation from first principles of the electron-phonon cou­ pling coefficient fJ is generally difficult to do because the exact expressions for V1 are not known. Thus the values of these parameters are determined from fitting theory to experimental data. However, it is possible to use group theory to determine when the matrix element in Eq. (4.2. 12) is zero or nonzero for a specific vibrational mode. Since the electronic eigenfunctions for the initial and final states of the transition transform as irreducible rep­ resentations of the group r ; and r1 and the coupling parameter V1 trans­ forms as rq for the qth vibrational mode, the reduction of the direct product r; X rq X r/ must contain the totally symmetric irreducible representation A1g for the matrix element to be nonzero. Thus only certain vibrational modes can cause nonradiative transitions between specific electronic states. Note that the rates of direct phonon emission and absorption between two specific states differ only in their temperature-dependent factors. If the transitions are fast enough between the two levels, the populations of the levels are said to be in thermal equilibrium. This condition can be described by writing the rate equations for the populations of the levels shown in Fig. 4.2 ( A ) , A

(4.2. 1 3)

where n , + n2 = N. In thermal equilibrium the time derivatives can be set equal to zero and these equations solved to give

n,

=

hw/kB T N +e ehw/kB T ' 1

(4.2. 14) (4.2. 1 5)

and thus

n2 hw/kB T . n, e

(4.2. 1 6)

The time it takes for a system to reach this equilibrium population distribu­ tion can be determined by defining a system relaxation time r ,

, 1

=

wnrem + wnrab

( )

3w3 ( el v, I el 2 coth hw 2npv5h I t/11 l t/1; ) 1 2knT The rate equations in Eq. (4.2. 1 3) can be solved to give n, ( t) n� e - tfr + NW��r( 1 e tfr ) , n2 (t) n�e -tfr + Nw:; r(1 e -tfr ) , =

·

(4.2. 17)

(4.2. 1 8)

1 32

4. Electron Phonon Interactions

where the quantities n? and ng are the initial populations of the levels, and the equilibrium populations given in Eqs. (4.2. 14) and (4.2. 1 5) can be expressed as

n1 ( ) NW�:,n r, n2( ) oo

oo

NW�� r .

(4 . 2 . 19)

Next consider radiationless transitions involving two phonons. There are two contributions to these processes, one that is described by the single­ phonon term in the electron-phonon interaction Hamiltonian and second­ order perturbation theory, and the other that is described by the two­ phonon term in the interaction Hamiltonian and first-order perturbation theory. These contributions add coherently. In general it is assumed that higher-order terms in the crystal-field expansion are very small and only the first contribution is considered. Since both contributions involve the same change in phonon occupation numbers, they have the same temperature dependence. The temperature dependence of the transition rate is the main measurable parameter, and thus it is difficult to distinguish between these two types of contributions. The physical processes of this type can be sepa­ rated into those involving two phonons being simultaneously absorbed or emitted, and those involving the absorption of one phonon and the emission of another phonon. These different cases are shown schematically in Fig.

4.3.

First consider processes involving the absorption or emission of two pho­ nons. In many cases, the ion couples strongly to one specific type of phonon mode, and in this case both phonons involved are of the same mode. The contribution to the transition rate for a two-phonon emission process de-

-'Vj'M f,l 'Vf.f

ljlel

(A) Direct Processes

6.

(B) Ram an Processes

(C) Orbach Processes

FIGURE 4.3. Two-phonon radiationless transitions.

4.2. Weak-Coupling: Radiationless Transitions

1 33

scribed by the first-order interaction Hamiltonian in Eq. (4. 1 .24) and second-order perturbation theory is

I

1 (2) ( em ) _ 2n (lfr%1 i (nw + 2 1 HfP i nw + 1 ) i lfr�) (lfr: i (nw + 1 IHfP I!fr�) l nw ) h Ea - (Ea + hw) 1 n + 1 lfrt) \lfr%1 l (nw + 1 IHfP I !fr� ) l nw ) 2 n +2 + \lfr% 1 ( w IHfP i wE -) l(E P! a b + hw) M;lM;� MglM;l 2 2n 1 n n + ) ( 2) + w h 2Mw w -hw + hw (4.2.20)

w nr

l

I

1

Note that the contribution described by the second-order interaction Ham­ iltonian and first-order perturbation must be added coherently to this ex­ pression. A similar expression can be written for the rate of absorption of two phonons of the same mode. For transitions involving three or more phonons, the procedure described above leading to Eq. (4.2.20) for two-phonon processes can be extrapolated by using higher-order perturbation theory. For a p-phonon process the expression for the transition rate will involve 2P l matrix elements. If the coupling is to a large number of phonons m, terms involving the emission of one of each type of phonon dominate terms involving the emission of many phonons into one highly excited mode. The general rate for a p-phonon emission transition of this type is given by

�r (em )

w

w; = x

� ,wm

+

1 )p / 2

( lfr%1 1 HfP llfr:1) ( lfr%1 1 H�P llfr:1) ( lfr�1 1 H�P llfr�1) (Ea - Eab - hw;) (Ea - Eab - 2hw;) [Ea - Eab - (p - 1 )hw;] ·

·

·

·

·

·

(4.2.21 ) Since the transition is sharply peaked about a particular frequency wo this expression can be simplified by defining an average matrix element - 1 such that = I M;l i P /(hwo)P

p ( em )

wnr

2n h 2Mwo

2 ) nw0 + 1 )P 22 (p - 1) m2p I M;l2 (p1 P l ) t5 (Ea _ ( Eb + hwo ) . ( hw0 ) (4.2.22)

Although it is difficult to calculate theoretical values for the radiationless decay rates using these expressions, it is possible to estimate the number of

1 34

4. Electron Phonon Interactions

phonons involved in the process. This can be done by comparing the transi­ tion rates of the p and p - 1 processes. From the preceding equations,

2 4m2 I M;i l ' n 1 ) + w o 2MWo ( hwo) 2

WKr( em) WKr I (em)

where the assumption has been made that the average matrix elements for the two processes are the same. For small values of the electron-phonon interactions,

wgr( em) wgr- I (em)

1

e «

.

Under these weak coupling conditions, the transition rate can be expressed in the form of an exponential,

WKr (em) = wgr l ( em ) = wgr- 2 ( em ) 2 = Wo el'ln (e) = Wo e ( !!E/ hw) ln ( ) . e

e

.

.

.

=

w�r (em)lf

e

(4.2.23)

In the final expression, the assumption has been made that the nonradiative decay process involves p phonons of equal energy hw crossing an electronic energy level gap of AE. This result predicts the well-known energy gap law for radiationless relaxation processes in rare-earth ions. This will be applied to specific cases in Chaps. 8 and 9. The prefactor to the exponent in Eq. (4.2.23) is difficult to calculate be­ cause of the lack of knowledge about the nature of the electron-phonon interactions involved in the matrix element. However, the temperature de­ pendence of the transition rate is contained in this factor through the pho­ non occupation numbers. From Eqs. (4.2.22) and (4.2.23), the temperature dependence of the process can be expressed explicitly as

WKr (T)

(

)p _

WKr (O) (nw + 1 )f ehw/kn T WKr (O) ehw jk8 T 1 '

(4.2.24)

where Eq. (4.2.7) has been used for the phonon occupation numbers. This expression shows that radiationless decay processes involving different numbers of phonons exhibit distinctly different temperature dependences. These can be measured experimentally and compared to the predictions of Eq. (4.2.24) to determine the value of p for the process of interest. This can then be used to determine the frequency of the phonon involved in the pro­ cess from the relationship phw AE and the known value of the energy gap. The treatment outlined above involving one effective phonon mode has been very useful in understanding the multiphonon radiationless decay pro­ cesses affecting the optical pumping dynamics of many rare-earth-doped =

4.2. Weak-Coupling: Radiationless Transitions

135

1 .0 0. 8

D ensity of S tates

10

20

Phonon

30

40

Frequency (x lo12 sec-1)

50

FIGURE 4.4. Density occupied phonon states in the Debye approximation.

laser materials. However, it must be remembered that the frequency spec­ trum of phonons for any specific type of solid has a significant amount of structure in its density of states, and the symmetry properties of different types of phonons can cause the matrix elements for electron-phonon inter­ actions to be quite different. Thus the above treatment is a very rough approximation to the real physical situation. For physical systems involving stronger coupling, the specific nature of the phonon spectrum must be taken into account as discussed in the following sections. The other types of two-phonon processes shown schematically in Fig. 4.3 involve the absorption of one phonon of frequency Wa and the emission of another phonon of frequency We . These processes can be important in non­ radiative transitions when the energy splitting between the electronic levels is so small that the density of states of phonons with the frequency necessary for a direct one-phonon transition is very small. Figure 4.4 shows a typical situation for the density of occupied phonon states, which can be seen from Eqs. (4.2.6) and (4.2.7) to be proportional to w2 [exp(hw/kBT) 1r 1 . The energy difference between the two electronic states is then made up by the difference in the energies of the two phonons. If the intermediate state of the process is a virtual state, these are called Raman processes; if the intermediate state is a real electronic energy level, these are called Orbach

processes.

Consider first the contribution of Raman processes to nonradiative relaxation between two electronic energy levels. In Sec. 4.4 the mathematical description of these processes is outlined in detail since they make the dom­ inant contribution to the broadening of many sharp spectral lines. For line­ broadening effects, the initial and final electronic states of the system are the

1 36

4. Electron Phonon Interactions

same, whereas for radiationless relaxation transttlons they are different. Other than this difference in the electronic matrix elements, the expressions for the Raman transition rates for line-broadening and nonradiative decay processes are the same. From Sec. 4.4, the Raman process transition rate is found to be

(I._)7JTD/T

WnRr A TD

o

X6 e x dx, ( ex - 1 ) 2

where the coupling coefficient is given by

(4.2.25)

1

Vt l l/1�1 ) (l/1}1 1 Vt l l/1�1 ) 2 (4.2.26) E;"l E1el In these expressions, a Debye distribution of phonons has been a,ssumed with a cutoff frequency of WD. This defines a Debye temperature TD hwD/kB. The integral in Eq. (4.2.25) is called the Debye integral and it can be found tabulated in tables of functions. For the condition T « TD the integral becomes 6 ! so the rate for the Raman relaxation transition is pro­ portional to T7 . Since the same set of phonons is involved in the absorption and emission parts of the transition, the Raman rate for excitation is ap­ proximately the same as the Raman rate for relaxation. Thus the rate at which the pupulations of the two electronic levels reach an equilibrium dis­ tribution through Raman processes is given by the characteristic time wRex + wdRec 2 WnrR · (4.2.27)

I

1 v A = 4n9w 3p2 v l 0 ( l/lJel l 2

+

_

� �

Thus the Raman relaxation rate is found from combining Eqs. (4.2.25)­ (4.2.27) and generally obeys a T7 temperature dependence around room temperatures. The final type of nonradiative decay processes to be considered are Orbach processes. These two-phonon processes are similar to Raman pro­ cesses except that the intermediate state is a real electronic level as shown in Fig. 4.3. The interest is in relaxation and excitation processes between levels 1 and 2 in the case when direct phonon processes are not effective because of selection rules, density of states, or other problems. For the situation of greatest interest, f5 « � < hwD, where f5 and � are the differences between the energy levels as shown in Fig. 4.3. The rate equations for the dynamics of the populations of ions in energy levels 1 and 2 are

(4.2.28) where the single-phonon transition rates given in Eqs. ( 4.2. 10) and (4.2. 1 1 ) can be written using Eq. (4.2. 12) as wl3 p l3 n(� + c5) ' w3 1 p l3 [n(� + c5) + 1 ] , W23 P23 n(�) , W32 P23 [n(�) + 1 ] .

4.3. Weak Coupling: Vibronic Transitions

1 37

For the case p 13 � P23 P and for temperatures kBT « Ll so that n 3 can be neglected with respect to n , and n2 giving N n 1 + n2 , the rate equations can be rewritten as

= pe- !!.fkB T (n, e-ofkB T n2) , where Eq. (4.2.7) has been used for n(E) . Setting this equation equal to zero n2

_

_

h,

gives the equilibrium populations

ne1

1

(4.2.29) 1' The solutions to the rate equations can be written in terms of these equilib­ rium populations as +

(4.2.30) where the characteristic time for relaxation to equilibrium is (4.2. 3 1 ) Thus the temperature dependence of radiationless relaxation by Orbach processes varies exponentially with the energy splitting to the intermediate electronic energy level. The measured temperature dependence of the radiationless relaxation can be used to determine the dominant type of transition involved in the pro­ cess. All three types of two-phonon processes, direct, Raman, and Orbach, have been found to be important in the optical pumping dynamics of differ­ ent types of solid-state laser materials. It should be noted that when the matrix elements of the electron-phonon interaction Hamiltonians repre­ sented by Eqs. (4. 1 .24) and (4. 1 .25) are zero, higher-order interactions or the presence of an external perturbation such as a magnetic field are required for the nonradiative transitions to occur. This can lead to a different tem­ perature dependence for the radiationless relaxation rate of Raman processes. 4.3

Weak Coupling: Vibronic Transitions

The sharp lines in the absorption and emission spectra of ions in solids due to electronic transitions with weak electron-phonon coupling are generally accompanied by low-intensity sidebands that display a distinct frequency structure. These are associated with phonon-assisted transitions and are re­ ferred to vibronic sidebands while the line associated with the purely elec­ tronic transition is referred to as a zero-phonon line. The vibronics on the high-energy side of the zero-phonon line in the fluorescence spectrum in­ volve phonon annihilation while those on the low-energy side correspond to phonon-creation transitions. The opposite is true in the absorption spectrum

1 38

4. Electron Phonon Interactions

;

- -

;

- -

HIGH ENERGY IBRO NICS

(A) ABSORPTION SP ECT RA

- -

nI

- -

HIGH ENERGY VIBRONICS

(B) FLUORESCENC E SPECTRA

FIGURE 4.5. Diagrams for vibronic transitions.

where the high-energy vibronics are associated with phonon creation of a phonon and the low-energy vibronics are involved with the annihilation of a phonon. The schematic energy-level diagrams for vibronic transitions are shown in Fig. 4.5. The mathematical expressions describing the transition rates for vibronic processes can be derived using second-order perturbation theory along with the electron-photon and electron-phonon interaction Hamiltonians given in Eqs. (3.2. 1 0), (4. 1 .24) and (4. 1 .25), where the strain Hamiltonian is used to describe the latter interaction since lattice phonons of various wave vectors and different branches can be involved. The quantum-mechanical diagrams describing the different types of vibronic processes are shown in Fig. 4.5. As an example, consider the low-energy vibronic transition in a fluorescence

4.3. Weak Coupling: Vibronic Transitions

1 39

spectrum that involves the emission of a photon of frequency Wt and polar­ ization nf along with the emission of a phonon of frequency wk . The matrix element describing this process is

M

nk + l , nt+ l iHe r lt/1}1 , nk + 1 , nt) (t/lj1 , nk + 1 , nti H? lt/1�1 , nb nt) E,.e1 _ ( �e1 +hwk ) (t/1/1 , nk + 1 , nt+ I I H? lt/1}1 , nb nt+ 1 ) (t/1}1 , nk , nt+ I I He r lt/1�1 , nk , + E,.e1 (Et + hwt) (4.3 . 1 ) ·

The sum runs over all possible intermediate states of the system. Substitut­ ing the expressions for the interaction Hamiltonians into Eq. (4.3. 1) and evaluating the nonzero contributions to the matrix element leads to

.

em x

Wk {f v

�

e

( t/1/1 1 2: e

)

f I t/1}1 ( t/JY I Vt I t/1�1 ) ' Ej1 (Et + hwk ) ik r P

·

1t

(4.3.2)

where the second term in Eq. (4.3. 1 ) has been neglected since the photon energy is much greater than the phonon energy. A similar expression can be obtained for the matrix element describing high-energy emission vibronic transitions involving the creation of a photon and annihilation of a phonon,

ff ffv

e em z. Wk2 nh MHE 2Mv Wt m t/J;t 1 2: e- rkr p . 1tf I t/1}1 ( t/1}1 1 Vt I t/1�1 ) x Ej1 - (Et - hwk )

�(

)

·

(4.3.3)

The expressions obtained for vibronic absorption transitions involving the annihilation of a photon and the creation or annihilation of phonons are

(4 . 3.4)

(4.3.5)

140

4. Electron Phonon Interactions

The matrix elements in Eqs. (4.3.2)-(4.3.4) can be used in the golden-rule equation (4. 1 . 1 ) to determine the transition rates for vibronic processes. Since the frequency structure of vibronic sidebands is due to the frequency distribution of the phonon density of states, the vibronic peaks will be sym­ metric in position about the zero-phonon line. The intensity of a specific peak depends on the occupation number of the phonon involved, and the temperature dependence of the transition rate is contained in the factor nk through the expression given in Eq. (4. 1 .8). Whether or not a specific vibra­ tional mode appears in a vibronic sideband of an electronic transition can be determined by group theory. The reduction of the direct product of the irreducible representations of the initial electronic state, the vibrational mode, and the radiation multipole operator must contain the irreducible representation of the final electronic state for the vibronic matrix element to be nonzero, ri X rq X r, ::J r/ . At very low temperatures, only single-phonon emission vibronics are present. As temperature is raised, single-phonon absorption vibronics also appear. At high temperatures two-phonon and higher multiphonon vibronics appear. The multiphonon spectrum can be treated as the convolution of single-phonon vibronics. An example of the analysis of vibronic spectra is given in Sec. 4.5.

4.4

Weak Coupling: Spectral Linewidth and

Line Position

There are several types of physical processes that contribute to the width and position of a spectral line associated with an electronic transition and how they change with temperature. Line broadening and line shifting mech­ anisms are described below. The width of a spectral line associated with a specific ion undergoing an electronic transition is the combined widths of the initial and final energy levels involved in the transition. At least one of these levels will be an excited state of the system and thus have a finite lifetime. Due to the ucer­ tainty principle relating the energy of a quantum-mechanical system with the time the system remains in the same energy state, a long-lived energy state will give rise to a very narrow energy level. This radiative lifetime con­ tribution to the width of a spectral line is called the natural linewidth for the transition. The measured width of a spectral line is almost always signif­ icantly greater than the natural linewidth due to the presence of other broadening mechanisms. When the ion is placed in a crystalline or glass environment, direct radiationless transitions and ion-ion interactions can shorten the lifetime of the energy levels and thus broaden the spectral lines. This type of contribution to the spectral linewidth is called lifetime broad­ ening. By measuring the fluorescence lifetime of a transition and using the

4.4. Weak Coupling: Spectral Linewidth and Line Position

141

usual transformation to energy, the lifetime broadening to the linewidth of the transition can be determined. Generally the width of a spectral line is broader than that predicted from lifetime measurements, implying that line-broadening mechanisms are pres­ ent that do not affect the lifetime of the levels involved in the transition. One mechanism of this type is the two-phonon Raman scattering process dis­ cussed in Sec. 4.2. In such processes, phonons of different frequencies are absorbed and emitted but the ion remains in the same electronic state and thus the lifetime of the energy level is not altered. The quantum-mechanical diagrams for phonon Raman scattering processes are shown in Fig. 4.3. These can be used with the electron-phonon interaction Hamiltonian in Eq. ( 4 . 1 . 2 5) to write the matrix element describing the Raman scattering of phonons:

MR (t/1�1 , nk - l , nk' + l i HR it/1�1 , nk , nk') nk - 1 , nk' + l i H? lt/1}1 , nk - 1 , nk') (t/1}1 , nk - 1 , nk' I H? lt/1�1 , nk , nk' ) J

Elel - (Pi -hWk ) j

(t/1�1 , nk - l , nk' + l i H? It/lj1 , nk , nk' + l ) (t/lj1 , nk , nk' + E�1 - (E�J 1 + hQ)k' ) + (t/1�1 , nk - 1 , nk' + l i H? l t/1�1 , nk > nk' )

+

[2.: (I

l

(t/1�1 1 Vi lt/1}1 ) ! 1 (nk - 1 , nk' + l l bl, bk I nk , nk') - 2Mv2 E;ei - ( El - hwk ) j (t/1�1 1 V! l t/1}1 ) ! 1 ( nk - 1 , nk' + l l bk bl, I nk > nk' + Efi - ( J0el + hwk' ) + (t/1�1 1 V2 lt/lj1 ) (nk - 1 , nk' + l l 2bk bl, l nk , nk')] _

h

I

(4 . 4 . 1 )

In the last step, the phonon energy has been neglected with respect to the photon energy. This expression can be rewritten in more compact form by factoring out the phonon frequency and occupation nubmer parts and de­ fining a coupling coefficient rx that contains the remaining constants and matrix elements, ( 4 . 4 . 2) where

(4.4.3)

142

4. Electron Phonon Interactions

The next step is to substitute this expression for the transition matrix ele­ ment into the golden rule transition rate expression given in Eq. (4.2.2). This calculation also reuqires an expression for the density of final states. For the case of the Raman scattering of phonons broadening a sharp electronic en­ ergy level, this can be expressed as the product densities of states of the two phonons involved in the process and a J, function to conserve energy,

(4.4.4) Thus the transition rate for Raman scattering of phonons is found from in­ tegrating over the phonon spectra to give

WR �� I I [ MR[ 2p (wk )p(wk')J (wk Wk') dwk dwk' 2n 2 h2 [ 0! [ I [p (wk )] 2wk2 nk(nk + 1 ) dwk .

If a Debye density of states is assumed for the phonons as discussed above, and Eq. (4.2.7) is used for the phonon occupation numbers, this ex­ pression becomes

2n 2 ( 3 v2 ) OJD w% ehwk /kB T d R W - h2 [0![ 4Jtlv6 Jo ehwk/kB T 1 wk [ 0! [ 2 2n9V3 h22v6 (kBTD)7 (TDT )7 ITn/T (exx6�1 ) 2 dx. _

0

(4.4.5)

Here the expression in Eq. (4.2.5) has been used for the phonon density of states and the Debye temperature is defined in terms of the Debye cutoff frequency as above, TD hwD/kB. Converting this rate to units of energy gives the contribution to the linewidth due to the Raman scattering of phonons,

a. (-TTD)7 JTn/T (exx6ex1 ) 2 dx,

ilv ( cm 1 )

(4.4.6)

0

where the new coupling coefficient has been defined to contain all the con­ stants in expression (4.4.5),

0!

(3.34 x x

1 0 ) 2n3p92 vl

(I: I

-II

0

(kBTD)7

1 2 + (t/1�11 V2 [ t/1�1)

)2

0

(4.4. 7)

Since the coupling constant is intrinsically positive, an increase in temper­ ature will cause an increase in the linewidth. Note that the Debye integral appearing in Eq. (4.4.6) is tabulated, and its values can be found in tables of functions covering a wide range of temperatures.

4.4. Weak Coupling: Spectral Linewidth and Line Position

143

Both the Raman scattering mechanism and the lifetime broadening mechanism have the same probability of occurrence for all ions in the en­ semble. Therefore they produce homogeneous broadening and result in a Lorentzian line shape. This is similar to collisional broadening of spectral lines in gases. In addition to these mechanisms, it is possible to have in­ homogeneous broadening mechanisms that produce a Gaussian contribution to the line shape similar to Doppler broadening of spectral lines in gases. These mechanisms have a different probability of occurrence at the site of each ion. For ions in solids, this is due to different local crystal-field envi­ ronments associated with a random distribution of microscopic strain fields in the material. Thus the same spectral transition has a slightly different fre­ quency position for each ion, and the observed spectral line is the super­ position of all of the individual ion lines. As discussed in Chap. 2, the superposition of Gaussian and Lorentzian contributions to the width of a spectral line results in a Voigt profile for the line shape. These concepts on the width of a spectral line of an ion in a solid can be summarized by the following expression:

+ L jJ ehwo /ka T

1

f>z

+ L jJ ehwo /ka T

1 f
+ higher-order terms.

ehwo /kaT

1 ( 4 . 4 . 8)

In this expression, jJ = 3 . 34 x 10 1 1 Pif in units of wave numbers, where the Pif are given in Eqs. (4.2. 1 1 ) and (4.2. 12). The first term on the right hand side of the equation is the contribution due to microscopic local strains. Only the single-phonon direct process terms are explicitly shown for the life­ time broadening processes since they usually dominate other contributions. The higher-order terms refer to two-phonon Raman and Orbach relaxation processes as well as multiphonon relaxation processes discussed in Sec. 4.2. It should be noted that the plus sign between the first term and the rest of the terms in Eq. (4.4.8) does not represent simple addition since the first term produces inhomogeneous broadening and the other terms produce homogeneous broadening. The Gaussian and Lorentzian contributions to the linewidth must be determined independently and then a convolution of these two shapes can be performed. Tables have been prepared to assist in doing this. 3 In general, the inhomogeneous contribution to the linewidth is independent of temperature and gives the dominant contribution to the observed linewidth at very low temperatures. As temperature is increased, direct radiationless relaxation processes begin broadening the line, and at high temperatures the Raman scattering term provides the strongest contri­ bution to the temperature dependence of the observed spectral linewidth.

144

4. Electron Phonon Interactions

Specific examples of the temperature dependences of the linewidths of laser transitions are given in Chaps. 6 and 8. The presence of the phonon field also causes the position of an absorption or fluorescence electronic transition of an ion in a solid to be temperature dependent. This is due to the continual absorption and emission of virtual phonons by the dopant ion. These processes differ from the Raman scatter­ ing of phonons discussed above in that the two phonons involved have the same frequency. Thus the presence of the phonon field makes a contribution to system of ions. The energy due to the electron-phonon interaction for the ion in the ith electronic energy level is given by

self-energy

(4 . 4 . 9) <>iii = (l/Ji i H? l l/Ji) + � I The first- and second-order electron-phonon interaction Hamiltonians are given by Eqs. (4. 1 .24) and (4 . 1 . 2 5 ) . They can be used in Eq. (4.4.9) to eval­ uate the temperature dependence of a spectral line position. This gives Jiii = - { ( l/J�1nk I v2 � h!)(bk' -bl, ) l l/J�1nk) + � [ ( ( ljl�1nk I Vt � foik(bk -bl) l l/Jj1nk+l) ( l/Jj1nk+1 1 Vt � fo)k(bk-bl) � l/J�1nk)) /(Ef1 - (El + hwk)) + ( ( l/J�1nk I Vt � foik(bk -bl) 1 1/JYnk - 1 ) ( .;:n, - I I v, � yli(b, -hi) I .;:'n, )) I (E,cl r (Ef - hwk))]} = [� Wk (l/J�1 1 V2 l l/J�1 ) (2nk + 1 ) + � Wk I (l/J�l l Vtl l/Jjl ) 1 2 (Eie! - hwk) + Ef! - -hwk)) ] · (4 . 4 . 10) x

X

Equation (4.4. 10) describes the shift in position of a spectral line asso­ ciated with an electronic transition due to the interaction with the phonon field. Note that there are two distinct parts to this line shift, one that is tem­ perature dependent and one that is not. The temperature independent part is given by

Jii'· (O) = 2Mv2 Wk V2

Wk EfI l(l/J-�l(iElVtl l/J+jln)wk)I Z ) . (4.4. 1 1 )

4.4. Weak Coupling: Spectral Linewidth and Line Position

1 45

The line-shift contribution from this term is due to self-energy and is similar to the Lamb shift associated with photon fields. The most important aspect of these considerations is the temperature de­ pendence of the line position. This is contained in the phonon occupation numbers. The expression for this effect can be found by subtracting Eq. (4.4. 1 1 ) from Eq. (4.4. 10): Jii; (T)

= x

{ � Wknk [2 (114 1 Vz lt/1�1) + � Wk I (t/1�1 1 Vt lt/lj1) J Z +

hwk )

)] }

(4.4 . 12)

·

The summation over all phonons can be evaluated by considering differ­ ent limiting cases with respect to the energy difference between the initial and intermediate states. First consider the contributions from intermediate states with energies such that 1 Ef1 - El1 1 » hwn . The contribution to the temperature-dependent line shift for this case is Jiiza (T)

= Mv2

z

V2

z

}'1- i

I ( t/I�I I Vt lt/lj'l ) 1 2 Eiel _ Ejel

)

k

k nk .

(4.4. 13)

Using the Debye model for the frequency distribution of phonons, the sum can be approximated by an integral over p(w) dw. Using Eqs. (4.2.56) and (4.2.7) for p(w) and n(w), Eq. (4.4. 1 3) becomes

where

rx'

X3 dx rx' (T-Tn)4 J TD/T -ex - 1 ' 0

(5.03 x 10 1 5 ) __].! 2n2pv5

(4.4. 14)

( h ) ( (t/le! I V2 knTn 4

1

1

)

I (t/1�1 1 Vt lt/1}1 ) i 2 Eel _ Ejel (4.4. 1 5) l

146

4. Electron Phonon Interactions

in units of wave numbers. Note that rx' can be either positive or negative depending whether the energy of the intermediate electronic level is greater than or less than the energy of the initial level. Thus this contribution can cause the position of a spectral line to shift to either higher or lower energy as temperature is increased. Next consider the contribution due to intermediate states with energies such that I E;el - El1 1 ::; hwD . For this case the summation in Eq. (4.4. 12) can again be approximated by an integral over the phonon density of states and the Debye approximation used. The second term in Eq. (4.4. 12) dominates due to the resonance leading to the result

(4.4. 1 6) where

(4.4. 17) and the pole in the integrand is treated by taking the principal value of the integral, denoted by P. Again these contributions can cause the position of a spectral line to move to either higher or lower energy as temperature is in­ creased depending on which of the two terms in Eq. (4.4. 1 6) is the largest. The final expression for the temperature-dependent shift of a spectral line away from its position at very low temperatures is found by combining the

4.5. Example: Spectral Properties of SrTi0 3 :Cr3 +

1 47

expressions in Eqs. (4.4. 14) and (4.4. 1 7), Jv(cm - 1 ) =

fn/T 1 dx 2 Tn/T � 1 (_I_ ) 2 PiJ' f1£. . P J ex - 1 2 (f1£ . . ) dx lJ lJ x knT 2 (T ) P Tn/T -1 x3 2 ' lJ f1£.lJ. J ex - 1 2 (11£ ) dx x o

-

j >i

o

_

__

_

__j

(4.4. 1 8)

knT

where the coupling constants a' and p� are given by Eqs. (4.4. 1 5) and

(4.4. 1 7).

In general when Eqs. (4.4.8) and (4.4. 1 8) are used to analyze experimental data the coupling constants and the Debye temperature TD are treated as adjustable parameters. It is usually found that the values for TD obtained from fitting thermal line-broadening and line-shifting data are not the same as the values obtained from thermodynamical investigations of the material. The reason for this discrepancy is that the electron-phonon coupling is not the same for all phonon modes and can be different for phonons contri­ buting to line-broadening processes compared to those contributing to line­ shifting processes. Except for this discrepancy, the perturbation approach described here has been very successful in providing an understanding of the thermal properties of spectral lines of ions in solids. 4 A more rigorous non­ perturbative approach has been developed to explain the temperature de­ pendences of spectral linewidths and positions5 that does not require treat­ ing TD as an adjustable parameter. However, the simpler mathematics of the perturbation approach provides better insight into the physical processes that are taking place. Examples of using these expressions to analyze data obtained on laser transitions are given in Chaps. 6 and 8. In the following section, an example is given of using the measured vibronic sideband as the effective phonon density of states. 4.5

Example : Spectral Properties of SrTi0 3 : Cr3 +

Because of the complex nature of the problem, very few attempts have been made to analyze the details of vibronic spectra of the types of materials that are useful for solid-state laser applications. One analysis of this type that was made on chromium-doped strontium titanate6 is summarized below as an example. SrTi0 3 has a crystal structure that is close to a cubic perovskite at room temperature. This is consistent with an oL symmetry group with one mole-

148

4. Electron Phonon Interactions

x e - Sr

·- TI Q - 0

(A)

(B)

FIGURE 4.6. (A) Unit cell and ( B) first Brillouin zone of SrTio 3 •

cule per unit cell. A phase transition to a structure with tetragonal symmetry occurs at 1 1 0 K and will be discussed further below_ The Cr3 + dopant ion replaces the Ti4+ because of their similarity in size. The charge compensa­ tion is found to be nonlocal so the chromium ion occupies a site with cubic symmetry. The unit cell and first Brillouin zone for this symmetry are shown in Fig. 4.6. The fluorescence spectra of SrTi0 3 :Cr3 + at two temperatures are shown in Fig. 4.7. At high temperatures the spectrum consists of a zero­ phonon magnetic dipole transition between the 2 E9 metastable state and the 4 A z9 ground state of Cr3+ in a cubic crystal field, along with vibronic side­ bands. Below the phase transition temperature the zero-phonon lines split 7930

;1o ..

zg :;)

�8

WAV E 8200

7940

•��••••Ur

.. , .... ,, ...

, • • • • 41 • •

: \ r� • s� .K

, .., . ,

r.

,, . .

..

I

880

\.,, \

I

..


�3 0

860

j .

c

;: �

2

'.•

840

SrTI o,,cr••

, .

:: 7 :6

z -

L E N GTH I l l

,-, '

I�

I�

I.MO

1.!520

L�

E N E R G Y ( ow )

1.460

L400

FIGURE 4.7. Fluorescence spectra o f Cr3 + i n SrTi03 at two temperatures [taken from Ref. 6(a)].

4.5. Example: Spectral Properties of SrTi0 3 :Cr3 +

149

into two lines and become stronger while more structure appears in the vibronic sidebands due to the lifting of the degeneracy of the electronic and vibrational energy levels due to the lower symmetry. These spectral proper­ ties can be understood through an analysis of the vibronic sideband as out­ lined below. The first step in analyzing the structure in the vibronic spectra is to use group theory to classify the vibrational modes of the system and determine the vibronic selection rules. This is done as described above in Sec. 4. 1 using the symmetry of the unit cell in real space. At high temperatures the site symmetry for the Cr3 + ions is Oh and this can also be assumed as a first ap­ proximation for the low-temperature symmetry. According to the symmetry selection rules for vibronic transitions discussed in Sec. 4. 1 , allowed tran­ sitions involve phonon modes that transform according to one of the irre­ ducible representations contained in the reduction of the direct-product rep­ resentation of the initial electronic state, the final electronic state, and the electric dipole moment operator. For this case Table 2.4 and Eq. (2.2.8) can be used to give Eg X A 2g X Tt u Tt u + T2u :: rv, where rv is the irreducible representation according to which the vibrational mode transforms. The next step in this procedure is to switch to reciprocal space and con­ sider the symmetry of the Brillouin zone. The symmetry representations for the vibrational modes at different parts in the Brillouin zone are represented by the reciprocal vector q, and can be determined from the Bloch basis functions 1 for an atom at position R,

uq (R) eiq R

f(R)

(4.5. 1 )

where uq (R) is the displacement of the atom from equilibrium. When a dopant ion is substituted for the central ion in the unit cell; it destroys translational symmetry so q is no longer a good quantum number and the relevant symmetry is the q 0 point group at the center of the Brillouin zone. In this case, the symmetry representations of all the phonons must be expressed in terms of these zone-center vibrational representations. SrTi03 has five atoms per unit cell, leading to 1 5 phonon modes at each point in the Brillouin zone. Since this is a symmorphic space group, it is possible to factor the translation and rotation operations and treat them separately. The irreducible representations of the phonon modes at a specific point in the Brillouin zone with wave vector q are expressed in terms of the group of the wave vector Go (q) at that zone point. This group contains all operations that either leave q invariant or transform it into q + Q, where Q is a primitive vector of reciprocal space. If a symmetry operation in Go (q) involving a rotation c and 0 translation is expressed as { c I 0}, its operation on a basis function of the type given in Eq. (4.5. 1 ) is given by

{ c i O}uq (R) eiq · R

{ cuq (R) } eiq · cR uq ( R')ei(cq) · R uq (R')eiq · R uq (R')eiQ · R. =

The character of this operation is the trace of the matrix that operates on the basis function multiplied by the factor eiQ · R. For points within the Brit-

1 50

4. Electron Phonon Interactions

louin zone, Q = 0 so the exponential factor is always one. This is not the case for some points on the surface of the Brillouin zone. As was found previously [see Eq. (4. 1 .28)], the trace of the symmetry operation matrix involving a proper or improper rotation ( ± 1 ) of an amount fjJ is given by twice the cosine of the angle ± 1 multiplied by the number of ions left unchanged by the operation. Thus the character for a symmetry operation in Go (q) is given by x(f/J, R) = L )2 cos f/J ± I )eiQ Ru . u

( 4.5.2)

This expression can be used to determine the character for each symmetry operation at each special point in the Brillouin zone and the resulting repre­ sentations can be reduced in terms of the irreducible representations of the group of the q vector at that point. These irreducible representations repre­ sent the symmetry modes of the phonons at that point of the Brillouin zone. The results of doing this are given in Tables 4.3-4.6. The special points of high symmetry shown in Fig. 4. 7 are generally those that make the greatest contribution to the phonon density of states and therefore play the most important role in thermal properties. Each special point can be considered separately. First consider the zone-center r point. Table 4.4 shows the characters of the vibrational representation rv reduced in terms of the irreducible representations of the point group at zone center, Oh . In this case rv = 4 Ti u + Tzu , which gives five triply degenerate vibra­ tional modes. The three acoustical modes transform like a vector and thus belong to one of the T1 u representations. The other three T1 u and the Tzu mode are associated with optical phonon modes. For the other points inside the Brillouin zone, Go (q) is a subgroup of Oh , and the easiest way to obtain the vibrational mode representations is through compatibility relationships between the irreducible representations of a group and one of its subgroups. These are found by determining how the representations of the vibrational modes at the r point reduce in terms of the irreducible representations of the subgroups at the � ' L, and A points, and at a general point q in the Brillouin zone. Table 4.5 shows the correla­ tion among the elements of the Oh symmetry group and its relevant sub­ groups, along with the compatibility relationships for their irreducible rep­ resentations. This analysis gives the symmetry designations for all of the phonon modes inside the Brillouin zone as summarized in Table 4.6. For points on the surface of the Brillouin zone, Q is not necessarily zero and must be determined for each operator. The results of doing this along with the characters found for the total vibrational representations are shown in Table 4.3 and 4.4. The reduction of these representations in terms of the irreducible representations of Go (q) at that point in the Brillouin zone are also given in Table 4.4. The phonon symmetries determined in this way are summarized in Table 4.6. The next step in obtaining the selection rules for vibronic transitions of

S4z

ad

Gh(xy)

and y axes)

3

c4z Czz q (4 5 ° to x

3 3

5 5

3

5

Nu

Operators

R = a(i + j + k) R = ak R=O R = ai + ak

R = ai R = 2ai + 2aj R = ak

Ti

R=O R=O R=O R=O

R=O R=O R=O

Sr

R = a(i + k) R = ak

R = aj

0]

R = a(k + i) R = ak

R = ai

Oz

R = a (i + j) R=O R=O R = ai

R = ai R = 2ai + 2aj R=O

03

Atoms contributing to the modified factor of the character and their vector change after applying the operator

+ ei(Q,+Q,+Q,)a + ei(Q,+Q,)a + ei(Q,+Q,)a + ei(Q,+Q,)a

2 + 3 eiQ, a 3 1 + ei(Qx +Q,) a + eiQ,a

1

Cl

.. VI

n .. +

9

�

"' 0 -, Cl

a. 0

0

'"d

,g

[

'"g

'E. �

I + 2eilba + 2ei2 (Q,+Q,)a + eiQ,a + eiQ,a 2 + eiQ,a

� !" tr:l �

8

N.

Modified x

L eiQ·R 1

TABLE 4.3. Modified part of the character of various operators at the surface of the Brillouin zone.

3 5 15

Ma x (ct) x ( r. )

-1 3 -3

-1 1 -1

-1 -1 1

-1 -1 1

°Contribution from the modified character factor.

3 5 15 -1 1 -1

-1 5 -5

2Cf

E

Ma X (ct) x ( r. )

c2

2q

2C4

0 0

5 -1 -1

3C2

-1 1 -1

3 1 3

6C4

1 -1 -1

3 -1 -3

6C2

-1 1 -1

Ma x (ct) x ( r. )

2 0 0

8C3

3 5 15

5 3 15

Ma x (ct) x ( r. )

E

i

-3 1 -3

-3 -1 3

-9

-3 3

5 -3 -15

i

R(Oh)

r (Oh)

0 0

2 0 0

-1 1 -1

1 -1 1 M(D4h ) 1 -1 5 -1 5 1

h

U

X (D4h )

2S4

-1 1 -1

3 -1 -3

6S4

8S6

1

2A t 9 + Bt 9 + 3E9 + 2A 2u + 2E,

A t 9 + A 29 + Bt 9 + B2g + Eg + A 2u + 2Bt u + 3Eu

1 3 3 1 -1 -1

T2g + A 2u + Eu + 2Ttu + T2u

4Ttu + T2u

Phonon modes

1 3 3

2ud

1 3 3

3 1 3

6ud

1 5 5

Uv

-1

-

1

5 1 5

3uh

TABLE 4.4. Characters of the vibrational representations at special points in the Brillouin zone.

"'

c. 0 t:l

�

..

ft

-

t:l

t:l 0 t:l

� 0

0 t:l I

.. ..

0

w

�

..

VI N

4.5. Example: Spectral Properties of SrTi03 :Cr3 + TABLE 4.5. Correlation of the oh A lg A 29 Eg T19 T2g A 1u A 2u Eu T1u T2u

__. -t

__. -t __. -t

-t

-t -t -t

group and its subgroups. C3v

D4h

C4v

A 19 B1 9 A 1 g + Big E9 + A 2g E9 + B29 A 1u B1u A 1u + B1u Eu + A 2u Eu + B2u

AI B1 A I + BI E + A2 E + B2 A2 B2 A 2 + B2 E + A1 E + B1

__.

oh

1 53

AI B1 E E + A2 E + A1 A2 AI E E + A1 E + A2

C2v

c1

AI B1 AI + BI A 2 + B 1 + B2 A I + A 2 + B2 A2 B2 A 2 + B2 A I + BI + B2 AI + A 2 + BI

A A 2A 3A 3A A A 2A 3A 3A

ions in solids is to express the vibrational representations of the phonons at all points in the Brillouin zone in terms of the representations of the zone­ center symmetry group. This is done in the usual way using Eq. (2.3.24) . The correlation between the irreducible representations of the Oh group and the irreducible representations of each subgroup are listed in Table 4.5. Ac­ cording to Sec. 4. 1 , if the subgroup representation appears in either the T, u or Tzu irreducible representation of Oh, then the phonon mode described by this symmetry representation can give rise to an allowed vibronic transition. Other phonons result in forbidden vibronic transitions. These selection rules are summarized in Table 4.6. At low temperatures specific vibronic peaks can be resolved in both the high-energy and low-energy fluorescence sidebands. 6 Using the selection rules given in Table 4.6 and comparing the vibronic spectra with infrared and Raman spectra and with the results of neutron-scattering experiments, it is possible to identify some of the spectral structure with specific phonon TABLE 4.6. Irreducible representations of the vibrational modes of SrTi0 3 at various points in the Brillouin zone ( BZ) at room temperature. ( The modes shown in bold are allowed for electric dipole vibronic transitions from the 2 E9 metastable state to the 4Az9 ground state.) Point in BZ r �

X

�

M

A

R

K

( qx, qy, qz ) (0 , (0 , (0 , (q,

0 , 0) 0 , q) 0 , !l q , 0) (! ' ! , 0) ( q , q , q)

(! , ! , !l

( ql ' q2, q3 )

Point group

oh C4v D4h C2v D4h C3v oh c1

Symmetry representations of the vibrational modes 4Tlu + Tzu 4AI + B1 + SE 2Aig + B 1 9 + 3Eg + 2Azu + 2Eu SA1 + A2 + SB1 + 4Bz A 1 9 + A 29 + B 1g + B2g + E9 + A2u + 4AI + A2 + 5E T2gA 2u + Eu + 2Tlu + Tzu ISA

2Biu + 3Eu

1 54

4. Electron-Phonon Interactions

modes. One important result of this procedure is the ability to identify both local-mode phonons and soft-mode phonons and follow their temperature dependence through the phase transition. 6 Vibronic spectroscopy can there­ fore be an effective tool for obtaining the properties of lattice dynamics for these types of materials. In addition, the vibronic sideband can be used as an effective phonon density of states for treating other processes involving electron-phonon interactions. The first step in doing this is to deconvolve the observed sideband into its one-phonon and multiphonon components. This can be accomplished with a computer analysis algorithm that divides the spectrum into 1023 equally spaced "phonon modes" with individual­ mode Huang-Rhys factors that are proportional to the corresponding spec­ tral intensity at the frequency of the mode. The density of states is then expressed as a series of equally spaced J functions normalized to the total Huang-Rhys factor (4.5.3) where

S(wq ) = L Sq. q The one-phonon spectrum can then be expressed as

h (w) = 2ne s L SqJ (wo - w - wq ) · q

(4.5.4)

The two-phonon and higher-order contributions can then be found from the self-convolution and cascaded self-convolution, respectively. Using a com­ puter iteration process, the best fit to an experimental sideband can be deter­ mined. The results of applying this type of analysis to chromium-doped strontium titanate is shown in Fig. 4.8. In this case an excellent fit to the experimental data is obtained for a one-phonon sideband with quadratic coupling with a local mode. 6 After a computer simulation of the effective one-phonon density of states is obtained, the expression in Eq. (4.4.8) for the temperature broadening of the zero-phonon line can be rewritten in terms of the individual mode Huang-Rhys factors as 1\v

= L\v0 + fi L (w� Sq ) 2 nq (nq + 1 ) + L {Jif n(wif )w�Sif q f> i + L fJif [n(wif ) + l]w�Sif. f> i

(4.5.5)

Since the case being considered here involves a strong local mode in the density of states, the coupling to this specific mode can be considered sepa­ rately. If only this mode is important, the contribution to the thermal line

4.5. Example: Spectral Properties of SrTi0 3 :Cr3 + "' .. z "

1 55

0.

"' • %

,_

0.1

a; 0: •

o•

,. ..

;; z "'

,, " • "

O.t

.. !

o.oz

o.o•

A[�,t,T I V E

o.oe

o . oe

/ �! \ �-

_ /..-- .. -. .... _/ . o.1o

0.11

PH O T O N E N E � G Y ( n l

o.1•

FIGURE 4.8. Results of a computer fit of the vibronic sideband of SrTi03 : Cr3 + using the one-phonon contribution with S 0.30 (solid line) and a contribution due to quadratic coupling of the one-phonon sideband to the local mode (broken line) [taken from Ref. 6(a)]. =

broadening becomes

Ails = Ailso + �s w;ns (ns + 1 ) .

(4.5.6) Figure 4.9 shows the temperature dependence of the width of the lowest energy zero-phonon line6 (designated R 1 as discussed in Chap. 6). Three different theoretical fits to the data are shown in the figure. The first is ob­ tained assuming a Debye distribution of phonons with the Debye temper>

•• • 0

..

J: 10

1 01 � 2

:. 1 o •

.. z �

!5 2 !5

I0

20

!50

100 200

TEMPERATUR E ( • K )

FIGURE 4.9. Variation of the width of the R 1 line of Cr3 + in SrTi03 . The circles are experimental points; the solid line is the best fit using a Debye distribution of pho­ nons, the dotted line is the fit obtained using one-phonon density of states obtained from the computer analysis of the vibronic sideband, and the dashed and dotted line is obtained using only the lowest-energy peak in the computer analysis of the vi­ bronic sideband for the effective phonon density of states [taken from Ref. 6(a)].

1 56

4. Electron Phonon Interactions

ature TD and the coupling parameters treated as adjustable parameters. As seen by the solid line in the figure, this procedure can give a reasonable fit to the data. However, the value of TD required to obtain this good fit is only 1 1 5 K, which is significantly less than the value of 400 K obtained from specific-heat measurements. In this model such a low value of the Debye temperature implies that the lower-frequency phonons make a greater con­ tribution to the thermal line broadening than the higher-frequency phonons in the density of states. The two broken lines in the figure were obtained from models using the effective density of states for phonons obtained from the vibronic spectra as described above. The results obtained using the entire one-phonon sideband (dotted line) give a poor fit to the data at high tem­ peratures. However, the results obtained using only the lowest-energy peak (dashed-dotted line) in the vibronic structure as the effective density of pho­ non states give an excellent fit to the experimental data at all temperatures. This again implies that there is much stronger coupling to the low-energy phonons for the line-broadening processes than there is to the high-energy phonons. Similar results are obtained for the thermal broadening of the Rz zero-phonon line and for treating the thermal shifts in the positions of these lines. 6 4.6

Strong Coupling

When the coupling between the electrons on the optically active ion and the lattice vibrations is strong, many phonons can be involved in the optical transitions. This leads to broad bands with strong temperature dependences of the intensities and decay times of the optical spectra of ions in solids. One method for theoretically treating this strong-coupling case is to use the N­ order perturbation theory approach leading to Eq. (4.2.21 ) for the emission rate of N phonons. This can be combined with the treatment of vibronic transitions given in Sec. 4.3. Although this works well for transitions in­ volving only a few phonons as was shown in Sec. 4.2, it becomes very cum­ bersome for transitions involving many phonons. Also, the characteristics of multiphonon radiationless transition rates such as their temperature de­ pendences may be accurately predicted by this approach, but the theoretical predictions of absolute transition rates are very inaccurate. In addition, this treatment utilized the harmonic approximation to describe the phonons and assumed the phonon modes to be the same for both electronic states in­ volved in the transition. The importance of allowing for anharmonic inter­ actions has been pointed out 7 but this is difficult to treat from first principles and will not be included here. However, some of the effects of anharmonic interactions on spectral characteristics will be mentioned below. One impor­ tant extension of the electron-phonon coupling theory is to allow for differ­ ent types of vibrational modes in the initial and final electronic states. This allows states to be connected whose quantum numbers differ by other than

4.6. Strong Coupling

1 57

± 1 . Several different approaches have been developed to treat the case of strong electron-phonon coupling, and one of the approaches commonly used is outlined below. However, before presenting this theoretical treat­ ment, the concept of configuration-coordinate models is discussed. This is helpful in obtaining a qualitative understanding of the effects of lattice vibra­ tions on transitions between electronic states of ions in solids. Configuration-coordinate diagrams are often used to describe transitions between electronic energy levels coupled to lattice vibrations. Although this model involves a one-dimensional displacement of a single vibrational mode, which is a great oversimplification of the true situation, it is still quite useful in explaining some aspects of the optical spectra of ions in solids. Schematic configuration-coordinate diagrams are shown in Fig. 4. 10. These depict the variation in the electronic state energy with respect to the dis.�

T

·� "

" .Q

Q (A)

S10kes Shift

j_

l'3 0: � Il

i

Qo Qo'

Q

(B)

" .Q fj

>-

"5

� Il

§

...

Qo Qo' Q

§' �

(C)

>"

�

(D)

Q

FIGURE 4. 10. Configuration-coordinate diagrams. (A) The case of no displacement between the excited-state and ground-state potential wells leads to sharp zero­ phonon lines in the absorption and emission spectra. ( B) Intermediate displacement between the excited- and ground-state potentials gives zero-phonon lines and strong vibronic sidebands with a Stokes shift between the absorption spectrum and emission spectrum peaks. The Huang-Rhys energies and vibrational wave functions are shown. (C) Large offset of the potential wells leads to a crossover between the ground and excited electronic states at an activation energy !l.E. This leads to a broad absorption band with no emission due to radiationless quenching. ( D) Anharmonic potentials enhance the Frank-Condon overlap factors and leads to enhanced radiationless quenching.

1 58

4. Electron Phonon Interactions

placement of the normal vibrational coordinate away from its equilibrium position. Results are shown for the ground state and excited state of the system for cases with different magnitudes of electron-phonon coupling. Since the electron charge cloud distribution of the excited state can be significantly different from that of the ground state, the equilibrium posi­ tions of the configuration coordinates can be different for the ground and excited states. In the harmonic approximation, the potential curves are de­ scribed by parabolas,

( 4.6. 1 ) The vibrational wave functions of the ground state and excited state are given by the usual harmonic oscillator expressions discussed further below, Xn9 ( Q )

= Nn e - ( Q/ag ) 212 Hn

Xme C Q )

= Nm e - [( Q

Qo ) fa,] 212 Hm

(4.6.2) where the Hn are Hermite polynomials with normalizing factors Nn, m · The force constants k; and zero-point vibrational amplitude factors a; are given by

h ag2 = -VJ(;M ' h 2 ae = v'J(;M "

(4.6.3)

Because the Frank-Condon approximation assumes that the vibrational motion of the ions is much slower than the motion of the electrons, elec­ tronic transitions involving absorption and emission of photons appear as vertical lines on configuration-coordinate diagrams. Since vibrational relax­ ation within an electronic state is very fast, electronic transitions at low temperatures will initiate from the ground vibrational level of the initial electronic state. Transitions initiating from higher levels can be present at higher temperatures when the presence of thermal energy causes these levels to be populated according to a Boltzmann distribution. It should be noted that the harmonic oscillator wave functions for the lowest vibrational level are peaked in the center of the electronic potential well while the wave functions for the higher-lying vibrational levels are peaked at the turning points. The Frank-Condon factor for the vibrational wave-function overlap appearing in the expression for nonradiative transition probability (dis­ cussed below) implies that nonzero transitions can occur from one specific vibrational level of the initial electronic state to several vibrational levels of the final electronic state. This leads to a spread in the allowed transition en­ ergies and thus to the appearance of broad spectral bands as shown in Fig. 4. 10. For the typical case of transitions initiating from the lowest vibrational level of an electronic state, the photons involved in an absorption transition

4.6. Strong Coupling

1 59

have higher energies than those involved in an emission transition. The en­ ergy difference of the absorption and emission spectral bands is called the

Stokes shift.

Both the widths of the spectral bands and the magnitude of the Stokes shift depend on the amount of offset of the excited-state potential well mini­ mum Q� with respect to the ground-state potential well minimum Qo . A large offset results in a very broad band that approximates a Gaussian shape, while a medium offset gives a band with a Pekarian shape, and a zero offset produces a sharp zero-phonon line. Many times it is possible to see both a zero-phonon line and a vibronic sideband as shown in Fig. 4. 1 0( B). One important situation is when there is a possibility for the two electronic potential wells to cross as shown in Fig. 4. 10(C). This leads to nonradiative decay from the excited state to the ground state. 8 If the potential crossover point occurs at an energy flE above the lowest vibrational state, the radia­ tionless decay rate will have an activation energy of this amount, leading to an exponential temperature dependence of both the fluorescence intensity and lifetime of the excited state. It should be emphasized that any anhar­ monicity associated with the potential curves will enhance the vibrational wave-function overlaps as shown in Fig. 4. 10( D) and thus increase the radi­ ationless decay rates. 7 The discussion of configuration-coordinate diagrams can be related back to the crystal-field model for transition-metal ions. Consider the case of an ion with a dn+m configuration in an octahedral crystal field with n electrons in the t2g level and m electrons in the e0 level. As shown in Fig. 4. 1 1 , the energy of an excited state with respect to the ground state depends on the magnitude of the crystal field 1 0Dq and on the difference in ground- and excited-state electron configurations. This can be expressed as (4.6.4)

_

E

_

_

_

E

lODq

FIGURE 4 . 1 1 . Crystal-field modulation by lattice vibration mode Q.

1 60

4. Electron Phonon Interactions

so the variation of the transition energy with respect to a normal vibrational coordinate Q is

8E9e 8Dq 8E9e 8Dq 6(m (4.6.5) a Q = a Q aDq = a Q [ e - m9) - 4(ne - n9)] . Here the partial derivative of the crystal-field strength Dq with respect to normal mode Q represents the electron-phonon coupling parameter. This expression and Fig. 4. 1 1 show that states having the same electron config­

urations have the same slopes on the crystal-field energy level diagram and therefore transitions between these states result in sharp spectral lines. If the electron configurations are different for the two states, the corresponding energy versus crystal-field strength slope of each state will be different and thus the phonon modulation of the crystal field results in a broad spectral band for the transition. Examples of these cases will be given in Chap. 6. One of the standard theoretical approaches for treating transitions be­ tween electronic states ions in solids that are strongly coupled to the host lattice vibrations was developed by Huang and Rhys. 9 In this treatment, it is assumed that the Born Oppenheimer approximation is valid so the wave functions can be expressed as products of the electronic part and the vibra­ tional part. Further, it is assumed that there is only a weak dependence of the electronic wave functions on the nuclear coordinates so the transition matrix elements can be factored into an electronic part and a vibrational part (Condon approximation). Finally, to make the calculations tractable, it is assumed that there is one dominant phonon mode in the electron-phonon interaction and that this mode has the same frequency but different normal coordinates in the initial and final electronic states of the transition. This last simplification implies linear coupling in the harmonic approximation. The kinetic energy of the phonons is initially neglected while the eigenstates of the Hamiltonian are found for fixed phonon coordinates. This produces the usual adiabatic potential surfaces. The transitions between these surfaces are due to the term containing the phonon kinetic energy or nonadiabatic term in the full Hamiltonian. With these assumptions, the wave functions for the system are

(4.6.6) where r is the coordinate of an optically active electron and Q is the coordi­ nate describing the positions of the surrounding nuclei. O;, v ( Q ) is the vibra­ tional wave function and r/J; (r, Q ) is the electronic wave function for a fixed position of the nuclei. This implies that the motion of the electron is very rapid compared to the nuclear motion. For linear electron-phonon coupling, the interaction Hamiltonian is ex­ pressed from Eq. (4. 1 .27) as

(4.6.7)

4.6. Strong Coupling

161

where Vs(r) is the electron-phonon coupling parameter for the s mode of vibration. The Schrodinger equations for the electronic and vibrational parts of the system are

[H;(r) + H?(r, Q ) ] l ¢; (r, Q )) = W;(Q) I Q>Jr, Q) ) , [H1 ( Q) + W;( Q)] I O;v ( Q ) ) = E;v i O;v ( Q ) ) ,

(4.6.8) (4.6.9)

where H; is the Hamiltonian for the electronic states of the ion in a static crystal field discussed in Chap. 2, and H1 is the lattice vibration Hamiltonian given in Eq. (4. 1 . 19). The solutions of Eq. (4.6.8) are the energy levels and wave functions for the electrons for a fixed value of the lattice vibrational coordinate. The electronic energy found in this way is treated as an effective potential for the lattice vibrations in solving Eq. (4.6.9). Since the electronic and vibrational wave functions are determined inde­ pendently from Eqs. (4.6.8) and (4.6.9) resulting in electronic wave functions that depend on Q , their product resulting in l l/1; v (r, Q ) ) given by Eq. (4.6.6) is not a stationary state of the total system. Th� vibrational state energy E;v given in Eq. (4.6.9) is the difference between the energy of the stationary states of the entire system and the energy for the nonstationary states. This can be used to construct a "nonadiabatic Hamiltonian" that provides the effective interaction needed to produce radiationless transitions between two vibronic states of an ion. The total Hamiltonian operating on the product wave functions gives

Hl l/J;, v (r, Q)) = ( H�E + H�E + He + Hev ) I Q>; (r Q ) O;, v ( Q) ) = ( H� + H�E + W;) I Q\; (r Q ) O;, v ( Q )) = I Q>Jr Q) )H�E I O;, v ( Q ) ) - I Q\; (r Q) )H� I O;, v ( Q )) + (H�E + H�E + W;) I Q\; (r Q) O;, v ( Q) ) = [H�E I Q>; (r Q) O;, v ( Q ) ) ( q); (r Q ) I H�E I O;, v ( Q ) )] + ( q); (r Q) I ( H�E + H�E + W;) I O;, v ( Q) ) = [H�E I Q>; (r Q) O;, v ( Q ) ) - ( Q\; (r Q ) I H�E I O;, v ( Q ) )] + E;v l l/J;, v (r, Q ) ) . This leads to the definition of a nonadiabatic Hamiltonian

HNA i l/J;, v (r, Q ) ) = H�E ( Q ) I Q\; (r, Q ) O;, v ( Q )) - ( ¢; (r, Q ) I H�E ( Q ) I O;, v (r, Q)) .

(4.6. 10)

Thus it is the kinetic energy of the lattice vibrations that is the key to non­ adiabatic Hamiltonian. Using the normal expression for quantum-mechan-

1 62

4. Electron Phonon Interactions

ical kinetic energy gives

Since the Condon approximation assumes that the electronic state varies slowly with tespect to the vibrational coordinate, the second term in this expression can be dropped. This leaves the expression for the nonadiabatic Hamiltonian as

HNA

( r , Q) )

=

h2

M

s

o i O;, v) o Qs o Qs .

(4.6. 1 1 )

The transition rate between two electronic states strongly coupled to the lattice is given by Fermi's golden rule with HNA used for the perturbation Hamiltonian causing the transition. Assuming thermal equilibrium so that there is a Boltzmann distribution for the population of the vibrational levels of the initial state, the transition rate is there is a Boltzmann distribution for the population of the vibrational levels of the initial state, the transition rate is

wif

=

Piv I (fv' IHNA i iv ) j 2t5 (EJv' - E;v ) , �L v, v'

(4.6. 12)

where Piv is the distribution function for the Boltzmann population of initial vibrational levels and a t5 function is used for the density of final states to ensure conservation of energy. Using the expressions for the phonon popu­ lations given in Eqs. (4.2. 14)-(4.2. 1 6) the Boltzmann distribution function can be written as exp

( -E )

(4.6. 13)

In order to evaluate the transition matrix elements for the case of strong coupling, it is necessary to have an exact expression for the vibrational wave functions instead of using the second quantized occupation number formal­ ism. It was shown in Sec. 4. 1 that the normal modes of vibration of the host material can be treated as an ensemble of harmonic oscillator. Thus the normal wave functions for quantum-mechanical harmonic oscillators can be used for the vibrational wave functions. These cna be found in any quantum

4.6. Strong Coupling

1 63

mechanics textbook and are given by1 0

(4.6.14) where the Hv, are Hermite polynomials. The eigenfunctions for the vibra­ tional system are product functions for N phonon modes given by

( Qs i B;v( Qs))

B;v

N

N

II X;v, ( Qs) II Xv, ( Qs - Qs(i)) ,

(4.6. 1 5)

where the changes in the normal coordinates are due to the electron-phonon coupling in the initial and final states. These are discussed below. The electronic wave functions can be expanded with respect to the normal coordinates Qs . Assuming that the electron-phonon interaction that mixes the pure electronic states can be represented by linear coupling as given in Eq. (4.6.7), time-independent perturbation theory gives

l �;( r, Q ))

� ��t o) (r)) + L Ni ) 1 �)0l (r)) + L I �J0 (r)) . (O) s,j-1-i E; - Ej

l �)ol (r)) (4.6. 1 6)

Here Ef0l represents the energy of the unperturbed electronic state and the interaction energy is given by

() - () Vsji - (�j0 (r) I Vs(r) l �i 0 (r)) .

(4.6. 17)

Similarly, the eigenvalues in Eq. (4.6.8) can be expanded to give

W; ( Q ) = EJO) + L Vsu Qs + L s s,s'j -1-i

- Ej

(4.6. 18)

The term in brackets on the left-hand side of Eq. (4.6.9) can be divided into the lattice kinetic energy Hamiltonian plus an effective adiabatic potential for the lattice vibrations. The latter is given by

U; ( Q ) = H�E ( Q ) + W; ( Q )

= Ei(O)

+2

s

2 2 Ws Qs +

L w; Q; + W; ( Q) s

s

V.

sii Qs +

Vsij Vs'ji Qs Q; + () ( ) ···' s,s',j -1-i E; 0 - Ej 0

(4.6. 19) where Eq. (4.6. 1 8) has been used. The first two terms in Eq. (4.6. 1 9) predict the frequency and equilibrium positions of each s normal mode as indepen-

1 64

4. Electron Phonon Interactions

dent of the electronic state i. This is consistent with the results of the Born­ Oppenheimer approximation. The electron-phonon coupling represented by the higher-order terms results in a shift in equilibrium position and fre­ quency of the normal modes that depends on the specific electronic state. The effect of the third term on the equilibrium position can be seen by grouping it with the second term to give

U;( Q) El0) + = E; + where

� w; ( Qs +

�

2:= w; [Qs - Qs (i)f, s

(4.6.20) V,;; Qs ( l") = 4� ws ·

(4.6.21 )

The second of these expressions gives the shift in the equilibrium for the ith electronic state. Note that this is linearly proportional to the diagonal elec­ tronic matrix element of the electron-phonon coupling interaction. The third term in Eq. (4.6. 19) does not introduce a change in vibrational mode frequencies. Including the fourth term in the equation for the effective potential does produce a change in frequencies. This term is generally negli­ gible for rare-earth ions but can be important for some transition-metal ions. Higher-order terms in the equation are needed to describe anharmonic effects. Substituting the nonadiabatic Hamiltonian and the wave functions given above into the expression for the transition rate in Eq. (4.6. 1 2) gives

Here the second term in the matrix element has been neglected with respect to the first term and the derivative of the electronic wave function with re­ spect to the normal coordinate has been assumed to be independent of the specific Qs . The validity of these assumptions can be seen by inspection of Eq. (4.6. 1 6). Taking the derivatives of the wave functions given in Eqs. (4.6. 1 5) and (4.6. 1 6) with respect to Qs , the expression for the transition rate

4.6. Strong Coupling

1 65

becomes

The electronic part of the matrix element has been written as

Rs (

=

h2 -M

(O)

V,if

Ei - Ef(OJ

(4.6.23)

with the interaction energy given in Eq. (4.6. 1 7). The specific phonons in­ volved in the electron-phonon coupling that are responsible for the elec­ tronic transition are called the promoting modes. They have nonzero matrix elements

The vibrational matrix elements not including the modes involved in the electron-phonon coupling are called the Franck Condon factors. The phonons that become excited through the conversion of electronic energy to vibrational energy are called accepting modes. They have nonzero matrix elements contributing to the Franck-Condon factor. The Franck-Condon factor, describing the overlap of vibrational wave functions in the initial and final states, determines the number of vibrational quanta of a specific mode that are excited by the transition. This depends on the modification of the normal · coordinates in the two electronic states given by Eq. (4.6.21 ) . The essential difference be­ tween promoting and accepting modes is that the former have an "allowed" vibrational matrix element and a nonzero off-diagonal electronic matrix element V,if while the latter have a "forbidden" vibrational matrix element and a nonzero diagonal electronic matrix elements ( Vsii - V,JJ ) . The next problem is evaluating the summation over all of the vibrational modes of the initial and final states with the Boltzmann distribution function for the population of the initial states. Several methods have been devised for doing this. Huang and Rhys9 used a series expansion of the harmonic oscillator wave functions while Miyakawa and Dexter1 1 used a generating function approach1 2 to evaluating the double sum. The latter procedure will be outlined here. This approach begins by defining a function FNA such that

FNA (E)

=

L I (Jv' IHNA i iv) I 2Pivo(E - Efv' + Eiv ) v, v'

(4.6.24)

1 66

4. Electron Phonon Interactions

with p;v given in Eq . (4.6. 1 3). Comparing this to the expression for the tran­ sition rate given in Eq. (4.6. 12) shows that (4.6.25) The generating function fNA ( A. ) is defined as the Laplace transform of the spectral function FNA (E) ,

J�oo FNA (E) e J.EdE.

/NA ( A. )

(4.6.26)

Recalling that the density operator is defined as

(4.6.27) where fJ

1 /kBT and the trace of P; ( fJ) is Tr [p; ( /J)]

Lv e pE,, ,

these expressions can be combined to calculate the generating function fNA ( A. )

J�oo �, I (fv' I HNA i iv) j 2p;vb(E - Ejv' + E;v )e J.EdE 1 j (fv' I HNA i iv) j 2e J.(EJ,' E,, ) e pE,, _ r[p; /J)] L

T

(

v, v '

This expression can be manipulated using the properties of the density op­ erator in order to evaluate the sum as follows: 1 i) l v) e ( P J.) E,, ( v i HNA ( if ) l v')e J.EJ, ' v' /NA ( A. ) = Tr [p; ( /3)] � ( I HNA ( / 1 Tr[HNA ( /i)p; ( /J - A. ) HNA ( if )p1 ( A. )] . [ Tr p; ( /J)]

(4.6.28)

The trace is over all possible vibrational wave functions. From Eq. (4.6.23), the electronic matrix can be expressed in terms of the electron-phonon inter­ action strength . HNA ( if )

= - Mh2

s

(0)

Vsif

E1

(0)

E;

a a Qs

=

.

z

a

s

Ssif a Q ' s

( 4.6.29)

where Ssif is known as the Huang-Rhys parameter,

Ssif -=

Vsif M Ef(0) - E;( 0) ·

. h2 - !-

(4.6.30)

Using the vibrational wave functions and interaction Hamiltonian dis-

4.6. Strong Coupling

1 67

cussed above it is possible to evaluate the traces of the density operator­ electronic matrix element products and thus determine the function fNA ( A. ) from Eq. (4.6.28). Then an inverse Laplace transform can be performed to obtain the spectral function 1 A.E (4.6.3 1 ) FNA (E) . /NA ( A. )e dA.. 211:1 oo Once the spectral function is known, Eq. (4.6.25) can be used to calculate the nonradiative transition rate. This procedure is a long mathematical pro­ cess and the details can be found in Ref. 1 1 . With the assumption that the density of states has only one singularity at a frequency Ws , the generating function for a N-phonon process is found to be iw N 2 N! [g2 n(n + 1 )] k ( A. ) exp l �h Eo g R [g (n + 1 ) e A. ] N! (N + 1 ) !k!

J

-

(·

X

_

+

oo

)

n(n + 1 ) + (2n + 1 ) 2

-

2n + 1

(n + 1 ) (2n + 1 ) 2 + (N + k + 1 ) (N + k + 2) gn

1 (N + k + 1 ) g 2 (N + k) (2n + 1 ) 2g + (2n + 1 ) + 2n + 1

]

)

·

At temperatures well below the Debye temperature, n « 1 , and this expres­ sion reduces to f/!A ( A. )

=

;y exp (iA.

)

iNwA. .

The inverse Laplace transform of this function is

This leads to the final expression for the transition rate at low temperatures as (4.6.32)

1 68

4. Electron Phonon Interactions

where the parameters in the above expressions are given by

and Eo is the electronic energy gap of the transition. Note that at zero tem­ perature the parameter g is equal to the Huang-Rhys parameter. The result given in Eq. (4.6.32) is identical to the result obtained by Huang and Rhys using the wave-function expansion approach to the problem. The temper­ ature dependence of the transition rate is contained in the phonon occupa­ tion numbers. The factor ( n + I t appears in the expression as expected, but there will be additional contributions to the temperature dependencies through the parameter R and g. For low temperatures and very weak coupling (g So < 1 ) , it is possible to manipulate Eq. (4 . 6.32) in such a way that it predicts an exponential en­ ergy gap law similar to the one obtained previously from the N-order per­ turbation theory approach to the problem. For these conditions 2n 2 So(N 2) wifN (0) � T R (Eo - Nhw) . (N 2) ! t5

Using Sterling's formula for N » 1 , In N! � N ln N N, this expression can be rewritten as (4.6.33)

(

where

)

1 N 2 AEeff Eo - 2hw, ct. hw In -- - 1 . This form of the exponential energy gap law is somewhat different from the form derived previously. The fact that the effective energy gap in this ex­ pression is less than the electronic energy gap by the energy of two phonons has been attributed to the fact that these phonons act as promoting modes and the other phonons as accepting modes. The temperature dependence of the nonradiative transition rate is con­ tained in the factor for the population of phonons. However, this not only appears explicitly in Eq. (4.6.32) but also in the R and g factors. If the ex­ pression for R is expanded and the assumption of small So is made, the ex­ pression for the decay rate becomes S 2n N 2 e - o N Wif (T) "' h So N!

()

S e - 2n o ( n +

l )N'

( 4 .6.34)

where the temperature dependence is in the last two factors. For small

4.7. Jahn Teller Effect s. =

Ss

0.1

=

1 69

10

E (mits of f! roJ

FIGURE 4. 12. Pekarian distribution function for different values of s• .

values of So the last factor dominates. This expression has been successful in predicting the relative change in the nonradiative decay rate with temper­ ature in a number of different materials. However, it should be recognized that many assumptions have been made in deriving this expression and the full expression and its variation with temperature is much more complicated. The treatment of multiphonon emission processes can also be applied to the vibronic sidebands of zero-phonon lines. The expression for the tran­ sition given in Eq. (4.6.32) approximates a Pekarian distribution in energy. The shape of the emission bands predicated by this expression varies signif­ icantly as a function of the Huang-Rhys parameter S. General examples of plots of this function are shown in Fig. 4. 1 2 for different values of s•. For strong coupling, multiphonon vibronic sidebands appear in the spectrum as smooth, broad bands, whereas for weak coupling the transition appears as a sharp line with weak one-phonon vibronic sideband. These spectral band shapes can be correlated with the configuration-coordinate diagrams in Fig. 4. 1 0. A large offset between the ground- and excited-state potential wells leads to a large value for the Huang-Rhys factor and a broad band that approximates a Gaussian shape. A medium offset gives a smaller value of S and a band with a Pekarian shape. For zero offset S = 0 and the transition is a sharp zero-phonon line. A detailed mathematical treatment of non­ radiative and vibronic transitions based on configuration-coordinate dia­ grams has been developed by Struck and FongerY 4.7

Jahn-Teller Effect

One important result of strong electron-phonon coupling can be a lifting of electronic state degeneracy through a splitting of the vibronic energy levels.

1 70

4. Electron Phonon Interactions

This is essentially a breakdown of the Born-Oppenheimer approximation and is described by the Jahn- Teller theorem. This states that any complex occupying an energy level with electronic degeneracy is unstable against a distortion that removes that degeneracy in first order. The vibronic coupling of ions in solids can cause a local distortion of the lattice in which the atoms move in the direction of normal-mode displacements to lift the electronic degeneracy. A new equilibrium position of the atoms is reached in which the local symmetry is lower than the point-group symmetry of the crystal. This splits the electronic energy levels to higher and lower levels with unchanged center of gravity. Thus the electronic degeneracy is replaced by vibronic de­ generacy. For ions such as the first-row transition-metal ions that have strong vibronic coupling and weak spin-orbit coupling, the J ahn-Teller splitting can be larger than spin-orbit splitting. The distortion can be due to either static or dynamic vibronic coupling, and thermal energy can allow the complex to jump between different equilibrium configurations at high temperatures. These processes have significant effects on the optical spectral features of materials with strong electron-phonon coupling. The most im­ portant feature in terms of solid-state laser materials is the splitting of the electronic energy levels due to the static Jahn-Teller effect. The theoretical description of this effect is outlined below, while a more detailed discussion plus a description of the dynamic Jahn-Teller effect can be found in the re­ view article by Sturge. 14 The effective Hamiltonian describing the vibronic states can be written as the sum of the harmonic oscillator Hamiltonian describing the normal modes of vibrations, Hvib , and the effective electronic part including both the static and dynamic contributions of the crystal field, Hij, H = Hij + Hvib

= E;Jij + L Gij(k ) Qk + Hvib, k

(4.7. 1 )

where (4.7.2) The first term represents the static crystal-field Hamiltonian for the elec­ trons. If it is g -fold degenerate with eigenvalue E and eigenfunctions trans­ forming as the symmetry representation r, the effective Hamiltonian for level r is a g X g matrix. The middle term is the electron-phonon inter­ action that describes the dynamic part of the crystal field given in Eq. (4. 1 .26) with the quantity Gij (k) being a square matrix of the order of the electronic degeneracy of the level. The Qk are the normal coordinates of the complex made up of the central ion and its surrounding ligands. They can be expressed in terms of creation and annihilation operators and ex­ panded in terms of lattice phonons as described in Sec. 4. 1 . The presence of the dynamic crystal-field term is what causes the breakdown of the Born­ Oppenheimer approximation so the eigenfunctions of the system can no

4.7. Jahn Teller Effect

171

longer be described as products of electronic and vibrational wave functions. Instead vibronic wave functions that are combinations of Born-Oppenheimer functions must be used. To illustrate the static Jahn-Teller effect, consider a dopant ion strongly coupled to its ligands in an octahedral configuration. The 1 5 normal-mode coordinates of an octahedral complex and their symmetry properties were discussed in Sec. 4. 1 . Since this complex is centrosymmetric and the matrix elements in Eq. (4.7.2) are between states of the same parity, only even­ parity vibrational modes will make nonzero contributions to GiJ(k) . In addition, group-theory considerations require that the irreducible repre­ sentation of the normal vibrational mode must be contained in the reduction of the direct-product representation of the square of the irreducible repre­ sentation of the electronic state being considered for GiJ(k) to be nonzero. For transition-metal ions of interest in solid-state laser applications, the most important degenerate electronic states in octahedral complexes trans­ form as doubly degenerate E or triply degenerate T1 or T2 representations. As an example, consider a 2Eg electronic state, which leads to eEg 2Eg) ::J 2A 1 g + 2Eg. Therefore, of the 1 5 vibrational modes of this complex, those transforming as OC J g and eg can couple with the 2Eg electronic state to give a nonzero contribution to GiJ(k) . Since the totally symmetric vibrational mode OCJ g only shifts the energy of the state and does not produce splitting, the vibronic coupling with the eg mode is the only one that must be considered to explain the Jahn-Teller splitting of the 2Eg level. The components of the 2 x 2 matrix GiJ(k) for the 2Eg level can be calcu­ lated using the single-electron d wave functions. To do this the Hamiltonian in Eq. (4.7. 1 ) is expanded as H Ho + GiJ( l ) Q 1 + GiJ(2) Q2 + (4.7 . 3) where Q 1 and Q2 are the two vibrational modes transforming as Eg. The operator equivalent technique can be used to evaluate the matrix elements in this equation. This technique takes advantage of the ability to express the matrix elements of interest in terms of angular momentum operators that have the same symmetry transformation properties. It is discussed further in Sec. 8.2. Since G(i) has to transform like Q; so the Hamiltonian will be totally symmetric, the G( i) can be replaced by a constant times their angular symmetry factors expressed in terms of Cartesian coordinates. Using the functions in Eq. (2.3. 1 3) with the common numerical factors absorbed in the constant A gives ·

·

H

Ho

+ Q1 A

(�

·

·

- /) + Q2 A ! (3z2 - r ) .

(4.7.4)

Using the angular momentum operators given in Eq. (2. 1 . 14), this can be rewritten as (4.7.5)

1 72

4. Electron Phonon Interactions

where /+ and /_ are the usual creation and annihilation (ladder) operators. The secular determinant for the perturbation Hamiltonian H (I) H - Ho is (4.7.6) The matrix elements are given by

J J

Hu = d;Ly1H (I) dx1 y1 dr , H22 = d;1H ( 1 ) dz1 dr , H12 = H21

J d;1_y1H( 1 ) dz1 dr

(4.7.7)

where the symmetrized e9 orbitals for the d electron are given in Eq. (2.3. 1 3) . The secular determinant is then given by c

-2 Q 1 - AE

c

- -2 Q 1 AE

0,

(4.7.8)

where c is a proportionality constant between the real matrix elements and the operator equivalent matrix elements. Solving this gives the perturbation energy as (4.7.9) which must be added to the harmonic potential energy to give the total en­ ergy of the system as (4.7. 1 0) Changing to polar coordinates by defining Q 1 energy of the system is given by

E = 2I

± l r. C

r cos rp and Q2 r sin rp, the (4.7. 1 1 )

This expression shows that the original twofold-degenerate potential energy surface is now double-valued and cylindrically symmetric. It is split into an upper potential surface represented by the plus sign in Eq. (4.7 1 1 ) and a lower potential surface represented by the minus sign. A cross section of the surface appears as two parabolas offset in opposite directions from the cen­ tral axis. This is rotated about the central axis to give the three-dimensional "Mexican hat" potential surface shown in Fig. 4. 1 3 . The energy minima are

4.7. Jahn Teller Effect

(A)

(B)

173

Oj

Potential energy curve for Jahn-Teller coupled 2E8 - '• vibrooic level.

Qj

Potential energy curve for Jahn-Teller coupled 2T 8 - '• vibronic level. 2

FIGURE 4. 1 3 . Schematic representation of the potential energy surfaces of (A) 2E9 and ( B) 2 T29 electronic states split by Jahn Teller coupling to a e9 vibrational mode.

found at

lei . ro = 2k

(4.7. 12)

The energy required to cross from one potential to the other is the Jahn­ Teller energy EJT. The triply degenerate electronic 2 T29 level is also split by the Jahn-Teller effect as shown in Fig. 4. 13. For this case the reduction of the direct-product representation of 2 T29 with itself results in selection rules that allow coupling to IXJg, e9, and <2g vibrational modes. In general, for transition-metal ions in octahedral complexes the e9 vibrations couple much more strongly to the

1 74

4. Electron Phonon Interactions

electronic levels of the central ion than the rzg vibrations. Since the coupling to the a 1 g vibrations does not split the degenerate energy level, only coupling to the eg vibrations must be considered. Following the procedure outlined above, the vibronic potential well has three minima as shown in Fig. 4. 1 3 . An example of the effect of Jahn-Teller splitting on the optical pumping dynamics of solid-state laser materials is given in Chap. 7. The Jahn-Teller interaction causes two different types of effects on opti­ cal spectra. First, the offset configuration coordinates modify the shape of the absorption and emission bands. Second, the splitting due to spin-orbit coupling is reduced. The latter is called the Ham effect and is most impor­ tant for T1 and T2 levels. This is due to the fact that the vibronically coupled energy levels can be expressed as products of the electronic components of the T1,2 level and one of the vibrational components of the eg mode. Using these product wave functions to calculate the matrix element of L · S as dis­ cussed in Chap. 2 results in a factored product involving the normal spin­ orbit interaction matrix element and the integral of the product of the two vibrational modes coupled to the electronic wave functions. This later factor has a magnitude less than 1 , thus reducing the spin-orbit splitting. The amount of the reduction varies exponentially with the Jahn-Teller energy. References

1 . B. DiBartolo and R.C. Powell, Phonons and Resonances in Solids ( Wiley, New York, 1 975). 2. C. Kittel, Introduction to Solid State Physics ( Wiley, New York, 1 976) . 3 . D.W. Posener, Aust. J. Phys. 12, 1 84 ( 1 959). 4. D.E. McCumber and M.L. Sturge, J. Appl. Phys. 34, 1 682 ( 1 963) . 5. J . L . Skinner, J. Chern. Phys. 36, 322 ( 1 988). 6. (a) Q. Kim, R.C. Powell, M. Mostoller, and T.M. Wilson, Phys. Rev. B 12, 5627 ( 1 975); (b) Q. Kim, R.C. Powell, and T.M. Wilson, Solid State Commun. 14, 541 ( 1 974); (c) Q. Kim, R.C. Powell, and T.M. Wilson, J. Phys. Chern. Solids 36, 61 ( 1 975). 7. M.D. Sturge, Phys. Rev. B 8, 6 ( 1 973). 8. D.L. Dexter, C.C. Klick, and G.A. Russell, Phys. Rev. 100, 603 ( 1 955). 9. K. Huang and A. Rhys, Proc. R. Soc. London A 204, 406 ( 1 950). 10. E. Merzbacher, Quantum Mechanics ( Wiley, New York, 1 96 1 ) . 1 1 . T . Miyakawa and D.L. Dexter, Phys. Rev. B 1, 2961 ( 1 970). 12. R. Kubo and Y. Toyozawa, Prog. Theor. Phys. 13 , 1 62 ( 1 955). 1 3 . C.W. Struck and W.H. Fonger, J. Lumin. 10, I ( 1 975). 14. M.D. Sturge, in Solid State Physics, edited by F. Seitz, D. Turnbull, and H. Ehrenreich (Academic, New York, 1 967), Vol. 21, p. 9 1 .

5

Ion-Ion Interactions

Since the gain of a laser depends directly on the concentration of lasing centers, it would appear to be beneficial to increase this concentration to as high a level as possible. However, empirically it is found that the optimum doping levels of different types of solid-state laser materials vary between a few hundredths of a percent and 100%. The latter are stoichiometric laser materials that result in high-gain, low-threshold minilasers. The optimum doping levels depend on the type of optical pumping that is used, the cavity configuration, and the mode of laser operation. As the concentration of doping ions in the host material increases beyond some level, the ions can no longer be treated as independent, isolated centers. Instead, the inter­ action between optically active centers must be considered. The concentra­ tion level at which this becomes important depends on whether the doping ions enter the host in a statistically random distribution or whether there is a correlated distribution with a tendency to form pairs and clusters of active ions. The way in which ion-ion interaction effects manifest them­ selves depends on the strength of their interaction. The interaction strength is a function of the separation of the two ions and the physical mechanisms of interaction. There are three different regimes that must be considered. In the strong-coupling regime, the exchange interaction between closely spaced ions causes them to behave as a new type of defect center with optical properties consistent with ion pairs. In the weak-coupling regime the ions maintain their independent optical properties and interact by the transfer of energy from one to another through nonradiative processes such as multi­ polar interactions. In the third regime the ions are not directly coupled but the radiative emission from one ion can be absorbed by another ion. These last two situations involving nonradiative and radiative energy trans­ fer are shown schematically in Fig. 5. 1 . In the case of radiative reabsorption, an ion is excited by photon absorption and another photon is emitted through radiative decay. This photon is then absorbed by another ion. Since the absorption and emission of real photons are well-characterized physical processes, there is no new physics involved in this process. However, radia­ tive energy transfer and energy migration produce observable effects on many 175

1 76

5. Ion Ion Interactions RADIATIVE REABSORPTION

PHOTOCONDUCTIVITY

RESONANT INTERACTION

EXCITON MIGRATION

h+

:: I

I

I

I

5. 1 . Schematic representations of electronic excitation energy-transfer mech­ anisms in solids.

FIGURE

experimental results and it is important to be able to recognize and under­ stand these effects. For example, since no direct ion-ion interaction occurs in this process, radiative energy transfer between different types of ions is not accompanied by concentration quenching of the fluorescence lifetime of the initially excited ion. If radiative energy migration takes place between ions of the same type, a lengthening of the fluorescence lifetime occurs, which is referred to as radiative trapping. The other types of energy transfer processes shown in Fig. 5.1 are radia­ tionless processes. The process of photoconductivity involves energy transfer accompanied by the transfer by an electronic charge. An incident photon creates a free electron-hole pair, either or both of which can migrate in the host material carrying with them both energy and charge. This is an impor­ tant process for microelectronic devices but generally is not of interest for solid-state laser materials. Therefore it will not be treated in detail here. The other two processes shown in Fig. 5. 1 involve the nonradiative trans­ fer of excitation energy without any charge transfer. The energy transfer takes place through a quantum-mechanical resonant process that can be treated as the absorption and emission of virtual photons. The first process of this type involves a single transfer step. The second process of this type involves many sequential steps and can be treated as an exciton. In this type of process, the excitation photon creates an electron-hole pair that is cou­ pled through a Coulomb interaction and migrates together as a quasiparticle carrying with it energy but not charge. If the Coulomb interaction is strong and the exciton is strongly coupled to the lattice through vibronic inter­ actions, lattice relaxation around the exciton will localize the electron and hole on the same ion. This is called a Frenkel exciton. If the interactions are

5. 1 . Exchange-Coupled Ion Pairs

1 77

weak enough to allow the exciton to be delocalized over many ions, it is called a Wannier exciton. These different types of ion-ion interaction mechanisms have very differ­ ent effects on the properties of laser performance. They can either enhance or degrade a specific type of laser performance characteristic. The details of these types of ion-ion interactions are presented in the following sections and their effects on specific laser systems are discussed in Chaps. 6- 10. 5. 1

Exchange-Coupled Ion Pairs

When two optically active ions interact strongly, it is no longer possible to describe accurately their optical spectroscopic properties in terms of the wave functions and energy levels of the individual ions. This type of strong coupling generally occurs when the concentration of active ions is high enough to produce a significant number of occupied near-neighbor sites around a central ion of interest. If C is the fractional concentration of dop­ ant ions, N the density of host ion sites that can be substitutionally occupied by the active ions, and h; the number of possible pair configurations for the ith nearest neighbors, then the concentration of ith neighbor pairs n ; is given by the concentration of dopant ions multiplied by the concentration of the ith pairs (5. 1 . 1 ) n ; = ( CN) (h;C) = h;NC2 . This shows that spectroscopic properties that vary with the number of active centers such as transition intensities will exhibit a quadratic concentration dependence for pair centers compared to a linear concentration dependence exhibited by single-ion centers. The new eigenfunctions and eigenvalues of the coupled-ion-pair system can be determined through diagonalizing the secular determinant using the ion-ion interaction as the coupling term in the Hamiltonian for the system. The isolated ion wave functions and energy levels are assumed to be the ini­ tial properties of the system used for calculating the coupled-pair properties. Thus the Hamiltonian for the pair system is (5. 1 .2) where H, and H2 are the Hamiltonians for the isolated ions and Hi�t is the ion-ion interaction that couples the system. The expression for Hi�t can be derived by considering the interaction potential between two electrons on two different ions (i.e., the special case of s, = ! and s = !l 2 where a , b designate the ion nuclei and 1 , 2 designate the electrons. The spin states o: ( l ) and /](2) are eigenfunctions of Sf, S�, S,z, and S 2z but not of

1 78

5. Ion Ion Interactions

S2 and Sz . The solution to the coupled system must be found to determine the eigenfunctions for the latter operators. The interaction matrix for Vab between all possible I S, Ms ) eigenfunctions is

1 0, 0 ) 1 1 , 0) 1 1' 1) 1 1, -1)

I O, 0 ) 1 1 , 0 ) 1 1 ' 1 ) l l ' - 1 ) 0 0 l12 Vi 0 0 v2 Vi! 0 0 0 E, 0 0 0 E2

This matrix can be diagonalized and the secular determinant solved to give the e:genvalues E E o + J12 ± K12 where J12 and K12 are the Coulomb and exchange integrals discussed in Chap. 2. The eigenfunctions are then found to be

'¥; '¥1 '¥k

1 1 J2 [t/Ja ( 1 ) t/Jb (2) + t/Ja (2) t/Jb ( 1 ) J J2 [tx ( 1 )P (2) - P ( 1 ) tx (2) ] , S O, Ms O 1 J2 [t/la ( l ) t/Jb (2) - t/la (2) t/Jb ( 1 ) ] [tx ( 1 ) tx (2)] , S 1 , Ms 1 1 ( l ) (2) 1 (2) t/Jb ( 1 ) ] J2 [tx ( 1 )P (2) + P ( 1 ) tx (2)] , [t/Ja t/Jb t/Ja J2 S - 1 , Ms 0

which are antisymmetrized singlet and triplet states written in terms of the single-electron orbital and spin functions. The interaction potential can now be put in a more useful form by making use of the spin operators, This operator has the singlet and triplet wave functions derived above as eigenfunctions with eigenvalues h2 S( S + 1 ). These wave functions are also eigenfunctions for Si and S� and thus for S, S 2 . The eigenvalue for Si and S� is i h2 and the eigenvalues for S 1 S 2 are - i h2 for S 0 and ;! h2 for S 1 . Using this information, the eigenvalue equation can be rewritten as E Ec - 2K12 I S 1 S 2 j , where the last factor is the eigenvalue for S, S2 and Ec Eo + J12 is the total Coulomb contribution. ·

·

·

·

5. 1 . Exchange-Coupled Ion Pairs

1 79

From the above discussion, the phenomenological Hamiltonian describ­ ing exchange interaction between two ions having spins sl and s 2 can be written in the form (5. 1 .3) The unperturbed eigenfunctions are the isolated ion wave functions 1 1/Jl ) and 1 1/1] ) while the pair eigenfunctions are expressed as a product: (5. 1 .4) In general, at short separation distances the strongest mechanism for ion­ ion interaction is through the exchange effect. For ferromagnetic coupling K > 0 while for antiferromagnetic coupling K < 0. The complete set of commuting operators for the couples pair is Hpair , Si , S� , S2 , and Sz. Thus the exchange interaction couples the spin angular momentum states of the individual ions to give pair states with total spin S. As discussed in Chap. 2, the results of the angular momentum coupling produces states with a range of total spin quantum numbers (5. 1 .5) The energy levels of the exchange-coupled pair can be found from the matrix element of the Hamiltonian

( l( I

E�:?r = ( 1/Jrir I Hpair 1 1/Jrir ) = 1/Jl 1/1] H1 + H2 -

· 1 ) 1 1/1] ) .

S 1 S 2 1/Jl

(5. 1 .6)

This can be evaluated by using (1/1/ I HI I I/1/ ) = Eiil , along with the relation

(1/1/ I Si i i/IJ ) = liS1 ( S1 + 1 ) ,

sI . s2 =

s2 - si - s�

2

Substituting these expressions into Eq. (5. 1 .6) gives

E�:fr = Ej + Ej

� [S ( S + 1 )

S1 ( S1 + 1 ) S2 ( S2 + 1 )] .

(5. 1 .7)

The ground state of the pair occurs when both individual ions are in their ground states 1 1/Jgair ) = 1 1/JJ ) 1 1/15) with energy Eo and spins So so

K ( 5 . 1 .8) Epa(O)ir 2Eo KSo ( So + 1 ) 2 S ( S + 1 ) with S = 2So, 2So - 1 , 2So 2 , . . . , 0. The first two terms shift the position _

of the energy level from its unperturbed position while the last term splits

1 80

5. Ion Ion Interactions LEYELS •

ENElil SflN •

•

•

•

•

•

•

•

•

LEVELS ENERGY

(6S0 3)K

2s0-3

(4So- l)K

2s0 2

2SoK

2s0 1

•

•

•

•

SPIN • •

•

•

6K

3

3K

2

K 2s0

0

0

(B) Antiferromagnetic Coupling (Z A given by Eq.

(A) Ferromagnetic Coupling (Zp given by Eq.

5. 1 . 1 5)

5. 1 . 14)

FIGURE 5.2. Energy-level splittings of exchange-coupled pairs.

the unperturbed level into a set of levels characterized by different values of S. The splitting between consecutive levels increases with increasing S value as (0)

LlEpair ( S)

0, K, 3K, 6K, . , So ( 2So + l )K. .

.

(5. 1 .9)

For ferromagnetic coupling K is positive so the state of lowest energy is the S 2So state. Energy splittings between successive levels are given by ilE( n + 1 ; n ) K ( 2So - n ) . For antiferromagnetic coupling K is negative so the S 0 state is the lowest in energy and the splitting of successive levels is given by il E (n + 1 ; n ) Kn. Figure 5.2 shows a schematic diagram of the ground-state splittings of coupled pairs. Since the exchange splitting of the energy levels is generally not large compared to thermal energy at ambient temperature, some of the higher levels of the exchange-split manifold are thermally populated. As discussed in Chap. 3, for a group of energy levels in thermal equilibrium the popula­ tion residing in the ith level is given by the Boltzmann distribution =

(5. 1 . 1 0) where E; is the energy of the ith level above the lowest level of the manifold, g; is the degeneracy of the level, T is the temperature, and kB is Boltzmann's constant. The factor A can be determined from the expression for the total occupation of the manifold (5. 1 . 1 1 )

5. 1 . Exchange-Coupled Ion Pairs

where Z is the partition function 1 z

L g; e i

E, fksT .

181

(5. 1 . 1 2)

Thus, (5 . 1 . 1 3)

For ferromagnetic and antiferromagnetic coupling the partition functions are given by z F zA

=

=

( 4So + 1 ) + ( 4So 1 ) e 2So K/kaT + ( 4So + e [So (2So+ l)] K/kaT , _

_

3) e (4So i ) K/kaT +

. . .

(5. 1 . 14)

1 + 3 e Kfka T + 5 e 3K/kaT + . . . + (4So + 1 ) e [So ( 2So+l )] K/ksT .

(5. 1 . 1 5)

Using Eqs. (5. 1 . 1 0) and (5 . 1 . 1 1 ) along with either Eq. (5. 1 . 1 4) or (5. 1 . 1 5) , the temperature dependence of the population of each of the levels of the exchange-split manifold can be determined. The population is important in determining the intensity of any transition originating on the level. The lowest excited state of a coupled pair occurs when one of the pair ions is in the ground state and the other is in the excited state l l/l fair ) l l/IJ ) l l/l t ) . The determination of the energy-level splitting of the excited state follows the procedure outlined above but the full expressions in Eq. (5. 1 .7) must be used since Ef , S1 and E] , Sz now have different values. Also it has been proposed that other phenomenological forms of the exchange inter­ action besides that given in Eq. (5. 1 . 3) might better describe pair interaction in some excited states. These include biquadratic exchange k(S, Sz) 2 , and individual electron spin coupling, (K' j h2 ) � i,j sf sJ . These can generally be treated by the same procedure described above. The magnitude of the exchange interaction is generally found by treating K as an adjustable pa­ rameter in fitting the observed energy-level splittings. First-principle calcu­ lations of K are generally not possible since evaluating the exchange matrix element requires knowledge of the exact expressions for the electronic wave functions of the two ions as discussed in Chap. 2, as well as knowing the exact positions and spatial orientations of the orbitals of the electrons on the two ions so the wave-function overlap can be determined. None of these parameters are generally known with the degree of precision required for accurate calculations. In addition, the exchange interaction between the two dopant ions can be significantly affected through the polarizability of the intermediate host ligands. This so-called superexchange further complicates fundamental calculations of the interaction strength. Ion pairs have been observed in the optical spectra of several solid-state laser materials. As the concentration of optically active ions increases to the level that they begin forming pairs, they decrease the lasing potential of the isolated ions. However, the pair centers themselves can provide new transi-

·

·

1 82

5. Ion Ion Interactions

tions for laser operation. The level at which ion-pair effects become important depends on whether the dopant ions enter the lattice with spatial random­ ness or whether they tend to be aggregated. If pairing takes place between two different types of ions, it has been found that aggregation is enhanced if the sum of the ionic radii of the two types of dopant ions equals the ionic radii of two host ions that they replace. 2 This can be used to enhance energy­ transfer pumping as discussed later. Ion pairs interact with the photon and phonon fields in the same way as isolated ions. Transitions between levels in an exchange-split multiplet generally take place by radiationless transitions and both direct and two-phonon Orbach processes have been found to be effective in relaxing a system to an equilibrium population distribution. Photon absorption and emission processes have been observed between the ground and excited state of pair systems and the widths of the observed spectral lines have been attributed to both Raman scattering of phonons and lifetime broadening processes. The most extensive investigation of pair spectra and lasing in solid-state laser materials has been performed on ruby and the results are summarized in Chap. 6. 5.2

Nonradiative Energy Transfer: Single-Step Processes

For the case of weak ion-ion interaction, the energy levels of the individual ions involved are the same as those of isolated ions. In this case a single ion is excited by the photon field and the interaction causes a nonradiative transfer of the electronic excitation energy to another ion that subsequently emits the energy. This energy-transfer phenomenon occurs in many different types of materials and plays an important role in a wide variety of physical properties. As stated in the introduction to this chapter, there are two funda­ mentally different types of energy transfer: photoconductivity, which involves the simultaneous transfer of electronic charge and energy, and energy trans­ fer with no accompanying charge transfer. The latter case is the one of importance to laser materials and is discussed below. There is no consistent convention for energy-transfer terminology. In this discussion the ion that absorbs the energy from the photon field is called the sensitizer and the ion that emits the energy is called the activator. In some discussions of energy transfer the terms donor and acceptor are used, but this terminology can be confused by the use of these terms for semiconductor dopants. If the sensi­ tizer is part of the host material, the term host-sensitized energy transfer is used, while impurity-sensitized applies to the case where the sensitizer is a dopant ion. If the energy moves from one sensitizer to another several times before emission occurs, the process is referred to as energy migration as opposed to single-step energy transfer directly from sensitizer to activator. Multistep energy migration is discussed in Sec. 5.4. The first step in treating energy transfer is to derive an expression for the ion-ion interaction Hamiltonian causing the processes to occur. A simple

'>

5.2. Nonradiative Energy Transfer: Single-Step Processes

Wra SENSI1ZER

(A)

ACTIVATOR

1 83

(B)

FIGURE 5.3. Single-step energy transfer between sensitizer and activator ions. (A) Energy-level model for a single-step nonradiative energy-transfer process. ( B) Feynman diagram for energy transfer.

energy-level diagram depicting the process of energy transfer between two ions is shown in Fig. 5.3(A). In the simplest analysis this process can be treated as a quantum-mechanical resonant interaction involving the exchange of a virtual photon as depicted by the diagrams in Fig. 5.3(B). The mecha­ nism of ion-ion interaction can be either an exchange interaction or an electromagnetic multipole-multipole interaction. As noted in the preceding section, exchange interactions can be very strong over a short range (a few fmgstroms), whereas electric dipole interactions can be effective over dis­ tances of tens of angstroms. The initial development of the theoretical treat­ ment of energy transfer through electric dipole-dipole interaction was done by F6rster3 and later extended by Dexter4 to include higher-order multipole interactions and exchange. Because of this, single-step nonradiative energy transfer is sometimes referred to as a Forster-Dexter process. As with any quantum-mechanical transition processes, the rate for energy transfer to occur is described by the golden rule of time-dependent pertur­ bation theory, given by ( 5.2. 1 ) The matrix element for the transition can be expanded as

(5.2.2) where Hf�t is the ion-ion interaction Hamiltonian and the l/1; represent the antisymmetrized product wave functions of the optically active electrons on

1 84

5. Ion Ion Interactions

the sensitizer and activator ions. The first term describes a resonant inter­ action while the higher-order terms are used to describe phonon-assisted energy transfer. The former is discussed below and the latter are discussed in the next section. There are two types of mechanisms for ion-ion interaction leading to energy transfer. The first is the exchange interaction discussed in Sec. 5. 1 . This occurs if the ions are close enough together that there is overlap of the orbitals of electrons on the sensitizer and activator ions. For the simplest case of isotropic Heisenberg exchange the Hamiltonian has a form similar to that in Eq. (5. 1 .3),

·

ntf�� = - L Kus; s1 , i,j

(5.2.3)

where KiJ is the exchange integral and the s; are the spins of the interacting electrons. The second type of interaction that can be responsible for energy transfer is electromagnetic interaction. The Hamiltonian in this case is expressed as the sum of all Coulomb interactions between the charge distributions on the two ions as shown in Fig. 5.4,

Here Ra and Rs are the positions of the activator and sensitizer ions, respectively, and ra and rs are the position vectors of the activator and sensitizer optically active electrons, respectively. The Coulomb interactions between the two nuclei and between the nuclei and the electrons result in zero matrix elements due to the orthogonality of the wave functions. This

FIGURE 5.4. Spatial geometry for the interaction between sensitizer and activator ions.

5.2. Nonradiative Energy Transfer: Single-Step Processes

1 85

leaves the Coulomb interactions between the electrons as the effective inter­ action. There are several useful ways of expressing this Hamiltonian. The ion-ion interaction can be expressed as a multipole expansion using a Taylor's series about the sensitizer-activator separation Rsa · The electric multipole part of the interaction Hamiltonian is given by mt

2 = :. e

Rsa + fa - fs , , e2 = eR3 [fs · fa - 3(fs · R sa) (fa · R sa)] sa , , e2 - · R, sa)] - · Rsa + R 4 [ 2s ( R' sa · Qs ) (fs · R sa) - (fa · Qs sa ' , e2 , + RS [( Rsa · Qa · Qs · Rsa ) + 4! ( Rsa · Qa · RsaRsa · Qs · Rsa ) sa + � ( Q a : Qs ) ] + . . . = Hi;�o + Ht!�Q + Hi�?D + Hi�?Q + . . . , e

_

_

_

,

,

_

,

e

(5.2.4)

where these Hamiltonians represent electric dipole-dipole, electric dipole­ quadrupole, electric quadrupole-dipole, and electric quadrupole-quadrupole interactions. The first part of each of these interactions refers to the transition on the sensitizer while the second part refers to the transition on the activa­ tor. The dielectric constant of the host crystal is given by e, the vectors fa and fs designate the positions of the electrons on the sensitizer and activator as shown in Fig. 5.4, and Qa is the quadrupole moment operator given by

(5.2.5)

Here Ms = I: fs is the dipole moment operator and i is the unit matrix. This interaction Hamiltonian can also be expanded in terms of spherical harmonics using the addition theorem for these functions. This results in the expression -

_

mt

e2 e

4n(ls + Ia ) ! ( - 1 ) 1" r�' r� ! [(21s + 1 ) (2/a + 1 ) ] 1 /2 , a l l R + + 1, 1 la O 00

00

��

( 5.2.6) where the dipole terms are found from setting I = 1 and the quadrupole terms from I = 2. The magnetic part of the ion-ion interaction can also be expressed as

1 86

5. Ion Ion Interactions

a multipole expansion by assuming each electron has a magnetic dipole moment J.l; ( ej2mc) (I; + 2s;) associated with its spin and orbital angular momenta. In this case, the lowest order magnetic dipole-dipole interaction term is the only one of importance. This is given by · J.lj 3 ( J.1; Rsa ) (J.lj · Rsa) HmMDD (5.2.7) · t 3 Rsas i,j Rsa

·

)

The terms in Eqs. (5.2.3), (5.2.4), (5.2.6), or (5.2.7) can be used in Eq. (5.2.2) to determine the matrix element for the energy-transfer transition for different types of interactions. Order-of-magnitude comparisons of the interactions show that the magnetic dipole-dipole interaction energy is w 9 smaller than the electric dipole-dipole interaction energy and therefore it will be neglected in the following discussion. The wave functions for the system of two ions must be expressed in anti­ symmetrized form, l l/1; )

[ l l/l�(rs, w�)) l l/la (ra, Wa)) lx�(s)) lxa (a)) - l l/l�(ra, w�)) l l/la (rs, Wa)) I X�( a)) lxa (s) )] ,

l l/11 ) =

( 5.2.8)

Ws)) l l/l� (ra, w�)) I Xs(s)) lia(a) )

- l l/ls(ra , Ws)) l l/l� (rs, w�)) I Xs(a)) lia(s) ) ] , where the subscript on the position vector refers to the ion where the elec­ tron is initially found. The parameter w represents the energy of the state. The matrix element has the form Msa

\ l/11 I Hi�t l l/1; ) x

( l/ls(r., Ws) I (l/l�(ra , w�) I Hi�t l l/l�(r., w:) ) l l/la (ra, Wa))

(xs(s) lx�(s)) (x� (a) lxa (a))

- ( l/ls(ra, Ws) I (l/l�(rs, w�) I Hi�t l l/l�(rs, w�)) l l/la (ra, Wa)) ( 5.2.9) The second term represents exchange interaction and the first term electro­ magnetic interaction. The first term will be considered here. Since it involves the selection rules x's Xs and x� Xa , the spin functions will no longer be written explicitly. To find the total energy-transfer rate, it is necessary to sum over all pos­ sible combinations of sensitizer and activator states for which energy is con­ served, w� - Ws w� - Wa E This can be accomplished by introducing probability distribution functions for the initial states of the sensitizer and activator ions, p� (w� ) and Pa (wa) . These distribution functions vary con­ tinuously over the energy of the excited state of the sensitizer and ground state of the activator ions. It is also important to include degeneracy factors

.

5.2. Nonradiative Energy Transfer: Single-Step Processes

1 87

for the excited state of the sensitizers g� and ground state of the activators 9a· The eigenfunctions for the sensitizer and activator ions are normalized as usual to an integral over all space. The probability distributions are nor­ malized over all possible energies as

I: p�(w�) dw� I: Pa (wa ) dwa

1.

(5.2. 10)

With the above considerations, the total transition rate is given by

I dw� I dws I Pa (wa ) dwa I p�(w�) dw�

ws�M X

I (t/lj I Hi�tt l t/li) 1 2 c5( (w� - Ws) - (w� - Wa )) ·

(5.2. 1 1 )

where the sum is over all quantum numbers other than the energy quantum number, and the c5 function is included in the density of final states to ensure conservation of energy. Integrating over w� gives

I dE I Pa (wa )dwa I p�(w�) dw� l ( t/IJ I Hi�� l t/li) l 2 ·

ws�M

(5.2. 12)

Using the first term in Eq. (5.2.4), the electric dipole-dipole interaction matrix element is

where the expectation value of the electron position vector is given by (r)

I t/J'* rt/1 dr .

This matrix element can be averaged over all possible orientations using Fig. 5.4. Assuming that there is no preferred orientation of the dipoles, the average over all orientation angles results in

Substituting these results into Eq. (5.2. 12) gives

w;oo

(g�ga ) 1

� � I dE I Pa ( Wa ) dwa I (ra) 1 2 I p� ( w� ) dw� I (rs) 1 2 .

(5.2. 1 3)

It is useful to reexpress Eq. (5.2. 13) using parameters related to the experimentally observed absorption and fluorescence spectra. For example,

1 88

5. Ion Ion Interactions

the Einstein A coefficient from Eq. (3.2.6), including a thermodynamic sta­ tistical average over all initial states, is expressed as

A(E)

2

L i

Ec ) l (ri[) l 2p' (w') dw', 3h c gi ( E J

f

(5.2. 14)

where the screening factor (Eel E) is the ratio of the electric field seen by the ion in the crystal to that "seen" by the ion in a vacuum. Because of the relationship between the Einstein A coefficient and the radiative lifetime, a normalized spectral function g is defined as (5.2. 1 5) g = rA. This is related to the integrated emission intensity of an ion as discussed in Chap. 3 . Similarly, the Einstein B coefficient from Eq. (3. 1 .37) can be used to express the absorption properties of the ions,

B(E) =

i

f

2ne2 Ec 2 i[ 2 I (r ) I p(w) dw. 3h g, E

( )J

(5.2. 1 6)

Since this is related to the absorption cross section by O"(E) = B(E)/(cjn), another normalized spectral function G can be defined as

G O"(E) ' Q

(5.2. 17)

where Q is the integrated absorption cross section, Q = J O"(E)dE. Using the definitions given in Eqs. (5.2. 14)-(5.2 . 1 7) in Eq. (5.2. 1 3) gives the energy­ transfer rate as (5.2. 18) where the oscillator strength has been used to describe the absorption tran­ sition instead of the parameter Q as given in Eq. (3.3.20). The expression in Eq. (5.2. 1 8) can be simplified by assuming that the electric field screening factor is close to unity and that the wave number in the region of spectral overlap does not vary significantly so an average value of Vsa can be used. The results of these simplifications are (5.2. 19) where the spectral overlap integral Q is given by (5.2.20)

5.2. Nonradiative Energy Transfer: Single-Step Processes

1 89

To simplify the expression further, it is useful to define a critical inter­ Ro given by

action distance

( 5.2.2 1 ) and a critical concentration Co given by (5.2.22) Substituting the expression for Ro into Eq. (5.2. 1 9) allows the energy trans­ fer rate for electric dipole-dipole interaction to be written as wsaEDD

(� RRo )6

r

= _.!._

sa

(5.2.23)

.

From this expression it is clear that the critical interaction distance is the sensitizer-activator separation for which the energy-transfer rate is equal to the intrinsic decay rate of the sensitizer. Similar expressions can be derived for electric dipole-quadrupole and quadrupole-quadrupole interactions. These energy-transfer rates can be expressed in terms of the electric dipole-dipole rate as3.4 2 / WsaEDQ = WsaEQD = Q � WsaEDD (5.2.24) fn

and WsaEQQ

=

(R ) sa

( /Q)2 (�R )4 fn

sa

WsaEDD '

(5.2.25)

where A. is the average wavelength of the transition in the spectral overlap region and/Q is the oscillator strength for a quadrupole transition given by 2m

!ifEQ - 3he 2 wiJ I ( (k · r) ( er)) if l 2 . Energy transfer by these higher-order multipole interactions can be impor­ tant when electric dipole transitions are forbidden on the sensitizer or acti­ vator ions. The energy-transfer rate for exchange interaction can be derived by using the interaction Hamiltonian given in Eq. (5.2.3) in the first term of Eq. (5.2.2) to evaluate the matrix element. In this case explicit expressions for the electronic wave functions must be used. To obtain a general expression, it is customary to use hydrogenlike wave functions, 4 which leads to

[(

)]

R 1 WsaEX = r exp 1 R o J ' sa �

(5.2.26)

1 90

5. Ion Ion Interactions

where

(5.2.27) and = 2n h 's

(5.2.28)

In these expressions, L is an effective Bohr radius and K is a constant involving the spatial overlap of the electron wave functions. For the important case of rare-earth ions in solids, expressions for ion­ ion interaction can be rewritten explicitly in terms of the Hamiltonian given in Eq. (5.2.6) and wave functions expressed as spherical harmonic func­ tions. 5 The Judd-Ofelt theory is used to express the strengths of the elec­ tronic transitions on the sensitizer and activator ions in terms of special pa­ rameters and reduced matrix elements (defined in Sec. 8.2). This theory is outlined in detail in Sec. 8.3. The results for the rate of electric dipole-dipole interaction are

w;oo

=

1 (2ls + 1 ) (2Ia + 1 ) x

(�

(;;J (� n;.s (ls II u(J.) I I J;) z) z

)

il;.a (Ja II U (J.) I I J�) 2 n,

(5.2.29)

which takes the place of Eq. (5.2. 19). Here the il;.s are the Judd-Ofelt intensity parameters, ls is the total angular momentum quantum number for the energy level involved, U is the unitary operator, and the double bars represent reduced matrix elements. Similar expressions can be written for the other multipolar interactions. The energy-transfer rates derived above describe a process occurring over a fixed sensitizer-activator separation Rsa· In a typical laser material in­ volving energy transfer, there are spatially random distributions of ensem­ bles of sensitizer ions and activator ions. Any specific sensitizer ion in the excited state has a probability of interacting with each unexcited activator ion in the host with the strength of the interaction varying with distance in accordance with the type of interaction mechanism. The effective energy­ transfer rate for the entire system of ions must reflect this probability distri­ bution of transfer rates for a given sensitizer ion, along with the random nature of the activator environment for each sensitizer site. This can be ac­ complished by dividing the sensitizers into classes having the same activator environment and then finding the average of the result for all classes. As an example of this procedure, consider the electric dipole-dipole energy transfer rate for a specific class of sensitizer designated as j. Using

5.2. Nonradiative Energy Transfer: Single-Step Processes

Eq. (5.2.23), the transfer rate is

w;nn =

� t (RRo )6 , 's i=l

,

191

(5.2.30)

where R; is the distance from the sensitizer of class j to the ith activator and the sum is over all Na activator ions. One experimental method for determin­ ing the energy-transfer rate of the system of ions is through measuring the fluorescent lifetime. This is associated with the time-dependent decay of the population of the excited state of the sensitizer ions. The rate equation describing the time rate of change of the population of the excited state of sensitizers in class j (designated nsj ) is

dnsj = 1 nsj - UJEDD nsj , dt 0 's -

assuming J-function excitation at time zero. The solution of this equation is

(5.2.3 1 ) To proceed further it is necessary to know something about the distribu­ tion of ions. If a spatially random distribution of activators is assumed, the number of sensitizers belonging to class j is given by

Na 4nR� dR; , Nsj _ Ns IT i=l V

(5.2.32)

where V is the volume of the crystal. Using this as the probability distribu­ tion function for a specific class of sensitizers with a time varying excited­ state population given by Eq. (5.2.30), the average of nsj over all classes then gives

(5.2.33) where Co is the critical concentration defined in Eq. (5.2.22). Differentiating this equation with respect to time gives the rate equation of the total pop­ ulation of excited sensitizers. The result has the same form as Eq. (5.2.30) with the total energy-transfer rate given by

-

wEDD 'J.n3 f2 R5 Na - 3 ( r� t) l /2

·

(5.2.34)

Thus the energy-transfer rate for electric dipole-dipole interaction among randomly distributed ions varies with time as r 1 12 and depends directly on the concentration of activator ions and on the cube of the critical interaction distance Ro .

192

5. Ion Ion Interactions

Other expressions can be derived in a similar way for different types of energy-transfer mechanisms. For electric multipole interactions the results can be expressed by the general equation ( 5.2.35)

where q 6, 8, or 1 0 for electric dipole-dipole, electric dipole-quadrupole, and electric quadrupole-quadrupole interactions, respectively. Similarly, for exchange interaction the sensitizer population is given by (5.2.36) where the function g (z) is given by (5.2.37) One important assumption in the derivation described above is that the distributions of sensitizers and activators are uniformly random throughout the sample with no correlation effects as reflected in Eq. (5.2.32) . As dis­ cussed in the previous section, effects such as local strains can enhance cor­ relation effects and cause preferential pairing of sensitizer and activator ions.6•7 This can be treated mathematically by modifying Eq. (5.2.32) to include correlation factors. The distribution of ions can be separated into several regions of space where nearest-neighbor shells have a high degree of correlation, whereas at long separation distances the random distribution of Eq. (5.2.32) is used. This result in much more efficient energy transfer at short times due to an enhanced concentration of near-neighbor sensitizer­ activator pairs. At long times the energy-transfer rate evolves toward the value obtained by assuming an uncorrelated, random distribution of both types of ions. 5.3

Phonon-Assisted Energy Transfer

The energy-transfer rates derived in the preceding section depend critically on having resonance between the emission transition of the sensitizer and the absorption spectrum of the activator. This is explicitly reflected in the spectral overlap integral factor appearing in the expressions for the energy­ transfer rates. Phonons play an important role in ensuring the conservation of energy. For resonant electronic transitions, phonons affect the widths of the spectral lines and thus the magnitude of the spectral overalap integral. Also the temperature dependences of the resonant energy transfer rates are contained in the spectral overlap integrals. For the case when the electronic

5.3. Phonon-Assisted Energy Transfer

•

.1Esa

1 93

FIGURE 5.5. Typical phonon-assisted energy-transfer process .

'

PA W sa

transitions are out of resonance with each other resulting in a value of zero for the spectral overlap integral, the rate of resonant energy transfer is negli­ gibly small. In this situation the energy mismatch between the sensitizer and activator transitions can be made up by the contributions of phonons re­ sulting in phonon-assisted energy transfer. A typical situation for phonon­ assisted energy transfer between two ions with a transition energy mismatch AEsa is shown in Fig. 5.5. The electron-phonon interaction can occur on either ion and in either the ground or excited state. The transition rate for phonon-assisted energy transfer can be calculated using the expressions in Eqs. (5.2. 1 ) and (5.2.2) with the interaction Hamil­ tonian containing factors for both ion-ion and ion-phonon interactions. Also the wave functions must now contain a factor for the occupation number of the phonon involved. Thus the matrix element associated with a one-phonon-assisted, ( PA) energy-transfer process is given by

MsaPA

(l/lsl/l:njk ± l i Hsa l l/l;l/la nik ± l ) ( l/l;l/lanik ± l i H? (m) l l/l;l/lanJk ) Es - (Es ± hWjk ) m ,a ( l/1sl/l:njk ± I I H? ( m) l l/1sl/l: njk) ( l/1sl/l: njk I Hsa l l/1; l/1anik ) + ' Es - Ea m=s,a

L =s

(5.3 . 1 ) where the sum accounts for the phonon emission or absorption occurring at either the sensitizer or activator site. Here Hsa is one of the resonant interaction Hamiltonians discussed in the previous section and H? is the electron-phonon interaction Hamiltonian expressed in terms of the strain e and crystal field V in Eqs. (4. 1 .22)-(4. 1 .24) . Since the spatial extent of the phonon is important in this case, the exponential factor appearing in Eq. (4. 1 .20) describing the ion displacement due to the phonon must be retained explicitly. ( This was suppressed in Chap. 4 where spatially localized pro-

194

5. Ion Ion Interactions

cesses were being considered.) This expression can be simplified to8

( 5.3.2)

k Msa

where is the ion-ion matrix element independent of the phonon state and is the phonon wave vector. The strain factor in the electron-phonon interaction can be expressed in terms of the phonon creation and annihila­ tion operators, and f and g are the electronic matrix elements of the crystal­ field operator in the ground and excited states, respectively. The difference in coupling strengths in the ground and excited states (f - g) is assumed to be equal for the sensitizer and activator ions. The exponential factors result in a phase factor exp( ± ik Rm) for the ion at position Rm. In addition, conservation of energy requires that the phonon energy hw1k is equal to With these conditions, the the electronic transition energy mismatch phonon-assisted energy transfer rate is expressed as ·

!:iEsa ·

(5.3.3) where the coherence factor is given by h( k,

Rsa )

Ie

ik R,.

1 12

4 sin2

(k .

(5.3.4)

This factor describes the degree to which a particular phonon mode causes the sensitizer and activator ions to move in phase with each other. This is a key consideration in determining how effective the phonon is in bringing the energy levels of the two ions into resonance. There are two different cases that can be considered. The first is the case when the energy mismatch between the electronic transitions is small com­ pared to the available phonon energies so that the relevant phonon modes are those with small wave vectors. Thus, in this case « 1 so the pho­ non wavelength is large compared to the sensitizer-activator separation. Using a Debye distribution of phonons to evaluate the sum over k in Eq. (5.3.3) and averaging the coherence factor over all angles, the phonon­ assisted energy transfer rate for this case becomes

k · Rsa

(5.3.5) where p is the density of the host, v1 is the phonon velocity, and r:x1 is the angular average of the strain parameter. The upper term in the last factor is for phonon emission and the lower term is for phonon absorption. The temperature dependence is contained in the phonon occupation numbers. The factor of in the numerator shows that the closer the sensitizer and activator are to each other, the less effective the phonon is in bringing their

R;a

1 95

5.4. Nonradiative Energy Transfer: Multistep Process

transitions into resonance. This is because, for the conditions of this case, the phonon tends to modulate the energy levels of both ions together . The second case is when the energy mismatch between the electronic transitions is large enough that the relevant phonons are those with large wave vectors so k R sa > 1 . In this case the wavelength of the phonons is shorter than the sensitizer-activator separation. Following the same proce­ dure described above, the energy-transfer rate for this case is given by ·

wPA sa ( 2 )

M}a (f -

nph4

J

r:l.j v5J

X

{

njk + nJ'k

1

},

( 5.3.6)

The energy-transfer rate for this case has the same temperature dependence as that for the first case. However, the dependence on the transition energy mismatch is quite different. In addition, there is no longer an explicit de­ pendence of the phonon modulation term on Rsa · This is because the pho­ non in this case modulates the transition energy of the sensitizer differently than it modulates the transition energy of the activator. For cases involving very small energy mismatches where the density of states of available phonons with the appropriate energy is also very small (as shown in Fig. 4.4), two-phonon-assisted energy-transfer processes may be­ come important. In this situation it is the difference in energy between one phonon that is absorbed and another phonon that is emitted that makes up the energy difference between the two electronic transitions. The energy­ transfer rate for this situation can be derived using the same steps outlined above but using the appropriate two-phonon matrix elements for the radia­ tionless decay processes discussed in Sec. 4.2. Similarly, for very large energy mismatches (that is, much greater than available phonon energies), multi­ phonon processes become important and the N-order perturbation approach to radiationless decay processes must be used. This again leads to an energy­ transfer rate exhibiting an "exponential energy gap law" and a temperature dependence described by [n (w ) + l] N as discussed in Sec. 4 . 2 . 5.4

Nonradiative Energy Transfer: Multistep Process

For some materials the concentration of sensitizers is high enough that ex­ citation energy can be transferred from one sensitizer to another several times before the final transfer to an activator occurs. In this type of multi­ step transfer process, the excitation energy can be viewed as a quasiparticle migrating on a lattice of sensitizers, and the mathematical description of the process is the same as that for exciton migration. This case involves a lo­ calized exciton with the electron and hole both located on the same ion and moving together. It is referred to as a Frenkel exciton. This view of multistep energy migration is especially applicable to host-sensitized energy transfer where the sensitizer is a constituent of the host lattice.

196

5. Ion�Ion Interactions s

s

s

s s

s

s

s

s

s

s

s

FIGURE 5.6. Multistep energy migration and trapping.

Figure 5.6 shows a schematic representation of multistep energy migra­ tion. Each energy-transfer step between sensitizers involves one of the ion­ ion interaction mechanisms discussed in the previous section. The final step between sensitizer and activator ions also involves one of these ion-ion in­ teractions mechanisms, but it does not have to be the same mechanism as the one for sensitizer-sensitizer interaction. The dynamics of the total energy­ transfer process are characterized by two distinct contributions: the migra­ tion of energy among the sensitizer ions and the trapping of the energy at activator sites. One of the difficulties in analyzing the effects of multistep energy transfer is separating the characteristics due to the migration process and those due to the trapping process. There are two mathematical ap­ proaches used to describe localized exciton migration with trapping, one based on a random-walk model and the other based on a diffusion model. In the limit of many steps in the random walk on a uniform three-dimensional lattice, the two approaches are equivalent. Both of these approaches are described below. The simplest situation is one in which the sensitizer-activator interaction is equivalent to the sensitizer-sensitizer interaction so the migrating exciton becomes trapped only when it happens to hop onto an activator site. As an example, consider a simple-cubic lattice arrangement of active ions (sensi­ tizers and activators) with electric dipole-dipole interaction between pairs of ions. Let the time for an excitation step be represented by the hopping time th, and the probability of sensitizer fluorescence per hopping time be repre­ sented by a. Also let the probability of activator trap fluorescence per hop­ ping time by represented by p, and the fraction of active ion lattice sites that are activator traps be represented by Ca. With these assumptions and nota­ tion, the probability for sensitizer fluorescence to occur on the nth step in a random walk is given by9 • 1 0 where Vn is the number of distinct sites visited on a walk before the nth step. The last factor in this equation is therefore the probability that none of the

5.4. Nonradiative Energy Transfer: Multistep Process

1 97

sites visited before the nth step are traps. For large n on a three-dimensional lattice, random-walk theory shows that Vn -+ ( 1 - F) n, where F is the probability the exciton eventually returns to the origin. The latter quantity varies with lattice symmetry and is found to be about 0.34 for a simple-cubic lattice. 8 Thus the sensitizer fluorescence intensity after n steps and an initial excitation intensity of ls(O) is

( 5.4 . 1 ) The number of distinct sites visited before step n has been approximated by

0.66n.

A similar expression can be written for the probability for fluorescence from an exciton trapped at an activator site after the nth step in the random walk. This can be divided into four time periods. First there are n 1 - 1 steps on normal lattice sites, then one step onto a trap, then a waiting period of n - n 1 - 1 steps, and finally emission of trap fluorescence. Thus the fluo­ rescence intensity of activator emission after n steps is r a

=

)

(

n ( 1 1X ) ( 1 - Ca ) (1 - F) ' P ( l - F) Ca ( 1 - Pt 1 -P ( 1 - IX) ( 1 - P) 0.66P Ca ( 1 - Ca ) 0.66 {( 0.66 n . n [( 1 IX 1 1 (1 - P) [1X - p + ( 1 - 1X)0.66Ca] - p) - - ) ( - Ca ) ] } ( 5.4.2)

In the limit of many steps, Eqs. (5.4. 1 ) and (5.4.2) can be rewritten as func­ tions of time expressed in terms of number of steps, t = n th ,

where

ls(t)

=

Ia (t)

=

-

a ls(O) 1 - a e (a+0.66 Ca) t/ th '

-0.66Ca + 1

( 5.4.3)

(e -btf th - e- (a+0 .66 Ca) t/ th ) ,

(5.4.4)

The exponential decay rate in Eq. (5.4.3) can be used to define the fluo­ rescence decay time of the sensitizers with and without any activator ions being present. These decay times are

rs 1

=

(a + 0.66Ca ) th

'

rsO1

a. th

= -

From these expressions the rate of energy transfer for a multistep random-

198

5. Ion Ion Interactions

walk process can be defined in the usual way,

rs 1

rs(}1 + wrw sa '

(5.4.5)

where

( 5.4.6) The ratio of the sensitizer fluorescence decay time in the undoped and doped samples is T s() 1 + sO Wrw . ( 5.4. 7) T sa Ts Since the ion-ion interaction mechanism has been assumed to be electric dipole-dipole, the expressions from the preceding section can be used to obtain an equation for the hopping time,

(

)6

R0 ( 5.4.8) rsO1 Rss ' where the critical interaction distance Ro was defined in Sec. 5.3 and Rss th 1

represents the nearest-neighbor sensitizer-sensitizer separation. The sensitizer and activator fluorescence intensities can be evaluated by integrating Eqs. (5.4.3) and (5.4.4) over time. This gives

Is Ia

oo J ls(t)dt oo Ia (t)dt Jo 0

a th ls(O) l _ a a + 0_66Ca ,

b -b

th - -; · + 0 66C a a . +l th

( 5.4.9)

0.66Ca From the first of these expressions, the ratio of the sensitizer fluorescence with and without traps present is / (0)

1 + T s() Wsr:' ·

(5.4. 10)

Using the same assumptions, an expression for energy transfer can be derived treating the energy migration as a diffusion processes. 1 1 If Pk is defined as the probability of finding an exciton on site k, then the time evo­ lution of this probability is given by

dPk dt

I

1 Pk , L Wkt (Pt - Pk ) - T I

s() where the summation runs over all active ion lattice sites. A continuum approximation can be used to change this summation to an integral and the probability can be expanded in a Taylor's series. Keeping only the lowest-

5.4. Nonradiative Energy Transfer: Multistep Process

1 99

order nonzero, rotationally invariant terms gives

(5.4. 1 1 ) The difusion coefficient is defined as

l Rkt 2 6 th '

_!_ Ro6

kl

I

(5.4. 12)

where Rkt represents the average step length in the random walk. This allows Eq. (5.4. 1 1 ) to be rewritten as

aPk DV2 1 . Pk - at rs0 Pk

(5.4. 1 3)

-

The average displacement from the ongm during the lifetime of an exciton is called the difusion length and is given by

Ln

�.

(5.4. 14)

The root mean square of the hopping distance can be found from Eq. (5.4. 13) to be l .07Rkt , which justifies the assumption of nearest-neighbor hops in the random walk. 9 To calculate the total rate of multistep energy transfer in the diffusion model, the rate equation for an ensemble of migrating excitons must be solved. If No excitons are created at time t 0 by a J-function excitation pulse W( t) , Eq. (5.4. 13) can be modified to give

oN(r ' t) at

W( t)

t

+ DV2 N(r , t) - r� N (r, t)

(5.4. 1 5)

with the boundary conditions

N(r , t 0) No, N(r RT, t)

0

=

(5.4. 1 6)

where R T is the effective trapping radius around an activator ion and N(r, t) NoPk (r, t) . This equation can be solved by making the substitution

u(r , t) r

N(r, t)

e tfr., ,

r > RT

(5.4. 17)

and then using Laplace transforms to obtain

[

( )]

RT erfc - RT N(r , t) No 1 --;:2 .Jl5i -

'

! e 1 r., .

( 5.4. 1 8)

The rate of energy transfer in the diffusion model is related to the flux of

200

5. Ion Ion Interactions

excitons crossing the effective trapping surface surrounding each activator:

F1 ( t) 4rr:DR2T aN

i

8, r=Rr

.

(5.4. 19)

Assuming a spherical trapping surface and a concentration of Ca non­ interacting traps, the total flux of excitons being trapped at time t is given by

(

FN (t) 4rr:DCa No e � tjr"' Rr + The total flux of excitons into traps can be viewed as the product of the exciton concentration far from a trap, No exp ( t/rsO), and the rate of energy transfer to traps,

(

Wfa 4rr:DRrCa 1 +

(5.4.20)

Any exciton created at time t 0 within the radius of a trap has an infinite rate of being trapped. For typical cases of interest in solid-state laser mate­ rials, the term Rr / VnJ5i « 1, which results in a time-independent energy­ transfer rate given by

Wfa 4rr:DRrCa .

( 5.4.21 )

In some situations the geometry of the sensitizer interactions and the ori­ entational dependence of the ion�ion interaction restricts the energy migra­ tion to one or two dimensions. The mathematical expressions for random walk and diffusion of excitons are quite different for these cases. Using the same procedure outlined above results for one- and two-dimensional sys­ tems can be analyzed. The point-trapping approximation used in the preceding mathematical development does not always adequately describe the true physical situa­ tion. Several models have been developed to account for the characteristics of exciton trapping for different limiting situations. One example is the extension of the random-walk formalism to include the effects of trapping regions of various sizes and geometries. 1 2 This can be important for mate­ rials involving large organic molecules, but is generally not significant for inorganic solid-state laser materials. A second approach to treating the effects of trapping is to explicitly include a term for sensitizer�activator interaction in the diffusion equation

t) -Psns(r, t) + DV2 ns(r, t) L Wsa(r - R; )ns(r, t) , ( 5.4.22) t i where R; is the position vector for a given activator and Wsa(r - R;) is the interaction rate for a given sensitizer-activator pair. No general solution to this equation has been obtained. However, solutions for limiting cases have been derived with the assumption of electric dipole-dipole interaction

201

5.4. Nonradiative Energy Transfer: Multistep Process

between sensitizer ions and between sensitizers and activators. For the case of weak diffusion perturbing a strong sensitizer-activator interaction, this equation has been solved using an operator expansion with a Pade approx­ imant technique to obtain1 3

ns(t)

_

exp

(

-

)

21 3 + 1 5.74x3 t413 f3s t Na13n3 /2 Ro3 .Jf 1 + l0.910xt 8_76xt ' 2/ 3 + (5.4.23) _

where X

nps l f 3 R0 2

(5.4.24)

·

As x __ 0 this reduces to the expression for single-step electric dipole-dipole energy transfer. The solution of Eq. (5.4.22) has also been obtained for the opposite case of fast diffusion perturbed by weak sensitizer-activator interaction. In this case an approximate potential approach with a propagator expansion in the first Born approximation was used. 14 The resulting expression for the energy-transfer rate is

( + 2nCa R} J: dr r 8nCa J: dr r

)

4nCaf3s R� Rr WsaDT 4nDRrCa 1 + v'7J5i + 3Rr =

[erfc

[erfc

2

J.

(5.4.25)

L ns(t - t')iis( t')e t '/to dt',

(5.4.26)

In order to use this expression numerical integration must be used to obtain the explicit time dependence of the energy-transfer rate. A third approach has been developed for treating the characteristics of exciton trapping based on treating the transfer rate as a random variable in a stochastic hopping process. 1 5 In this model the sensitizer luminescence is proportional to the sensitizer excited-state population density described by

ns(t)

iis(t) e tf to + t0 1

where to is the average value of the hopping time and iis( t) is given by the general expressions for the time evolution of the excited sensitizer pop­ ulation in the absence of sensitizer-sensitizer interaction. The solution of this equation in the limits of fast diffusion and no diffusion are equivalent to those obtained by solving Eq. (5.4.22). However, this formalism is ideal for computer simulations. Monte Carlo techniques have been used to simulate energy migration on a random lattice. 1 6 Instead of using an average value for the hopping time, a weighted set of random numbers is used to describe

202

5. Ion Ion Interactions

the variation in hopping time in each step in the random walk due to the randomness of the lattice site distribution and the randomness in the ion­ ion interaction rates. A standard set of random numbers is first generated and then weighted by a Hertzian distribution to account for the spatial site randomness. Then the numbers are weighted again to account for the ion­ ion interaction (such as for the electric dipole-dipole interaction) . This procedure has been used to fit experimental data on energy transfer and the results show that the average hopping time obtained using the Monte Carlo procedure to simulate the randomness of the physical situation is signif­ icantly smaller than the hopping time obtained from assuming a uniform lattice. 1 6 At high concentrations of sensitizers, percolation theory may be more accurate than this Monte Carlo approach. 1 7 In this situation the dis­ tribution of sensitizers in not truly random, but instead there are regions of high densities where energy transfer is very efficient, and weak transfer occurs from one of these regions to another. It is common to write the ion-ion interaction rate in the form Ws�) = Cs�) /Rn , where Cs�) = R0jr� is a microscopic energy-transfer parameter in­ dependent of concentration and the power n depends on the nature of the multipole interaction. For an exciton diffusing by electric dipole-dipole in­ teraction on a cubic lattice, the trapping radius can be defined as the dis­ tance at which the sensitizer-activator energy transfer rate is equal to the rate of sensitizer-sensitizer transfer on the lattice,

,-6

( )1/4

R T - 0 . 676 CDsa

( 5.4.27)

The diffusion coefficient in Eq. (5.4. 12) is then given by

( 5.4.28) and the energy-transfer rate associated with diffusion and trapping from Eq. (5.4.21 ) is

(5.4.29) This provides a means to predict the sensitizer concentration dependence of the diffusion coefficient and the sensitizer and activator concentration dependencies for the energy transfer rate. If the hopping model is used, the average value for the hopping time found by assuming a random distribu­ tion of sensitizers is1 8

( 5.4.30)

5.4. Nonradiative Energy Transfer: Multistep Process

203

If the trapping radius is defined as the distance at which the rate of sensitizer-activator transfer is equal to the hopping rate, the overall energy transfer rate becomes

(5.4.31 ) This predicts the same concentration dependencies as Eq. (5.4.29) but dif­ ferent dependencies on the ion-ion interaction parameters. As pointed out above, computer simulations have shown that the rate of energy transfer predicted by assuming an average hopping time is significantly different from the value obtained when the true randomness of the exciton migration is taken into account. The differences in Eqs. (5.4.29) and (5.4.3 1 ) may reflect this same problem. In the localized hopping model for exciton migration used in the above discussion, the excitons move incoherently. That is, phase memory is lost at each step of the random walk and the wave vector for the motion is not a good quantum number. The excitons are considered to be self-trapped due to lattice relaxation after each step. This takes the transition energy out of resonance with neighboring unexcited sensitizers and thus requires phonon activation to move from one site to another. Each step in the random walk can be treated as a phonon-assisted energy-transfer processes as described above. In this case the diffusion coefficient and thus the energy-transfer rate varies with temperature as exp ( - !:iE/kBT), where !:iE is the activation energy required for hopping. For situations where the electron-phonon coupling is weak enough that the self-trapping energy is very small, the excitons can move coherently over several lattice spacings before being scattered. In this case the diffusion coefficient is expressed in terms of the group velocity of the excitons v9 and the time between scattering events r,

( 5.4.32) where A is the mean free path of the exciton motion. The energy-transfer rate for long mean-free-path exciton motion is similar to a kinetic gas scat­ tering expression,

(5.4.33) where (J is the trapping cross section. The temperature dependence of the energy-transfer rate is determined by the exciton-phonon scattering time that limits the mean free path of the exciton motion. The exact expression for this scattering time is difficult to calculate due to the unknown details of exciton-phonon coupling. However, the most generally accepted results1 9 assume that scattering by acoustic phonons dominate other scattering mechanisms and this predicts that D oc

r-112 •

204

5. Ion Ion Interactions

5.5

Connection with Experiment: Rate Equation

Analysis

There are several specific questions that must be answered in characterizing energy transfer in a specific material. The first question is whether the transfer is a single-step or multistep process. Then it is important to know if the interactions involved in the energy transfer are resonant or phonon­ assisted. Next the types of interaction mechanisms must be identified and the strength of the interactions must be determined. The latter is usually characterized by the critical interaction distance R0 . If the energy transfer is a multistep process, the properties of both the energy migration process and the trapping processes must be determined. The former can be characterized by the diffusion coefficient, diffusion length, hopping time, and number of steps in the random walk, as well as the nature of the ion-ion interaction mechanism generating the random walk. The trapping can be characterized by parameters such as trapping cross section and trapping rate as well as the nature of the sensitizer-trap interaction mechanism. If enough information is known about the material, it is possible to obtain theoretical estimates of all of the relevant parameters from the theoretical models described in the previous two sections. Experimental measurements of spectral properties such as the fluorescence intensities and fluorescence lifetimes as functions of variables such as temperature, active ion concentration, and time can be used to obtain independent estimates of these same parameters. The com­ parison between theoretical and experimental estimates be used to answer these questions about the properties of energy transfer. The most common procedure has been to study the concentration quenching of the fluorescence intensity or lifetime of sensitizer ions. The major problem with this tech­ nique is that it requires accurate knowledge of the concentration of active ions in a series of samples and this is generally is not available. Measure­ ments of the time evolution of the fluorescence is more difficult but the ex­ periment is done on one sample, thus eliminating concentration differences from sample to sample. The most general expression for the time evolution of energy away from an initially excited ion is

where Pi ( t) is the probability of finding the excitation on the ith sensitizer ion at time t , f3 is the intrinsic fluorescence decay rate of the sensitizer ions, Wy is the energy transfer rate from sensitizer i to activator j and fV.ii is the transfer in the opposite direction (backtransfer), and Win describes the energy migration among sensitizer ions before fluorescence or transfer to an activa-

5.5. Connection with Experiment: Rate Equation Analysis

205

tor ion occurs. This equation must be solved and the results related to ex­ perimental observables such as the fluorescence intensity. This requires per­ forming a configuration average over the distribution of all possible ion-ion interactions and the inclusion of the initial conditions. Although attempts have been made to develop a general solution to this equation, 20 this is a difficult task since a double configuration average is required to account for both spatial disorder (random location of ions) and spectral disorder (vari­ ation of transition energies from site to site as reflected in inhomogeneous broadening of spectral lines) . Knowing the details of these distributions is critical in understanding the physics of energy transfer in a particular case. For example, the time dependence of the energy transfer is significantly dif­ ferent if sensitizer and activator ions are located in pairs all having the same separation, randomly separated pairs, or distributed with all of the sensi­ tizers on one side of the sample and all activators on the other side. The ini­ tial excitation conditions can excite sensitizer ions in centain spatial regions and not those in other regions of the sample. However, for most practical cases with solid-state laser materials, it is sufficient to assume a random spatial distribution of activators with the uniformly excited sensitizers either having a similar random distribution or, for host-sensitizer cases, being dis­ tributed in a known lattice configuration. Also, in general the spectral dis­ tribution is most important at low temperatures and can be ignored at room temperature where phonons are available to bring transitions of neighboring ions into resonance with each other. Some of the important exceptions to these statements are discussed below and in the following chapters. Even with the simplifying assumptions discussed above, it has proven to be very difficult to use the master equation in Eq. (5.5. 1 ) based on micro­ scopic energy-transfer parameters to analyze experimental resuHs. Although theoretically this should be the most direct method for obtaining the desired information about the physics of energy transfer in a material, in practice it has been found to be easier to use models based on macroscopic parameters for the initial step in data analysis. The primary parameter obtained from experimental data is generally the macroscopic energy transfer rate. The measured results provide information on the variation of this transfer rate with the variables mentioned above, and these properties allow the identi­ fication of the microscopic ion-ion interaction mechanism. The most com­ mon method of analyzing experimental results to obtain the energy-transfer rate under specific experimental conditions is to use a phenomenological rate equation model describing all of the energy levels and transitions in­ volved in the system. The equations describing the time evolution of the populations of the various levels of the system can then be written down and solved assuming the appropriate experimental conditions. The expressions obtained for the populations of the metastable states are directly propor­ tional to the measured fluorescence intensities from transitions originating on these levels with the proportionality constant being the radiative decay rate.

206

5. Ion Ion Interactions

Ws k

J}a

FIGURE 5. 7. Phenomenological model for the rate-equation analysis of energy transfer.

A simple example of a two-level system for both sensitizers and activators is shown in Fig. 5.7. The rate equations for the populations of the excited states are dt =

dns

Ws - flsns - kns ,

( 5.5.2)

where ns and na are the concentrations of excited states of the sensitizers and activators, fls and fla are their fluorescence decay rates, Ws is the rate of ex­ citation, and k is the energy transfer rate. For experiments involving con­ tinuous excitation Ws is a constant and the time derivatives of the pop­ ulations are zero. For pulsed excitation these derivatives are no longer zero and Ws can be expressed as a J function. The solutions to the rate equations for steady-state excitation are

k ns = fl Ws k ' na = fl (fJWs+ k) ' s+ a s

(5.5.3)

whereas the solutions for pulsed excitation conditions are

(

L

ns(t) ns(O) exp -fls t - k(t') dt' n a (t)

)

= exp( fla t) t k(t') exp(fJf)ns( t') dt'.

(5.5.4)

The expressions for the energy-transfer rate with its time dependence ex­ plicitly included must be used to obtain the final solution to Eqs. (5.5.4). The magnitude and properties of the phenomenological energy-transfer rate parameter k are determined by analyzing experimental data using the expressions derived from the rate-equation model described above. Due to the difficulty of making absolute intensity measurements, it is more common to make relative measurements. These can be relative measurements of the fluorescence properties of the sensitizers in samples with and without acti-

5.5. Connection with Experiment: Rate Equation Analysis

207

vators present, or measurements of the sensitizer fluorescence properties relative to the activator fluorescence properties. The parameters that are usually measured are the total fluorescence intensities, the fluorescence life­ times, and the time evolution of the sensitizer and activator fluorescence emissions. As an example of the use of a rate-equation analysis, consider the case of a system with a time-independent energy-transfer rate under pulsed excita­ tion. The solutions to Eqs. (5.5.4) for a constant k are

ns( t) = ns(O) exp [ (Ps + k)t] , na (t) = kns(O) k { exp [-( Ps + k)t] exp( -Pa t) } . Pa Ps

( 5.5.5)

The energy-transfer rate can be determined from these equations by mea­ suring the fluorescence decay rate of the sensitizer ions in a sample with no activators present r;o1 Ps and comparing it with the fluorescence lifetime of the sensitizers in a sample with activator ions present r_;- 1 • According to the first expression of Eq. (5.5.5), the energy-transfer rate is then given by (5.5.6) k = <s ! - - 1 7:.s{}

•

A second method for determining k is to measure the integrated fluorescence intensities of the sensitizers in samples with (Is) and without (!�) activators present. The initial expression in Eq. (5.5.3) can then be used to find the energy-transfer rate (5.5.7) A third method for obtaining the energy-transfer rate is to measure the integrated fluorescence intensities of both the sensitizers and the activators in the sample. Using the two expressions in Eq. (5.5.3), the energy-transfer rate is expressed as

k

/J� la . Is

(5.5.8)

Each of these methods for determining the energy transfer rate has advan­ tages and disadvantages depending on the characteristics of the system being investigated and the experimental capabilities that are available. However, it is generally advantageous to perform measurements on a single sample instead of making different measurements on more than one sample. One important method for obtaining the maximum amount of information from experiments on a single sample is to perform time-resolved spectroscopy measurements in which the relative intensities of the sensitizer and activator ions are monitored as a function of time after pulsed excitation. For the

208

5. Ion Ion Interactions

1ft

FIGURE 5.8. Sensitizer fluorescence decay after pulsed excitation normalized to the fluorescence decay time in the undoped (no activator) sample, r. (A) Samples with no activator ions exhibit exponential decay as predicted by Eq. (5.5.4) with k 0. ( B) Single-step electric dipole-dipole interaction between randomly distributed sen­ sitizer activator pairs resulting in an energy-transfer rate varying as t 1 12 as given in Eq. (5.4.20) . (C) Electric dipole dipole energy transfer in the presence of weak dif­ fusion among the sensitizers as predicted by Eq. (5.4.23). =

example of a time-independent k, Eqs. (5.5.5) can be solved to give p�

k { 1 - exp [(Ps + k - Pa )t] } . P� Pa Ps k _

( 5.5.9)

This experimental technique is very important when the energy-transfer rate is time dependent and the functional dependence on time must be determined. Typical examples of time-dependent energy-transfer data are shown in Figs. 5.8 and 5.9. The first of these figures shows the predicted decay curves for the fluorescence emission of the sensitizer ions as a function of normal­ ized time. With no energy transfer present the decay is exponential. In the presence of single-step energy transfer between randomly distributed pairs of ions the decay is nonexponential. In the presence of sensitizer-activator energy transfer modified by diffusion among the sensitizer ions, the initial decay is nonexponential. However, this evolves into an exponential decay at long times as the diffusion process distributes the excitation energy uniformly so all activators "see" the same excited sensitizer environment. The second of these figures shows how the ratios of the fluorescence intensities of the activator and sensitizer ions evolve with time after pulsed excitation for two typical cases. The important aspect of these examples is the effect of

5.5. Connection with Experiment: Rate Equation Analysis

209

0.8

0.6

0. 4

0. 2

t (arbitrary units)

FIGURE 5.9. Time evolution of the ratios of the fluorescence intensities sensitizer and activator transitions for two typical cases. (A) Single-step electric dipole dipole in­ teraction between sensitizer activator pairs having a fixed distance and with equal rates for transfer and back transfer. ( B) Single-step electric dipole dipole interaction between randomly distributed sensitizer activator pairs with no back transfer.

backtransfer. Without backtransfer the fluorescence intensity ratio increases continuously with time, while the presence of backtransfer causes the fluo­ rescence intensity ratio to reach an equilibrium condition that is time inde­ pendent. The types of curves shown in these figures are found· in sensitized solid-state laser materials discussed in the following chapters. The magnitude of the energy transfer rate can be determined as a function of parameters such as temperature, activator concentration, hydrostatic pressure, or uniaxial stress using one of the procedures described above. Once the properties of the phenomenological energy-transfer rate parameter are known, they can then be compared to the predictions of the various theories discussed in the last section to identify k with one of the Wsa energy­ transfer rates associated with these theories. It should be emphasized that in most cases this procedure is more complicated than indicated by the simple example used here. Effects such as backtransfer of energy from the activator to the sensitizer ion, direct excitation of some of the activator ions, non­ random spatial distributions of ions, and random distributions in transition energies from site to sight can cause the energy-transfer dynamics to vary greatly from the predictions of the simple rate-equation model. 20 However, models have been developed to include these effects and the general analysis for these cases is the same as the one described above even if the equations are more complicated. After the primary parameter, i.e., the energy-transfer rate, is determined, all of the secondary parameters characterizing energy

210

5. Ion Ion Interactions

transfer such as Ro , D, etc., can be obtained from the various equations given in the preceding section. It should be noted that the discussion thus far has focused on nonradia­ tive energy transfer. As mentioned at the beginning of this chapter, energy can also be transferred radiatively. In this case the initially excited ion simply emits a photon and another ion absorbs it. This can result in both energy migration among sensitizers and sensitizer-activator transfer. In this case the emitting ion does not alter its properties because of the presence of the absorbing ion. Thus the energy transfer does not quench the fluorescence lifetime of the sensitizer. In fact, radiative reabsorption among the sensitizer ions can produce an apparent increase in fluorescence lifetime and is a major problem in obtaining an accurate value for this parameter. In the techniques discussed so far, the sensitizer ions are assumed to be spectrally different from the activator ions. Thus the activator ions act as a probe for energy transfer through providing differences in the spectral prop­ erties of the sample's fluorescence emission. This is obviously the case when the sensitizers and activators are two different types of ions. However, in some cases there is interest in understanding the properties of transfer or migration of energy among the sensitizer ions without perturbing the system by introducing activator ions into the sample. The spectral differences of sensitizer ions can be exploited to study this type of energy transfer using high-resolution laser-excitation techniques. If there are distinctly different types of crystal-field sites occupied by the sensitizer ions, one type can be selectively excited while ions in the other type of site play the role of activa­ tors. This experimental technique is generally referred to as site-selection spectroscopy. 2 1 If the differences in transition energies in different crystal­ field sites are too small to distinctly resolve the different spectral lines, it will cause inhomogeneous broadening of the absorption or emission lines of the sensitizers. If the linewidth of the laser used for excitation is significantly smaller than the inhomogeneous linewidth, it can be used to selectively excite only a subset of ions within this line. The initial spectral characteristics then appear as hole burning in the absorption transition and fluorescence line nar­ rowing in the emission transition. The widths and intensities of the spectral hole or the narrowed emission line can be followed in time as. they evolve into the normal inhomogeneously broadened line shapes through energy transfer that randomizes the distribution of excitation energy among the distribution of sites. 22 This is an especially important technique for rare­ earth ions in glass hosts that have significant inhomogeneous broadening of their spectral lines. However, the overlap of transitions between different Stark components can complicate the interpretation of data. A typical example of site-selective, time resolved, fluorescence line­ narrowing spectroscopy is shown in Fig. 5.10. A narrow-line laser-excitation source is used to excite a specific subset of ions within the inhomogeneously broadened spectral profile. At very short times after the laser pulse the fluo­ rescence comes only from the subset of ions initially excited and thus the

5.5. Connection with Experiment: Rate Equation Analysis

21 1

Laser excited subset of ions {A)

( B)

FIGURE 5. 10. Time-dependent fluorescence line narrowing. (A) Inhomogeneously broadened absorption line with one of the homogeneously broadened subsets of ions to be selectively excited by the laser. ( B) Time evolution of the fluorescence emission after site-selective, pulsed excitation.

spectrum appears as a sharp line. As time increases, energy is transferred to ions in. other subsets within the inhomogeneous profile. This causes a rela­ tive decrease in the fluorescence intensity associated with the initially excited ions and a relative increase in the fluorescence from the other ions. This trend continues as time increases until the energy is spread uniformly to all subsets of ions across the inhomogeneous profile and the emission appears as the inhomogeneous band that is observed after normal broad-band exci­ tation. In general it has been found in these types of experiments that the entire spectral profile grows uniformly as the selected subset fluorescence decreases, as opposed to a spreading of the energy from the selected subset outward across the band profile. This implies that the transfer of energy between two ions in this case is relatively independent of small spectral dif­ ferences in transition energies. This probably indicates the dominance of nonresonant interactions involving two phonons, as discussed in Sec. 5.3. The type of behavior exhibited schematically in Fig. 5. 10 is typical of many solid-state laser materials that have strong inhomogeneously broadened lines, as discussed in the following chapters . Measuring spatial energy transfer without any spectral difference is of significant interest in understanding the physics of energy-transfer processes. For materials with high concentrations of sensitizers, one powerful experi­ mental technique that has been developed to directly monitor spatial migra­ tion of energy without spectral energy transfer is laser-induced grating spec­ troscopy. 23 In this technique, coherent laser beams are crossed inside the

212

5. Ion Ion Interactions

sample to form an interference pattern in the shape of a grating. This creates a spatial pattern of excited sensitizer ions with the same shape. As discussed in Sec. 3.4, ions in the excited state have a different polarizability than those in the ground state, and this changes the refractive index according to Eq. (3.4.8). Thus the excited-state population grating looks like a refractive index grating to light beams in the material. This grating pattern decays away by two processes: the fluorescence decay of the excited sensitizers and the migration of excitation energy from the peak to valley region of the grating. A third laser beam acting as a probe beam incident on the grating undergoes Bragg diffraction from the spatial variation in the refractive index. The measured decay of the scattered probe beam (signal beam) has the same time dependence as the decay of the excited-state population grating. Thus by monitoring the grating decay rate as a function of grating spacing, the energy diffusion coefficient can be obtained. The grating decay rate can easily be changed by changing the crossing angle of the two beams writing the grating. The theoretical description of the decay dynamics of laser-induced grat­ ings was developed from the generalized master equation by Kenkre.24 The most general solution gives the normalized signal beam strength as

S(t)

(

e Z t/r Jo(bt) e ca

+a J�

du e �> ( t u) Jo (b( r - u2 ) 1 12 )

a

y,

(5.5 . 1 0)

where r is the fluorescence lifetime, is the exciton scattering rate, lo is the Bessel function of order zero, and the parameter b is

b

=

(5.5. 1 1 )

where V is the nearest-neighbor interaction rate, a is the average distance between active ions, and the grating spacing is given by A A./ [2 sin(B/2)] where A. is the excitation wavelength and B is the crossing angle between the write beams (both in air). This general expression must be used for con­ ditions when the excitons have long mean free paths ( partially coherent migration). This predicts a grating decay that is nonexponential and with temporal oscillations. As conditions change so the migration becomes less coherent, the grating decay becomes exponential and the expression for the signal decay rate simplifies to

S(t)

�

S(O) exp

( { [ (a+�)'+ t� ) r • } ) 21

1 6 V2 'in

(5.5. 1 2)

For completely incoherent hopping motion, the expression can be written in terms of the diffusion coefficient D, (5.5. 13)

References

213

As discussed in the previous section, the exciton dynamics can be charac­ terized in terms of the diffusion coefficient, the mean free path Lm, the dif­ fusion length Ld, the coherence parameter (, and the number of sites visited between scattering events Ns. With the parameters of this theory, these are given by

Ld (

(2Dr) 1 /2 ,

(5.5. 14)

0( '

� Lm Ns a . Equation (5.5. 1 0), (5.5 . 1 2), or (5.5. 1 3) can be used to fit the experimental data to obtain the microscopic parameters V and a , and these can be used in Eq. (5.5. 14) to obtain the macroscopic parameters describing the migration. In addition, the theoretical expressions for ion-ion interaction rates devel­ oped in Sec. 5.2 can be used to interpret the value obtained for V . Energy-transfer properties have been investigated in many types of solid­ state laser materials. Single-step, diffusion-limited migration, trap-limited migration, and partially coherent long-range migration cases have all been observed. The results of using the techniques described here for some impor­ tant materials are discussed in the following chapters. References 1 . K. Huang, Statistical Mechanics, 2nd ed. ( Wiley, New York, 1 987), p. 1 44. 2. J. Rubio, H. Murrieta, R.C. Powell, and W.A. Sibley, Phys. Rev. B 31, 59 ( 1 985). 3. 4. 5. 6. 7.

T. Forster, Naturfurshung 12, 233 ( 1 950) . D.L. Dexter, J. Chern. Phys. 8, 144 ( 1 960) . T. Kushida, J. Phys. Soc. Jpn. 34, 1 3 1 8 ( 1 973). A.L.N. Stevels and J.A.W. van der Does De Bye, J. Lurnin. 18/19, 809 ( 1 979) . S.R. Rotman, A. Eyal, Y. Kalisky, A. Brenier, C. Pedrini, and G. Boulon, Opt.

Mater. 4, 3 1 ( 1 994) . 8. T. Holstein, S.K. Lyo, and R. Orbach, Phys. Rev. Lett. 36, 89 1 ( 1 976); T. Hol­ stein, S.K. Lyo, and R. Orbach, in Laser Spectroscopy of Solids, edited by W.M. Yen and P.M. Selzer ( Springer-Verlzg, New York, 1 98 1 ), Chap. 2. 9 . E.W. Montroll and G.H. Weiss, J. Math. Phys. 2, 1 67 ( 1 965). 10. A. Blumen and G. Zumofen, J. Chern. Phys. 75, 892 ( 1 98 1 ) . 1 1 . S . Chandrasekhar, Rev. Mod. Phys. 15, 1 ( 1 943) .

214 12. 13. 14. 15. 1 6. 17. 18. 19. 20. 21. 22. 23. 24.

5. Ion Ion Interactions Z.G. Soos and R.C. Powell, Phys. Rev. B 6, 4035 ( 1 972). M. Yokota and 0. Tanimoto, J. Phys. Soc. Jpn. 22, 779 ( 1 967) . H.C. Chow and R.C. Powell, Phys. Rev. B 21, 3785 ( 1 980) . A.I. Burshtein, Sov. Phys. JETP 35, 882 ( 1 972) . C.M. Lawson, E.E. Freed, and R.C. Powell, J. Chern. Phys. 76, 4 1 7 1 ( 1 982). R. Kopelman, in Radiationless Processes in Molecules and Condensed Phases, edited by F.K. Fong (Springer-Verlag, Berlin, 1 976), p. 297. R.K. Watts, in Optical Properties of Ions in Solids, edited by B. DiBartolo ( Ple­ num, New York, 1 975), p. 307. V.M. Agronovich and Yu.V. Konobeev, Phys. Status Solidi 27, 435 ( 1 968); Soviet Phys. Solid State 6, 644 ( 1 964); 5, 999 ( 1 963). D.L. Huber, in Laser Spectroscopy of Solids, edited by W.M. Yen and P.M. Selzer (Springer-Verlzg, New York, 1 98 1 ), Chap. 3. R.C. Powell, in Energy Transfer Processes in Condensed Matter, edited by B. DiBartolo ( Plenum, New York, 1 983), p. 655. W.M. Yen, in Spectroscopy of Solids Containing Rare Earth Ions, edited by A.A. Kaplyanskii and R.M. Macfarlane ( Elsevier, Amsterdam, 1 987), Chap. 4. J.R. Salcedo, A.E. Siegman, D.D. Dlott, and M.D. Payer, Phys. Rev. Lett. 41, 1 3 1 ( 1 978). V.M. Kenkre and D. Schmid, Phys. Rev. B 31, 2430, ( 1 985); V.M. Kenkre, Phys. Rev. B 18, 4064 ( 1 978) .

6

Al20 3 : Cr 3 + L aser Crystals

Trivalent chromium ions have played a central role in the development of solid-state lasers. Cr3 + was the active ion in the first laser (ruby) and has been the most successful transition-metal ion used for laser applications in other host crystals. With the proper choice of host, chromium lasers can operate either pulsed or continuous wave with either sharp line or broad­ band tunable emission between about 6900 and 12 500 A. This versatility provides the ability to study the effects of the host environment on the spec­ troscopic properties of the active ion and determine how these alter the las­ ing characteristics of the material. In the following sections the properties of the electronic energy levels and transitions of Cr3 + ions are discussed, and the effects of changes in the local crystal field, electron-phonon interactions, and ion-ion interactions are de­ scribed. The spectroscopic properties of ruby are presented as an example of a strong field Cr3 + laser material. Since ruby is a three-level laser material, it is not extensively used in practical applications. However, for historical rea­ sons, the spectroscopic and laser properties of ruby have been extensively characerized, and the results are used as a basis for comparison and under­ standing other solid-state laser materials. The properties of weak-field Cr3 + materials are discussed in Chap. 7. The fundamental concepts used in this discussion were outlined in Chaps. 2-5. 6. 1

Energy Levels of Cr3 +

The electronic configuration for the 24 electrons of a chromium atom is l s 2 2s2 2p6 3s 2 3p6 3d 5 4s. The first five sets of orbitals make up the filled core, while the optically active electrons are in the half-filled 3d orbitals. The lat­ ter are shielded by the electron in the outermost 4s orbital. The trivalent chromium ion has given up three electrons from the outer two sets of orbi­ tals, leaving three unshielded electrons in the 3d orbitals. These are the electrons that play the dominant role in determining the optical properties of the ion. The Russell-Saunders coupling approach discussed in Sec. 2. 1 215

214 12. 13. 14. 1 5. 16. 17. 18. 19. 20. 21. 22. 23. 24.

5. Ion Ion Interactions Z.G. Soos and R.C. Powell, Phys. Rev. B 6, 4035 ( 1 972). M. Yokota and 0. Tanimoto, J. Phys. Soc. Jpn. 22, 779 ( 1 967) . H.C. Chow and R.C. Powell, Phys. Rev. B 21, 3785 ( 1 980). A.I. Burshtein, Sov. Phys. JETP 35, 882 ( 1 972) . C.M. Lawson, E.E. Freed, and R.C. Powell, J. Chern. Phys. 76, 4 1 7 1 ( 1 982) . R. Kopelman, in Radiationless Processes in Molecules and Condensed Phases, edited by F.K. Fong (Springer-Verlag, Berlin, 1 976), p. 297. R.K. Watts, in Optical Properties of Ions in Solids, edited by B. DiBartolo ( Ple­ num, New York, 1975), p. 307. V.M. Agronovich and Yu.V. Konobeev, Phys. Status Solidi 27, 435 ( 1 968); Soviet Phys. Solid State 6, 644 ( 1 964); 5, 999 ( 1 963). D.L. Huber, in Laser Spectroscopy of Solids, edited by W.M. Yen and P.M. Selzer (Springer-Verlzg, New York, 1 98 1 ), Chap. 3. R.C. Powell, in Energy Transfer Processes in Condensed Matter, edited by B. DiBartolo ( Plenum, New York, 1 983), p. 655. W.M. Yen, in Spectroscopy of Solids Containing Rare Earth Ions, edited by A.A. Kaplyanskii and R.M. Macfarlane ( Elsevier, Amsterdam, 1 987), Chap. 4. J.R. Salcedo, A.E. Siegman, D.D. Dlott, and M.D. Fayer, Phys. Rev. Lett. 41, 1 3 1 ( 1 978). V.M. Kenkre and D. Schmid, Phys. Rev. B 31, 2430, ( 1 985); V.M. Kenkre, Phys. Rev. B 18, 4064 ( 1 978).

6

Al20 3 : Cr 3 + Laser Crystals

Trivalent chromium ions have played a central role in the development of solid-state lasers. Cr3 + was the active ion in the first laser (ruby) and has been the most successful transition-metal ion used for laser applications in other host crystals. With the proper choice of host, chromium lasers can operate either pulsed or continuous wave with either sharp line or broad­ band tunable emission between about 6900 and 12 500 A. This versatility provides the ability to study the effects of the host environment on the spec­ troscopic properties of the active ion and determine how these alter the las­ ing characteristics of the material. In the following sections the properties of the electronic energy levels and transitions of Cr3+ ions are discussed, and the effects of changes in the local crystal field, electron-phonon interactions, and ion-ion interactions are de­ scribed. The spectroscopic properties of ruby are presented as an example of a strong field Cr3 + laser material. Since ruby is a three-level laser material, it is not extensively used in practical applications. However, for historical rea­ sons, the spectroscopic and laser properties of ruby have been extensively characerized, and the results are used as a basis for comparison and under­ standing other solid-state laser materials. The properties of weak-field Cr3 + materials are discussed in Chap. 7. The fundamental concepts used in this discussion were outlined in Chaps. 2-5. 6. 1

Energy Levels of Cr3 +

The electronic configuration for the 24 electrons of a chromium atom is l s2 2s2 2p6 3s2 3p6 3d 5 4s. The first five sets of orbitals make up the filled core, while the optically active electrons are in the half-filled 3d orbitals. The lat­ ter are shielded by the electron in the outermost 4s orbital. The trivalent chromium ion has given up three electrons from the outer two sets of orbi­ tals, leaving three unshielded electrons in the 3d orbitals. These are the electrons that play the dominant role in determining the optical properties of the ion. The Russell-Saunders coupling approach discussed in Sec. 2. 1 215

216

6. A h0 3 : Cr3 + Laser Crystals

can be used to determine the electronic terms and multiplets of the free ion with three electrons, each having quantum numbers n = 3 and l = 2. The types of spectroscopic terms available to the three optically active electrons can be determined by making use of the Pauli exclusion principle, which states that no two of the electrons can have the same values for the set of four quantum numbers n, l, mt , ms . For Cr3 +, all of the optically active electrons have the same values of n and /, and thus there are only certain combinations of the values of mt and ms that are allowed. 1 In general, the values of the total angular momentum quantum number of the multielectron terms can run from L = 0 to L = 'I.; l; = 6 in integral steps. However, L = 6 is not allowed by the Pauli principle since this would re­ quire that at least two of the electrons have a multiplet with identical sets of quantum numbers. Thus the largest range for the ML quantum number is between +5 and 5 in integral steps. Similarly, the total spin quantum number of the multielectron term can run from S = ! to S = 'L;si in integral steps. For Cr3 + this gives two allowed spin states, S = � or ! · Since the sum of the mt quantum numbers for the three electrons must add to one of the ML quantum numbers, and the sum of the ms quantum numbers must add to one of the Ms quantum numbers, a table of single­ electron states that contribute to the multiplets of the multielectron terms can be constructed. The results are shown in Table 6. 1 . The single-electron TABLE 6. 1 . Single-electron states for a 3d 3 . electron configuration. a Ms 3

2

5 4 3

2

0

•

I

2

a(2+ , 2- , I + ) b(2+ , 2- , o+)c(2+ , I + , J - ) e(2+ , 2- , I + )J(2+ , I + , o- ) g(2+ , I - , o+ )h(2- , I + , o+) j(2+ , 2+ , 2- )k(2+ , I + , I - ) /(2+ , 1 - , I + )m(2- , I + , J + ) n(2+ , o+ , O;- )o(l + , I - , o+ ) ( I + , o+ , o- ) ( I + , I - , I + ) ( 1 + , 2+ , 2- ) (2+ , I + , o- ) (2+ , I - , o+ ) (2- , I + , o+ ) ( J + , 2- , 2+ ) ( 1 - , 2+ , 2+) ( J + , I + , o- ) ( 1 + , 1 - , o+ ) ( 1 - , J + , o+ ) (2+ , 2+ , o- ) (2+ , 2- , o+ ) (2- , 2+ , o+) (2+ , J + , I - ) ( I + , I-, 2+)

Implied terms

4p 2p

Data for table: n; = 3 , I; = 2, s; = !· Single electron states: (m�'' , m�'' , m7,'' ) . The table is

symetric for ( ML ,

Ms ) values.

6. 1 . Energy Levels of Cr3 +

217

states are represented by (m11 , m1� , m 1� ), where the + or superscript des­ ignates spin up ( s = ± !) or spin down ( s = !), respectively. All possible allowed combinations of single-electron quantum numbers are organized into single-electron states and placed in the appropriate ML row and Ms column of a multielectron term. Since the largest values of ML and Ms equal L and S, respectively, for a specific term, Table 6. 1 can be used to determine the spectroscopic terms of a 3d3 ion. The highest ML row in the table has single-electron states, occu­ pying only the cells in the Ms = ± ! columns. These must belong to a 2 H term since ML = L = 5 and Ms = S = !· One of the single-electron states in each of the other cells in the Ms = ± ! columns also will belong to this term. Subtracting one single-electron state from each cell in the center two columns of the table leaves the occupied cells with the largest values for ML those with Ms = ± ! · These must belong to a 2 G term, and again one state in each cell in the two center columns will belong to the same term. Elimi­ nating these states from the table leaves the occupied cells with the largest value for the spatial orientation quantum numbers those with ML = 3 and Ms = 1· This indicates the presence of a 4 F term. Single-electron states as­ sociated with this quartet term will be present in each of the cells with smaller quantum numbers, so one state must be subtracted from the cells in each of the four columns in the table. Two occupied cells still remain in the ML = 3 row of the table, those with Ms = ± !· These are associated with a 2 F term along with a single-electron state in each of the other cells in the two center columns. Subtracting the states belonging to this doublet from the cells in the two center columns leaves two states left in each of the two cells with ML = 2, Ms = ± !· These must be associated with two different 2 D terms, which allows the subtraction of two single-electron states from each remaining cell in the Ms = ± ! columns. The ML = I cell with Ms = 1 is now the occupied cell with the highest quantum numbers, which indicates the presence of a 4 P term. Subtracting one single-electron state from each cell leaves one state in each cell with ML = 0 , ± 1 and Ms = ± !· These are associated with a 2 P term. Thus, in this way all of the single-electron states can be accounted for in relation to the multielectron terms. The next problem is to order the eight terms with respect to their energies. The ground state can be determined by Hund's rules. These require that the ground-state term have the greatest multiplicity possible and greatest orbital angular momentum value consistent with this multiplicity. Therefore, the ground state for the 3d3 configuration of Cr3 + is the 4 F term. In Sec. 2. 1 it was shown that for ions with Russell-Saunders coupling the sum of the roots in the secular determinant for the energy levels is equal to the sum of the diagonal elements (diagonal sum rule), and there are no matrix elements connecting states of different ML and Ms. Using this in­ formation along with the fact that in Russell-Saunders coupling the energy of the terms is independent of mt and ms , sets of linear equations in terms of

218

6 . Ah0 3 : Cr3 + Laser Crystals

single-electron functions can be written for the energy of levels belonging to each cell in Table 6. 1 :

EeH) E(2+, 2 , 1+), EeG) + EeH) E(2+, 2 , o+) + E(2+, 1+, 1 ) , E(4F) E(2+, 1 +, o+), E(4F) + EeH) + EeG) + EeF) E(2+ , r , 1 +) + £(2 +, 1 , o+) + £(2+ , 1 +, o ) + E(2 , 1+, o+) , EeH) + EeG) + E(4F) + EeF) + EeD) + E( 2D' ) E(2+, o+ , o ) + E(2+, 1 , - 1 +) + E(2+, -2+, 2 ) + E(2+ , 1 + , - 1 ) + E(2 , 1+, - 1+) + £( 1+ , 1 , o+), E(4F) + E(4 P) E(2+, - 1 +, o+) + E( 1 + , 2+ , -2+), EeH) + EeG) + E(4F) + EeF) + EeD) + EeD' ) + E(4 P) + EeP) E(1 + , o+ , o ) + £( 1+, 2+ , -2 ) + E(2+ , - 1 , o+) + £( 1 + , 2 , 2+) + £(1 +, 1 , 1+) + £(2+, - 1 +, o ) + E(2 , - 1 +, o+) + £( 1 , 2+, -2+) . Only these eight cells need be considered since all of the terms can be de­ termined from these cells. The above equations can be solved for the term energies in terms of the energies of the single-electron states:

EeH ) EeG) E(4F) EeF)

E(2+ , 2 , 1+), E(2+ , 2 , o+) + E(2+, 1 + , 1 ) - E(2+ , 2 , 1+), E(2+ , 1+ , o+), E(2+, r , 1+) + E(2+, 1 , o+) + E(2+ , 1+ , o ) + E(2 , 1+, o+) E(2+ , 1 + , o+) - E(2+ , 2 , o+) - E(2+ , 1+ , 1 ) , EeD) + EeD' ) E(2+, o+ , o ) + E(2+ , 1 , - 1 +) + E(2+ , -2+ , T ) + E(2+ , 1+, - 1 ) + E(2 , 1+ , - 1+) + £( 1+, 1 , o+) - E(2+, r , - 1 +o) - E(2+ , 1 , o+) - E(2+, 1 +, o ) - E(2 , 1 +, o+) , E(4P) E(2+ , 1+ , o+) + E(1+, 2+ , 2+) - E(2+ , 1 +, o+) ,

6. 1 . Energy Levels of Cr3 +

E( 2P)

219

£( 1 + , o+, o ) + £( 1 +, 2+ , -2 ) + E(2+, - 1 , o+) + £(1 +, 2 , -2+) + £( 1+, 1 , - 1+) + E(2+, - 1 + , o ) + E(2 , - 1+, o+) + £(1 , 2+ , -2+) - E(2+, o+, o-) - E(2+, 1 - , - 1+) - E(2+, -2+ , T ) E(2+, 1 +, - 1 ) - E(2 , 1+, - 1 +) - £( 1+, 1 - , o+) .

The energies of the single-electron states can be written in terms of the Coulomb and exchange integrals as defined in Eq. (2. 1 .32). The term en­ ergies are then combinations of these integrals:

E(2H) EeG) E(4F) EeF)

1 (2, 2) + 21 (2, 1 ) - K(2, 1 ) , 21 (2, 0) - K(2, 0) + 1 (1 , 1 ) , 1 (2, 1 ) + 1(2, 0) + 1 ( 1 , 0 ) - K(2, 1 ) - K(2, 0 ) - K( 1 , 0) , 21 ( 1 , 0) + 21 (2, - 1 ) - 1 ( 1 , 1 ) - K(2, 1 ) - K(2, 0) - 2K ( 1 , 0) - K(2, - 1 ) , E(4P) 1 (2, -2) + 1 ( -2, 1 ) + 1(2, 1 ) + 1( - 1 , 0) - 1 ( 1 , 0) - K(2, -2) - K( -2, 1 ) - K(2, - 1 ) - K( - 1 , 0) + K( 1 , 0) , EeD) av 1( 0, O) + 1 (2, - 1 ) + 31( 1 , - 1 ) + 21(2, -2) + 1 ( 1 , 1 ) - 1 (2, O) - 1 ( 1 , 0) - 1 (2, 2) - 21(2, 1 ) - 3K(2, - 1 ) - 3K( 1 , - 1 ) - K(2, -2) + 2K(2, 0) + 2K( 1 , 0) + K(2, 1 ) + K (2, - 1 ) , EeP) 1 (0, 0) + 21( - 1 , 0) + 21( -2, 1 ) - 21 (2, 2) + 1 (2, 0) - 21(2, 1 ) - 1(2, - 1 ) - 31 ( 1 , - 1 ) - 1 ( 1 , 1 ) - 2K (1 , 0) - 2K(2, 0) - 2K( - 1 , 0 ) - K(2, -2) - 2K( -2, 1 ) + K(2, 1 ) + K(2, 1 ) + 3 K( 1 , 1 ) . Note that only the average energy for the two 2D terms is obtained in this way. Using definitions given in Eqs. (2.33)-(2.41 ) and Tables 2.2. and 2.3,

the term energies can be rewritten in terms of either the Slater-Condon pa­ rameters or the Racah parameters. The results are 1 •2

EeH) 3Fo - 6F2 - 12F4 3A - 6B + 3C, EeG) 3Fo - 1 1F2 + l 3F4 3A - 1 1B + 3C, E(4F) 3F0 - 1 5F2 - 72F4 3A - 1 5B , EeF) 3Fo + 9F2 - 87F4 3A - 1 5B ,

220

6. Aiz0 3 : Cr3 + Laser Crystals

E(4P) 3F0 - 147F4 3A , EeD) av 3Fo + SF2 + 3F4 3A + 5B + 5C , EeP) 3Fo - 6F2 - 12F4 3A - 6B + 3C. Finally, the energies of the individual 2D terms must be determined. This

can only be done by determining specific linear combinations of the single­ electron states that make up the ML 2, Ms ! multiplets of these two terms along with solving the secular determinant for their eigenvalues. To do this we follow the treatment developed by Condon and Shortley, 2 which makes use of the fact that the multiplets associated with different terms having the same values of ML and Ms are orthogonal. Starting with the 2H (5, !) a multiplet in Table 6. 1, the L - , 1 - , s , and s lowering oper­ ators can be applied to determine the linear combinations of single-electron states that make up the other 2H (ML, Ms) multiplets. From Eqs. (2. 1 . 16) and (2. 1 . 17), L [2H (5, !)J 1 [a] ,

v'102H(4 , !)

so

- v'4(2+, 1 , 1 +) + v'6(2+ , 2 , o+) - v'4c + v'6b,

2H (4, �)

10 1 12 [ v/6b - 2c] .

The lowering operators applied to the other single-electron states give:

r [b] - 2h + 2g + v'6e , 1 [c] -v'6g + v'6J, 1 [e] -2m + 21 + 2j, r [!] v'6n + v'6k , r [g] 2o - v'6n + v'61 , r [h] -2o + v'6m, s [d] h + g +f, s [i] m + l + k. These expressions can be used when the lowering operator L - is applied successively to the 2H (4, !) multiplet given above to obtain expressions for the remaining 2H (ML, Ms) multiplets of interest:

2H (3, !) 2H (2 -I ) '2

1 (v'6e 2f 4g 2h) , - + 1

v'30

·

6. 1 . Energy Levels of Cr3 +

221

According to Table 6. 1 , the 2 G ( 4, !) multiplet is some linear combination of single-electron states b and c . This combination is arbitrary except that it must be orthogonal to the linear combination associated with the 2H (4, !) multiplet. One satisfactory choice is 2G (4, !) = [2b + v'6c] /.Jf0, which has been normalized. Applying the lowering operator L twice gives the other two relevant multiplets for this term, 2G ( 3, 2I )

=

1

+ 3f - g - 2h) ,

Next consider the 4 F term. According to Table 6. 1 , 4F ( 3, !) = d . Apply­ ing the s - lowering operator gives 2 F ( 3, !) = 3 1 12 (! + g + h) . Then apply­ ing the L - lowering operator gives

Finally, the 2 F term must be considered. Appropriate combinations of single-electron states for the other three terms having multiplets with ML = 3 and Ms = ! have been derived above. The linear combination of e , f , g, and h associated with 2 F ( 3, !) is arbitrary except that it must be orthogonal to the similar multiplets of the other three terms. One normal­ ized combination satisfying this is 2 F ( 3, !) = ( v'6e + f + g - 2h) / ,;ri. Ap­ plying the orbital angular momentum lowering operator gives 2F (2, !) = ( -2j + k -

l + v'6o)j,;ri.

Linear combinations of single-electron states for multiplets of four of the six terms having ML 2 , Ms = ! have now been derived. The orthogon­ ality condition then allows the determination of the appropriate linear com­ binations of the j, k , l , m, n, and o states associated with the multiplets of the remaining two terms, 2 D and 2D'. One normalized combination that fits the orthogonality requirements is 2 D ( 2, !) = (-j - k + l + n)/2. The second combination must be orthogonal to this multiplet as well as the oth­ erfour. Constructing 2D' ( 2, !) = Aj + Bk + Cl + D m + En + F o, where A , . , F are arbitrary coefficients, using en') (8L) 0 for all other terms and eD ' ) eD' ) = 1, gives six equation with six unknowns. Solving these gives one satisfactory combination 2D' ( 2, !) = ( - 5j + 3k + l - 4m 3n - 2v'6o)jv'84. Now that expressions have been derived for the two multiplets of the 2 D and 2D' terms in terms of linear combinations of single-electron states, their energies can be determined as before using the Coulomb and exchange integrals and then writing the results in terms of Slater-Condon parameters or Racah parameters. Using the Slater-Condon parameters =

.

.

=

222

6. Ah0 3 : Cr3 + Laser Crystals

gtves,

EeD) = 3Fo + ?F2 + 63F4 , EeD') 3Fo + 3F2 - S?F4 . The secular determinant for the Coulomb repulsion among the electrons can now be formed. Since the matrix for the Coulomb interaction is diagonal with respect to (L, S, ML, Ms ) and independent of ML and Ms, only a 2 x 2 matrix made up of the 2D(2, !) and 2D'(2, !) multiplets is required,

I

eD i g I 2D) eD' I g I 2D)

Expressing each matrix element in terms of Slater parameters gives

I

3J2f(F2 - SF4 ) (3Fo + 7F2 + 63F4 ) - E 3v'2f(F2 - SF4 ) (3Fo + 3F2 - 57F4 ) - E

Solving this determinant for the energy eigenvalues gives

EeD)

3Fo + SF2 + 3F4 = 3A + 5B + 5C -

I

O.

l 6SOF2 F4 + 8325Ff + SBC + 4C2 ,

l 6SOF2 F4 + 8325Ff + SBC + 4C2 • Expressions have now been derived for all eight terms of the 3d3 electron EeD') = 3Fo + SF2 + 3F4 + 3A + 5B + 5C +

configuration of a free ion in terms of the Racah parameters. It is difficult to determine these parameters from first-principle calculations. Thus they are generally determined empirically by fitting the theoretical expressions to measured experimental data. Since all energies are measured relative to the ground state, the 3A energy contribution can be subtracted from each term to give

12

8

�

)'D

6. 1 . Energy Levels of Cr3 +

223

cr3 + (B/C=9 1114 1 33=0. 222 )

4 2n -2 2 2G p, H - 4p

0 -4

0

0. 1

0.2

BIC

0.3

0.4

0.5

FIGURE 6. 1 . Free-ion terms of a 3d3 ion and energy levels for Cr3 + (reprinted from Ref. I with the permission of Cambridge University Press) .

( c) + 8 c + 4 , ( c) + 8 c + 4 . 1 93 1 93

EeD') 5 B + 5 + c = c

--

B

B

2

2

B

B

(6 . 1 . 1 )

These expressions are plotted1 in units of E/C vs B/C as shown in Fig. 6. 1 . By measuring the optical absorption and emission spectra of a specific type of 3d3 ion such as Cr3 +, the energy-level splittings shown on the right side of the figure have been found and identified with the designated terms. This fixes the value of B/C for Cr3 + at 0.222 as shown by the broken vertical line. Comparing the measured transition energies with the E/ C scale gives C 41 33 and thus B = 9 1 8 for this ion. Table 6.2 lists values for the Racah B and C parameters for the other 3dn transition-metal ions. 1 The numbers quoted for the exact values of the Racah parameters vary in the literature due to variations in the accuracy of both theoretical calculations and ex­ perimental measurements. The general trends in relative magnitudes of B and C in Table 6.2 are consistent with other reported values. These param­ eters change when the ion is placed in a crystalline environment as discussed in the following section. It is the values of B and C in a specific host envi­ ronment that are importance to solid-state laser applications. The free-ion terms are further split by spin-orbit interaction. However, for 3d3 ions this is generally a smaller effect than the crystal-field splitting,

6. Ah0 3 : Cr3 + Laser Crystals

224

TABLE 6.2. Racah parameters for 3dn ions. Ion Parameter

B (crn- 1 ) 2+ 3+ C ( crn 1 ) 2+ 3+ C/B 2+ 3+

Ti

v

Cr

Mn

Fe

Co

Ni

695

755 862

810 918

860 965

917 1015

971 1065

1030 1 1 15

29 10

3257 381 5

3565 4133

3850 4450

4040 4800

4497 5120

4850 5450

4. 19

4.31 4.43

4.40 4.50

4.78 4.61

4.41 4.73

4.63 4.8 1

4.71 4.89

and thus either the free-ion terms derived above or the strong-field d orbitals form the starting point of discussing the energy levels of Cr3 + ions in a crystal field. 6.2

Crystal-Field Splitting

The next step in understanding the optical spectral properties of Cr3 + ions in solids is to describe the crystal-field splitting. As discussed in Sec. 2.3, the effects of an octahedral crystal field are generally considered as a first-order perturbation with contributions from lower symmetry crystal fields treated as higher-order perturbations. It is useful to correlate the results of the weak-field and strong-field treatments so information can be obtained on the variation of the energy levels as a function of crystal-field strength. The re­ sults of doing this type of analysis for Cr3 + ions are shown here. Using Eq. (2.3.6), the characters of the symmetry operators for the octa­ hedral point group can be determined for the reducible representations of each of the eight free-ion terms derived in the last section. Then using the character table for the 0 group given in Sec. 2.2 and the reduction formula given by Eq. (2.2.8), the following table shows how the irreducible repre­ sentations of the free ion terms reduce in terms of the irreducible repre­ sentations of the 0 group. 0

2y 2G 4p 2p 2D 2D ' 4p 2p

E

3C2

8 C3

11 9 7 7 5 5 3 3

1 1 1 1 1

1 0 1 1 1 1 0 0

1 1

6C4

1 1 1 1 1

6q 1 1 1 1 1 1 1

2E + 22TI + 2T2 2A I + 2E + 2TI + 2 T2 4A 2 + 4T1 + 4T2 2A 2 + 2TI + 2T2 2£ + 2T2 2£ + 2T2 4T1 2T1

6.2. Crystal-Field Splitting

225

As an example of these calculations, consider the 4 F term. The value of the orbital angular momentum quantum number is L = 3 so the characters of the irreducible representation for this term are

x (E) = 2 X 3 + 1 = 7 , . 7n sm x ( C2) � - 1 , sm 2 14n .sm x ( C3 ) = . n = 1 , sm 3 x ( C4 )

. 7n sm-

n

. sm 4

-1.

Using these characters along with those of the five irreducible representa­ tions of the 0 group from Sec. 2.2 gives

n(A 1 ) -b (7 X 1 X 1 + 1 X 1 X 8 - 1 X 1 X 3 - 1 X 1 X 6 - 1 X 1 X 6) 0, n(A2) -b (7 X 1 X 1 + 1 X 1 X 8 - 1 X 1 X 3 - 1 X - 1 X 6 - 1 X 1 X 6) 1 , n(E) -b (7 X 2 X 1 + 1 X - 1 X 8 - 1 X 2 X 3 - 1 X 0 X 6 - 1 X 0 X 6) 0, n( T1 ) = -b (7 X 3 X 1 + 1 X 0 X 8 - 1 X - 1 X 3 - 1 X - 1 X 6 - 1 X 1 X 6) 1 , n( T2) -b (7 X 3 X 1 + 1 X 0 X 8 - 1 X - 1 X 3 - 1 X 1 X 6 - 1 X 1 X 6) 1 . This shows that the 4 F free-ion term splits into one orbital singlet crystal field term (A 2 ) and two orbital triplets (TI and T2 ) all with the same spin

multiplicity of the free-ion term. Similar calculations can be carried out for the other free-ion terms with the results shown in the table above. Thus the eight free-ion terms of a Cr3 + ion split into 20 crystal-field terms in an octa­ hedral crystal field. Next consider the strong-field approach to the crystal-field splitting. As seen from Fig. 2.6, four electron configurations must be considered, t�9 , t�9 e9 , t29 e�, and e�. In Sec. 2.3 the reduction of the direct-product repre­ sentations of two d electrons in an octahedral crystal field was considered. Now the effects of a third electron must be determined. First consider the t�9 configuration. In Sec. 2.3 it was shown that t29 x t29 A 1 9 + E9 + T1 9 + T29, and it is now required that the reduction of the direct-product representation formed by t29 and each of these four irreducible representa­ tions be found. Using the normal procedure,

t2g X A l g = T2g, t2g X Eg Tl g + T2g,

226

6. Ah0 3 : Cr3 + Laser Crystals

t2g X Tl g A 2g + Eg + Tl g + T2g, t2g X T2g = A ,g + Eg + T,g + T2g, so the lowest-energy configuration results in the crystal-field states t�9 = A,9 + A 29 + 2E9 + 3T, 9 + 4T2g · Similarly, the configuration t�9 e9 t2g x t29 x e9 A,9 + A 29 + 2E9 + 2T,9 + 2T29, the configuration t29e� t29 x e9 x e9 2T, 9 + 2T29, and e� = e9 x e9 x e9 A ,9 + A 29 + 3E9. Which of these crystal-field states are allowed as doublets or quartets can be determined using the statistical expression for total degeneracy given by Eq. (2.3.12). For the t�9 configuration the total degeneracy is given by 6 C3 6!/ [ 3!3! ] 20. This can be obtained from the available crystal field states with the terms 4A 29 + 2E9 + 2 T, 9 + 2T29. For the t�9 e9 configura­ tion, 6 C2 x 4 C1 6!/[4!2!] x 4!/ [3! 1 !] 60 for the total degeneracy, which can be obtained from the available crystal-field states with the terms 2A 1 9 + 2A 29 + 22 £9 + 22 T,9 + 22 T29 + 4 T,9 + 4 T29. For t29e� , 6 c, x 4 C2 36, which can be obtained by 22 T1 9 + 4 T1 9 + 22 T29. For e� , 4C3 4, which can be obtained by 2 E9. Note that the 20 crystal-field terms obtained by this strong-field analysis are the same as those obtained above using the weak­ field analysis. For each individual strong-field configuration there is rp.ore than one pos­ sible set of crystal-field terms that fulfill the total degeneracy requirement. However, there is only one complete set of these terms for all the config­ urations that is consistent with the terms obtained from the weak-field anal­ ysis. It is generally necessary to make several guesses at the right combina­ tion for each configuration until the correct set that matches the weak-field multiplets is found. It is possible to derive the correct combination of terms through a detailed consideration of the wave functions. This was described in Sec. 2.3 for a t�9 • The next step in this procedure is to couple the third electron into this configuration. Using group-theory procedures, the direct product of the single-electron t29 orbital representation with the representations of the four terms of the t�9 configuration can be formed and the result reduced in terms of irreducible representations of Oh . The possible terms with respect to symmetry considerations found in this way are 4g

1A 1 1£1 1 Tz 3 T1

__

__ __

__

t�g 2Tz 2T1 , 2 Tz 2A 1 , 2 E, 2 T1 , 2 Tz zA z , z E, z T1 , z Tz , 4A 2 , 4 E, 4 T1 , 4 Tz

Each of these must be checked to see if it is one of the terms allowed by

6.2. Crystal-Field Splitting

227

the Pauli exclusion principle. The wave function for the t�9 crystal-field terms can be written as a linear combination of products of the wave func­ tions of the G9 wave functions found in Sec. 2.3 and single-electron t29 functions, 'I' ( G9 [Soro] tz9 [Sr] ) =

L

Mom3 YoY3

I/I(G9 So roMo y ) tp (tz9m3 y3 ) ( SoMo ! m 3 I SM) (roy0 Tz y3 l ry) ,

where is the wave function for a state of the G9 configuration and

tp ( tz9m 3 Y3 ) is the wave function for the third tz9 electron. The last two factors in this

expression are the Clebsch-Gordan coefficients for the spin angular mo­ mentum coupling and the orbital angular momentum coupling in the cubic basis. These were both tabulated in Chap. 2. 'I' is required to be anti­ symmetric for the exchange of all three electrons. This can be accomplished by forming Slater determinants in the expansion 'I' ( 49 [So ro] tz9 [Sr] ) =

1

L { I 1Pt ( 1 ) tpz (2) I IP3 (3) - 1 1Pt ( 1 ) tpz (3) I IP3 (2)

Mom3 YoYJ

+ l iP I (2) tpz (3) I IP3 ( 1 ) } ( SoMo ! m 3 l SM ) (r o Yo Tz Y3 j ry) , where the subscripts on tp designate the single-electron orbital and the num­

ber in parentheses designates the electron. To simplify the notation, it is common to use the expressions for the two-electron wave functions given in Table 2.7 where the states in the terms are designated A z( e) , E(u, v ) , T1 ( fJ, y ), and Tz ( c;, 1J, () with a minus superscript designating spin down and no superscript for spin up. First consider as an example the 4A 29 term with 3 Tt g parentage. The rele­ vant Clebsch-Gordan coefficients for coupling the angular momentum are found from Table 2. 1 to be

rx,

( Mo ! m3 1 H) J(Mo 1 )J(md). The Clebsch-Gordan coefficients for the orbital angular momentum in the cubic basis are found from the section of Table 2.6 for the A z representation in the reduction of the T1 x T2 product representation. The results can be expressed in terms of the notation for the two-electron wave functions in

228

6. Ah0 3 : Cr3 + Laser Crystals

Table 2.7. The final wave function for this three-electron term is

The spins of all three electrons are aligned to give an M quartet term. This can be rearranged to give

� multiplet of a

which shows that 4 A 1g is an allowed crystal-field term for the t�g electron configuration in an octahedral environment. Next consider the 4 Eg term with 3 T1 g parentage. Using the same proce­ dure described above, the wave function is found to be

'P (�i Tig] tig 4EgM �) =

1

v'2 0,

[ 1 1!( 1 (1, 2) � (3 ) - I C� 1 (1, 2) '1 (3)] I 11C� I +

1

I I v'2 C�11

where a cyclic permutation of orbitals leaves the sign of the determinant unchanged. Thus, 4Eg is not one of the allowed terms. Now try the 1 Eg term with 3 T1 g parentage. Using the tables for spin and orbital angular momentum coupling and the expression given above for the expansion of the wave function in terms of products of single-electron orbi­ tals, the multielectron term wave function is

'P (t�g [3 Ti g] tig 1EgM = !) =

-1

(3 ) - l 'lC 1 (1, 2) � (3 ) [i 2y'3 1 '7 C ( I , 2) � + I C� I o , 2J '7 (3 ) + I CC l o , 2J '7 (3 )J 1 + v'3 [ 1 '7C I ( I , 2J C (3 ) + I C� I o , 2J 'l (3 ) J

=

1

1

( I- I v'2 1 �'7 C v'2 C11W ·

Thus 1Eg i s one of the allowed terms. As a final example, try the 2 T1 g term with 1 Eg parentage. The relevant Clebsch-Gordan coefficients are found from Table 2. 1 to be given by

Using these factors along with the orbital angular momentum coupling co-

6.2. Crystal-Field Splitting TABLE 6.3. Wave functions for the terms of the I 2Ad e2 ) = l ¢11(1

I 2E ! u)

t�g configuration lt�/SrMy) (after Ref. 4).

¢ii( I

I �11( I )

I 2 T1 !P) =

I 2 T1 ! tX) = 1

v'2

-

( 1 ¢111JI + WW

I 2 Td 11) =

1

v'2

229

(111((1 + 111¢�1)

I 2 E ! v) =

I 2 T1 !J') = I 2 Td 0 =

1

v'2

(1(¢�1 + 1(11iil)

efficients found from Table 2.6, the wave function is

'¥ (�iE9] ti9 2 T1 9M !) - v'3 1 J6 [ 1 �� 1 (1, 2) � (3) 1 '7'7 1 (1, 2) � (3) + 2 1 (( 1 (1, 2) � (3) 1 '7'1 1 (1, 2J � (3) I CC I ( I , 2J � (3) + 2 I �� 1 (1, 2J � (3) I CC I ( I , 2J � (3) 1 1

- I �C I ( I , 2J � (3) + 2 1 ,, 1 (1, 2) � (3)] - 2 -/2 [ 1 �� 1 (1, 2) � (3) 1 '7'7 1 (1, 2) � (3)

+ 1 '7'1 1 (1, 2J � (3) I CC I ( I , 2J � (3) + I CC I ( I , 2J � (3) = 0. Thus this 2 T1 9 is not one of the allowed terms.

� �� 1 (1, 2J � (3)]

This type of procedure can be followed for each of the symmetry-allowed terms and the results are those listed in Table 6.3 for the t�9 configuration. Similar procedures can be followed for the other configurations for 3d elec­ trons. Since this is a time-consuming procedure, for configurations with more than two electrons it generally is more efficient just to guess at appro­ priate combinations using the degeneracy calculation until the one con­ sistent with the weak-field analysis is found. Figure 6.2 shows a correlation diagram for a d3 ion in an octahedral crystal field. The free-ion terms are shown on the left side of the diagram and the strong-field configurations are shown on the right. The results of crystal-field splitting are shown for both of these sets of levels, and the one­ to-one correspondence among weak-field and strong-field terms is shown explicitly. The variation of the energy levels with the strength of the crystal field is not meant to be exact in this type of diagram and is discussed in de­ tail below. The first step in determining the quantitative value for the energy levels of the crystal-field terms is to determine the crystal-field energy, n ( 4Dq) + m(6Dq) for a f29 e; configuration. Thus for the configurations of a d3 ion, 2Dq, E(t29e�) 8Dq, and E(e�) l 8Dq. E(t�9 ) = l2Dq, E(t�9 e9) The next step is to add to these energies the energies due to the Coulomb and exchange interactions among the electrons. In Chap. 2 it was shown that the Coulomb and exchange energies for electrons in the (xz) and (yz)

230

6. Alz0 3 : Cr3 + Laser Crystals FREE ION TERMS

WEAK FIELD

SINGLE ELECTRON CONFIGURATIONS

2E

- Dq --

eg3 ( 1 8�

S TRONG FI ELD

FIGURE 6.2. Correlation of strong-field and weak-field energy levels of a 3d3 ion in an oh crystal field.

orbitals of the form given in Eq. (2.3. 1 3) are J(xz, yz) Fo - 2F2 - 4F4 and K(xz, yz) 3F2 + 20F4 , where the Fi are the Slater parameters. The interaction energies for electrons in the remaining combinations of orbitals can be calculated in the same way to give3

J(z2 , z2 ) J(� - i, � - i) J(xy, xy) J(xz, xz) Fo + 4F2 + 36F4 , J(x2 - i, xz) J( � - i, yz) J(xy, yz) = J(xy, xz) J(xz, yz) Fo - 2F2 - 4F4 ,

6.2. Crystal-Field Splitting

TABLE 6.4. Matrix elements for r!l for 3d electron wave functions.

a (xz) (yz) (xz) (yz) (z2 ) (z2 ) (z2 )

b

c

d

(ab 1 1 /rn l cd)

(z2 ) (z2 )

(xz) (yz) (z2 ) (z2 )

(x2 y2 ) (x2 - y2 ) (xZ - Y. l (xZ - Y. l

2 J3Fz + IO J3F4 2 J3Fz IOJ3F4 J3Fz 5 J3F4 -J3Fz + 5J3F4 J3Fz 5 J3F4 J3Fz 5J3F4 2J3Fz IO J3F4 3Fz I SF4 -Fz + I 5F4

(x' ) (y')

(x2 y2 ) (xZ - Y. l

23 1

(xy) (xy) (xz) (xy) (xy)

(x' )

(yz) (xy) (xz) (yz)

J(z2 , xy) J(z2 , x2 - i)

(yz) (xz) (yz) (yz) (xz)

Fo - 4F2 + 6F4 ,

J(x2 - i, xy) Fo + 4F2 + 34F4 ,

( 6.2. 1 )

and

K(x2 - i, xz) K(x2 - i, yz) K(xy, yz) K(xy, xz) K(xz, yz) 3F2 + 20F4 , K(z2 , xz) K(z2 , yz) F2 + 30F4 , K(z2 , xy) K(z2 , x2 - i) 4F2 + 1 5F4 , K(x2 - i, xy) 35F4 .

(6.2.2)

The other matrix elements for r1l are given in Table 6.4. These results can now be used to find the energy of each term. Consider as an example the ground-state term energy. For 4A 29 there must be one electron in each of the t29 orbitals, (xy) , (xz) , and (yz) in order to have all of the spins aligned and not violate the Pauli exclusion principle. The term energy must therefore include contributions from all possible two-electron combinations for Coulomb and exchange matrix elements obtained from expanding the Slater determinant. Using the above expressions shows this contribution to the energy to be 3J(xz, yz) - 3K(xz, yz) 3Fo 1 5F2 - 72F4 • Thus the total ground-state energy is E(4 A 29) - 12Dq 1 5F2 - 72F4 - 12Dq + 3A - 1 5B, where the Racah parameters have been used in the final expression. Similar calculations can be made for each of the terms. The most important results for chromium-doped laser materials

6. Ah0 3 : Cr3 + Laser Crystals

232

t� Configuration

Pze Configuration

E(4A 2 ) 3A I 5B I2Dq EeE) 3A 6B + 3C I2Dq Ee T1 ) 3A 6B + 3C I2Dq Ee T2 ) 3A + 5C I2Dq

E( 2A I ) E(4 T1 ) E(4 T2 )

=

=

= =

3A 3A 3A

l i B + 3C 2Dq 3B 2Dq I 5B 2Dq

=

The energies of the other crystal-field terms are so high that they are gen­ erally above the ultraviolet absorption band edge of the host material and thus do not play a role in the laser properties of Cr3 + laser materials. Since 3A is a common contribution to all of these energies, it can be subtracted out of these expressions and the resulting crystal-field term energies plotted as a function of crystal-field strength. Generally the energies are normalized with respect to the Racah B parameter so the energy levels expressed as Ej B are plotted versus Dqj B. The results of doing this are referred to as Tanabe­ Sugano diagrams and these are extremely useful tools in analyzing the opti­ cal spectra of transition-metal ion laser materials. 4 Figure 6.3 shows the Tanabe-Sugano diagram for a 3d3 ion. All three of the Racah parameters are associated with the Coulomb interaction between same configuration electrons. For a Cr3 + ion, B 9 1 8 cm- 1 for a free ion

�

40

WMWH/M

2T 2

30

2Tl 2E

WkMWAJ

10

ENERGY LEVELS

FIGURE 6.3. Tanabe Sugano diagram for a 3d3 ion with energy levels for ruby with Dq/B 1 720/765 2.25 (after Ref. 4) =

=

6.3. Spin Orbit Splitting and Selection Rules

233

and this reduces to 765 cm- 1 in an octahedral crystal-field environment. This difference is associated with the change in the radial wave function between the free ion and the ion in the crystal and is especially sensitive to covalency effects. In general C 4B is a good approximation for transition­ metal ions. The effect of A is to reduce the crystal-field lODq. For an alu­ minum oxide host crystal Dq = 1 720 cm- 1 • Thus Dq/ B 2 2 5 for ruby. B alone determines the energies of the 2E and 2T2 levels and the magnitude of configuration mixing. In this case the 2 E term is the lowest excited state that becomes the metastable state from which the laser transition is initiated. This is called a strong-field material. For host materials with smaller values of Dq, the 4T1 term becomes the lowest excited state and thus the initial state of the laser transition. These are referred to as weak-field materials. Since the laser transition for strong-field materials is a spin-flip transition between levels of the same t� configuration, it is a sharp line transition. In a weak-field material the laser transition is between different crystal-field con­ figurations t�e to t� and thus is a broad-band transition. These differences are discussed in detail in the Sees 6.4 and 6.5. .

6.3

Spin-Orbit Splitting and Selection Rules

For transition-metal ions, the energy-level splitting due to the spin-orbit in­ teraction is much smaller than the splitting due to the electrostatic inter­ action of the crystal field. Thus spin-orbit coupling can be considered as a perturbation on the octahedral crystal-field terms derived above for a 3d3 ion. To determine the qualitative nature of the energy-level splitting due to the spin-orbit interaction, it is again possible to use group theory. This is done by forming the direct-product representation of the spin and orbital parts of the crystal-field wave functions and then using Eq. (2.2.8) to reduce these results in terms of the irreducible representations of the crystal-field symme­ try group. For the Oh symmetry group, the spin functions for the doublet terms are represented by the E1 ;29 irreducible representation with the char­ acters given in the Oh character table shown in Sec. 2.2. The spin functions for the quartet terms are represented by the G9 irreducible representation. The spin-orbit splittings for six lowest-energy terms of a 3d3 ion in an Oh crystal field are shown in the following table: (3

t2 e

4A 2 2£ 2T1 2T2 4 T1 4T2

--+

--+

--+

--+ --+

G G

E112 + G £3;2 + G E112 + £3;2 + 2 G E112 + £3;2 + 2 G

6. Ah0 3 : Cr3 + Laser Crystals

234

As an example of how the results in this table were obtained, consider the 2T29 terms. The direct-product representation is found from the character table as shown below. E

oh

E1 ; 2 £3/2 G £1 ;2

x

2, 2, 4, T2 6,

i 2 2 4 6

2, 2, 4, 6,

2 2 4 6

8C3 I, 1 I, 1 1 I 0, 0 ,

8C3 i I, 1 I, 1 1, I 0, 0

3C2 0, 0 0, 0 0, 0 0, 0

3C2 i 6C4 6C4 i 6C� 0, 0 v'2, -v'2 v'2, - v'2 0, 0 v'Z, v'Z v'Z, v'Z 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 0, 0 v'Z, v'Z v'Z, v'Z 0, 0

6c'2 i 0, 0 0, 0 0, 0 £3;2 + G

These results are then used with Eq. (2.2.8) and the character table to give

n(£1;2 ) fs ( 6 X 2 X 2 + 0 X 1 X 16 + 0 X 0 X 6 + - .Ji X .Ji X 12 + 0 X 0 X 12) 0, n(£3 /2 ) fs ( 6 X 2 X 2 + 0 X 1 X 16 + 0 X 0 X 6 + - .Ji X - .Ji X 12 + 0 X 0 X 12) 1 , n(G) fs (6 X 4 X 2 + 0 X - 1 X 1 6 + 0 X 0 X 6 + - .Ji X 0 X 12 + 0 X 0 X 12) 1 . The example of spin-orbit splitting of a 4 T1 9 term was described in Sec. 2.3.

This analysis shows that there is no splitting of the ground-state term as a result of spin-orbit coupling. In the strong-field limit, the lowest excited state 2E9 also remains unsplit by the spin-orbit interaction. However, all of the doublet and quartet T1 9 and T29 terms are split by spin-orbit coupling. The quantitative values for spin-orbit splitting can be found using the Lande formula, Eq. (2. 1 .23). As discussed in Sec. 2.3, the magnitude of the parameter A, is determined by comparing theoretical predictions with ex­ perimentally observed spectra. For ruby the spin-orbit coupling parameter ( 170 cm 1 . The radiative transitions for the free Cr3 + ion are electric dipole allowed only if they take place between energy levels have different electron config­ urations, such as 3d3 __ 3d1 4s. This type of transition generally falls in the ultraviolet spectral region. The transitions of interest for laser operation fall in the visible spectral region and take place between energy levels having the same 3d3 electron configuration. These Laport forbidden transitions take place through magnetic dipole or electric quadrupole interactions, and thus result in weak spectral lines. Table 6.5 lists the transitions that can take place between Russell-Saunders terms through these multipole interactions. As discussed in Sec. 3.2, the spin quantum number remains unchanged between these transitions while the orbital angular momentum quantum number changes by 1 or 0 for magnetic dipole and 2 , 1 , or 0 for electric

6.3. Spin-Orbit Splitting and Selection Rules

235

TABLE 6.5. Radiative transitions between Cr3 + terms. Electric quadrupole

4 p <- 4 p z p <-> z F z D <- z o z p ._ z H

Magnetic dipole or electric quadrupole

2p ,_ 2 D z D ,_ z D 2 D <- zp zp ,_ z 0 z o ,_ z H

quadrupole transitions. The selection rules for transitions between specific spin-orbit multiplets were discussed in Sec. 3.2. When the Cr3 + ion is placed in a crystal field, the relevant transitions are still strictly Laport forbidden, but the total orbital angular momentum selection rules for the radiative transitions are now determined by the site symmetry. The strongest transitions in the spectra are those that obey crystal-field electric dipole selection rules and are thus "forced electric dipole" transitions. In the example of an Oh crystal field discussed in the preceding section and in Sec. 3.2, the ground-state energy level for Cr3+ is 4A 29. Spin selection rules allow transitions only to other quartet levels and, since the electric dipole operator has odd parity, only transitions to odd parity states are allowed. However, all of the state in Oh site symmetry have even (g) parity so no absorption transitions will be allowed. Similarly, no emission transitions from 2£9 or 4 T29 excited states to the ground state will be allowed. The reason that transitions are seen is that the crystal-field states are not completely pure 3d3 configurations. They are formed from an admixture of states from all possible configurations. Thus, although the states are predominantly even parity 3d3 , they have enough odd-parity con­ figuration mixed in to permit forced electric dipole transitions to occur. The greater the admixing, the stronger the transitions. Similarly, transitions be­ tween quartets and doublets can take place because of spin-orbit mixing of the levels. The admixing interactions that are important for permitting tran­ sitions to occur are shown in Fig. 6.4. If the parity of the levels is ignored because of configuration admixing, then the selection rule table in the preceding section shows that only a 4 A 2 --+ 4 T2 absorption transition will be both spin and forced electric dipole allowed. The transition to the 2 Tz will be spin forbidden but forced electric dipole allowed. Similarly, in emission the transition from the 2£ excited state to the ground state is both spin and electric dipole forbidden, the transition from the 4 T2 is spin and forced electric dipole allowed, and transitions from 2 T2 are spin forbidden but forced electric dipole allowed. Thus the absorp­ tion spectrum of Cr3 + in Oh symmetry should exhibit one relatively strong absorption band and a weak line. The emission spectra will depend on the nature of the lowest excited state as dictated by the strength of the crystal

236

6. Alz03 : Cr3 + Laser Crystals

FiGURE 6.4. Admixing inter­ actions for Cr3 + energy levels in a site with primarily oh crystal-field symmetry due to spin orbit inter­ action and a trigonal distortion.

Eg,A2g)

Hso

Hso

2 H so H so

Hso

Hg

V(f ! g)

H so

field. For a strong-field material, a week emission line with a long fluo­ rescence lifetime is expected, while for a weak-field material a relatively strong, broad-band emission with a fast decay time is expected. In most cases, the crystal-field site symmetry is lower than Oh, which re­ sults in more transitions being forced electric dipole allowed. Specific cases are discussed in detail in the following sections. As an example, in C3v sym­ metry, the z component of a vector transforms as the A 1 irreducible repre­ sentation while the x and y components transform as the E representation. The character table for this representation and reduction of all possible TABLE

6.6. C3v Character table and direct-product representations.

C3v A1 A2 E E1 ; 2 Ei;2 EJ/2 A1 x A1 A 1 x A2 A1 x E A2 x A2 A2 x E ExE E1 ;2 x Ei;2 El /2 x EJ/2 El/2 X El/2 Ej/2 x Ei;2 EJ;2 x Ei; 2 EJ;2 x EJ;2

E

2 2 1

2 I 2 4 2 2 4 1

R

2 1 1

C3 , R C3 1 1 1 1

C�, RC�

1 1

1 1 I 2 2 4

1 I

1 I

3av 1 0 0

3av R z

0 i

i 1 I 0 0 0 0 0 0 1 1 I

x, y

0 0 0 1

A1 A2 E A1 E A 1 + A2 + E E E A 1 + A2 + E A2 A1 A2

6.4. Strong-Field Laser Materials

237

TABLE 6.7. C3 v Selection rules.

AI

C3 v

AI A2 E EI/2 E"J;2 E)J2

II

0 j_

A2 0 II

j_

E j_ j_ j_ + II

EI/2

j_ + II

j_

j_

E"J;2

EJ/2

j_ 0

j_

II

II

0

product representations are given in Table 6.6. (See Chaps. 2 and 3 for the reduction of direct-product representations.) From this information the se­ lection rules for forced electric dipole transitions can be determined. Because of the anisotropic nature of the crystal, the selection rules are different for light polarized parallel and perpendicular to the crystallographic z axis. These are summarized in Table 6.7. The admixing interactions due the pres­ ence of a trigonal field that allow additional transitions to occur are shown in Fig. 6.4. 6.4

Strong-Field Laser Materials

Ruby is the standard example of a specific example of a strong-field Cr3 + laser material. The host crystal is a-phase corundum, generally referred to as sapphire. It has the chemical formula Ah03 and the lattice structure5 shown Fig. 6.5. The planes of oxygen ions are almost hexagonal close-packed. However, the angular distortion of some of the oxygen bonds prevents per-

FIGURE 6.5. Lattice structure of a sapphire crystal ( Ref. 5).

6. Aiz0 3 : Cr3 + Laser Crystals

238

feet close-packing resulting in an R3 C space group for the crystal. The alu­ minum ions fit between the oxygen planes with an A B C stacking with every third cation site along the c axis being vacant. The Cr3 + ions enter the host lattice substitutionally for the Al3 + ions with little size distortion and no charge mismatch. They are surrounded by six oxygen nearest-neighbor ions in almost octahedral coordination. Thus the local site symmetry can be approximated by the Oh point group. However, the oxygen octahedron is stretched along the threefold axis (corresponding to the c axis of the crystal) producing a trigonal distortion that lowers the site symmetry to C3 v· The angular displacement of the oxygen bonds further lowers the symmetry to C3 . The concentration of chromium doping for standard ruby laser crystals is of the order of 0.05 wt. % ( 1 .58 x 10 1 9 cm- 3 ) . Some of the important material properties of ruby include a specific heat of 0. 1 8 g cal K - 1 ; a thermal diffusivity of 0. 1 3 cm2 s- 1 , a thermal expansion coefficient of 5.8 X I Q 6 c J , a room-temperature thermal conductivity of 0.42 W cm- 1 K - 1 , and refractive indices at the laser wavelength of 1 .763 for E .l c and 1 .755 for E II c. It is a very hard material that is environmentally stable and can be polished to a laser-quality finish. All of these properties contribute to ruby's success as a laser material. The major features in the optical spectra of ruby can be explained by the Tanabe-Sugano diagram shown in Fig. 6.3 and the C3v selection rules given in Table 6.6. The reduc­ tion in symmetry from C3v to C3 coupled with the spin-orbit interaction produces additional energy-level splittings and changes in the selection rules for the transitions compared with the results for a cubic crystal field. Figure 6.6 summarizes the energy-level splittings for ruby. The major features of the absorption and fluorescence spectra are explained by the octahedral crystal-field states that are grouped into quartets and doublets associated -

2a 4F

2a

4F

2G

2G

t22e

2 t2 e t2 3

t2

2e

t 23 t 23

4-r l 4T

2

2A t

t>'@lf�.

G

2T 2 2T

--

l

2E

t2 3

4A

< <

G

Ew G

B

- G 2 Q UARI'ETS DOUBLErS

CRYSTAL FREE ION CONFIGURATION TERM FIGURE

OCTAHEIRAL FIELD

<

El f2

-

y

4F

--

-

El l

E l/2

- El l <.2E 3f2 El / 2 RJ 2 L

SPIN-ORBIT INTERACTION

�12 El f2

1RIGONAL FIELD

6.6. Ruby energy levels. (Spin orbit and trigonal field splittings are shown only for sharp levels.)

6.4. Strong-Field Laser Materials 0.6

PA RA LLEL

0.4

>

l-

�

TO C3

AXIS

en o.2 z

.. < 0

�

0

0.2

3000

4000 (A)

0.8

1-

en

z w 0

..

< 0 i= Q,

0

FIGURE 6.7. Absorption spectra of ruby ( Ref. 8). The transitions associated with the labeled spectral features are shown in Fig. 6.6 .

P E R P E N D I C U LAR TO C 3 AXIS

0.4

>

239

0.6' 0.4 0.2

5000 A.(A)

7000

6000

ROOM TEMPERATURE

PARALLEL TO

C3 AXIS

0

0.6

0.4

PERPENDICULAR

0.2 0 3000

TO

C3 AXIS s .

7000 (8) T

=

77°K

with the ground-state and the first excited-state crystal-field configurations. The effects of the spin-orbit interaction and trigonal crystal-field distortion produce fine structure in the spectra. The room-temperature absorption spectra of ruby for two different polar­ ization directions are shown in Fig. 6.7. The major features are two intense broad bands with oscillator strengths of the order of 1 o 4 , and three groups of weak sharp lines with oscillator strengths of the order of 1 0- 6 to 1 0 7 . The bands are associated with the spin-allowed transitions from the 4A 2g ground state to the 4T2g excited state ( U band) and to the 4T1 g excited state ( Y band). As seen in Fig. 6.6, these transitions involve an electron moving from a t2g orbital to an eg orbital, and thus the thermal modulation of the crystal field results in the large bandwidth. The position of the 4T2g band can be used with the equations in Sec. 6.2 that are plotted in the Tanabe­ Sugano diagram to determine a value for the crystal-field strength. For ruby Dq is found to be 1 720 cm- 1 . The fit between the observed spectral tran-

240

6. A1z0 3 : Cr3 + Laser Crystals

sition energies and the theoretically predicted energy levels shows that the value of the Racah parameter B for Cr3 + ions is reduced from its free-ion value of 9 1 8 cm- 1 to a value of 765 cm-1 in a ruby crystal. The factor C/B is found to equal 4. If only octahedral crystal-field symmetry is considered, the transition to the lower energy 4 T2g should be much stronger than the transition to the higher energy 4T1 g band, and there should be no polar­ ization dependence. The presence of a trigonal component to the crystal field predicts that the transitions to both levels should be equally allowed for both polarizations. The fact that the 4T1 g band is stronger than the 4 T2g band is associated with the strength of the configuration mixing. This inter­ action strength has a factor in the denominator that is the difference in energy between the level involved and the 3/ level. This splitting is smaller for the higher energy level. The polarization dependence is associated with the fact that these are vibronic bands and thus their strength depends on their coupling to specific phonon modes. Coupling to a mode of specific symmetry can result in polarization selection rules. In addition, the parity of the phonon mode can lift the parity restrictions on the transitions without the presence of configuration mixing. This is discussed further below. As temperature is lowered, these bands become sharper and structure appears on the low-energy side of the bands. This structure is associated with the zero-phonon line for the transitions and specific one-phonon vibronic transitions. In addition, the position of the 4A 2 -- 4 T2 transition shifts significantly to lower energies with temperature. Since the energy of this transition is the cubic crystal-field strength lODq and this is propor­ tional to the inverse of the fifth power of the lattice parameter, this shift in energy has been interpreted 6 in terms of the thermal lattice dilation ( 11/ /o) . The latter factor can be found for sapphire from handbooks of thermody­ namic parameters of materials. This interpretation is consistent with the ex­ perimental results and is confirmed by the results of measurements of ruby spectra under hydrostatic pressure. 7 This lowering of the 4 T2 energy level as temperature is raised enhances the admixing of the 2E and 4 T2 levels at high temperatures. These thermal effects are discussed further below. The weak, sharp lines in absorption are associated with spin-forbidden transitions to the doublet levels. As seen in Fig. 6.6, these are between states of the same crystal-field orbital configuration and take place through single­ electron spin flips. The lowest energy set of lines terminates on the 2E term of Oh symmetry and are called the R lines. These fall below the two broad bands at R 1 = 14 399 and R 2 1 4 428 cm- 1 and are the lowest-energy feature in the absorption spectrum. The combined effects of the spin-orbit interaction and trigonal crystal-field distortion split this level into two com­ ponents, resulting in the observation of two lines designated R 1 (lowest energy) and R 2 (higher energy). The separation of these two lines is about 29 cm- 1 • Under high-resolution spectroscopy each of these lines is split into two components due to the ground-state splitting shown in Fig. 6.6. Since

6.4. Strong-Field Laser Materials

241

this splitting is only 0.38 cm- 1 , it is generally not resolved at room temper­ ature. With even higher resolution at very low temperatures, these sharp lines exhibit additional structure due to different isotopes of chromium. The next higher energy set of sharp lines are called either R' or S lines. They are associated with transitions to the split components of the lowest 2 T1 9 level. The S lines are still on the low-energy side of the absorption bands. The third set of sharp lines are called the B lines. Energetically these fall between the two broad absorption bands. They are associated with transitions termi­ nating on the split components of the 2T29 level. The relative differences in intensities of these lines and their change with polarization direction can be explained using the selection rules for C3 v symmetry. Most important is the change in the two R lines. The transitions from both ground state compo­ nents to the lowest level of the 2E manifold are allowed in the E j_ c spectra but only one of these is allowed in the E II c spectra. Thus the Rr line ap­ pears in the absorption spectra for both polarization directions but is more intense in the E j_ c spectrum. The intensity of the R 2 line is less dependent on polarization since the transition to the upper component of the 2£ mani­ fold from one of the split components of the ground state is allowed in the E j_ c spectra and the transition from the other ground-state component is allowed in the E II c spectra. The charge-transfer bands and host band edge for ruby are in the near­ ultraviolet spectral region with strong absorption beginning about 270 nm . The transitions to the higher-energy levels of the Cr3 + ions shown in Fig. 6.6 are masked from view by this band-edge absorption. Charge-transfer band peaks are near 40 500 and 41 500 em I . The fluorescence spectra of ruby at room and liquid-nitrogen temper­ atures are shown in Fig. 6.8. They consist of the R-line zero-phonon tran­ sitions and their vibronic sideband. Essentially all of the energy pumped into the strong absorption bands relaxes rapidly to the 2£ metastable state, which becomes the initial state for the fluorescence transitions and laser emission. The excited-state absorption from this metastable state is very small and thus is a neglegible loss mechanism for laser operation. The populations of the two split components of this level are in thermal equalibrium. Thus the Rr line originating from the lowest component is the most intense spectral line, and as temperature increases the R 2 line originating from the higher component of the split manifold increases in intensity relative to the R 1 line. At high temperatures, weak emission lines are observed on the higher­ energy side of the R lines. This is associated with emission from the S lines originating from the split components of the 2 T1 9 level. The temperature dependence of the relative intensities of these lines follows the exponential expression discussed in Sec. 4.2 for levels connected by fast nonradiative transitions. This implies that populations of the 2£ and 2T1 9 levels are in thermal equilibrium with each other. 9 If temperature is lowered to very low temperatures such as 2 K, there is a bottleneck in the phonon transitions be-

242

6. Alz0 3 : Cr3 + Laser Crystals

� 10 c ::

� .d

T= 3 0 7

GAIN REDUCED B Y FACTOR OF 3 7 . 5 EXPA N D E D A SCALE

K

5

GAIN REDUCE D BY A FACTO R OF 390 EXPAN DED A S C A L E

A

(A)

FIGURE 6.8. Fluorescence spectra of ruby ( Ref. 8).

tween the two split components of the 2E level that stops the populations of these levels from reaching thermal equilibrium before fluorescence emission occurs. At high temperatures, the fluorescence emission appears as a relatively smooth broad band approximately symmetric about the position of the R lines, which are broadened to the extent that R 1 and R 2 cannot be resolved. The broad bands are associated with multiphonon vibronic transitions orig­ inating on the 2 E level. As temperature is lowered, the vibronic sidebands decrease in intensity and width and exhibit structure associated with specific single-phonon vibronic transitions. At very low temperatures the high­ energy vibronics associated with phonon absorption disappear completely, leaving only the low-energy vibronic band associated with phonon emis­ sion. 1 0 The detailed structure of the vibronic transitions can be seen in Fig. 6.9. This has been analyzed with the techniques described in Sees. 4.3 and 4.5. Some of the structure can be correlated with the vibrational modes of the sapphire host lattice and compared to phonon spectra in the infrared spectral region. It was found that all phonon modes give allowed vibronic transitions for this case for E j_ c polarization, while there are some symme­ try selection rules for phonons taking part in vibronic transitions with E II c polarization. The fluorescence emission of ruby after pulsed excitation exhibits a single exponential decay. 8 • 1 1 The decay time is shown in Fig. 6. 10 as a function of temperature. Both R lines and the vibronic sidebands have the same decay time since they originate from metastable states that are thermalized. At

6.4. Strong-Field Laser Materials I N T E N S I TY

13 60 I N TE N SI TY 11

10

R,

13 800 1400 14200 WAV E N U M B ER (cm- 1 )

243

FIGURE 6.9. Vibronic side­ band of ruby (from Ref. 1 0 with copyright permission from Springer-Verlag) .

lit@

E II C

9

R,

8

3

2

1380 14200 1400 WAV E N U M BER (cm-1)

1440

very low temperatures where the emission is dominated by radiative tran­ sitions, the lifetime is about 3 ms. This decreases slightly above room tem­ perature due to populating the 2 T1 level. At very high temperatures the 4T2 level is populated, leading to nonradiative decay to the ground state and a shorter fluorescence lifetime. This is discussed further below. Part of the de­ crease in fluorescence lifetime above room temperature is also associated with the temperature-dependent increase in vibronic emission probability. At low temperatures the lifetime becomes independent of temperature. As shown in Fig. 6. 10(A), a longer lifetime is measured for a thick sample versus a thin sample. 1 1 The shorter time is the intrinsic decay time of the fluorescence emission, while the longer time reflects the effects of radiation trapping due to reabsorption energy transfer. This will be larger or smaller depending on the concentration of Cr3 + ions and the size of the sample. The millisecond time scale of the fluorescence lifetime reflects the spin- and parity-forbidden nature of the emission transition. The fluorescence decay in

10.0 5. 0

2.0

i

J!

LL

1 .0

0.5

0.2

..

0. 1

0.05

0.02

10

o

LIFETIME FOR A 0.03% RUBY SAMPLE

.o.

LIFETIME WITHOUT REABSORPTION

(9.4 X 7.J X 3.8 nun)

50

30

20

100 T (K)

500

200

1 000

(A)

1 0. 0

5. 0

2.0 u; "0 c 0 u Q) "'

:2:

..u.

1 .0

0. 5

0

0

lJ.

0

0

l!J c

r#'f !J. 0�o i� D

o

(>)

0. 2 0. 1

0.05

o R l i ne o N1 line tJ. N2 l i n e

0

0. 02 0.01

10

20

50

100

T (° K )

200

500

(B)

FIGURE 6. 10. Temperature dependence of the fluorescence lifetime of ruby. (A) Lightly doped sample (from Ref. 8 with data included from Ref. 1 1 ) . ( B) Heavily doped sample [from Ref. 14(a)]. 244

6.4. Strong-Field Laser Materials u; a z 0

fd

(J)

12

FIGURE 6. 1 1 . Dependence of the fluorescence lifetime of ruby on Cr3 + concentration. The dashed line is for thin samples with no reabsorption. ( From Ref. 1 2 with data included from Ref. 1 3. )

10

:J

..J

:E

1.IJ

::! i=

1.IJ u.

:J

245

� - - - · - - - - - -.

2 0 0.0 0 1

0.01

CONCENTRATION

0.1

( WT %

.., 0

Cr2 0

3

1 .0

)

ruby with a high concentration of Cr3 + ions becomes nonexponential with a shortened decay time. 8 This is due to energy transfer as discussed below. The concentration dependence of the decay time of ruby12 • 1 3 is shown in Fig. 6. 1 1 . The differences in measured lifetimes with and without the effects of radiative trapping are shown. At high chromium concentrations, many new lines appear in the absorp­ tion and emission spectra of ruby in the region of the R lines. These have been attributed to transitions involving exchange-coupled pairs14 and are referred to as N lines. The properties of the spectral lines have been used to identify them with four different types of pairs, each having separations of about 3.5 A or less. These are associated with first-, second-, third-, and fourth-nearest-neighbor positions whose numbers of equivalent lattice sites are 1 , 3 , 3, and 6 respectively. The first three types of pairs are coupled anti­ ferromagentically and the fourth-nearest-neighbor pairs are coupled ferro­ magnetically. Other spectral structure in the ultraviolet region may be asso­ ciated with higher-energy pair lines. Figure 6. 12 shows an example of the fluorescence spectra of heavily doped ruby at two temperatures in the region of the R and N lines. The ratio of the intensity of the N2 line to that of the R1 line as a function of chromium concentration1 5 is shown in Fig. 6. 13. The temperature dependence of the intensities o f two of the N lines1 6 in the absorption spectrum is shown in Fig. 6. 14. The solid lines are the theoreti­ cal predictions assuming N1 originates from an S 1 state of an anti­ ferromagnetically coupled pair and N2 originates from an S 1 level of a ferromagnetically pair as discussed in Sec. 5. 1 . This type of temperature­ dependent study along with investigations involving spectral changes due to uniaxial stress have been useful in developing an energy-level diagram for each of the types of exchange-coupled pairs and associating observed spec­ tral lines with transitions between sets of these energy levels. The two types of pairs producing the most prominent spectral lines are the third- and fourth-nearest-neighbor pairs. The energy-level diagrams for these pairs are shown in Fig. 6. 15. From these diagrams and the theory of exchange

246

6. Ah0 3 : Cr3 + Laser Crystals

�

7050

7000

>. t A l

7050

6950

690 0

6950

6900

FIGURE 6. 1 2. Fluorescence spectra of heavily doped ruby in the region of the R and N lines [from Ref. 14(b)].

FIGURE 6. 1 3 . Variation of the ratio of the intensity of the N2 line to the R1 line as a function of Cr3 + (after Ref. 1 5) .

FIGURE 6. 14. Temperature dependence o f the intensities of the N1 (7041 A) and N2 (7009 A) lines in the absorption spectrum of heavily doped ruby. ( Reprinted from Ref. 1 6, copy­ right 1 964 by Columbia University Press. Re­ printed with permission of the publisher.)

0 o

0. 2

0.4

0.6

1 .0

0.8

CONCENTRATION OF CHROMIUM IN WT t.

8

2

1/T

4

6

8

10-t

6.4. Strong-Field Laser Materials 14, 387. 5 e m '

1 1

14, 385. 2 em '

1

14, 380. 9 e m ' 14, 37&. 5 em '

247

1

14. 365. 5 em ·I 14, 363. 2 em ·I

14, 34. 2 e m ·I

14, 341. 8 em ·I

14, 232. 0 em -I

67. 9 cm 1

S·3

32. 7 cm" 1

S·2

10. 3 cm" 1

S•1

0 cm" 1

TH I RD NfAREST NE IGHBOR PAI R SYSTEM

(A) 14,477. 5 cm 14, 451. 8 cm

1 1

14, 446. 0 cm-l 14. 4 1 8. 2 cm-l 14, 416. 3 cm-l 14. 411. 2 cm-l

14,379.9 cm'1 14, 32!.2 cm'

14, 327.2 cm'

1

1

14,324. 1 cm· l 14, 299.4 cm-

1

14, 298. 7 cm-l

RlURTH NEAREST NEIGHBOR PAIR SYSltM

(B)

FIGURE 6 . 1 5 . Energy-level diagrams for third- and fourth-nearest-neighbor pairs of Cr3 + ions in ruby [after Ref. 14(a)].

248

6. Ah0 3 : Cr3 + Laser Crystals

coupling discussed in Sec. 5.1, values for the coupling parameters can be determined. The general trends in energy splittings observed for these anti­ ferromagnetically and ferromagnetically coupled pairs are consistent with the predictions of the simple exchange theory. However, adding a biqua­ dratic exchange term improves the agreement between theory and experi­ ment, and to explain the magnitude of the interactions it is necessary to in­ volve superexchange interactions. The widths and positions of the R and N lines change with temperature as predicted by the theoretical expressions developed in Sec. 4.3. Examples1 7 of these temperature dependences are shown in Figs. 6. 16 and 6. 17. The shapes of the lines at high temperatures were Lorentzian at high temperatures but at low temperatures they exhibited Voigt profiles. Figure 6. 1 8 shows a typi-

· � 10 �

T (K)

FiGURE 6. 1 6. Linewidths of the R lines in ruby with 0.94% chro­ mium ( Ref. 17). The theoretical curves are the predictions of the theory described in Sec. 5.3.

T (K)

6.4. Strong-Field Laser Materials

80 70

60

)' 50 40 ii 50 20 10

FIGURE 6 . 1 7 . Shift in the posi­ tions of the R lines of ruby be­ low their values at O K ( Ref. 1 7) . The theoretical curves are the predictions of the theory de­ scribed in Sec. 5.3.

o'\,l.l 'l Cr

o"- OJI4CJI> Cr

0

0

50

60

100

150

20

249

250

50

350

400 . 450

500

250

50

350

400

50

T (K)

o'\, Z.l 'l Cr a'\o 0.94CJI> Cr

30 20 10

0

50

150

10

20

T (K)

450

- EXPERIMENTAL UNE

o

VOIGT PROFILE

--- GAUSSIAN UNE

-- LORENTZIAN LINE

0.5

-0.5

-0.1 0.1

C.-Uo)6w

0.5

FIGURE 6. 1 8 . Shape of the N1 line of a ruby sample with 2. 1 % chromium at 21 K (after Ref. 1 7) .

250

6. Ah0 3 : Cr3 + Laser Crystals

cal example1 7 of a measured line shape, the shapes of pure Gaussian and Lorentzian line shapes having the same linewidth, and the theoretical fit of a Voigt profile to the measured line shape. From this type of analysis the temperature-independent contribution to the linewidth due to strain broad­ ening is determined. For lightly doped ruby, the strain-broadened linewidth can be less than 0. 1 cm - 1 and it is usually significantly greater for heavily doped samples. The dominant broadening mechanism at high temperatures is Raman scattering of phonons. The theoretical lines in Fig. 6. 1 6 are de­ rived assuming a Debye distribution of phonon states as discussed in Sec. 4.4. For the R lines, the fit between theory and experiment in the temper­ ature region near 1 00 K can be improved slightly by including a term due to direct phonon processes between the split components of the 2 E level. For the R lines in heavily doped ruby it is necessary to include phonon processes between the exchange-split levels of the ground split manifold. The domi­ nant processes are Orbach processes obeying a selection rule !1S = ±2. The temperature-dependent shifts in positions of the lines can be explained by theoretical expressions for phonon scattering without including any con­ tributions for direct processes. Similar measurements have been made on the S and B lines/ 8 and both direct and Raman phonon processes have been found to contribute to the linewidths. At low temperatures, the quantum efficiency of radiative emission from the 2E level in ruby is approximately 0.96. As temperature increases, this quantum efficiency decreases and is approximately 0. 7 at room temperature. This is due to increased vibronic emission and population of the 2 T1 level. Above 450 K the total radiative quantum efficiency of ruby decreases due to radiationless decay to the ground state, which is associated with thermal activation of population to the 4 T2 level as shown in the configuration­ coordinate diagram1 9 in Fig. 6. 19. The detailed configuration-coordinate model for ruby developed in Ref. 1 8 uses Franck-Condon overlap integrals to account for the positions and shapes of the optical bands as well as the nonradiative decay processes and fluorescence quenching in ruby. The Huang-Rhys factors for the transitions from the 4A 2 level to the 4T2 and 4T1 levels are found to be 3 and 6, respectively. This analysis predicts emissions rate of 2 E -- 4.4 2 = 1 . 4 x 102 s- 1 and 4 T2 -- 4.4 2 = 2.3 x 1 04 s 1 with a non­ radiative crossover rate of about 104 s- 1 at 0 K. These rates increase with temperature. The radiationless relaxation times between excited levels in this model are found to be 4 T2 -- 2E � 10- 1 3 s and 4T1 -- 2E � 1 0- 1 2 s. It should be noted that other analyses based on configuration-coordinate models have resulted in much smaller estimates for the nonradiative decay rates. It has been difficult to measure these rates directly with any degree of accuracy. Picosecond pulse-probe experiments have set an upper limit of 7 ps on the decay time of the 4T2 level. 2 0 Another possible loss mechanism for emission from the 2 E metastable state is excited-state absorption. However, for ruby there is no good energy match for transitions from the metastable state to higher energy levels (see

6.4. Strong-Field Laser Materials

251

FIGURE 6. 19. Configuration-coordinate dia­ gram for ruby showing radiationless decay channels ( Ref. 1 9) .

CONF I G U RATIONAL COORDINATE

Fig. 6.3). Thus excited-state absorption of photons emitted in the R 1 line is not an effective loss mechanism for ruby. As discussed in Sec. 3.4, there is a change in the polarizability of the Cr3 + ions in the 2 E level as compared to the 4A 2 level. However, with normal optical pumping conditions the change in the refractive index associated with populating the metastable state [see Eqs. (3.4.8) and (3.4. 16)] is not large enough to produce serious problems with optical beam distortion. Studies of the temperature and chromium concentration dependences of the fluorescence intensities and lifetimes of heavily doped ruby show that nonradiative energy transfer takes place between single ions of Cr3 + and ion pairs. Above a critical concentration of 0.4%, the excitation diffuses among the single ions via a superexchange interaction. This can be described 14 as localized exaiton migration with a hopping rate of about 106 s- 1 for samples with a chromium concentration of 1%. The final step of energy transfer from single ions to pairs also takes place through exchange interaction but the measured rate for this processes is much smaller. For a sample with 1% chromium the transfer rate i s about 103 s- 1 . This transfer step i s best de­ scribed as a two-phonon assisted process. The effect of this efficient ion-ion interaction in heavily doped ruby can be seen in Fig. 6. 10( B). At temper­ atures above about 75 K, the fluorescence lifetimes of the single ions and pairs are the same. This indicates that the distribution of excitation energy among these different types of opeically active centers is thermalized. For this to occur there must be fast energy transfer back and forth between the ions in the different types of sites. At lower temperatures the energy transfer becomes less efficient. In this region the R, N1 , and N2 lines all exhibit dif­ ferent fluorescence lifetimes, showing that there is no thermal equilibrium in

252

6. Aiz0 3 : Cr3 + Laser Crystals

the distribution of excitation energy among the different types of centers. Excitation by intense laser light can induce a photocurrent in ruby2 1 with electrons hopping between Cr3+ and Cr4+ ions. This can build up an inter­ nal Stark field that causes changes in the observed spectra of ruby. The spectroscopic properties of ruby described above make it an excellent laser material. It operates as a three-level system with sharp line laser emis­ sion at the R 1 line position of 6943 A. The energy density in the metastable sate at threshold is 2. 1 8 J/cm3 . Laser action has also been obtained from the R 2 line and the N1 and N2 lines. The strong, broad absorption bands result in efficient absorption of fl.ashlamp pump energy, and the fast relaxation to the 2E level, which has a high quantum efficiency and low excited-state absorption, results in efficient overall pumping of the laser emission. The narrow linewidth of the laser transition gives a high stimulated emission cross section, which becomes even greater at low temperatures. At room temperature the R 1 line stimulated emission cross section is fJR 1 2.5 x IQ- 20 cm2 and the absorption coefficient at the peak of the lasing transition is !XR 1 0.2 cm- 1 . The long lifetime of the metastable state provides the ability for a high level of optical storage leading to good pulsed and Q­ switched operation. For complete inversion, the maximum energy density of the 2E level is 4.52 J jcm 3 and the maximum extractable energy is 2.35 J jcm3 . The spectroscopic and lasing properties of Cr3 + ions vary significantly from host to host. Examples of the optical properties of other chromium-doped materials are given in the next chapter. References

1 . J.S. Griffith, The Theory of Transition Metal Ions (Cambridge University Press, London, 1 96 1 ) . 2 . E.U. Condon and G.H. Shortley, The Theory of A tomic Spectra (Cambridge University Press, London, 1 935). 3. C.J. Ballhausen, Introduction to Ligand Field Theory ( McGraw-Hill, New York, 1 962) . 4. Y . Tanabe and S. Sugano, J. Phys. Soc. Jpn. 9 , 766 ( 1 954); S . Sugano, Y . Tanabe, and H. Kamimura, Multiplets of Transition-Metal Ions in Crystals, (Academic, New York, 1 970) . 5. S. Geschwind and J.P. Remeika, J. Appl. Phys. 3 3 (Suppl.), 370 ( 1 962) . 6. C.J. Donnelly, S.M. Healy, T.J. Glynn, G.F. Imbusch, and G.P. Morgan, J. Lumin. 42, 1 1 9 ( 1 988). 7. H.G. Drickamer, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic, New York, 1 965), vol. 1 7, p. 1 . 8. R.C. Powell, Doctoral thesis, Arizona State University Department of Physics, 1 967 (also published as Physical Sciences Research Paper No. 299, Air Force Cambridge Research Laboratories, Hanscom Field, Bedford, MA, 1 966) . 9. A. Misu, J. Phys. Soc. Jpn. 19 , 2260 ( 1 964) . 10. U. Rothamel, J. Heber, and W. Grill, Z. Phys. B 50, 297 ( 1 983). 1 1 . D.E. Nelson and M.D. Sturge, Phys. Rev. 137, Al 1 1 7 ( 1 965).

References

253

12. G.F. Imbusch, Phys. Rev. 153, 326 ( 1 967) . 1 3 . N.A. Tolstoi and Liu Shun'-Fu, Opt. i Spektroskopiy 13, 403 ( 1 962) . 14. (a) R.C. Powell and B. DiBartolo, Phys. Status Solidi A 10, 3 1 5 ( 1 972); (b) R.C. Powell, B. DiBartolo, B. Birang, and C.S. Naiman, Phys. Rev. 155, 296 ( 1 967); (c) R.C. Powell, B. DiBartolo, B. Birang, and C.S. Naiman, in Optical Proper­ ties of Ions in Crystals, edited by H.M. Crosswhite and H.W. Moos ( Inter­ science, New York, 1 967), p. 207. 1 5. A.L. Schawlow, D.L. Wood, and A.M. Clogston, Phys. Rev. Lett. 3, 27 1 ( 1 959) . 16. P. Kisliuk, A.L. Schawlow, and M.D. Sturge, in Advances in Quantum Elec­ tronics, edited by P. Grivet and N. Bloembergen (Columbia University Press, New York, 1 964), p. 725. 17. R.C. Powell, B. DiBartolo, B. Birang, and C.S. Naiman, J. Appl. Phys. 37, 4973 ( 1 966) . 1 8 . T. Kushida and M. Kikuchi, J. Phys. Soc. Jpn. 23, 1 333 ( 1 967) . 19. W.H. Fonger and C.W. Struck, Phys. Rev. B 1 1 , 325 1 ( 1 975). 20. S.K. Gayen, W.B. Wang, V. Petricevic, R. Dorsinville, and R.R. Alfano, Appl. Phys. Lett. 47, 455 ( 1 985). 21. A.A. Kaplyanskii, J. Lumin. 48-49, 1 ( 1 99 1 ) .

7

Transition-Metal-Ion Laser M aterials

Trivalent chromium has been made to lase in a wide variety of different oxide and fluoride host crystals. The strength of the crystal field varies sig­ nificantly from host to host and in some cases is much weaker than the crystal field for ruby. In these weak-field materials, broad-band fluorescence occurs from the 4 T2 level and this can be used for tunable lasers. Although trivalent chromium has been the dominant ion used for solid-state laser based on transition-metal ions, several other ions in the same row of the periodic table have been made to lase and several of them are beginning to find important applications. All of these have similar electronic config­ urations and their optical spectra are based on electronic transitions between levels of the unfilled d shell. As with Cr3 +, the emission spectra and lasing of these ions can either be a sharp line or a broad band, allowing for tunable operation, and can occur in the visible to near-infrared spectral regions, de­ pending on the host material. In some cases laser operation has been suc­ cessful only at temperatures well below room temperature. The availability of lasers based on this variety of ions offers an extension of operational characteristics including coverage of a broader spectral range, shorter pulse widths, and reduced excited state absorption losses compared to some of the Cr3 + -based systems. In the following two sections the general spectral properties of Cr3 + ions in a variety of different hosts are discussed and compared to those of ruby described in Chap. 6. Then the properties of other transition-metal ions in different host materials of interest for laser applications are discussed. The subsequent sections describe the details of the spectral and lasing character­ istics of several of the most important transition metal ion lasers. The fun­ damental concepts needed to understand these properties were outlined in Chaps. 2-5. 7. 1

Broad-B and Cr3 + Laser Materials: Alexandrite

In host materials with weak crystal-field strengths, the 4Tz9 is the lowest excited state as seen in the Tanabe-Sugano diagram of Fig. 6.3. This results 254

7 . 1 . Broad-Band Cr3 + Laser Materials: Alexandrite M

e

•

M

255

FIGURE 7. 1 . c-axis view of chrysoberyl structure where M denotes mirror planes [from Ref. l (a)].

Be AI

Qo

in broad-band emission and thus tunable laser output. The first solid-state laser material to reach commercial importance as a tunable laser was alex­ andrite (BeAh04 : Cr3 +) . Strictly speaking, the crystal-field strength in alex­ andrite is in the intermediate range with the 2E level lying just below the 4 T29 in energy. However, at room temperature there is significant thermal activation from the 2E level to the 4T29 level, and the spin-allowed transition from the quartet state is much stronger than the spin-forbidden transition from the doublet state. Thus the fluorescence and lasing characteristics are essentially those of a strong-field material with vibronic emission. The crystal structure of the chrysoberyl host material is hexagonal-close­ packed with the Pnma orthorhombic space group1 shown in Fig. 7. 1 . The Al3 + ions are octahedrally coordinated by the oxygen ions and occupy two inequivalent crystal-field sites, one with mirror symmetry belonging to the point site group Cs and one with inversion symmetry belonging to the point site group C;. The Cr3 + ions substitute for the Al3 + ions with 78% going into mirror sites and 22% going into inversion sites. The energy levels of the mirror-site chromium ions can be determined by considering the main con­ tribution to the crystal field from oxygen ligands with Oh symmetry that is slightly distorted to c•. The procedure for determining the energy levels is the same as that described in Sections 6. 1-6.3 for ruby and the results are shown in Fig. 7.2. The electric dipole transitions determined by symmetry selection rules are also shown in the figure, assuming that configuration mixing lifts the parity restrictions. A similar analysis can be done for the inversion-site ions that have a lower concentration. The absorption spectrum of alexandrite is dominated by ions in mirror sites. 2 It is very similar to that of ruby, consisting of two intense broad bands and three sets of sharp lines as shown in Fig. 7.3(A). The ftuo-

256

7. Transition-Metal-Ion Laser Materials

FIGURE 7 .2. Energy-level diagram and allowed transitions for Cr3 + ions in mirror sites in alexandrite.

E(lb E(la,c I

4A 2

..'

I

I

rescence spectrum consists of the two zero-phonon R lines and the broad vibronic band from the 4 T19 level. Emission from mirror-site ions dominate the spectrum. In addition, weak lines associated with the R lines from inver­ sion site ions can be observed as shown in Fig. 7.3(B). The optical pumping of alexandrite laser crystals assumes very fast non­ radiative relaxation processes between the 4T19 and 2£9 levels. Several dif­ ferent types of laser spectroscopy experiments have been used to study these processes and they have been found to have radiationless transition times of a few picoseconds. In order to explain this fast time for these processes, anharmonic interactions must be included in the theoretical treatment. 2 Be­ cause of the efficiency of these processes, the populations of the 4 T19 and 2 E9 levels are thermalized. Radiative transitions occur from both levels at room temperature, but because the transitions from the quartet level are spin allowed, they are stronger. Thus the lower-lying 2 9 level acts as a pop­ ulation storage level for the 4 T19 level. As the population of the quartet level decreases through fluorescence emission, it is replenished by radiationless transitions from the 2 9 level. The fluorescence lifetime of the emission of the mirror site ions decreases from about 2.3 ms at 10 K to about 290 J.lS at 300 K, as shown in Fig. 7.4. The long lifetime at low temperatures is the intrinsic lifetime of the 2 level and the thermal quenching of the fluorescence decay time is associated with thermal activation into the shorter lived 4 T29 level. Following the discussion in Sec. 4.2, the coupled lifetime of these two levels in thermal equalibrium is given by

E

E

E

r 1

rE 1 +
(-AE ) knT '

(7. 1 . 1 )

7. 1 . Broad-Band Cr3 + Laser Materials: Alexandrite

AI nm

� §

(B) 4

R, m -�

257

1

(A)

R2m -�:\

:· : R2i R , i :l I ;',� .. - -\./···· � ,,

E

II b

I

3

.ci .. .!. 2

I\•. =488.0 nm ,i\ ,. =514.Snm

)\ ,.= 579.1 n m ,i\ 1 n m 1

(B)

FIGURE 7.3. (A) Absorption and ( B) fluorescence spectra of alexandrite [from Ref. 2(a)].

where 11E is the splitting between levels and
7. Transition-Metal-Ion Laser Materials

258

(A) 4.0

(A)

FIGURE 7.4. Temperature dependence of the fluorescence lifetimes in alexandrite ( Ref. 2) . (A) Mirror-site ions [reprinted from Powell et al. , Phys. Rev. B 32, 2788 ( 1 985)]; ( B) inversion-site ions [reprinted from Suchocki et al. , J. Lumin 37, 29 ( 1 987), with permission of Elsevier Science NL].

The temperature dependences of the widths and positions of the R lines in mirror symmetry can be explained in the same way as described in the pre­ vious chapter for ruby2 using the theoretical models developed in Sec. 4.4. For samples with high concentrations of Cr3 + new lines appear in the ab­ sorption and fluorescence spectra due to exchange coupled pairs. Spectral transitions associated with six different types of pairs have been identified. 2 These can be seen in Fig. 7.5, which shows the fluorescence emission in the region of the R lines that occurs after narrow-line excitation at four closely spaced wavelengths between 670 and 680 nm. Using the theoretical models described in Sec. 5. 1 , two of these pair systems are found to be ferro­ magnetically coupled and the other four antiferromagnetically coupled. 2 One interesting feature about alexandrite compared to ruby is the strong energy transfer among the chromium ions. Laser-induced population grat­ ing techniques have been used on both types of crystals to characterize the properties of long-range spatial migration of energy. No indication of this

7. 1 . Broad-Band Cr3 + Laser Materials: Alexandrite

10K

259

FIGURE 7.5. Site-selective fluo­ rescence spectra of heavily doped alexandrite for different excitation wavelengths showing transitions associated with ex­ change coupled pairs of Cr3 + ions ( Ref. 2) .

..

�=

.t:i

type of process was observed in ruby, indicating that the transfer observed between single ions and pairs takes place over distances much shorter than the wavelength of light used in the grating experiments. However, in alex­ andrite long-range diffusion of energy among the Cr3+ ions in mirror sites was observed. 2 The mechanism for ion-ion interaction causing the exciton migration is consistent with exchange theory, and the temperature depen­ dence of the measured diffusion coefficient indicates that the exciton moves over several chromium ions before being scattered by a phonon. The theory of long-mean-free-path energy migration discussed in Sec. 5.5 has been used to interpret the data obtained from transient grating spectroscopy. The ex­ citon diffusion coefficient is different for different crystal directions, and no long-range migration was observed for chromium ions in inversion sites. The reason that ruby and alexandrite crystals have very different long-range en­ ergy migration properties may be associated with the distribution of Cr3 + ions in the host crystals. The distribution in ruby is generally random, whereas in alexandrite there are preferred planes of chromium ions and banding within these planes. Thus local concentrations of Cr3 + ions can be much higher than that predicted by a statistically average distribution. This leads to having Cr3 + ions nearer to each other in specific directions and thus enhanced ion-ion interaction and energy transfer in these directions. The presence of long-range energy migration among mirror-site chromium

260

7. Transition-Metal-Ion Laser Materials

ions was confirmed by site-selective Stark shifting techniques. 3 Additional radiative energy transfer between these ions has also been observed at low temperatures. 4 Alexandrite has been operated as a laser under a variety of different con­ ditions. One is as a three-level laser with narrow line emission from the R 1 line of mirror site ions. This type of laser is very similar to ruby. A second mode of operation is as a quasi-four-level laser with emission from the vibronic emission band. This configuration is important since it allows for continuously tunable laser emission from about 700 to about 825 nm . Alexandrite was the first tunable vibronic laser developed to the extent of commercial availability. This type of laser has been operated in pulsed, Q-switched, and continuous-wave configurations. Alexandrite has a high stimulated emission cross section and a high power density at gain satura­ tion and thus a high extraction energy for Q-switched operation. To understand the gain characteristics of alexandrite lasers, it is necessary to extend the standard stimulated emission model described in Chap. 1 to include the vibronic nature of the transitions. The theory for doing this was developed by McCumber5 and applied specifically to alexandrite by Walling et al. 6. Thermal equilibrium is assumed to exist for the populations of all of the vibronic levels of the ground and excited electronic states. Since the final state of the emission transition is an upper vibronic level of the electronic ground state, this is referred to as a phonon-terminated laser transition. The room temperature emission cross section for alexandrite at 750 nm is mea­ sured to be 5.0 x w- 20 cm- 2 , which is consistent with the prediction of the McCumber model. The quantum efficiency of alexandrite has been mea­ sured to be 0.95 and excited-state absorption has been measured to be small in the laser band. As temperature is raised above room temperature, the gain increases and the peak of the gain curve shifts to longer wavelength. This is associated with the thermal change in the population distribution that determines the properties of the vibronic laser transition. These proper­ ties are opposite the usual temperature effects of decreased gain at high temperatures due to thermal broadening of sharp zero-phonon lines. Alex­ andrite is similar to ruby in possessing the chemical stability, hardness, and thermal properties that make it a good laser material. 7.2

Spectral Properties of Cr3 + in Different Hosts and

Their Laser Characteristics

The optical properties of trivalent chromium have been studied in many different types of host materials. The spectral and lasing characteristics are similar to either the strong-field or weak-field examples discussed above. A list of some of the chromium-doped crystals that have been made into lasers is given in Table 7. 1 . Here 11EET is the energy difference between the 2E and

7.2. Spectral Properties of Cr3 + in Different Hosts TABLE

261

7 . I . Laser host crystals for Cr3 + ions.

Name Ruby Alexandrite Emerald B orate Tungstate LGS YGG YSGG LLGG GGG GSAG YSAG GSGG LiCAF LiSAF GFG Pentafluoride Perovskite

Host material A!z03 BeAb04 Be3Aiz (Si03 ) 6 ScB03 ZnW04 La3Gas Si0 1 4 Y3GasOu Y3 SczGa3 0 1 2 La3LuzGa3 0 1 2 Gd3 Gas0 1 2 Gd3 SczAI30 1 2 Y3 SczAI301 2 Gd3 SczGa30 1 2 LiCaAIF6 LiSrAIF6 Na3 GazLh F 1 2 SrAlF 5 KZnF3

!l.EET

(cm- 1 )

2300 800 400 3000 600 250 <0 300 0

500

Apeak

(run)

r,

(f.lS )

695 3000 752 240 768 60 843 115 1035 0.5 5.4 968 740 241 750 1 39 830 68 769 160 784 1 50 767 1 15 785 170 780 825 67 791 310 932 95 820 176

ase

z ( l o - o cm2 )

2.5 0.6 1 .9 1 .0 43.0

Dq

B

/lO(

(cm- 1 )

(cm- 1 )

3 (A )

1 720

765

o.Ol5EIIb 0.047EIIc

0.36 0.6 1.6 0.6 0.7

1630 1613 1480 1 597

639 630 619 621

0.9 1 .3(n) 4.8(n) 0.63 2.1 1 .3

1 563

638

4 T2 levels, A peak is the peak wavelength of the laser transition,

0.062 0. 1 14 < 0.01

r, is the radiative lifetime of the laser transition, (Jse is the stimulated emission cross section of the laser transition, B is the Racah parameter, and �IX is the dif­ ference in the polarizability of the Cr3 + ion in the ground and excited states. The properties listed are those measured at room temperature. This repre­ sents a wide variety of different types of oxide and fluoride materials. Al­ though the spectral properties of Cr3 + ions in glass materials have also been investigated, there have been no practical lasers developed of these materials because of low quantum efficiencies. Figures 7.6 and 7.7 show examples of the absorption spectra and fluorescence spectra of several chromium-doped laser crystals varying from strong to weak crystal fields. The key parameter in determining the important spectral properties is the energy splitting be­ tween the 4 T2 and 2E levels after lattice relaxation occurs in the excited state. This a measure of the strength of the crystal field. The relaxation process has been described in various ways including tunneling between vibronic sublevels, 7 admixing of the levels through the spin-orbit interaction, 8 and anharmonic potentials. 2 According to Eq. ( 1 . 1 . 14), the efficiency of laser operation is enhanced when the quantum defect is small and the ratio of the excited-state absorp­ tion ESA cross section to stimulated emission cross section is small. The former situation is achieved when the splitting between the 4 Tz pump band and the emitting state (either the 2E level or the relaxed 4 Tz level) is small.

262

7. Transition-Metal-Ion Laser Materials

FIGURE 7.6. Absorption spectra for Cr3 + in several laser host crystals ( Ref. I 0).

Diode laser pumping on the low-energy side of the pump band can minimize the quantum defect. The latter situation can be a problem for chromium­ dope laser materials since there are many high lying levels of the 3d3 con­ figuration plus charge-transfer bands that can give rise to significant ESA. This can be an important factor in limiting the tuning range of vibronic laser materials on the high-energy side of the emission band. Thus having materials with high stimulated emission cross sections can overcome prob­ lems with ESA. However, high emission cross sections give the potential for amplified spontaneous emission and parasitic oscillations, thus limiting allowable energy storage. LLGG

FIGURE 7.7. Fluorescence spectra for Cr3 + in several laser host crystals.

7.2. Spectral Properties of Cr3 + in Different Hosts

263

The laser performance tends to be better for materials where Cr3 + sub­ stitutes for small ions such as Al3 + at sites with relatively high crystal fields and large elastic constants than it is for materials where chromium sub­ stitutes for large-size ions. 9 With these conditions, there is less lattice relax­ ation and the laser emission occurs at shorter wavelengths than in other materials. This enhances performance through smaller quantum defects, smaller ESA, and higher intrinsic quantum efficiencies. In terms of ESA, for these conditions the higher-energy absorption bands and laser emission band are not shifted into resonance for ESA transitions. Materials such as ruby, alexandrite, emerald, and LiCAF that fit these conditions have very low levels of ESA, while crystals such as YSAG, GFG, and ScB03 , which do not meet these conditions, have high ESA to stimulated emission cross section ratios. Having a high quantum efficiency of the metastable state is another im­ portant property for a good laser material. For Cr3 + in hosts with high crystal fields where the 2E level is the lowest excited state, this condition is generally satisfied. However, for chromium ions in hosts with low crystal fields where the relaxed 4T2 level contains much or all of the metastable­ state population, nonradiative decay processes leading to low quantum efficiencies can be a problem. From the discussion of the theories of multi­ phonon nonradiative decay processes in Chap. 4, it is clear that the proba­ bility for these transitions generally increases when the energy separation between the initial and final states decreases, since fewer phonons are required for conservation of energy. Also, in the simple configuration­ coordinate model, having a crossover point for initial- and final-state poten­ tial wells that is an energy of less than the thermal energy of the system will enhance the radiationless decay channel over the radiative decay channel. These conditions occur in materials having low crystal fields and large Stokes shifts so that the relaxed 4T2 potential well has its minimum at low energies and shifted significantly from the equilibrium minimum of the ground-state potential well. Note that these conditions are similar to those leading to enhanced ESA discussed above. Because of this, most good Cr3 + lasers have emission wavelengths of less than 1000 nm. The final column in Table 7. 1 lists the change in polarizability of the Cr3 + ion when it is in the metastable state versus the ground state. 1 1 These were determined by four-wave-mixing measurements as discussed in Chap. 3. This change in polarizability is greater for oxide hosts than fluoride hosts because of the higher-energy position of the charge-transfer level in the latter class of materials. 1 2 For oxide crystals, the polarizability change is greater for weak field materials where the 4 T2 level is populated than for strong-field materials where the 2E level has the dominant excited-state population. The relationship connecting the peak stimulated emission cross section, radiative lifetime, and bandwidth of the lasing transition imposes some re­ strictions on developing optimized laser materials. This is especially impor-

264

7. Transition-Metal-Ion Laser Materials

tant when broadband tunable emission is required and flashlamp pumping is needed for high-energy storage. From Chap. 3, the important expression is a , r,

8

nn

•

(7.2. 1 )

For efficient laser operation, the flashlamp pulsewidth should be close to the value of the radiative decay time and the laser should operate near the satu­ ration fluence level. If the decay time is too short, the lifetime of the flash­ lamp is reduced. Generally values of r, ?: 100 J.lS are desirable for this rea­ son. For a fixed wavelength and bandwidth of laser operation, an increase in radiative lifetime requires a decrease in stimulated emission cross section. For maximum energy storage the lifetime must be as large as possible, but this results in a small emission cross section leading to a high satura­ tion fluence. If the saturation fluence level is too high, optical damage can become a problem. To minimize damage problems, values of a1 ?: 2 X 10 20 cm2 are desirable. The properties discussed above are demonstrated in Fig. 7.8. 1 3 The emis­ sion cross sections, radiative lifetimes, and saturation fluences are plotted for several Cr3 + -doped laser crystals. As seen in the figure, none of the typi­ cal samples used in this example have combined lifetime and cross section values that fall in the preferred range of values. One important observation from the data presented here is that the fluoride hosts generally come closer to achieving the desired lifetime-cross-section combination than the oxide hosts. This is due primarily to the smaller index of refraction of fluorides compared to oxides, since appears in the denominator of Eq. (7.2. 1 ) . The

n1

F luorescence li fetime

T1

(I' sec)

FIGURE 7.8. Properties of several Cr3 + -doped laser crystals and plots of Eq. (7.2. 1 ) (see text for explanation of theoretical lines). (After Ref. 1 3 . )

7.2. Spectral Properties of Cr3 + in Different Hosts

265

solid lines in the figure represent plots of Eq . (7.2. 1 ) using typical parameters of )q 750 nm and Llv 1 700 cm- 1 . The values of n for oxide and fluoride crystals were chosen to be 1 .85 and 1 .44, respectively. One method of im­ proving the possibility of obtaining the required combination of values for the lifetime and cross section is to use an anisotropic crystal. For this sit­ uation Eq. (7.2. 1 ) is rewritten separately for the components of the tran­ sition cross sections in the a and n polarization directions. For the equations the radiative lifetime of the metastable state must be replaced by the spon­ taneous emission rates for the specific polarization, Acr or A n . The radiative decay rate is the weighted sum of these two Einstein A coefficients, 1 1 (7.2.2) 3 A n + 32 Acr . r, Thus if a material can be found that has Acr « A n , Eq. (7 .2. 1 ) is increased by 3, nn . The broken lines in Fig. 7.8 represent plots of this expression using the same typical values given above. Therefore polarized laser operation in the n direction in an anisotropic material can be advantageous. From the discussion above, it appears to be advantageous to use aniso­ tropic fluoride crystals with Al 3+ sites as host materials for Cr3 + lasers. One important class of materials of this type includes LiCaAlF6 (LiCAF) and LiSrAlF6 (LiSAF) . 14 The crystal structure of these materials is shown in Fig. 7.9. The chromium ions substitute for the aluminum ions in this col­ quiriite structure. 1 5 This site has close to octahedral symmetry and thus the major features of the optical absorption and emission spectra are similar to those of other materials involving Cr3 + ions with sixfold coordination of ligands. However, the bond angles in the AlF6 cluster are slightly distorted from a perfect octahedron. The crystal-field contributions associated with

eca

®AI . LI

OF

FIGURE 7.9. Crystal structure of LiCAF and LiSAF (from Ref. 1 5).

266

7. Transition-Metal-Ion Laser Materials

FIGURE 7. 1 0. (A) Absorption and ( B) fluorescence spectra of LiSrAlF6 : Cr3 + . [Reprinted from Ref. 14(a), Payne et a!. , J. Lumin. 44, 1 67 ( 1 989) with per­ mission of Elsevier Science NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands].

.-.

8

es 0 2 ..-

ou

'7

4

lil s �0 4 c) 2 0 ;: Il

Absorption at T = 20 K

LiSrARF6 : Cr 3+

LISrARF6 : Cr 3+

Emission spectra T : 20K

650

\

(A)

LiSrAeF6 : Cr3+

(B)

800 750 Wavelength (nm)

these distortions, along with the strong coupling to specific lattice vibra­ tions, are responsible for the spectral details of these materials. The absorption and fluorescence spectra of LiSrAlF6 :Cr3 + are shown in Fig. 7. 10. 14 The major features of the absorption spectra are the broad bands associated with the transitions from the 4A 2 ground state to the 4T2 level ( peaked at 650 nm) and the 4T1 level ( peaked at 450 nm). Because of the uniaxial nature of the crystal field, the peak cross section for the former transition is over three times as large in the n(E I I c) spectrum than in the (J(E j_ c) spectrum, while the peak cross section of the latter transition is twice as large in the E j_ c spectrum than in the E II c spectrum. The sharp dips in the 4 T2 band are associated with interference of sharp line transitions to the 2£ and 2 T1 states. The fluorescence emission appears as a broad band associated with the 4T2 -- 4A 2 transition. It is peaked near 780 nm and shows a significant amount of vibronic structure at low temperature. The spectra of LiCaAlF6 :Cr3 + exhibit similar spectral characteristics with the bands shifted to slightly higher energies and the peak absorption cross sec­ tions about half as large as Cr-LiSAF. The vibrational fine structure is also more resolved in Cr-LiCAF than it is for Cr-LiSAF.

7.2. Spectral Properties of Cr3 + in Different Hosts

267

One of the physical mechanisms causing differences in spectra associated with vibronic transitions in different materials is the electron-phonon inter­ action that leads to lattice relaxation in the excited state. Using the simple single-configuration-coordinate model, the Huang-Rhys factor S deter­ mines the strength of the electron-phonon interaction and thus the offset of the excited-state parabola with respect to the ground-state parabola. As dis­ cussed in Chap. 4, the value of S can be determined from the ratio of the total intensities of the zero-phonon emission line and the vibronic sideband, exp( -S) /r(OO)/ fr(vibronic). In this case, the values of S for Cr-LiCAF and Cr-LiSAF are approximately 4.2 and 5.9, respectively. 14 Thus Cr­ LiSAF has a slightly stronger electron-phonon coupling interaction, leading to a greater relaxation of the fluorine nearest neighbors in the excited state, although Cr-LiCAF has a more pronounced vibronic spectral structure. Using a more detailed crystal-field analysis of the spectral properties of these materials shows that the observed features are associated with the de­ viation from a pure octahedral environment at the site of the Cr3 + ion with both static and dynamic lattice distortions making important contribu­ tions. 14 The major contribution to the odd-parity crystal field distortion of the CrF6 octahedron is a decrease in some bond angles and an increase in others, leading to a symmetry change that is characterized by a tzu repre­ sentation in the Oh point group. The operator describing this odd-parity potential in terms of symmetry characteristics is Vu

J3 ( tzuc; + tzu1'f + tzu() , c

(7.2.3)

where c is a normalization coefficient. The operator describing the dipole moment for the photon field (see Chap. 3) is written in terms of its a and n components as

(7.2.4) where a and b are constants. The products of the polarized dipole moment operators with the odd-parity potential operator is used as the interaction operator to determine the strengths of the transitions between the 4A z ground state and the 4T1 and 4Tz excited states. The transition matrix ele­ ments are calculated by using the single electron tz9 and e9 orbitals of the t� and t�e configurations and the tables in Griffith1 6 as described in Sec. 6.2 and in Chap. 3. The results show that the transition to the 4T2 level should be significantly stronger in the n-polarized spectrum than in the a-polarized spectrum, while the opposite is true for the transition to the 4 T1 level. This is consistent with the experimental observations for both Cr-LiCAF and Cr-LiSAF. However, the relative magnitudes of the polarized transition strengths and the differences between the two types of materials is not

268

7. Transition-Metal-Ion Laser Materials

explained by treating only the static odd-parity crystal-field contribution. Assuming that the measured oscillator strengths of the transitions can be expressed as simple sums of the static and dynamic contributions to the odd­ parity crystal-field distortions, the difference between the measured oscillator strengths and those predicted by the static contribution calculated above can be used to obtain estimates of the dynamic contributions to the transition strengths. The results show that the static contributions to the oscillator strengths are significantly larger in Cr-LiSAF than in Cr-LiCAF while the dynamic contributions are only slightly larger for Cr-LiSAF than for Cr­ LiCAF. Thus in Cr-LiCAF the static distortions are significantly greater than the dynamic contributions, whereas in Cr-LiSAF the dynamic distor­ tions make equal or greater contributions to the spectral transition strengths. The temperature dependences of the fluorescence lifetimes of Cr-LiSAF and Cr-LiCAF are shown in Fig. 7. 1 U4 These results can again explained by considering the effects of the static and dynamic odd-parity crystal-field distortions discussed above. Recalling from Chap. 3 that the fluorescence lifetime is related to the integrated absorption cross section by r

1 'Y! �_2 9e -8ncn2 g9 f (Ja dv qe

(7.2.5)

--

where 9e and g9 are the degeneracies of the excited and ground states, respectively, A is the peak wavelength of the transition in air, 'Y/qe is the quantum efficiency, and the cross section must be averaged over all polar­ ization directions. Using the measured integrated absorption cross sections in Eq. (7.2.5) predicts the fluorescence lifetimes for Cr-LiSAF and Cr-LiCAF to be 1 56 and 280 f.1S, respectively. These values are greater than the mea­ sured low-temperature fluorescence lifetimes of 67 and 250 f.1S. This may be due to the fact that the trigonal distortion splits the 4T2 level and emission occurs from only the lowest component at low temperatures, whereas the Emission lifetime data

FIGURE 7. 1 1 . Temperature de­ pendences of the fluorescence lifetimes of Cr-LiSAF and Cr­ LiCAF. [Reprinted from Ref. 14(a), Payne et a!. , J. Lumin. 44, 1 67 ( 1 989) with permission of Elsevier Science NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.]

-; �

�

e

•

•

1 50

6

: 100

w

/

USrARF1: er 3+

Tempentun (KJ

•

7.2. Spectral Properties of Cr3 + in Different Hosts

269

integrated absorption cross section includes transitions to all of the un­ resolved components. This is consistent with the fact that the discrepancy is greatest for Cr-LiSAF where the trigonal distortion is greatest. The lack of temperature dependence for r in Cr-LiSAF implies that the fluorescence decay rate of 12 890 s- 1 is dominated by the static contribu­ tion to the trigonal distortion. For Cr-LiCAF there is a weak temperature dependence of r with the static distortion contribution given by the low­ temperature value of 2960 s- 1 • The additional contribution to the decay rate for this material at high temperatures is associated with the dynamic contribu­ tion, r (kstat + kdyn) - 1 . From the expression for the strong electron-phonon coupling-induced radiative decay rate derived in Chap. 4, the dynamic con­ tribution to the decay rate can be expressed in terms of a constant k�yn and the phonon energy of the odd-parity mode causing the distortion, hw, kdyn

( )

hw 0 kd n coth y kB T .

(7.2.6)

Using a typical value for the phonon energy of 310 cm 1 , the best fit to the Cr-LiCAF lifetime data gives k�yn 2040 s- 1 . Note that this gives a value for the dynamic contribution to the decay rate that is of the same order as the static contribution for Cr-LiCAF but is significantly less than the static contribution to the decay rate of Cr-LiSAF. The fine structure in the Cr-LiCAF spectra has been analyzed in terms of spin-orbit coupling. 14 Four zero-phonon lines are observed in the emission spectra at low temperatures consistent with group-theory prediction of the splitting of the 4T2 level. The magnitude of the splitting of these lines is sig­ nificantly less than that predicted by the simple spin-orbit interaction, due to the consequences of the Jahn-Teller effect as described in Sec. 4.7. If Ga is substituted for Al in LiSAF, the crystal becomes LiSrGaF6 ( LiSGF ) . This has the same crystal structure as LiSAF, but Cr3 + ions now substitute for Ga3 + ions instead of Al3 + . This results in a slightly lower emission cross section, slightly longer emission lifetime (88 ,us), and a small shift to higher energy of the peak in the emission band. 1 7 Thus changing the substitutional site from Al3 + to Ga3 + makes a much smaller difference in spectral characteristics than a change in cations from s�+ to Ca2+ . This is probably due to the lattice relaxation that occurs around the Cr3 + ions that compensates for the size differences between host and dopant ions. 1 6 It is interesting to compare the spectral properties described above for Cr3 + in LiSAF with Cr3 +-doped Na3 GazLhF 1 2 , which has a garnet struc­ ture and is referred1 8 to as GFG. The chromium ion substitutes for the gallium ion, which is in a site of octahedral symmetry. The point-group symmetry is very close to Oh although there is a slight distortion to C3 ;. Be­ cause of this inversion-site symmetry, the fluorescence lifetime of GFG is very long (3 10 ,us) compared to Cr-LiSAF, and the peak0 emission cross sections is about an order of magnitude smaller (0.6 x 102 cm2 ) . The fluo-

270

7. Transition-Metal-Ion Laser Materials

rescence lifetime decreases as temperature is increased while the quantum efficiency remains unchanged. This implies that the radiative transition is enabled by coupling to odd-parity vibrational modes and radiationless decay processes are not effective in quenching the fluorescence lifetime. The gain and slope efficiency of laser operation are lower than theoretically pre­ dicted, and this discrepancy is attributed to excited state absorption. 7.3

Transition-Metal Ions and Host Crystals

The chromium-doped laser crystals listed in the preceding section demon­ strate the wide variety of types of host crystals available for transition­ metal-ion lasers. Research continues on developing new materials with the ideal mechanical strength and thermal properties required for laser applica­ tions. This includes improved methods of crystal growth leading to opti­ mum solubility and distribution of dopant ions and minimum defects that produce scattering and absorption losses. Some of these materials have new types of crystal structures while others are isomorphic variations of known laser host materials. Even relatively small changes in lattice spacing or ionic radii can produce significant changes in crystal-field strength or ion-ion interaction, leading to different spectral properties, improved quantum effi­ ciencies, and decreased ESA. By choosing the appropriate combination of dopant ion and host crystal, it is possible to tailor-make a laser material with characteristics desirable for specific applications. The transition-metal ions that have been made to lase in addition to Cr3 + are Ti3 + ' y2+ ' Cr4+ ' c.-2+ ' Mn5+ ' Fe2+ ' Co2+ ' and Ni2+ . The first three of these ions have laser emissions that overlap the spectral region covered by Cr3 + materials and extend it slightly further into the infrared, while the last five of these ions have laser emissions at significantly lower energies than Cr3 + materials. Some of the most important transition-metal-ion laser materials and their properties1 9 are listed in Table 7.2. The electronic configurations of the eight different first-row transition­ metal ions of interest all involve 1 8 electrons in a filled core ending with six electrons in 3p orbitals. They differ in the number of electrons they have in the outermost 3d orbitals. The electronic terms and multiplets are de­ termined through the Russell-Saunders coupling approach for the number of equivalent electrons with n 3 and I 2 as was done for two electrons in Sec. 2.2 and for three electrons in Sec. 6. 1 . Using the single-electron states, the energies of the terms can be expressed in terms of Coulomb and exchange integrals and then written in terms of Slater-Condon or Racah parameters. The Racah parameters for all of the 3dn ions are listed in Table 6.2. The results of this analysis gives the free-ion energy-level diagram for the transition-metal ions of interest. Then using crystal-field theory in either the strong-field or weak-field limit, the energy levels of the ions in the host material can be determined. As was done for Cr3 + ions in the last chapter, it

7.3. Transition-Metal Ions and Host Crystals

27 1

TABLE 7.2. Properties of transition-metal-ion laser materials ( Ref. 1 9) . Ion 3+

Ti (3d 1 ) Cr4+ (3d2 ) Mn5 + (3d2 ) y2 + (3d3 ) cr + (3d4 ) Fe2 + (3d6 ) Co2 + (3d7 ) Ne+ (3d8 )

Host Ajz03 BeAiz04 YA103 Mg2 Si04 Y3Als 0 1 2 Ba3 (V04h MgF2 (80 K) CsCaF3 (80 K) ZnSe n InP(2K) KMgF3 (80 K) KZnF3 (80 K) MgF2 KMgF3 (80 K) MgF2 (80 K) Mg0 (80 K) CaY2 Mg2 Ge3 0 1 2 (80 K)

A. (tmi)

0.66 1 .06 0.73 0.95 0.6 1 1 6 1 . 1 67 1 .345 1 .2 1 .7 1.181 1 .07 1 . 1 6 2.0 2.8 1 .24 1 .33 3.53 1 .62 1 .90 1 .65 2. 1 5 1 .5 2.3 1 .59 1 .61 1 .74 1 .3 1 8 1 .46

7:(J.LS )

3.15 4.9 14 15 3.6 430 2 400 2 500 8

11

36.5

1 1 000

2 200

is common to begin by assuming Oh symmetry for the crystal field and treat any deviations from this situation as a perturbation. As described in Sec. 6.2, this gives Tanabe-Sugano diagrams of the energy levels in an octahe­ dral environment as a function of crystal-field strength. Figure 7. 12 shows the simplified Tanabe-Sugano diagrams for the electronic configurations of interest. 20 These can be used to explain the gross features of the absorp­ tion and fluorescence characteristics of 3dn ions in host crystals. Lower­ symmetry crystal-field contributions, the spin-orbit interaction, and vibronic coupling all can play important roles in determining the details of the optical spectra and transition strengths of these ions. The properties of several of the most important laser materials of this type are described below. In general the crystal-field strength Dq has an order of magnitude for tri­ valent transition-metal ions that is twice as large as for divalent ions for the same ligands. The crystal-field strengths for the major lasing ions can be ordered as Ni2+ < Co2+ < Fe2+ < y2+ < Cr3 + . The variation of Dq with different ligands around the same metal ion can be ordered as I < Br < Cl < S < F < 0 < N < C. The variations in the magnitude of Dq for different ligands are of the order of 100 em- 1 •

7.4

Laser Materials Based on Ti 3 + Ions

Trivalent titanium has a 3d 1 electron configuration. As was discussed in Chaps. 2 and 6, a single d electron in an octahedral crystal field splits into

Dq/B

2T 2

2E

'F

'G

'P '0

s

�

-

30

50

0

1

2

Dq/B

' A , (e ' )

3

' E (e 2 )

4

... 3T , (t �)

' E (t j ) ' T 2 (t j ) )

' A , (t j)

,

' T 2 (t 2e)

' T , (t 2e)

3A ' (e 2 )

FIGURE 7. 1 2. Simplified Tanabe-Sugano diagrams for ions with 3dn electron configurations [from Ref. 20(b)].

�

d'

·

70

N

&:

..

"" S" ::1 . e:. Cl

�

..

Cl

(1)

t" ""

0 ::

..

� -

� (1)

I

0 ::

::

�

-.

N

-.

'F

'F

�

10

30

50

70

d'

1

Dq!B

3

.,

'A , (t j)

'I

'F

�

FIGURE 7 . 1 2 (continued)

4

' E (t l)

'T , (t l)

4T , (t �e )

2 A , (t �e )

30

70

0

1

'A,

DqiB

2

'A ,

I

3

4

'T

,

'E 'T ' 'E

' A,

'A 2

'T , 'A ,

'T ,

g.

§:

N -.} w

"'

n

Q "'

::I: 0 "' -

Il = 0.

"'

0 =

..

s: (I)

§:

'

=

i!l .

=

�

w

;-

__

50

__

'5 --

•G

•o

•F

70

0

d•

1

Dq/8

2

3

I I I

�

--

50

• o __

'I

' F --

FIGURE 7. 1 2 (continued)

4

• T , (t�o) 6A , (t �o 2 )

•r ,

'E

•A : . • e

'F

70

0

d6

1

2

Dq/8

3A 2 1 A 2 3A ,

3

'A,

,

'E

4

' T,

'E

N

¥-

:: .

�

�

Il C1>

t" Il "' C1> ..

0 =

..

-

C1>

�

t:t. 0 ::s

Il ::s

�.

;.. ..

-. .,.

10

30

' F --

�

0

1

2

Dq/B

2A 2

3

4A 2 ,

4

,

2E

'T

�

•s

40

FIGURE 7. 1 2 (continued)

'A

0

1

Dq/B

2

3

'T ,

4

'A 2

'E

'T 2

'T ,

w

N -. VI

.. a "'

� "'

::I: "' .. \.l

0

I» = Q.

= "'

0

("p .. a -

�

g. =

�f!l .

..

-.

276

7. Transition-Metal-Ion Laser Materials

two levels, giving 2£2 and 2T2 terms. As shown in the Tanabe-Sugano dia­ gram in Fig. 7. 1 2, this leads to only one d-to-d transition in absorption or emission. Since this transition involves a change in crystal-field config­ uration, its energy is very sensitive to the strength of the crystal field and therefore the transition results in broad spectral bands. The width of the absorption band is favorable for efficient absorption of pump radiation, while the width of the emission band provides broadly tunable laser oper­ ation. The absence of additional excited states of the 3d 1 configuration minimizes the effects of ESA. As seen from Table 7.2, the fluorescence life­ time of titanium-doped crystals is only a few microseconds. This is mostly associated with radiative decay and thus results in a high stimulated emis­ sion cross section, leading a to low lasing threshold. However, it also makes flashlamp pumping difficult and limits the energy storage capacity of the materials due to amplified spontaneous emission. One of the major prob­ lems encountered with Ti 3+ laser materials is valance state stability. There is a strong tendency to form Ti2+ -Ti4+ pairs that reduce laser performance. Residual defect absorption in the lasing band has also been a signifi­ cant problem. A detailed description of the spectroscopic properties of Ah03 : Ti 3 + crystals is given below, and then these properties are compared to those of Ti3 + in other host materials. The room-temperature absorption and emission spectra of Ti-sapphire are shown in Fig. 7. 1 3 along with the temperature dependence of the fluo­ rescence lifetime. 2 1 The unresolved double bands in absorption are asso­ ciated with Jahn-Teller splitting of the 2E level as discussed below. The broad emission bandwidth associated with strong vibronic coupling provides excellent tunability for laser emission and the ability to obtain ultrafast pulses from mode-locked operation. The 2 T2 ground state is split by the spin-orbit interaction and the crystal-field effects of the deviation of the site symmetry from cubic as discussed in the previous chapter considering Cr3 + in sapphire. The inset in the figure shows the low-temperature emission spectrum where the zero-phonon lines resulting from transitions to the three split components of the ground-state manifold can be resolved. The two lines highest in energy are labeled R 1 and R 2 analogous to the R lines in ruby. The lowest-energy line is mixed in with the vibronic transitions on the edge of the broad emission band. Figure 7. 14 shows the energy-level dia­ gram for Ah0 3 : Ti3 + with the observed ground-state splittings. As temperature is varied, the spectroscopic properties of Ah0 3 : Ti3 + change significantly. As seen in Fig. 7. 1 3(C), the fluorescence lifetime de­ creases with increasing temperature above about 300 K. The total integrated emission intensity exhibits a similar decrease in this temperature range. These properties are consistent with having a quantum efficiency close to unity at very low temperatures and a quenching of the quantum efficiency above 200 K due to the increasing probability of nonradiative decay pro­ cesses. A model for describing this is discussed below. The shape of the broad emission band also changes with temperature and the full width at

FiGURE 7. 1 3. Spectroscopic properties of Ah0 3 : Ti3 + (after Ref. 2 1 ) . (A) Absorp­ tion at room temperature. ( B) Fluorescence emission at 10 K. (C) Temperature de­ pendence of the fluorescence lifetime.

0.5

A(IJ.m)

0.6

0 .7

(A)

x 400

c ::

.. ii

::

630

640

800

II( n m )

(B)

1

0

10 200

300

T(K)

400

(C)

278

7. Transition-Metal-Ion Laser Materials E ll_l--

2E

20

FREE ION

/: H� CTA

NAL

S.O. & J.T. INTERACTIONS

t E1f� --

2J?;--/

��2;-

FIGURE 7 . 1 4. Energy-level diagram for Ah0 3 : Ti 3 + .

half maximum becomes greater at high temperatures. In addition, the zero­ phonon lines exhibit temperature-dependent behavior. They broaden, shift in position, and decrease in intensity as temperature is increased. The temperature-dependent line broadenings and intensity decreases of the R 1 and R 2 lines are shown in Fig. 7. 1 5. The shifts in peak positions are not shown because the interference of the broad band masks the details of this spectral property. The theoretical explanations for these properties are given below. The temperature dependences of the linewidths of the R lines can be in­ terpreted using the theory derived in Chap. 4 and applied to ruby in Chap.6. The temperature-independent contribution observed as the linewidth at very low temperatures ( T < 20 K) is dominated by inhomogeneous broadening associated with microscopic strains in the crystal. The amount of inhomo­ geneous broadening varies from crystal to crystal depending on material quality, but is generally of the order of a few wave numbers for both the Rt and the R 2 lines. Above about 60 K the linewidths increase proportional to T7 as predicted if the dominant broadening mechanism is the Raman scat­ tering of phonons. Between about 20 and 60 K there is a weaker temper­ ature dependence for the line broadening. This can be described by the equation

(7.4. 1 )

where kn is Boltzmann's constant, Avo is the low-temperature linewidth due to inhomogeneous broadening and spontaneous phonon-emission processes, and p is the electron-phonon coupling parameter for the process. This equation describes the contribution to the linewidth associated with the stimulated emission or absorption of phonons of energy AE, as discussed in Sec. 4.2. The solid line in Fig. 7. 1 5(A) represents the best fit to the

7.4. Laser Materials Based on Ti 3 + Ions

279

12 10

;;? a

' I I I


.

6

�6

I I I I I

4 2

T(K)

(A)

T(K)

(B)

FIGURE 7. 1 5. Temperature dependences of the (A) linewidths and ( B) integrated fluo­ rescence intensities of the R lines in Ah 0 3 : Ti 3 + (after Ref. 2 1 ) .

data treating !.iE and p as adjustable parameters. This gives a value of !.iE 38 cm- 1 , which is the same as the spin-orbit splitting of the ground state. Thus, the mechanism for line broadening of the zero-phonon lines at low temperatures appears to be direct phonon processes connecting the two lowest spin-orbit split components of the ground-state manifold. The decrease in intensity of the R 1 and Rz lines as temperature is raised can be explained by an equation of the form (7.4.2)

which describes the fluorescence emission from a metastable state including nonradiative coupling to a state lying above it energy by an amount !.iE. The solid lines in Fig. 7. 1 5( B) are the best fits to the data using this ex­ pression and treating A, B, and !.iE as adjustable parameters. This proce­ dure gives values for !.iE of 125 cm- 1 for R 1 and 83 cm- 1 for R 2 . These values represent the additional energy required for vibronic emission and the largest value is consistent with the distance between the R 1 line and the beginning of the vibronic sideband seen in the spectrum. The difference be­ tween the values found for the two zero-phonon lines is associated with dif­ ferences in the Frank-Condon overlap factors for transitions to the different levels of the spin-orbit split ground-state manifold.

280

7. Transition-Metal-Ion Laser Materials

The temperature dependence of the fluorescence lifetime can be inter­ preted using the expression p (7.4.3) r 1 r, 1 + eAE/kB T - 1 ' where r;:- 1 represents the radiative decay rate of fluorescing level and the second term describes radiationless quenching due to phonon absorption to a higher level. p is the rate constant for the radiationless transition and 1'1E is the potential barrier for the process. The best fit of Eq. (7.4.3) to the measured decrease in the fluorescence lifetime with increasing temperature is shown as a solid line in Fig. 7.1 3(C). The adjustable parameters used to ob­ tain this fit are 3 JlS for the radiative lifetime, a radiationless quenching con­ stant of 2.62 x 106 fl.S- 1 , and a potential barrier of 4000 cm- 1 . As described below, the offsets of the two different components of the Jahn-Teller poten­ tial wells in the excited state place their minima at different positions with respect to the ground-state potential-well minima. This leads to one Jahn­ Teller component that has a high probability of radiative emission and another component that has a high component of nonradiative decay. As temperature is raised there will be thermal activation from the strong fluorescing level to the weak fluorescing level, thus quenching the observed fluorescence intensity and lifetime. The measured activation energy for quenching will be associated with both the process of thermal crossover from one Jahn-Teller component to the other and the crossover from the weak fluorescing potential well to the ground-state potential well. The mea­ sured value of 4000 cm- 1 is slightly higher than the Jahn-Teller energy of 3072 cm - 1 determined from spectroscopic characteristics. This indicates that most of the activation energy for quenching is associated with the crossover between Jahn-Teller components but that some additional energy is re­ quired to accomplish the second crossover to the ground-state potential well. Figure 7. 1 6 illustrates the configuration-coordinate model required to de­ scribe the spectroscopic properties of Ah0 3 : Ti3 + . The single d-electron configuration produces electronic energy levels that are strongly coupled to the lattice, leading to an excited-state potential well whose minimum is shifted compared to that of the ground-state potential well, leading to a large Stokes shift between absorption and emission bands. Since both the ground and excited states exhibit electronic degeneracy, the strong coupling with the lattice allows the degeneracy to be lifted through vibronic coupling in accordance with the Jahn-Teller theorem discussed in Sec. 4.7. The vibrational modes transforming as E9 play the most important role in the vibronic coupling. Thus the configuration coordinates shown in Fig. 7. 1 6 for the 2E excited state correspond to the potential-energy curve shown in Fig. 4. 1 3(A). The ground-state 2 Tz level is also split by the Jahn-Teller effect as shown in Fig. 4. 1 3 ( B) and discussed in Chap. 4. However, in the case of Ah0 3 : Ti3 + the trigonal crystal field also partially lifts the degeneracy of

7.4. Laser Materials Based on Ti3 + Ions

28 1

Q FIGURE 7 . 1 6. Configuration-coordinate model for explaining the spectroscopic prop­ erties of Ah0 3 : Ti3 + .

the 2 Tz level and the Jahn-Teller effect in the ground state does not play as important a role in determining the spectroscopic properties of the system as it does in the excited state. From the discussion above leading to the configuration-coordinate model in Fig. 7 . 1 6, the spectroscopic properties of A}z0 3 : Ti3 + at low temperatures can be understood. The two unresolved broad bands seen in the absorption spectra shown in Fig. 7. 1 3(A) are associated with transitions from the ground-state potential minimum to the two Jahn-Teller split components of the 2E excited-state potential well. The ground-state splitting is too small to affect the absorption transitions. The absorption transitions appear as broad bands because the minima of the excited-state potential curves are offset from the ground-state minimum resulting in vibronic transitions. The mea­ sured properties of the absorption spectrum can be used in conjunction with the above equations to determine the Jahn-Teller energy of the excited state. The energy difference of the two absorption peaks is given by llE- era , where ra is the configuration-coordinate value for one of the ground-state potential minima. The sum of the energies of the two absorp­ tion peaks is given by f:lE+ 2Ecr + kr�/2, where Ecr is the crystal-field splitting between the ground and excited state in the absence of any Jahn­ Teller interaction. Finally, the energy of the zero-phonon transition is Ezp Ecr - EJT. Combining these three expressions, the Jahn-Teller energy of the 2E level can be found from the properties of the absorption spectrum through

(7.4 .4)

28 2

7. Transition-Metal-Ion Laser Materials

Note that the highest-energy zero-phonon line is used for this calculation since the various processes leading to the ground-state splitting have been neglected. The fluorescence emission is associated with transitions from the lowest vibrational levels of the metastable state originating at the minimum of the cylindrically symmetric potential well. Again the different offsets of the potential minima of the ground and excited states cause the terminal level of the transition to be an upper vibrational component of the ground state. Since the ground state has three paraboloids with their own minima, it is not cylindrically symmetric and this causes the emission band to be very broad. This can be seen in Figs. 7.13(B) and 7. 16. In addition, the ground-state splitting leads to three zero-phonon lines with slightly different energies as seen in Fig. 7. 13(B). The spectroscopic properties of Ti3 + in other oxide crystal hosts are sim­ ilar to those described above for Ti-sapphire. The absorption peak, emission peak, and zero-phonon line all shift to higher energies for hosts such as YAl03 with a higher crystal-field strength, and shift to lower energies for garnet hosts with lower crystal-field strengths as expected. The crossover energy between the excited-state and ground-state configuration-coordinate potentials that is critical to nonradiative relaxation processes also varies di­ rectly with crystal-field strength. However, the Jahn-Teller energy of the 2E level is not directly correlated with the crystal-field strength. In general there are no special advantages for using hosts other than sapphire for solid-state lasers based on Ti3 + ions. 7.5

Laser Materials Based on Ions with 3d

2

Configurations

Several transition-metal ions have a 3d 2 configuration for their optically ac­ tive electrons including v3+, Cr4+ , and Mn5+ . The latter two ions have been made to lase in host crystals where they are in sites with tetrahedral coordi­ nation with their ligands. Note that the tetrahedral crystal-field environment leads to a different energy-level structure than that shown in the Tanabe­ Sugano diagram for 3d 2 ions in an octahedral environment shown in Fig. 7. 12. The correct crystal-field splittings are found from applying the tech­ niques described in Sec. 2.3 to d electrons in a crystal field with tetrahedral symmetry instead of octahedral symmetry. Weak crystal-field systems again lead to broadly tunable laser output, whereas strong-field systems provide sharp line emission. The major interest in solid-state lasers based on these ions is their ability to generate laser emission in the near-infrared spectral region as listed in Table 7.2. The first host material to exhibit laser emission with a 3d 2 ion was tetra­ valent chromium doped forsterite22 (Mg2 Si04 ) . The most promising host crystal for Cr4+ lasers appears to be Y3 Al5 0 1 2 ( YAG).23 Since chromium

7.5. Laser Materials Based on Ions with 3d 2 Configurations

283

generally prefers being in the trivalent state and having octahedral coordi­ nation, significant research effort has been expended in identifying the lasing center in these materials. The effects of having ions in multiple valence states and occupying different types of crystal-field sites as well as perturbations from defect centers limits the complete understanding of the spectroscopic properties of these materials. A detailed model describing the defect chem­ istry of the relative concentrations of fourfold coordinated Cr4+ and sixfold coordinated Cr3 + centers in YAG has been proposed. 23 The relative amounts of each of these types of centers depends on crystal-growth con­ ditions and post-growth treatments, and thus the optical properties of these materials can vary significantly from sample to sample. These tetravalent chromium systems involve weak crystal fields resulting in broad vibronic emission bands. Yttrium aluminum garnet is an excellent host for rare-earth-ion lasers and the crystallographic properties of this material are described in Chap. 8 and the relevant part of the crystal structure is shown in Fig. 8.4. The Cr4+ ions replace Al 3+ ions in a site with four oxygen ions as ligands. The resulting tetrahedron is stretched along the s4 symmetry axis giving a site symmetry of D2d. The required change compensation is generally proveded by another substitutional ion such as Mg2+ and is nonlocal. The crystal-field splitting of the free-ion terms of the 3d 2 -electron configuration in a site with D2d sym­ metry leads to the energy-level diagram shown in Fig. 7. 1 7. The allowed electric dipole transitions are also shown in this figure. For this symmetry group, the x and y components of the electric dipole vector operator trans­ form according to the E representation and its z component transforms ac­ cording to the B2 representation. With the z direction chosen parallel to the 3E 3A2

3A2

3E 3B2 3Bl

El/x,y

!:RYSTAL FIELD

E//z

FIGURE 7. 1 7. Energy-level diagram for an ion with a 3d2 -electron configuration in a crystal-field site with D2d symmetry. Allowed electric dipole transitions are shown (after Ref. 23) .

284

7. Transition-Metal-Ion Laser Materials

[001] crystallographic axis, there are three different orientations for these optically active centers. Thus the use of polarized excitation can lead to selective absorption by different types of centers and result in polarized emission spectra. This is demonstrated in Fig. 7. 18( B). By comparing the observed spectra with the energy-level diagram and predicted selection rules it is possible to assign specific spectral features to transitions. For example,

9

10

v(l03 em· ' )

11

(A)

6

v(103 em ' )

8

9

(B)

FIGURE 7. 1 8. Absorption and emission spectra of YAG : Cr4+ . (A) Absorption spec­ trum at 5 K. ( B) Fluorescence spectra at 77 K (curves I and 2) and 300 K (curves 3 and 4) for excitation polarization parallel to the [001] crystallographic direction and the fluorescence polarized either parallel (curves I and 3) or perpendicular (curves 2 and 4) to the [00 1 ] axis. [Reprinted from Ref. 23, Okhrimchuk and Shestakov, Opt. Mat 3, I ( 1 994) with permission of Elsevier Science NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.]

7.5. Laser Materials Based on Ions with 3d 2 Configurations

285

the transitions between the ground state 3B 1 and lowest excited states 3B2 and 3E produce the absorption band observed in the 8000-1 3 000-cm- 1 re­ gion (0.77- 1 .25 J.If) and the fluorescence band in the 6000-9000-cm- 1 region ( 1 . 1 7-1 .67 pm) . The two higher absorption bands at 14 000-26 000 cm- 1 (385-774 nm) and 29 000-38 000 cm- 1 (263-345 nm) are associated with transitions from the ground state to the two sets of 3A 2 , 3E level split from the free-ion 3F and 3P terms, respectively, as shown in Fig. 7. 1 7. The sharp line features in the low-temperature spectra shown in Fig. 7 . 1 8(A) are the zero-phonon lines and these can be used to determine the positions of the energy levels involved in these transitions. From these results, 23 the crystal­ field strength was found to be Dq 885 cm- 1 • As temperature is increased from 77 to 400 K, there is a continual de­ crease in the integrated fluorescence intensity and a shortening of the fluo­ rescence lifetime of YAG : Cr4+ emission while the strength of the absorp­ tion band remains constant. 23 The thermal effects on the spectral dynamics of this system are explained by a model that assumes that the nonradiative processes between the 3B2 and 3EeT2 ) levels (see Fig. 7. 1 7) are fast enough that a Boltzmann population distribution is established in these two levels in a time short compared to the fluorescence lifetime. The total emission rate is the sum of the radiative and nonradiative transitions from this set of coupled levels to the 3B 1 ground state. The nonradiative decay rate is given by the Huang-Rhys expression written as Eq. (4.6.33) . The radiative decay rate is a weighed sum of the radiative decay rates of the two coupled excited states with the weighting factors determined by the relative populations of the two states. In addition, a phonon coupling factor must be included to account for the dynamic removal of the symmetry-forbidden nature of the transition. This model can provide a good fit to the measured experimental results. 23 At room temperature and above the thermal characteristics are dominated by the nonradiative decay process with a Huang-Rhys factor of about So 0.9 and an effective phonon energy of about 435 cm- 1 , which is close to an intense phonon line observed in the infrared spectrum. At low temperatures the phonon-assisted radiative rates appear to make a non­ negligible contribution to the spectral dynamics. From these results a value for the stimulated emission cross section of 8 x I Q 1 9 cm2 was determined.23 This is less than twice as large as the excited-state absorption cross section at the lasing wavelength. Thus, excited-state absorption is a serious problem in this laser material. These thermal effects limit the quantum efficiency at room temperature to only about 1 1% with a fluorescence lifetime of 3.6 ps. In addition, it should be noted that chromium dopant concentrations greater than about 1 0 1 9 cm- 3 result in new spectral features and a decrease in the fluorescence lifetime due to the introduction of new types of optically active centers and ion-ion interaction. 23 All of the general spectral charac­ teristics described here for YAG : Cr4+ are similar for other tetravalent chromium based materials such as forsterite. Another 3d 2 ion of interest is Mn5 + . Laser emission of this ion has been observed in a Ba3 (V04 h host crystal.24 This material has a stronger crystal

286

7. Transition-Metal-Ion Laser Materials

field than the chromium-based systems discussed above and thus the 1E state is below the 3 T2 state in the tetrahedral environment. This means that the lowest-energy transition between a metastable state and the ground state is spin-forbidden and thus will have a long fluorescence lifetime. The Mn5 + ions in this material substitute for y5 + ions and thus and thus require no charge compensation. The crystallographic site is tetrahedrally coordinated with oxygen ligands and has a point-group symmetry of C3 v· The absorption and emission spectra of Ba3 ( V04 h : Mn5 + are shown in Fig. 7. 1 9. The transition between the 1E metastable state and the 3A 2 ground state can be seen as a sharp line in both spectra. The strong absorption bands between 600 and 800 nm are attributed to symmetry-allowed tran­ sitions from the ground state to the 3 T1 excited state, while the weaker bands T

�

:

JOO K

O.J

0.2

0. 1

/. ...

OJ I

. . .. .

0.0 < 0>

WAV E L E N G T H

(A)

L<0

< <>

< >0

< "

(J'.m )

296 K

wavelength (nm)

(B)

FIGURE 7. 1 9. Absorption and emission spectra of Ba3 (V04 h : Mn5 + with 0.25 at. % Mn. (A) Absorption spectrum at 300 K. ( B) Fluorescence spectra at room tem­ perature and low temperature (after Ref. 24) .

7.6. Laser Materials Based on Ions with 3d3 Through 3d8 Configurations

287

extending from about 800 to 1000 nm are attributed to transitions to the 3 T2 state that are forbidden in tetrahedral symmetry. 24 The fluorescence emis­ sion exhibits a single exponential decay with a lifetime that decreases from about 1 . 1 ms at 1 1 K to about 0.43 ms at room temperature. This temper­ ature quenching of the lifetime is much smaller than the quenching observed in tetravalent chromium materials indicating a weaker electron-phonon coupling in the metastable state. From this spectral information, the peak stimulated emission cross section is estimated to be of the order of w - 19 cm2 . This material operates as a three-level laser system on the 1E -- 3A 2 transition at 1 1 8 1 nm. The pump bands are favorable for either diode laser pumping or efficient flashlamp pumping. The demonstration of this laser system suggests the possibility of Mn5 + lasing in other similar oxyvanadate or oxyphosphate crystals that may provide stronger crystal­ field environments and stronger electron-phonon coupling leading to four­ level laser operation with broad-band emission. Laser Materials Based on Ions with 3d 3

Through 3d8 Configurations

7.6

Divalent cobalt has a 3d7 -electron configuration. As can be seen from the Tanabe-Sugano diagram, this provides a situation in which the ground state changes from a 4T1 level in the weak crystal-field regime to a 2E level in the strong crystal-field regime. The free-ion value of the Racah B parameter is 971 cm -1 . The host materials for Co2+ lasers are fluoride crystals that provide a weak crystal-field environment with laser emission based on the 4T2 -- 4T1 transition. Since this transition involves an electron changing from an e to a t2 single-electron crystal-field orbital, it is strongly coupled to the lattice and provides a broad band for tunable laser emission in the near­ infrared spectral region. There is one strong absorption band in the visible region of the spectrum due to the 4T1 -- 4A 2 transition. The spectral prop­ erties of MgF2 : Co2+ are shown25 in Fig. 7.20. The 4T1 ground state is split by noncubic crystal field contributions and the spin-orbit interaction into six levels and the zero-phonon lines associated with transitions terminating on these levels can be resolved at low temperatures. The emission is a mix­ ture of magnetic dipole transitions and vibrationally induced forced electric dipole transitions. The fluorescence lifetime decreases with increasing tem­ perature as shown in Fig. 7.20(C). This can be attributed to the increasing rate for nonradiative decay from the metastable state. The decreased quan­ tum efficiency makes lasing difficult at room temperature and thus many Co2+ lasers only operate at low temperatures. One positive aspect of Co 2+ laser materials is that ESA does not appear to decrease the effective cross section for laser operation or limit the tuning range for laser emission. Divalent vanadium has a 3d3 electron configuration and thus is associated

WAV E L E N GTH (nm)

(A) ZERO-PHONON LINES

�"' z

�

�

w u z w u "' w a: 0

3 ..

o;

�

2.8

(B)

1 .0

w

::E i=

w u.

:J

0. 1 0

T E M P E R ATU R E (K)

(C)

FIGURE 7.20. Spectroscopic properties of MgF2 : Co2+ . (A) Absorption at 300 K. ( B) Fluorescence emission. (C) Temperature dependence of the fluorescence lifetime. [Reprinted from Ref. 25, Laser Handbook, Moulton, p. 203 ( 1 985) with permis­ sion of Elservier Science NL, Sara Burgerhartstraat 25, 1 055 KV Amsterdam, The Netherlands]. 288

7.6. Laser Materials Based on Ions with 3d3 Through 3d8 Configurations

289

with the same Tanabe-Sugano diagram as Cr3 + for describing its energy levels in an octahedral crystal field. The lower charge on the divalent ion generally puts V2+ in the low crystal field regime compared to trivalent chromium ions in the same host. This shifts the absorption and emission spectra to lower energies and enhances nonradiative decay rates. The free­ ion Racah B parameter for V2+ is 755 cm- 1 , which is less than the value of B for Cr3 + . Laser action has been observed in fluoride hosts at low temper­ atures. For MgF2 : V2+ the wavelength of laser emission26 is peaked near 1 . 12 �tm. It is associated with the broad fluorescence band of the 4T2 -- 4A 2 transition. For this host the V2+ ion is at a site with inversion symmetry and thus the vibronic emission is dominated by phonon-induced electric dipole transitions. The fluorescence lifetime decreases sharply as temperature is raised above 200 K due to the increasing probability of radiationless decay from the 4T2 level. This material has the same two broad absorption bands seen in the spectra of trivalent chromium materials. This provides the op­ portunity for efficient flashlamp pumping. It has been found that severe problems with ESA limit the extent of the tuning range and the effective cross section for laser operation26 of MgF2 : V2+ . This is associated with the 4T2 __ 2 T1 transition, with will cause similar problems for y2 + in other host crystals with different crystal field strengths. Divalent nickel has a 3d8-electron configuration. The Tanabe-Sugano di­ agram for this configuration is similar to that of the 3d3 configuration and thus the spectral properties of Ni2+ are similar to those of V2+ . The free ion Racah B parameter for divalent nickel is 1030 cm- 1 . Figure 7.21 shows the absorption spectrum, the fluorescence emission, and the temperature de­ pendence of the fluorescence lifetime of Ni2+ in both oxide and fluoride host crystals. 25 There are three absorption bands spread out through the visible and near-infrared spectral regions. Co-doping with Mn2+ enhances pumping efficiency through providing additional pump bands in the visible spectral region and effective energy transfer from Mn2+ to Ni2+ ions. Lasing has been observed in both oxide and fluoride host crystals in the low-crystal­ field regime. This results in tunable laser emission from the broad vibronic band associated with the 3 T2 -- 3A 2 transition. The site symmetries for the laser host crystals for Ni2+ all have inversion. As seen from the data in Table 7.2, the fluorescence lifetimes for the lasing transitions are quite long. In oxide hosts, these lifetimes exhibit almost no decrease with temperature and in fluoride hosts the temperature dependence is weak. Thus the tran­ sition is probably magnetic dipole in origin instead of phonon-assisted elec­ tric dipole. The decreases in the decay times at high temperatures are asso­ ciated with increases in the nonradiative decay rates. Both the problem with nonradiative decay from the metastable state and problems with ESA have limited the laser performance of materials doped with Ni2+ ions. Although both pulsed and cw laser operation has been observed, lasing has been re­ stricted to low temperatures. Divalent iron is an ion with a 3d6 -electron configuration. Since lasing has

290

7. Transition-Metal-Ion Laser Materials

TEMPERATURE IKI

(C)

FIGURE 7.21 . Spectroscopic properties of Ni2+ . (A) Absorption, ( B) fluorescence, and (C) temperature dependence of the fluorescence lifetime. [Reprinted from Ref. 25, Laser Handbook, Moulton, p. 203 ( 1 985) with permission of Elsevier Science NL, Sara Burgerhartstraat 25, 1 055 KV Amsterdam, The Netherlands].

only been observed at very low temperatures for one special type of host crystal, this has not become a laser ion of practical interest. A result of recent interest is the report of spectroscopic properties and lasing of transition-metal ions in chalcogenide host crystals.27 The emission spectra of cr-2+, Co 2+, and Ni2+ in ZnS, ZnSe, and ZnTe are broad bands between 2 and 4 Jl. The substitutional sites have tetrahedral symmetry with

7.6. Laser Materials Based on Ions with 3d3 Through 3d8 Configurations

29 1

no center of symmetry giving rise to transitions with high radiative cross sections and quantum efficienies. These have the potential for developing into tunable lasers in the infrared spectral region. However, crystal quality is currently a problem. So far Cr2+ has been observed to lase in both ZnS and ZnSe. This ion has a 3d4 electron configuration with a 5D free-ion ground­ state term that splits into a 5 T2 ground term and a 5E excited term in a tet­ rahedral crystal field. The Jahn-Teller effect and local distortions produce minor splittings of the energy levels. The major absorption and emission transitions are between these two levels. Spin-forbidden transitions to higher-lying triplet or singlet levels are very weak, which results in no excited-state absorption. With anion ligands such as selenium, the non­ radiative decay rate of the metastable state is significantly less than that of a material like MgF2 : Co. Figure 7.22 shows the absorption and emission E

u

E

u

t

i..

-

c 0

"i Gl

ZnS•:Cr ... = 8 11$.C

..

.. ..

n = 2.44

e

u c 0

� g

.a <

Wavelength (nm)

Wavelength (nm)

(A)

0" 1o

1

(B)

ZnSe:Cr

.. . .

100

20

. . .. .

\

30

Temperature (K} (Cl

FIGURE 7.22. (A) Absorption spectrum and ( B) fluorescence spectrum of ZnSe : c�+ . The temperature dependence of the fluorescence lifetime is shown in (C). (After Ref. 27.)

292

7. Transition-Metal-Ion Laser Materials

spectra of ZnSe : Cr2+ and the temperature dependence of the fluorescence lifetime. Note that thermal quenching begins to occur only above room temperature, and thus the quantum efficiency is effectively one at room temperature. Preliminary lasing measurements demonstrated a slope effi­ ciency of 20% for this material. However, the sample used in these mea­ surements had significant passive losses, so the lasing properties should im­ prove as the crystal quality improves.

References I . (a) E.F. Farrell, J.H. Fang, and R.E. Newnham, Am. Mineral. 48, 804 ( 1 963); (b) E.F. Farrell and R.E. Newnham, ibid. 50, 1 972 ( 1 965). 2. (a) R.C. Powell, L. Xi, X. Gang, G.J. Quarles, and J.C. Walling, Phys. Rev. B 32, 2788 ( 1 985); (b) A.B. Suchocki, G.D. Gilliland, R.C. Powell, J.M. Bowen, and J.C. Walling, J. Lumin. 37, 29 ( 1 987); (c) R.C. Powell, J. Phys. ( Paris) Coli. C7, Supp. 1 0, 46, 403 ( 1 985). 3. S.A. Basun, S.P. Feofilov, and A.A. Kaplyanskii, J. Lumin. 48 & 49, 1 66 ( 1 99 1 ) . 4 . Z . Hasan and N.B. Manson, J. Phys. C 21, 3 3 5 1 ( 1 988). 5. D.E. McCumber, Phys. Rev. 134, A299 ( 1 964) . 6. J.C Walling, O.G. Peterson, H.P. Jenssen, R.C. Morris, and E.W. O'Dell, IEEE J. Quant. Electron. QE-16, 1 302 ( 1 980) . 7. M. Yamaga, B. Henderson, and K.P. O'Donnell, Phys. B 46 , 3273 ( 1 992) . 8. C.J. Donnelly, S.M. Healy, T.J. Glynn, G.F. Imbusch, and G.P. Morgan, J. Lumin. 42, 1 1 9 ( 1 988). 9. S.A. Payne, L.L. Chase, H.W. Newkirk, L.K. Smith, and W.F. Krupke, IEEE J. Quant. Elect. QE-24, 2243 ( 1 988). 10 R.C. Powell in Proceedings of the International Meeting on Laser Materials and Laser Spectroscopy ( World Scientific Publishing, Singapore, 1 989), p. 6. I I . S.C. Weaver and S.A. Payne, Phys. Rev. B 40, 1 0727 ( 1 989). 12. R.C. Powell and S.A. Payne, Opt. Lett. 15, 1 233 ( 1 990).

1 3 . W.F. Krupke, presented at the Conference on Lasers and Electro-Optics (CLEO), Anaheim, CA, 1 984. 14. (a) S.A. Payne, L.L. Chase, and G.D. Wilke, J. Lumin. 44, 1 67 ( 1 989); (b) H.W.H. Lee, S.A. Payne, and L.L. Chase, Phys. Rev. B 39, 8907 ( 1 989); (c) L.L. Chase and S.A. Payne, presented at the SPIE meeting in Los Angeles, January 1 989. 15. V.W. Viebahn, Z. Anorg. Allg. Chern. 386, 335 ( 1 97 1 ) . 1 6 . J.S. Griffith, The Theory of Transition Meta/ Ions (Cambridge University Press, London, 1 96 1 ) . 1 7 . L.K. Smith, S.A. Payne, W.L. Kway, L.L. Chase, and B.H.T. Chai, IEEE J. Quant. Electron. 28, 1 1 88 ( 1 992) . 1 8 . J.A. Caird, S.A. Payne, P.R. Staver, A.J. Ramponi, L.L. Chase, and W.F. Krupke, IEEE J. Quant. Electron. 24, 1 077 ( 1 988). 19. J.A. Caird and S.A. Payne, in The CRC Handbook of Laser Science and Tech­ nology, edited by M.J. Weber (CRC, Boca Raton, FL, 1 989), p. 23. 20. (a) Y. Tanabe and S. Sugano, J. Phys. Soc. Jpn. 9, 753 ( 1 954); (b) S. Sugano,

References

21. 22. 23. 24. 25. 26. 27.

293

Y. Tanabe, and H. Kamimura, Multiplets of Transition-Metal Ions in Crystals (Academic, New York, 1 970). R.C. Powell, G.E. Venikouas, L. Xi, J.K. Tyminski, and M.R. Kokta, J. Chern. Phys. 84, 662 ( 1 986). V. Petricevic, S.K. Gayen, and R.R. Alfano, Appl. Phys. Lett. 53, 2590 ( 1 988); H.R. Verdun, L.M. Thomas, D.M. Andrauskas, T. McCollum, and A. Pinto, ibid. 53, 2593 ( 1 988). A.G. Okhrimchuk and A.V. Shestakov, Opt. Mater. 3, 1 ( 1 994) . L.D. Merkle, A. Pinto, H.R. Verdun, and B. Mcintosh, Appl. Phys. Lett. 61, 2387 ( 1 992). P.F. Moulton, in Laser Handbook, edited by M. Bass and M.L. Stitch ( Elsevier, Amsterdam, 1 985), p. 203. L.F. Johnson and H.J. Guggenheim, J. Appl. Phys. 38, 4837 ( 1 967). L.D. DeLoach, R.H. Page, G.D. Wilke, S.A. Payne, and W.F. Krupke, Advanced Solid State Lasers, Technical Digest (Optical Society of America, Washington, DC, 1 995), p. 286.

8

Y 3 Als 0 1 2 : Nd3 + Laser Crystals

Trivalent neodymium ( Nd3 +) is the most successful type of active ion for solid-state lasers and thus far has been made to lase in more types of crystal and glass hosts than any other ion. It can operate as either a pulsed or continuous-wave laser with a sharp emission line. The most common emis­ sion wavelength is near 1 ,urn, but there are several possible laser transitions in the near-infrared spectral region, and in addition a near-ultraviolet laser line. Although the effects of different host environments on the spectroscopic properties of Nd3 + are more subtle than those for transition-metal ions, they can cause significant differences in lasing characteristics through changes in physical processes such as radiative transition strengths, radiationless decay probabilities, excited-state absorption, and cross relaxation quenching. In the following section the properties of the electronic energy levels and transitions of Nd3 + ions are discussed, and the properties of neodymium in yttrium aluminum garnet crystals ( Nd : YAG) are described. This is cur­ rently the most important solid-state laser material, and its characteristics serve as an example for understanding rare-earth laser materials. The effects of changes in the local crystal-field, electron-phonon interactions, and ion­ ion interactions are described. The fundamental concepts used in this dis­ cussion were outlined in Chaps. 2-5. In the next chapter, examples are given of the spectroscopic properties of Nd3 + ions in several different crystal and glass hosts and the laser characteristics of these materials are summarized and compared to Nd : YAG. 8.1

Energy Levels of Nd 3 +

The energy levels of rare-earth ions are discussed in detail in several books and review papers. 1 - 5 The electronic configuration for the 60 electrons of a neodymium atom is:

1?2s2 2p6 3s 2 3p6 3d 104s24p64d 10 4f4 5s2 5p6 5d 0 6s2 .

The first nine sets of orbitals make up the filled core, while the optically ac­ tive electrons are in the partially filled 4/ orbitals. The latter are shielded by 294

8. 1 . Energy Levels of Nd3 +

295

the electrons in the outermost 5s through 6s orbitals. The trivalent neo­ dymium ion has given up three electrons, two from the outer 6s orbitals and one from the 4/ orbitals, leaving a configuration of ( Xe) 4f3 5s2 5p6 • The three 4/ electrons are the ones that play the dominant role in determining the optical properties of the ion. Although the transitions giving rise to the optical spectra take place between different levels of the 4f3 configuration, it is also important to know: the positions of the energy levels of the 4fN- 1 5d configuration since configuration interaction is critical in determining spec­ tral line strengths as discussed below. The Russell-Saunders coupling ap­ proach discussed in Chap. 2 can be used to determine the electronic terms of the free ion with three electrons, each having quantum numbers n 4 and I 3. Due to the shielding of the outer-shell electrons, the crystal-field splitting is always treated in the weak-field limit, so spin-orbit coupling is applied first to determine free-ion multiplets. The radial charge distributions of the electrons shown in Fig. 8.1 demonstrate the shielding effect of the outer-shell electrons. The types of spectroscopic terms available for Nd3 + ions can be de­ termined by using appropriate combinations of single-electron orbitals and applying the Pauli exclusion principle as was done in Chap. 6 for trivalent chromium. Since all of the optically active electrons have the same values of n and I, there are only certain combinations of the quantum numbers m1 and ms that are allowed. Since I 3 for each of the electrons, the largest value of the total angular momentum quantum number could be 9 so ML should run from 0 to ± 9 in integer steps. However, the only way for the orbital angular

1.3

I 1 I I I I I I I 1 I I I I

1.2 1. 1

l .o

s:

_. �

.9 .8

.7

I I I

.6

I I

.s

.4

.3 .

2

I I

I

0 .2

'

.6

!,�' - .. , 1.0

\,

l4

", -. / 1.8

•/

I

I

/

,.

-

.f.!• -..

'

/ / ·-_,

2.2

��- 26

30

3.4

3.8

4.2

r(au)

'

4.6

'

'

5.0

5.4

5.B 6.2

6.6

7.0

74

FIGURE 8. 1 . Approximate charge distributions of electrons in different orbitals for rare-earth ions demonstrating the shielding of 4f electrons by outer-shell electrons ( Ref. 6).

296

8. Y3 Al501 2 : Nd3 + Laser Crystals

momenta of the three electrons to couple to give L = 9 would result in two of the electrons having an identical set of quantum numbers, which is not allowed. The spin angular momenta of the three electrons can couple to give quartets or doublets, so Ms = ± � , ± !· A table of single-electron states that contribute to the multielectron terms can be constructed and the results are shown in Table 8. 1 . Only the positive ML-positive Ms quarter of the table is shown since the table is symmetric and thus all of the required in­ formation can be obtained from this quarter. Here the single-electron states are represented by (mi'f , m/i_ , m /j ) where the + or superscript designates spin up or spin down, respectively. These are placed in the ML row and Ms column of an appropriate term. Since the largest values of ML and Ms equal L and S, respectively, for a given term, Table 8.1 can be used to determine the spectroscopic terms of a 4/3 ion. The highest ML row in the table has single-electron states occupy­ ing only the cells in the Ms = ± ! columns. Since this cell represents ML = 8, these must belong to a 2L term. One of the single-electron states in each of the other cells in the Ms = ± ! columns will also belong to this term and thus can be eliminated. This leaves the highest occupied cells those with ML = L 7 and Ms S = !- These are associated with a 2K term which has other single-electron states in each of the cells of the two center columns of the table. Eliminating these states leaves the highest occupied cells those with ML = 6, and Ms ± �- The states in these cells belong to a 4/ term with additional states in all the remaining cells. This procedure can be fol­ lowed until all of the single-electron states have been associated with multi­ electron terms resulting in the identification of 17 terms: 2L, 2K, 4!, 2/, 2H, 2H, 4G, 2G, 2G, 4F, 2F, 2F, 4D, 2D, 2D, 2p, 4s_ Next these terms must be placed in order of ascending energy. Using Hund's rules discussed in Sec. 2. 1 implies that the 4/ term is the ground state, and the energies of the other terms with respect to the ground state must be determined. As with transition-metal ions, perturbation theory is used to determine the energy levels with the zero-order perturbation being the electrons in the 4/ orbitals moving independently of each other in the central field of the nucleus and the inner-shell electrons. Under these con­ ditions, all energy levels of a given 4r configuration are degenerate. The Coulomb interaction between the optically active electrons is treated as one perturbation and the spin-orbit interaction of the electrons is treated as an­ other perturbation. This gives a good approximation for calculating the en­ ergy levels. However, for rare-earth ions the interactions between electrons in different configurations can also be important in some cases. The Coulomb interactions between electrons is treated as described in Sec. 2. 1 . The radial integrals arising in the interaction matrix elements are expressed as Slater parameters (see Table 2.2), p4 p6 (8 . 1 . 1 ) Fo , F2 = � F4 = F6 = ' 7361 . 64 ' 1089 or in terms of the Racah parameters given by Eq. (2. 1 .42). The parameter

s

p

D

L K I H G F

L

=

1 /2

(3+3-2+) (3+3- 1 +) (3+2-2+) (3+3-o+)3(3+2- 1 +) (3+3- - 1+)3(3+2+o-) (3+ 1 - I +) (2+2- I+) (3 + 3- - 2+)3(3 + 2+ - I -)3(3+o- I+) (2+2-o+) (2+ I - I+ ) (3+3- - 3+ )3(3+2+ - 2-)3(3+ I - - 1 +) (3+o-o+ ) 3(2+ I+o- ) (2+2- - I+) 3(3+2- - 3+)3(3+ I+ - 2- )3(3+o- - I+) (2+o-o+ ) 3(2+ I+ - I - ) (2+2- - 2+) ( I+ I -o+) 3(3+ I - - 3+ )3(3+o+ - 2-) (3+ - I - - I+ ) (2+2- - 3+) 3(2+ I+ - 2-)3(2+o- - I+ ) ( I + 1 - - I + ) ( l +o-o+ ) 3(3+o- - 3+)3(-3+ I +2- )3(2+o- - 2+) (2+ - I - - 1+) 3(3+ - I+ - 2- )3(I +o- - I+ ) ( l + I - - 2+ )

s

TABLE 8. 1 . Single-electron states for a 4/3 -electron configuration. =

3/2

(3 + 2+ - 3+) (3+ - 2+ I+) (o+3+ - I +) (2+ I+ - 1+) (3+ I+ - 3+) (3+ - 2+o+ ) (o+2+ - I+) (2+ I+ - 2+ ) (3+o+ - 3+ ) (2+ - 3+ 1+) (0+2+ - 2+) (o+ I+ - 1+) ( -2+3+ - 1+)

(3+2+ 1 +) (3+2+o+) (3+ 1+o+) (3+2+ - 1+) (3+2+ - 2+) (3+ - I+ 1+) (o+2+ I+)

s

4s

2p

4 D, 22 D

zL zK 4I , 2 I 22 H 4 G , 22 G 4F, 2z F

Term

m

N '-0 --.1

+

0.

z

"' 0 ..

�

(I> <

r

i:l (I> .. OCl '<

00 ..

8. Y 3 Al501 2 : Nd3 + Laser Crystals

298

Fo is common to all energy levels and thus is not important in determining their relative splittings. For 4fn configurations the magnitudes of the Slater parameters are approximately related by

(8.1 .2) so only one parameters F2 , is required to characterize the energy levels. For trivalent rare-earth ions this can be approximated by

Fz = 12.4(Z - 34) ,

(8.1 .3)

where Z is the nuclear charge. If all three electrostatic parameters are used, it is helpful to form the linear combinations1 5B = 5Fz + 6F4 - 91F6 , 5C = 7F4 - 42F6 , D = 462F6 . Following the same procedure used in Sec. 6. 1 , the energies can be ex­ pressed in terms of the Slater-Condon parameters or the Racah parameters [given by Eqs. (2. 1 .42)]. As was the case with the two 2D terms of Cr3+, the secular determinant must be solved for each of the four sets of duplicate terms. The first step is to write the energy of each box in Table 8.1 in terms of the Coulomb and exchange integrals and use Eqs. (2. 1 .33) and (2. 1 .34) and Table 2.3 to express these in terms of the radial integral parameters. The first few are EeL) = £(3+3 2+) = J(33) + 21 (32) - K(32) =

3F0 - 63 F4 - 736118.64 F6 ' 1089

TABLE 8.2. Terms of a 3/3 -electron configuration ordered with respect to energy relative to the ground term (from Ref. 2). Term

E (4I) E (4F) E (4S) E eH) E eG) E eK) E (4G) EeD) E eP) E el) E (2L) E (4D) E eH' ) E (2D' ) E eF) E eG' ) E e F' ) 4cc2 4r

5056B2 2176B2

Racah parameters

Slater parameters

F2 parameter

0 21£3 21£3 3£ 1 + 24£3 3£ 1 + 780£2 / 7 + 9£3 / 7 3£ 1 1 35£2 + 10£3 33£3 3£ 1 858£2 / 7 + 1 14£3 / 7 3£ 1 + 10£3 3£ 1 30£2 + 24£3 3£ 1 + 105£2 + 1 8£3 54£3 3£ 1 + 21£2 3£3 3£ 1 + 1 1 31£2 / 7 + 1 59£3 / 7 9£ 1 + 21£3 3£ 1 1 683£2 / 7 + 334£3 / 7 3£ 1 + 195£2 + 54£3

0 35B 35B 42B + 3 1 .5C + 3D cc 72B + 56.5C + 3D p 25B + 80C + 3D 55B 58B + 28.5C + 3D y 40B + 35C + 3D 60B + 45C + 3D 65B + 3D 90B 42B + 3 1 .5C + 3D + cc 58B + 28.5C + 3D + y 120B + 37.5C + 6D .5 72B + 56.5C + 3D + p 120B + 37.5C + 6D + .5

0 3 1 . 1 73F2 3 1 . 1 73F2 32.928F2 43.243F2 48.51 5F2 48.987F2 58.799F2 58.889F2 77.367F2 78.834F2 80. 161F2 87.949F2 90. 182F2 107.768F2 134.393F2 194.732F2

1 8816BC + 291 69C2 4P2 12676B2 3 1 676BC + 361 69C2 1 8096BC + 74529C2 4.52 9700B2 1 80B(45C + 2D) + 9 (45C + 2D) 2

8. 1 . Energy Levels of NdH

299

E e K)

E(3+3 - 1 +) + E(3+2+2- ) E(3+ r 2+) = 21(3 1 ) + 1 (22) K(3 1 ) 38 p6 + 3Fo � F2 + , 225 1089 736 1 .64 E(4 I) E(3+2+ 1 +) 1(32) + 1 (3 1 ) + 1(21 ) K(32) K(3 1 ) 141 4 221 0 65 2 = 3F F F F6 225 1089 7361 .64 '

K(2 1 )

The results are listed in Table 8.2 and shown schematically in Fig. 8.2 where the energy of the 4/ ground-state term is subtracted from all terms so the ground state is the zero of energy. The spin-orbit interaction is the same order of magnitude as the Cou2F 712

2n 5/2 2u9 /2 .. 113/2 " 4L1512 _ n5/2 - 4ny 2

;

·�

.,:;-

�

e ;. � "'

�

r.l 2

2F 5/2

2p 1/2' G91

-

2�2 17" 2� 17" 2u n t2

::

/f" 12""

2

41

1312

� 1112

- 2n 5/2 _ 2ny2 _4Gn t 2 t 3/ 2 �/,l 4G,6 /2

f

b a

11512 11427

8 7 6 5 4 3 2 1

6734 6639 6583 6570 5970 5936 5814 5758

7 6 5 4 3 2 1

4498

6 5 4 3 2 1

2521 2468 2147

2m

5

857

4442

4435 4047 4032

���

202

308

199 130

FIGURE 8.2. Energy levels of the lowest terms of the Nd3 + ions, and the crystal-field splitting of the lowest levels in a YAG host crystal.

300

8. Y3 Al501 2 : Nd3 + Laser Crystals

lomb interaction for rare-earth ions and therefore must be treated before treating the crystal-field perturbation. The qualitative splitting of the terms into multiplets follows the description of angular momentum coupling in Sec. 2. 1 with the energy of the splitting given approximately by Eq. (2. 1 .23). The values of the J quantum numbers for the multiplets of a term with quantum numbers L and S run in integer steps from L S to L + S. Thus the ground term of Nd3 + has the four multiplets 4/9 ;2 , 4 /1 1 ; 2 , 4 /1 3 ;2 , and 4/15 ;2 with the first of these being lowest in energy because the electronic configuration of this ion is less than a half-filled shell. The four multiplets of the metastable state range from 4F3;2 to 4F9 ;2 • Some of these splittings are shown in Fig. 8.2. If the F2 parameter is used to characterize the Coulomb interaction strength and ( is sued to characterize the spin-orbit interaction, the effects of spin-orbit coupling can be seen by plotting the energy of the multiplets as a function of (/F2 as shown3 in Fig. 8.3. The values of the Slater-Condon or Racah parameters and the spin-orbit coupling parameter

FIGURE 8.3. Spin-orbit splittings of the lowest terms of Nd3 + (from Ref. 2).

8.2. Crystal-Field Splitting

301

are generally treated as adjustable parameters to be determined by fitting to experimental data. For Nd3 + in an aqueous solution these are found to be F2 = 321 cm- 1 , F4 = 46.3 cm- 1 , F6 = 4.69 cm- 1 , ( 884 cm- 1 , and the appropriate value of the ( / F2 ratio for Nd3 + is shown in Fig. 8.3. The devi­ ation from pure Russell-Saunders coupling can be seen in this figure as deviations from straight lines for large values of (. 8 .2

Crystal-Field Splitting

There are several mathematical approaches that have been developed for expanding the crystal-field potential. The best choice of these approaches depends on the strength of the crystal-field perturbation compared to cou­ lomb and spin-orbit interactions as discussed in Chap. 2. For the case of the shielded f electrons of trivalent rare-earth ions, the perturbation treatment is performed in the weak-field scheme. For this situation, it is convenient to express the crystal-field potential as a spherical harmonic expansion of a charge density p( R), V:

cf

-

e;p( R) d r I R r; I

_

-

_ =

�

4n ( J e;p(R) 4k + l

B'kq Ckq ( O; , t/J; ) , L kq

r� q 1 ) Ykq ( O; , ¢; ) Yk q ( O , t/J) k 1 dr G+ (8.2. 1 )

, ,i

where (8.2.2)

q

Here Bicq = ( 1 ) Bk, q· The sum is over the three 4f electrons with orbital radii r; . For a point-charge model the integral is replaced by a sum over the ligands and r� -- r� < R. The Bkq expansion coefficients are directly related to the productions of the expectation values of the radial positions and the Akq parameters discussed in Chap. 2, Bkq = (rk )Akq, and these are listed in Table 8.3. The Ckq are tensor operators defined by 1/2 Ykq · Ckq = 4n (8.2.3) 2k + l Thus,

( )

302

8. Y3 Al501 2 : Nd3 + Laser Crystals

TABLE 8.3. Crystal-field parameters and equivalent operators for YAG : NdH ( Refs. 1 3). Operator equivalent constants Crystal field parameters

A 2o(r2 ) = 0.500B2o = 208 A 22 (,.Z ) = 1 .220B22 = 305 339 A 4o(r4 ) 0.125B4o A 42 (r4 ) = 0.791B42 = 410 A44(r4 ) = 1 .046B44 = 1065 A6o(r6 ) 0.0625B60 = 70 A62 (r6 ) 0.6404B62 = 197 A64 (r6 ) = 0.701 6B64 = 1 1 16 A66 (r6 ) = 0.9499B66 = 41

Manifold

er: ( x 103 )

P ( x 103 )

419/2 411 1 /2 411 3/2 411 5/2 4F3/2 4F5/2 4F7/2 4F9/2 4S3;2 4Gs;2 4G112

36.8162 4.05020 1 8.6096 7.8483 1 5 1 .40 28.2323 36.5130 40. 1760 1 5.200 2.5915 1 1 .0249

23.3573 1 1 .4408 3.3908 2.6371

1 04. 1088 21 .9549 3.8778 28.4666

82.5740 10.8066 17.7146

1 1 1 .3777 5.7358

249.2206 1 .4783

43.4223

y ( x 103 )

Equivalent operators

02o = 3if J(J + 1 ) Ozz = 0.5(J� + J: ) 04o = 35J; 30J(J + 1 )J} + 25J} 6J(J + 1 ) + 3J2 (J + 1 ) 2 042 = 0.5{ (7J; J(J + 1 ) 5) (J� + J: )}s" 044 = 0.5(J! + J�)

06o = 23 1 5J; 3 1 5J(J + 1 )J; + 735J; +105J(J + 1 ) 2 J; 525J(J + 1)J; +294J; 5J3 (J + 1 ) 3 +40J2 (J + 1 ) 2 60J(J + 1 ) 062 = 0.5{ (33J:[1 8J(J + 1 ) + l23]J; +J2 (J + 1 ) 2 + 10J(J + 1 ) 1 02) (J� + J: ) }s 064 = 0.5{ [1 \Jz J(J + 1 ) 38J (J! + J�)L 066 = 0.5(J� + J� )

a{ABL = ! (AB + BA).

and

As discussed in Chap. 2, the electronic wave functions can also be expressed in terms of spherical harmonics YZ with k corresponding to the angular momentum quantum number of the electron configuration and q its projec­ tion. Using the relationship in Eq. (8.2.3), these wave functions can also be expressed in terms of Ckq tensor operators. A mathematical formalism has developed around calculating the matrix elements of tensor operators and their products. This is called Racah algebra and is useful for describing the effects of the crystal-field interaction. Since this is the formalism that is generally used to treat crystal-field splitting and radiative transitions of rare-earth ions in solids, some of the relevant aspects of Racah algebra are summarized here. One important relationship is the commutation relationship between a tensor operator and the angular mo­ mentum operator J. Since the Ckq operators do not involve spin, only the

8.2. Crystal-Field Splitting

303

orbital part of J is important. By considering the effects of lz = ( 1 I i) 8 I 81/J and / ± = e ± itft ( ± oloB + i cot oloi/J) on spherical harmonic functions ap­ pearing in Ckq , the commutation relations are determined to be

[Jz, Ckq]

=

qhCkq , [J ± , Ckq] = [k(k + 1 ) - q(q + 1 ) ] 1 /2 hCkq± l ·

(8.2.4) Then the Wigner-Eckart theorem can be used to write matrix element of Ckq between two states as

(yJm I ckq I r 'J 'm' )

-m

i', ) ,

k i ( - 1 ) 1 (yJ I ck II r'J' ) -m q m

(

(8.2.5)

where y represents all of the quantum numbers not affected by Ckq· The last factor in Eq. (8.2.5) is a 3-j symbo l, defined in Sec. 2. 1 , which contains spa­ tial orientation part of the matrix element. These are tabulated in several reference books 7 and some values can be found in Table 2. 1 . The middle factor in the equation is a reduced matrix element that does not depend on the spatial orientation quantum numbers. These are also tabulated for d and f electrons. 8 The method for calculating these matrix elements is outlined below. The unit spherical tensor in orbital space for single-electron wave func­ tions is defined in terms of its reduced matrix element ( I' I uk Il l ) = Jl ' ,

so

Ckq

( /' II ck ll l) ukq ·

It is standard to normalize the reduced matrix elements such that ( /' II Ck ll /) 1 . Here

( I ll ck Il l)

+

1 ( /(O)k(O) l l (O) ) ,

where ( /(O)k(O) I (0)) is a Clebsch-Gordan coefficient. The unit spherical tensor is so

"L ckq (i) i

( I ll ck Il l) ukq·

( rxL' M{ S' M� I Ukq I rxLMLSMs ) 1 = Jss'JMs Ms' ( L(ML)k(q) I L'(M{) ) ( rxL'S' II Uk ll rxLS) . +1

The matrix element of the unit spherical tensor in orbital space is

To include spin angular momentum, a double unit tensor is defined by

(/ 's' ll vKk (i) ll ls)

1 ) ( 2s + 1 )JwJss'

,

8. Y3 A!501 2 : Nd3 + Laser Crystals

304

where K is the rank in spin space and k is the rank in orbital space. Then

VKk = L VKk (i) i

and

Li Ii

·

si =

+

1 ) (2/ + 1 ) L ( - 1 ) -l V�.u· .!

Here K and k have both been set equal to 1 for the spin-orbit interaction. This can be used in the expression for the spin-orbit interaction replacing � ( ri ) by its expectation value (. The matrix element of the double unit tensor is

( aL'M� S'M� I V't l aLML SMs) 1 (L(ML ) k ( q ) I L' (M� )) (S(Ms) K(.�.) I S'(M�)) + l lS + l )( ' ) k x ( aL'S' II VK ll aLS ) . For the specific case of crystal-field matrix elements, using Eqs. (8 2 . 1 ) and (8.2.5) with the normalization convention described above leads to .

(JN SLJM1 I V lfN S'L'J'M;) Bkq (JN SLJMJ I Ukq iJN SL 'J'M; ) (fll Ck ll /) , L kq

( 8 .2. 6)

where the last factor is for normalization and

( 8. 2 . 7 ) with

( 8 . 2 .8 ) The matrix elements and large-bracketed symbols are tabulated and can be simply found in tables instead of determined through mathematical evalua­ tion. The last expression involves a doubly reduced matrix element. This is useful when LS coupling can be employed to designate the electronic states and the crystal-field operator only acts on one part (L ) of the wave function. The large brackets in this case are 6-j symbols and these are also tabulated.

8.2. Crystal-Field Splitting

305

Using the formalism described above, the crystal-field interaction Hamil­ tonian can be expressed as

Hi�t = L B'kq L Ckq (i) = v'2T+l(l ll Ck ll l) L B'kq Ukq · �

�

The first step in evaluating crystal-field matrix elements is to determine which k , q terms in the expansion of the crystal-field operator are nonzero. Using this mathematical formulation, the orthogonality relations for spher­ ical harmonics requires that the only nonzero matrix elements of the crystal­ field operator between two 4/ states (I + I ' :s; k; parity requires k even) will have expansion coefficients B2q , B4q, or B6q· For configuration admixing with 5d states the orthogonality conditions require that the only nonzero matrix elements of the crystal-field operator will be those with expansion coefficients having odd values of k :s; 5. Generally it is not possible to cal­ culate the Bkq parameters from crystallographic data. Therefore the matrix elements of the Ckq are calculated and the Bkq are determined by treating them as adjustable parameters in fitting theoretical predictions to experi­ mentally measured energy levels. The requirement that the Hamiltonian of the ion in the crystal be in­ variant under the symmetry operations of the site group further limits the number of nonzero components appearing in the crystal-field expansion. The spherically symmetric term in the crystal-field expansion k q 0 gives a uniform shift to all of the free-ion multiplets without causing any splitting. Other terms in the expansion will mix some of the free-ion multi­ plets so that J and MJ are no longer good quantum numbers and are re­ placed by the irreducible representations of the site group according to which the wave functions transform. These are listed for D2 site symmetry in Table 8.4. The free-ion levels are split into J + ! Stark components. The states are all doubly degenerate since Kramer's theorem predicts that the time-reversal degeneracy cannot be lifted by a crystal-field perturbation. The crystal-field states in Table 8.4 are classified in terms of the irreducible representations of the D2 symmetry group according to which they trans­ form. There are other designations of crystal-field quantum numbers that are used, but these are not always unique. The results of this type of analysis are the same for all other noncubic symmetry groups. For cubic crystal-field symmetry there may be both doubly and quadruply degenerate energy levels, TABLE 8.4. Crystal-field splittings for a 3/3 -electron configuration in D2 symmetry. r

D 112 m 2 S (�)

l

2

0

3

2

5

2

2 1

7

2

2 2

J

2

2 2 3

ll

T

3 3

11

2 4 3

15

T

4 4

306

8. Y3 Als01 2 : Nd3 + Laser Crystals

and the splittings for each free-ion J state are different. Note also that the crystal-field splitting results are different for ions with an even number of optically active electrons. The Coulomb interaction and spin-orbit parameters for the free ion are somewhat reduced when the ion is placed in the crystal due to the charge cloud overlap of the electrons on the surrounding ligands (called the neph­ elauxetic effect). However, for trivalent rare-earth ions the crystal-field energy levels can generally be associated with their parent free-ion levels. There has been a significant amount of work done in calculating the Bkq crystal-field parameters. The simplest models utilize only one-electron operators for C(k) in Eq. (8.2.2) and treat the ligands as point charges. More sophisticated models taking into account such effects as exchange interaction through the use of two-electron operators, multiple charge distributions, and covalency have also been attempted. However, these calculations have generally not been very successful and the Bkq are better determined empirically from iterative fitting of the data obtained from experimentally observed spectra. Once these parameters are known, a complete set of energy levels and eigen­ functions can be determined for the ion in the crystal. Since first-principle calculations of the crystal-field matrix elements can be quite difficult, it is common practice to simply look up the relevant 3-j symbols and the reduced matrix elements in tables. If calculations must be carried out, an operator-equivalent technique has been developed to simplify the process. This gives good order-of-magnitude estimates for the positions of the crystal-field energy levels and is especially useful when the free-ion terms are well separated. To use this technique Eq. (8.2. 1 ) is rewritten in the form

v�r

L AkqPkq (q � o) , k,q,i

(8.2.9)

where the Pkq are the Legendre polynomials and the Akq parameters are re­ lated to the Bkq parameters as shown in Table 8.3. Since these polynomials and the angular momentum operators J, Jz and J ± have the same symme­ try properties, the matrix element of the crystal-field expansion expressed in Eq. (8.2.9) can be evaluated by replacing x, y, and z with Jx, Jy, and Jz taking into account the fact that Jx and Jy do not commute. For example, the first term in the expansion (k 0 , q 0) will have a matrix element written in operator-equivalent form as

(1NJM1 I � A20 (3z� - ?;) kNJM� )

where

A 2o (? 'zr!:.! (JN JM1 I 3 J; - J ( J + l ) I JN JM� ) A 2o (?)rx1(JN JMJ I 02o l fN JM� ) ,

(8.2. 10)

8.2. Crystal-Field Splitting

307

(8.2. 1 1 )

and A 2o (r2 ) is the usual crystal-field parameter with rx1 the factor relating the normalization difference between the standard matrix element and the operator equivalent matrix element. Some of the important operator equiv­ alent relationships and matrix elements are listed in Tables 8.3 and 8.5. For example, Nd3 + ions in Y3 As 0 1 2 (YAG) are at a site symmetry of D2 and the crystal-field Hamiltonian has the form

V�r rx1(A2o (r2 ) 02o + A22(r) 022) + /h(A 4o (r4 ) 04o 6 6 + A 42 (r4 ) 042 + A44 (r4 ) 044) + y1(A6o (r ) 06o + A62 (r ) 062 + A 64 (r6 ) 064 + A66 (r6 ) 066) · (8 .2. 12) M, = ± !

TABLE 8.5. Matrix elements of the equivalent operators

J 3 5

2 2 7

2 9

2

11 123

2 15

2

5

2 7

2 9

2

11 123 li 2

2

1 29 2

11 l3

2

2

J

¥ 7

2

11

2

13 2 15 2

pa

+ J2

+ �2

+ 12

okq

+ 22

( Refs. 1 3).

+ ll2

+ 112

+ li2

02o 311 J(J + 1 ) 1 1 4 1 5 3 1 5 7 4 3 1 2 6 35 29 17 25 55 2 8 7 6 2 5 13 7 21 1 3 19 15 9 35 21 9 040 = 35J: 30J(J + l )J; + 25J; 6J(J + 1 ) + 3J2 (J + 1 ) 2 2 3 1 60 13 3 60 9 7 17 18 22 84 3 18 120 12 33 13 28 27 33 60 13 1 32 63 92 77 108 143 23 129 60 1 89 101 201 221 91 273 2 060 = 23 1/� 3 ! 5J(J + I )J: + 735J: + !05J (J + I ? 1; 525J(J + I )J; +294/; 5J3 (J + 1 ) 3 + 40J(J + 1 ) 2 60J(J + 1 ) 1260 5 I 9 5 2 5040 8 11 10 6 20 7560 4 11 11 25 31 2160 200 25 227 11 143 319 185 F < ± � II + � > < ± ¥ II + � > < ± ¥ II + ! > < ± ¥ II ± ! > < ± ¥ II ± � >

3 2 3 6

360 360 360 720 360v'IT

../7

14 28v'3 2 1 v'IO 42v's

066

2v'TI 7v'30 7 v'6 84

= (J! + J� )/2 v'462

7 v'3 4 v'273

v'429

7 v'39

y'35)(T3

8. Y3 Al501 2 : Nd3 + Laser Crystals

308 TABLE

J

2 3 4 5 6 7 8 2 3 4 5 6 7 8 3 4 5 6 7 8

8.5 (continued) F

M1

=0

±I

±2

±3

±4

±5

±6

±7

±8

02o = 3Jf J(J + I ) 2 3 I 2 2 3 3 4 5 0 17 8 7 28 -20 I 3 10 6 15 9 6 13 2 10 11 22 14 5 3 44 56 29 19 8 52 91 53 24 12 23 20 8 15 25 3 04o = 35J: 30J(J + I )J} + 25J} 6J(J + I ) + 3J2 (J + 1 ) 2 12 I 4 6 I 7 3 60 6 21 11 60 18 14 9 1 6 420 6 4 6 6 99 96 11 54 84 64 60 66 12 621 294 704 1001 756 25 1 429 869 13 420 31 24 3 36 39 17 39 060 = 23 1J: 3 1 5J(J + ! )J: + 735J: + 105J2 (J + I? J} 525J(J + ! )J} +294J} 5J3 (J + I ) 3 + 40J(J + I ) 2 60J(J + I ) I 20 1 80 6 15 20 22 1260 4 17 2520 48 36 40 12 15 29 43 40 22 22 20 55 7560 8 3780 176 143 200 55 - 125 286 197 50 1 20 2 13860 85 78 128 1 69 65 93

J

F

3 4 5 6 7 8

360 360 360 360 360 360

< 3 11 3 )

<4 11 +2 )

< 5 11 + I )

< ± 6 11 0 )

< ± 7 11 ± 1 )

40

52

104

< ± 8 11 ±2 )

066 = (J! + J� )/2

7 28 84 210 462

2../7 14v'3 14v'30 42 Vl 42v'ITO

y'2[0

7 ..;6 28v'3 X 77

2v'231 X 143 X 143

X

!43 55

X

X

143

" Multiplying factors common to all elements in a row.

The various parameters and forms of the operators in this equation are given in Tables 8.3 and 8.5. This analysis provides a good fit between theo­ retical predictions and the energy-level splittings for this system. The operator-equivalent method provides a simplified method for calculat­ ing approximate values of crystal field splittings when interactions between different multiplets can be ignored. However, with tables of 3-j coefficients readily available, these calculations are generally unnecessary. The lowest-energy electric dipole-allowed transitions for Nd3 + are 4f3 --+ 4j2 5d transitions in the spectral region. The transitions of direct

8.3. Radiative Transitions: Judd Ofelt Theory

309

TABLE 8.6. Selection rules for Nd3 + in D2 crystal-field symmetry.

D t;z

zS

x, y, z x, y

x, y x, y,

relevance for solid-state lasers are Laport forbidden transitions between levels of the 4/3 -electron configuration. These occur by the magnetic dipole interaction or forced electric dipole interaction through admixing with the levels of the 5d configuration. The selection rules for transitions between free-ion spin-orbit multiplets require that 111 0 , ± 1 . When the Nd3 + ion is in a crystal field of D2 symmetry, the selection rules are determined from group theory. The x and y vector components of the electric dipole inter­ action operator transforms as the E representation in D2 while the z com­ ponent transforms as the A representation. The direct-product analysis and Eq. (3.2. 10) leads to the results shown in Table 8.6. 8.3

Radiative Transitions : Judd-Ofelt Theory

The theoretical formalism that has been the most useful for interpreting the intensities of spectral lines of lanthanide ions in crystal fields was developed independently by Judd 5 and Ofelt. 9 The transitions between levels of the 4r configurations can be caused by the forced electric dipole radiation induced by odd terms of the crystal field, forced electric dipole radiation induced by lattice vibrations, allowed magnetic dipole radiation, and allowed electric quadrupole radiation. For solid-state laser materials, the majority of the transitions are forced electric dipole, and thus the treatment outlined below will focus on these transitions. Following the tensor operator formalism outlined in Sec. 8.2, the position vector r is a tensor of rank 1 and thus related to the irreducible tensor oper­ ator c O l given in Eq. (8.2.3), r = rcO l . The three components are Y( i ) Y (O)

r(

1)

()

1 4n /2 = rei 1 ) = r 3 Yu = rC61) =

()

1

- y'l ( x + iy) ,

Yw = z ,

4n '/2 _ (I) _ 1 . . - rC 1 - r 3 Y1 _ , - y'2 ( x - zy)

(8.3. 1)

Note that the notation conventions of having the order of the tensor appear

310

8. Y3 Als01 2 : Nd3 + Laser Crystals

as a subscript or superscript are equivalent. The electric dipole moment op­ erator is expressed as P

e L r1 j

e L r1( c ( l l)1

eD ( l ) ,

(8.3.2)

j

where the sum over j is over all optically active electrons, the tensor com­ ponents are found from Eq. (8.3. 1 ), and - eD (I) is the tensor expression for the electric dipole moment operator with - eD�' ) components given through Eq. (8.3.2). A similar procedure for the crystal-field operator given in Eq. (8.2. 1 ) results in the tensor expression t,p

t,p

(8.3.3)

where the crystal-field parameters A1p were defined in Sec. 8.2. Forced electric dipole transitions take place because odd harmonics in the static or dynamic crystal field mix states of opposite parity. Thus the parity of a wave function for a specific energy level is no longer determined solely by the parity of the single-electron orbital angular momentum quantum number, and the Laport selection rule of AI ± 1 no longer holds. For this to occur through static interactions requires a host lattice with a local site symmetry that does not have a center of inversion (such as C3 v , D2d , Td, and 0) . Thus the terms in Eq. (8.3.3) with odd values of t are important in admixing states. The states of the 4fn configuration are expressed as linear combinations of Russell-Saunders terms. The crystal field acts as a perturbation to mix states with opposite parity, j

( 8.3. 4) where

The parameter ct represents all of the quantum numbers needed to designate the state other than those listed explicitly, and p designates the even-parity configuration. The strength of a forced electric dipole transition between two states l A ) and I B) can be determined from the oscillator strength expression in Eq. (3.3.1 ) , (8.3 . 5) where the Lorentz field correction factor has now been included,

8.3 . Radiative Transitions: Judd Ofelt Theory

x

311

n(n2 + 1 ) 2 9

Expression (8. 3.2) is used for the electric dipole moment operator in the matrix eiement, and the initial and final states of the transition are given by (8.3.4) . The oscillator strength then becomes fed

n2

[-

(

e

( q)a i D�l ) l q)p ) ( q)p i D�t) l q)b ) p X 3he A (Eb Ep ) p, t,p t 2 1 + ( q)a i D�l l q)p ) ( q)p i D� l l q)b ) (Ea - Ep ) 8 2 mc

)]

(8.3 .6)

To evaluate this matrix element, consider the first term in Eq. (8.3.6). The same procedure will then apply to the second term. Using Eq. (8.3.2), the first term can be rewritten as

where the parts of the matrix elements depending on the radial parts of the wave functions have been factored into separate terms. The next sim­ plification is to use the closure relationship in the sum over p and to make the assumption that the splittings of the multiplets in the excited-state con­ figuration are small compared to the energy splitting of the configurations. The latter assumption means that Ep is independent of the J quantum num­ ber of the intermediate state. The angular part of the above expression then becomes

L ( q)a I ( C�1 l ) ; I q)p ) ( q)p I ( c�tJ ) ; I q)b )

d'S" L" J"M7i

(

-

l )P+q+" ( 2A

+

1)

(�

A

-p

q

�) ( q)a I � [( C�1 l c�tJ )�p q] ; l q)b )

·

Here the 3-j symbol has been used for the angular momentum coupling. Note that A in this expression is being used as one of the indices and does not represent wavelength. At this point it is possible to use the tensor oper­ ator formalism discussed above to define a product operator (8.3.7) where Vkq is the normalized angular tensor operator and Ckq is a normal-

ization matrix element. The above expression can then be written as

L (¢a i ( C�1 ) ) ; I ¢p) ( ¢p i ( C�t) ) ; l ¢b )

rl'S" L" J"M;i

( - l )p+q+f+t (2A + x

1 ) (2/ + 1) (2/ + 1 )

(1

q

A -p - q

t P

)

{ ; ; � }u1 1 c(J) ll l) (l ll c(t) llf) (¢a ll u��-q ll ¢b ) .

Assuming that the energy denominators in Eq. ( 8 . 3 .6) are approximately equal Ea - Ep Eb - Ep 11E, the two terms in the expression for the os­ cillator strength are the same except for their 3-j symbols. However, because of the symmetry of 3-j symbols,

(1

q

A

-p - q

t

p

)

(

( - 1 ) 1 +2+ t pt

A

-p - q

)

1 ,

q

(8 . 3 . 8)

the two terms are equal if A is even and they cancel each other if A is odd. As stated above, for 4f electrons A 2, 4, and 6 (which also follows from the 6-j symbol in the above expression. The oscillator strength is then

8n2 mc fed -_ X 3hA x

[

L A 1p (2A + 1 ) ( - 1 )P+q p, t 2=2,4,6

c

q

-p - q

where

( t , A)

2 L (- 1 l+ 1 (2f + 1 ) (2/ + 1) l,n

X

{ f1 AI ft }

(! II c (l) Il l) (I ll c (t) II!) (4r II r ll nl) (nl l l r1 l l 4r ) . 11E

(8 . 3 . 10)

In dealing with room-temperature spectroscopic measurements of ions in solids, it is generally not possible to resolve transitions between individual Stark components. For this case it can be assumed that all Stark com­ ponents of the ground-state manifold are equally occupied and the sum over the Stark components can be evaluated. For unpolarized light, the sum over the components of the dipole moment operator and the crystal-field operator can be performed. The sum over the components of the 3-j sym­ bols gives (�) 1/2 [! J + 1] 1/2(! t + 1 ) 1/2. Thus the expression for the oscil­ lator strength of a transition between two multiplets in the Judd-Ofelt

8.3. Radiative Transitions: Judd Ofelt Theory

313

formalism is

(8.3. 1 1 ) where n, = (2A + 1 ) L 1A tr i 2
p ,t

(8.3 . 1 2)

Using the triangular inequality for angular momentum coupling,

1 1 - A I :: t :: I I + A I . Therefore, for A = 2, t = 1 , 3; for A = 4 , t = 3 , 5; and for A = 6, t = 5 , 7. Since 1.1 = ± 1 the intermediate configurations must in­ volve the 5d or 4g single-electron states. The spin selection rule AS = 0 and the conditions I AJ I , I AL I :: A also hold. The J = 0 J ' = 0 transition is forbidden and if either J or J ' are 0 then I AJ I must be even. The 3-j symbol involving M1 and M; give the selection rule I AM1 1 = p + q. For polarized light, q = 0 for polarization and q = ± 1 for n polarization. The value of +-+

p

a

is determined by the crystal-field point-group symmetry. For example, for D3h symmetry the relevant crystal-field parameters are A 33 , A 5 3 , and A73 so IAMJ(rr) l = 3 or IAMJ( n ) l = 2 , 4. The requirements on the L and S quantum numbers only hold rigorously for Russell-Saunders states. Since the energy levels are more accurately described by a linear combination of Russell-Saunders states, the conditions on the total angular momentum quantum number J are more accurate. The n, are intensity parameters that are found by fitting the predictions of the Judd-Ofelt theory to experimentally obtained absorption spectra as discussed below. This procedure requires knowing the values of the matrix elements of the unit angular tensor operator. These values are calculated theoretically using known wave functions for the rare-earth ions. The results of this procedure have been tabulated and the values of the required matrix elements can generally be found in the published literature. 6 Since they de­ pend only on the wave functions of the initial and final states, the values of (a ii U (") lib) do not vary significantly from one host to another or from one type of crystal-field site to another. The n, parameters reflect the strength of the configuration admixing and thus depend on the radial integrals of the configuration interaction, the energy separation of the configurations, and the odd harmonic terms in the crystal-field expansion. These parameters are sensitive to the local crystal-field environment. Therefore the n, parameters vary significantly from host to host and from one type of site to another within the same host. Values for (a ii U (") l i b) are given in Tables 8.7. The values of the Judd-Ofelt parameters for Nd : YAG are given in the follow­ ing section and compared to the values of n, for Nd3 + in other host mate­ rials in Chap. 9. Equation (8.3. 1 1 ) can be used with the various expressions for the oscil­ lator strength to describe other spectroscopic parameters in the Judd-Ofelt

314

8. Y3 Als01 2 : Nd3 + Laser Crystals

2 TABLE 8.7. Values of (a I I U (J.) l l b) for several important 3 transitions of Nd + ions ( Ref. 6) .

Transition

1 Uz l 2

I U4 1 2

I U6 l 2

%;z to 4F3;2 4Fs;z 2H9;2 4F111 4S3;2 4F9;2 2H1 112 4Gs;2 2G112 4G112 2Kn;2 2G9;2 4G9;2 4G1 1 ;2 2K1s;z 2D3;2 2Pl/2 1Ds;2 2P3/2 4D 3/2 4D 112 4F312 to %12 4/11/2 4/13/2 4/1 5 /2

0 0.0008 0.0091 0.0010 0 0.0010 0.0001 0.8970 0.0740 0.0550 0.0068 0.0010 0.0046 0 0 0 0 0 0 0 0 0 0 0 0

0.2296 0.2355 0.0076 0.0424 0.0024 0.0094 0.0026 0.4094 0.1 803 0.1 570 0.0002 0.0 1 53 0.0605 0.0054 0.0053 0.0196 0.0398 0.0002 0.0010 0.1960 0.2580 0.2296 0.1423 0 0

0.0563 0.3977 0. 1 17 1 0.4245 0.2337 0.0398 0.0103 0.0353 0.03 1 0 0.0550 0.03 1 2 0.0141 0.0405 0.008 1 0.0144 0.0002 0 0.0020 0.0010 0.01 70 0 0.0563 0.4070 0.21 1 7 0.0275

formalism. For electric dipole transitions, the line strength parameter is given by

sed

1 1 (! 1 P i i) l 2 e2

f ed 8n mcx

L = 4

-' 2 , ,6

Q_,

l (4r aJ I I U (-') II 4r a'J') I 2 2J + 1 (8.3. 13)

The integrated absorption coefficient is pn e2 2 d J.J. 2 �3 r

J a(J.)dJ.

3c m n

8pn3 lx

=

3chn 2 =L -' 2 ,4 ,6 8pn3 AX sed 3chn2

n,

l (4r aJ I I U (-'l ll 4r a'J') I 2 (2J + 1 )

·

The coefficient for spontaneous emission can be expressed as

(8.3. 14)

8.3. Radiative Transitions: Judd Ofelt Theory

315

(8.3. 1 5) Using the relationship between the Einstein A coefficient and the radiative lifetime of the transition,

1

A ab· r� L b

-

(8.3 . 1 6)

The branching ratio for a transition between a metastable state and one of several specific final states is given by the expression fJ

ab

A ab A ab raA ab· b

2:

r

(8.3 . 1 7)

The summation is over possible final states of transitions from the meta­ stable state. The spectroscopic quality parameter is given by1 0

(8 . 3 . 1 8) This is related to the intermultiplet branching ratios. For small values of x the 4F3;2 -- 4/1 1 ;2 branching ratio is large, while for large values of x the 4F3;2 -- %;2 branching ratio is large. Values of x have been found to vary from 0.22 to 1 .5 for neodymium in different host materials as discussed in Chap. 9. Equations (8.3. 1 1 )-(8.3. 1 8) can be used to analyze experimental spectro­ scopic data (absorption spectra and fluorescence lifetime measurements) to obtain the parameters needed to theoretically describe the laser properties of rare-earth-ion laser materials. The absorption spectra analyzed by Judd­ Ofelt theory are generally obtained at room temperature because of the as­ sumption that all MJ levels of the ground state are equally populated. In some cases the crystal-field splitting of the ground-state multiplet can be quite large, thus causing an uneven population distribution of Stark compo­ nents even at room temperature. However, since each Stark component is a linear combination of MJ states, the Judd-Ofelt theory works well in most cases. The situations in which the Judd-Ofelt theory does not give pre­ dictions consistent with experimental observations are those in which the 4/N 5d configuration is at low energy (like Pr3 + ) so the assumption that the energy difference denominators in Eq. (8.3.6) are the same is not very good, or those in which higher-order electron-radiation or electron-lattice inter-

8. Y3 Al50 1 2 : Nd3 + Laser Crystals

316

actions make important contributions. Some of these exceptions can still be treated within the formalism of the Judd-Ofelt theory as discussed below. For most important laser transitions, the forced electric dipole treatment described above is appropriate. However, some transitions of rare-earth ions in solids cannot be described by forced electric dipole transitions, and in these cases higher-order electron-radiation interaction operators must be used. For example, the magnetic dipole moment operator is defined by

eh -2me

M

I

+ 2Si ) ·

This is an even-parity operator and thus transitions between states of the

4fn configuration are allowed. The expression for the oscillator strength for fmct

magnetic dipole transitions is

4n2x' I (A I M I B) I 2 me e 2 2n 1 1 r (4 a1MJ I L + 2S I4r a 111M; ) 1 2 hme X

(8.3. 1 9)

The Lorentz local-field factor for magnetic dipole interactions is given by n3 . Applying the Wigner-Eckart theorem to the matrix element gives

X1

fmct

=

[(

(

)

]

2 n2 x 1 k 11 (r a1 I I L 2S II r d111) 2 ' - 1 / M' -M + J q MJ1 hme I

As before, assuming isotropic light allows the summation over the three po­ larization directions and assuming equal populations allows averaging over all values of MJ. This reduces the 3-j symbol to (W /2 [ 1 /(21 + 1 )] 1 /2 . The magnetic dipole oscillator strength is then given by 1mct

2 n2 x 1 l ( r a1 II L + 2S I Ir d111) 1 2 21 + 1 - 3hme

(8.3.20)

Following the procedure given above for electric dipole transitions, the ex­ pressions for spectroscopic parameters involving magnetic dipole transitions are srnct

=

2me

(r a1 II L + 2S I Ir d111) 1 2 (21 + 1 )

(8.3.2 1 )

and

(8.3.22)

8.3. Radiative Transitions: Judd Ofelt Theory

317

The reduced matrix elements for magnetic dipole transitions have been tabulated. Generally these transitions are more than an order of magnitude weaker than forced electric dipole transitions. The selection rules for magnetic dipole transitions are 11 = 0, tiS = 0, tiL = 0, til = 0, ± 1 ( no 0 +- 0 ) , tiMJ (a) = 0, tiMJ (n) = ± 1 . In some cases there are contribu­ tions to a transition from both electric dipole and magnetic dipole inter­ actions. For these cases the integrated absorption cross section and peak induced emission cross section are given by )

8 n2 m v- 1 ed d (8.3.23) mc2 3h(2J + 1 ) n2 (xS + X sm ) , 64n2 v3 Ap ed . d (8.3.24) a ( Ap ) 8 ncn2 tiA.err 3hc3 (2J + 1 ) (xS + X sm ) where v is in wave numbers, Ap is the peak emission wavelength, and tiA.err

J

_

a(v

_

dv =

ne2

1

I

=

is the effective linewidth of the transition. Similar procedures could be followed for higher-order interactions such as electric quadrupole. How­ ever, these transitions would be extremely weak and not important for laser applications. Another mechanism for mixing configurations of even parity into the 4fN configuration is through the electron-phonon interactions produced by lattice vibrations that are nontotally symmetric. This can be expressed in terms of the modulation of the crystal-field parameters by the normal mode of vibration represented by the coordinate Qj vi ! = mt

oA tp Q · D ( tl . o Qj J p

(8.3.25)

The wave functions are now represented by product functions I A n ) = l A) I n), where l A) represents the electronic part as before and I n) represents the vibrational part with n being the phonon occupation number. Substitut­ ing these expressions in Eqs. (8.3.3)-(8.3.5) instead of the static crystal-field interaction operator and electronic wave functions gives the oscillator strength for a vibronic-forced electric dipole transition with isotropic light and averaged over MJ as

(8.3.26) where nvib = 2

(2A. + 1 )

I 1

oA tp 2 <1>2 (t A.)p (n) l (n 1 Qj l nl) l 2 ' (2t + 1 ) ' , aQj

(8.3.27)

where p (n) is the probability that the system is in vibrational state n. These expressions for vibronic transitions have the same form as those in Eqs. (8 . 3 . 1 1 ) and (8.3. 1 2).

For host materials where the transitions among individual Stark com­ ponents cannot be resolved in room-temperature spectra, the Judd-Ofelt parameters n,� are treated as phenomenological parameters determined from least-squares fits of the line strengths measured in the absorption spec­ trum. For this procedure the reduced matrix elements are taken from the published values that have been calculated theoretically. 1 0 This works well since in most cases they show little dependence on the local environment. The major uncertainties in the values of n,� are associated with the problems of knowing accurately the doping concentration. In some cases the dopant ion can occupy more than one crystal-field site in the host material and this complicates the problem further. Additional errors can be present when the refractive index and its frequency dependence are not well known. Another problem in determining values of n,� is separating the contributions from static and vibronic interactions. This can be done through temperature­ dependent measurements of the line intensities since static interactions are generally independent of temperature while vibronic interactions exhibit strong temperature dependences. Comparative studies of different types of materials having different vibronic coupling strengths can also be useful for this. Theoretical calculations of Judd-Ofelt parameters have generally not been very successful for the same reasons that first-principle crystal-field calculations are not very accurate. First, there is a lack of exact knowledge of the radial wave functions for the ions in a host matrix environment, and second, the point-charge model for ligands is not a complete description of the electron-ligand interaction. Empirically it is found that the magnitudes of the Q,� parameters decrease with increasing numbers of 4f electrons. The Judd-Ofelt parameters also vary with local environment due to changes in the A1p crystal-field parameters associated with changes in the charges and positions of the ligand ions. Q2 is found to be more strongly dependent on the environment than Q4 or Q6 . This is due in part to the fact that the mag­ nitudes of the Judd-Ofelt parameters vary with electron-ligand distance as R- (2'+2) where t = A. ± 1 . Thus the A. = 2 parameter involve stronger ligand interactions that the A. = 4 or 6 parameters. There are certain transitions associated with specific host-ion material combinations that are much more sensitive to changes in local environment than other transitions. These are called hypersensitive transitions. These are transitions that have large values of the reduced matrix element (A I I U (2) II B) and obey the selection rules I AJ I � 2 , I ALI � 2, and AS = 0. This results in the value of Q2 being very sensitive to the local environment, but not the values of Q4 or Q6 . One example of a hypersensitive transition is the 4!912 .. 4G5; 2 transition of Nd3 + occurring near 1 7.3 x 1 0 3 cm- 1 . One ob­ vious physical property that can cause hypersensitive transitions is the pres­ ence of the A 1p crystal-field parameter that occurs in Q2 but not in Q4 or Q6 . This parameter is nonzero only in certain site symmetries: Cs, C1 , C2 , C3 , C4 , C6 , C2v, C3 v, C4v, and C6v · Thus hypersensitive transitions should be very strong in hosts where the ion substitutes for a host constituent in a position

with one of these symmetries, and should be weak in other hosts. As an ex­ ample the hypersensitive transition for Nd3 + cited above is very strong in Y2 0 3 where the ion is at a site with C2 symmetry, compared to its intensity in Y3 Al s 0 1 2 , where the ion is at a site of D2 symmetry. Unfortunately, there are a number exceptions to this simple explanation of hypersensitivity. These may be associated with dynamic coupling to vibrational modes hav­ ing A 1p-type symmetry, or with more complex electron-ligand interactions leading to enhanced quadrupole transition strengths. It is not possible to completely understand all hypersensitive transitions within the Judd-Ofelt formalism. When the transitions between specific Stark components can be resolved, the full crystal-field approach must be used treating the product A1p as phenomenological parameters. The local crystal-field symmetry determines the nonzero A1p components. There have been some cases for which the parameters have been determined theoretically. 1 1 This approach has the same problems as other crystal-field theory calculations.

As an example of the optical spectra of a Nd3 + laser material, consider the properties of Nd : YAG. The garnet host is a cubic crystal with space group Ia3d. It has high mechanical strength, good chemical stability, and the abil­ ity to be synthesized in large sizes with high optical quality. It has high thermal conductivity and is hard enough to hold a high optical polish. Typ­ ical Nd concentrations for laser crystals is 1 .0 at. % or 1 .38 x 1020 cm- 3 . Stress rupture occurs between 1 .3 and 2.6 x 10 3 kg/cm3 . The main laser transition occurs at 1 064 nm due to the 4F3 12 � 4Iu;2 transition. Since the terminal level is well above the ground-state multiplet, Nd : YAG operates as a four-level laser system. The width of the laser transition is 4.5 A and the stimulated emission cross section is 6.5 x 10- 19 cm2 . The index of refraction at the wavelength of the laser transition is 1 .82, and the scattering loss co­ efficient for a typical laser crystal is 0.002 cm- 1 . There is a second prominent laser transition for Nd : YAG that originates on the same metastable state but terminates on the 4!1 3 ;2 level. This occurs at 1 338 nm and has a cross section that is about four times smaller than that of the main laser line. 1 2 The slope efficiency of this laser transition i s somewhat less than that of the main laser line which is consistent with the lower energy of the photons in­ volved in the transition. The threshold for lasing is higher, the gain lower, and the maximum output power is lower for the 1 338-nm emission as com­ pared to 1064-nm emission. Both lasing transitions can be operated in cw, pulsed, and Q-switched modes. For Q-switched operation, the 1 064-nm line typically provides pulses with temporal widths between about 12 and 75 ns while the 1 338-nm line typically provides pulses between about 100 and 400 ns.

FIGURE 8.4. Local site symmetries for a Y3 Als01 2 crystal. (from Ref. 1 3) .

Figure 8.4 shows part of the garnet crystal structure. 1 3 The yttrium ion is surrounded by eight oxygen ions in the shape of a distorted cube. The Nd3 + ions substitute for Y3 + in YAG without the need for charge compensation. The larger size of the Nd3 + ion results in polyhedra with sides that are greater in length than those of the Al3 + polyhedra. This distorts the lattice and thus limits the maximum doping concentration to several atomic weight percent. The lattice strains that are introduced by doping affect the prop­ erties of the optical spectra as discussed below. The local site symmetry is Dz . The energy-level diagram for Nd3 + in YAG is given in Fig. 8.3. The absorption spectra of Nd : YAG in the visible and infrared spectral re­ gions14<•>· 1 5· 1 6 are shown in Figs. 8.5 and 8.6. The fluorescence spectra14<•>· 1 7 are shown in Figs. 8.6 and 8.7. The spectral properties are tabulated in Tables 8.8-8.12. The energy levels are listed in Table 8.8 and the reduced normalized tensor matrix elements for the transitions are given in Table 8.7. There is a relatively good fit between the experimental results and the pre­ dictions of parametrized crystal-field calculations as described above. 1 8 The Nd3 + absorption transitions of interest for optical pumping appear as a series of lines throughout the visible spectral region. At wavelengths less than about 3500-A strong host absorption takes place. The average wave­ length, integrated absorption coefficient, and line strengths calculated from

>- O.J !:

�0 "' z �

g0,1

FIGURE 8.5. Absorption spectrum of Nd3 + in YAG 2> in the visible spectral region [Ref. 14(a)].

Y AG : Nd

1-b

T

=

(ass Ofo)

295 K

0.6

�

·

0.2

16

1a

20

22

24

38

40 44

).(103 nm)

y (102cni1)

42

4 r 13/2

411312

sa

eo

4 I1512

ee

•1,$1

6B

(A)

(B)

FIGURE 8.6. (A) Absorption and ( B) fluorescence spectra of Nd3 + in YAG in the infrared spectral region. ( Reprinted from Ref. 1 6 by permission of Springer-Verlag.)

�

..

.c

§

i'

0.6

al 0: 0.4

z

::;)

�

aa

-

w t-.

+

0.

z

;:

�

c:;< � 0

i

><

tTi

00

�

Flu oresc ence 'F312 ( 1 ) -- ' 1 912 ( 1 to 5)

I

?;­

I

I

4

Fluorescence

ii c

�

.5 ., u c .. u "'

I I 2 3

I 4

I

?;­ 'ii c !

4 FJn l2 1 - 1 91 2 11 to 5 1

.!:

0

0

:2

:2

.. >

.. >

�

.ii

-.; a:

-.; a:

9 20 9000 Wavelength. A

I I

I I

?;­

'

I

I

Fluorescence

ii c ..

1 2

4 F312 (1 1

I 4

Fluo rescence

., u c .. u VI ..

� 0

II

14 axl

14 50

Fluofoscenco 4F312( 1 l - 4 11512 I I

Auorasconce 4 F312 l 2 1 - 411512

l

I

5 6 7

34

Wavelength, A

c ..

-�

4 F312 ( 1 1 - 4 1 1512 (1 to 7 1

13 50

?;­ 'ii

.. u c .. u

II

II 12

I

.5

I

__;U1\)l"JL

13 00

4 1 1112 (1 to 5 1

II

c�

9400

Fluorescence 4 F312 1 2 1 -'I,112( 1 to 6 1

4 F312121-4 1 1'.1/2 ( 1 to71

II

.. u c .. u "' ..

�

8800

F l u o r escence

D

II

l

I

(1 to 51 II I

;\

(1 to 51

0

:2

:2

-.;

-; =

..

.. . >

.� ..

-� a:

10 60

10 BO

Wavelength, A

11 20

11axl

Wavelength, A

(A)

1 1 .4

1 1 .1

YAG : Nd T = 15 K

Aex = 5 8 8 9

A

Aex= 5 8 84

A

>­ I-

Il z lJ.J 1-­ z

0

9 0 00

FIGURE 8.7. Fluorescence spectrum of Nd3 + in YAG. (A) Fluorescence emtsston from the 4F3;2 level ( Ref. 1 7) . ( B) Low-temperature site-selective excitation of 4F3;2 .. 4 19;2 fluorescence [Ref. 1 4(a)]. 322

TABLE 8.8. Experimentally observed energy levels for YAG : Nd3 + at 77 K ( Refs. 1 3 , 1 7) . Free ion multiplet

%;2 4/l l/2 4/13/2 4/1 5 /2 4F3/2 4Fs;2 2H9 /2 4F112 + 4S3;2 4F9/2 2Hl l/2 4Gs;2 2G112 4G112 2Kn;2 + 2G9;2 4G9/2 4Gl l/2 2KI 5 /2 2D 3;2 2PI /2 2Ds;2 2P3/2 4D3/2 4D 112

Crystal field levels (cm- 1 ) (crystal quantum number "ji)

0@ , 1 30(�) , 199@ , 308(�) , 857(�) 2002(�) , 2029(�), 21 10{!) , 2147(�) , 2468(!) , 2521 (�) 3922(!) , 3930@ , 4032(!) , 4047(�) , 4435 (!) , 4442@ , 4498 @ 5758 (!) , 5814(�) , 5936(!) , 5970@, 6570(!) , 6583@ , 6639(!) , 6734@ 1 1427(�) , 1 1 5 1 2(!) 12370(�) , 12432(!), 12519@ 12575 (!) , 12607(!) , 12623@, 128 1 9(!) , 12840@ 1 3363(!) , 1 3433 @, 1 3563(!) , 1 3572@ , 1 3596(!) , 1 3633G) 14626(!) , 14678@, 14793 (!) , 148 1 9Gl , 149 1 6(�) 1 5743 GJ , 1 5838(�) , 1 5870Gl , 1 5957(!) , 16103Gl , 1 6 1 1 9(!l 16849@, 1 6992(!) , 17047@ 17241 (!) , 17258{!) , 1 7322@ , 17575 @ 1 8723 (!) , 1 8822(!) , 1 8843Gl , 1 8986@ 1 9 1 54(!) , 19194(!) , 19294(!) , 19470(!) , 19543 (!) , 1 9596(!) , 1 9620(!) , 19651 (!) , 19814(!) , 20048(!) 20730(!) , 20773(!) , 20790(!) , 20803(!) 20962(!) , 21029(!) , 21080(!) , 21 1 10{!) , 21 1 59(!) , 2 1 1 62(!) 21 522(!) , 21 593(!) , 21661 (!) , 21 697(!) , 21767(!) , 2179 1 (!) , 21 872{!), 21906(!) 22036(!) 23 1 55(!) 23674(!) , 23764(!) , 23849(!) 25994(!) 275 17(!) , 27670(!) 27809(!)

the equations given above and the experimental data are listed in Table 8.9. The infrared absorption characteristics are important in understanding the terminal state decay of the laser transition and cross relaxation interactions with other Nd3 + ions. The positions, widths, cross sections, branching ratios, and line strengths of the fluorescence transitions are given in Table 8.10, while the fluorescence lifetime and Judd-Ofelt parameters are given in Table 8. 1 1 . The fluorescence spectrum consists of four groups of transitions in the near-infrared originating on the two crystal-field split components of the 4F3;2 metastable state and terminating on the various Stark levels of the 4/9;2 1 5; 2 multiplets. Several of these transitions can provide laser emission. The fluorescence lifetime varies with temperature and neodymium concen­ tration as discussed below. Excited-state absorption ( ESA) from the metastable state has been ob­ served to several terminal levels. Table 8.12 gives the line-strength parame­ ters and wavelengths of some of the important transitions. 1 9 Note that photons in the main lasing transition at 1 .06 pm as well as photons from

324 TABLE

8. Y3 Al501 2 : Nd3 + Laser Crystals

.<. (A)

Term

411 1/2 4113/2 411 5/2 4F3/2 4Fs;2 + 2H9/2 4F1;2 + 4S3f2 4F9/2 2H1 112 4Gs;2 + 2G1;2 2K13;2 + 4G112 + 2G9/2 2K1s;2 + 2G9;2 + 2D3/2 4G11;2 2P 1/2 2Ds;2 2P3/2 4D 3f2 + 4D s;2 + 2!1 112 + 4Dr;2 + 2L1 5 /2 2113/2 + 4D112 + 2!1112 + 2L1112 2H9/2 + 2D3/2 2Hr r;2 + 2Ds;2 2Fs;2 2F7/2 TABLE

J k(-'.) d.<. ( A/em)

s��, ( lo- 21 cm2 )

890 89 109 493 42 1 33 6.0 222 1 57 19.6 1 6.6 5.8 0.70 0.46 1 16 3.9 3.0 1 .6 2. 1 0.92

21.6 3.6 7. 1 35.0 33.0 2.7 0.54 2 1 .7 1 6.6 2.4 2.0 0.74 0.09 0.06 1 7.9 0.64 0.51 0.30 0.48 0.20

8.9. Room-temperature absorption intensites for YAG:Nd3 + ( Ref. 1 9) .

50,000 24,000 1 6,300 8,800 7,900 7,500 6,800 6,250 5,800 5,300 4,750 4,600 4,300 4,200 3,850 3,580 3,330 3,050 2,950 2,630 2,550

8. 10. Spectral parameters for YAG : Nd3 + ( Ref. 19).

0.2 X 1020

2.7 X 1020

5.0 X 1020

X

250

0.54

other possible laser transitions can cause excited-state absorption as can pump beam photons at 532 and 800 nm . Note that 532 nm is the frequency­ doubled wavelength of the 1 .06-,urn laser emission. The line strengths were calculated using the Judd-Ofelt procedure. The strength for the excited-state absorption transition at the wavelength of the major laser line is close to the line strength for the fluorescence transition at this wavelength. It has been found that the gain and efficiency of cw lamp-pumped YAG : Nd3 + lasers are increased if pump radiation at wavelengths less than 500 nm is filtered out. 20 One possible explanation for this observation is decreased excited­ state absorption, but this has not been proven. Excited-state absorption of pump photons has been shown to alter the slope efficiency of YAG : Nd3 + lasers for the case of monochromatic pumping. 2 1 This effect depends strongly on the wavelength of the exciting light and can involve both vi­ bronic transitions from the metastable state and electronic transitions from higher pump levels. Fluorescence emission originating on the 2P3 ;2 and 2P1 ;2 levels occurs in the blue-green spectral region due to these ESA processes.

8 .4. Example: Y3 Al501 2 : Nd3 +

325

TABLE 8 . 1 1 . Positions, widths, peak cross sections, and branching ratios for roomtemperature fluorescence transitions of YAG : Nd3 + ( Refs. 1 3, 1 6) . Transition

4F3/2 -- 4/9/2 b -- 1 a -> ! b -- 2 b -- 3 a -- 2 a -- 3 b -- 4 a -- 4 b -- 5 a -- 5 4F3 /2 -- 4/l l /2 b -- I b -- 2 a -- 1 b -->3 a -- 2 b -- 4 a -- 3 a -- 4 b -- 5 b -- 6 a -- 5 a -- 6 4F3/2 4/13/2 b -- 1 b -- 2 a -- 1 a -- 2 b -- 3 b -- 4 a -- 3 a -- 4 b -- 5 b -- 6 a -- 5 a -- 6 b -- 7 a -- 7 4F3/2 -- 4/15/2 --

). ( nrn )

�ii (cm - 1 )

up

( lo- 19 cm2 )

PJ.J'

pij

0.30

0.89

13 10 18 19 19 10 31 31 10 9

0.42 0. 1 1 0.13 0.42 0.35 0.24 0.05 0.12 0.48 0.51

1052. 1 1054.9 106 1 . 5 1064.2 1064.4 1068.2 1073.7 1077.9 1 105.5 1 1 1 1 .9 1 1 1 5.8 1 122.5

4.5 4.5 4.6 5.0 4.2 6.5 4.6 7.0 1 1 .0 10.2 10.6 9.9

0.95 0.06 2.50 3.00 1 .45 0.60 1 .65 0.77 0.16 0.36 0.42 0.40

1 3 1 8.7 1 320.3 1 333.5 1 335. 1 1 338.1 1 341 .9 1 353.3 1 357.2

4.0 4.6 3.0 3.3 4.0 6.0 4.0 4.0

0.95 0.23 0.44 0.54 1 .00 0.36 0.28 0.73

0.0183 0.0061 0.0073 0.0100 0.0243 0.0120 0.0062 0.0214

141 5.0 1427. 1 1432.0

8.5 7.0 10.0

0.20 0.08 0. 1 3

0.0099 0.0028 0.0066

1444.4

9.0

0.28

869.0 875.4 879.2 884.4 885.8 891 . 1 893.4 900.0 938.6 956. 1

SJ.J' ( 10- 20 cm2 )

0.032 0.012 0.0 1 5 0.035 0.065 0.037 0.004 0.035 0.026 0.037 2. 1 8

0.56 0.0382 0.0023 0.0799 0. 1275 0.0533 0.0340 0.0657 0.0463 0.0145 0.0297 0.0356 0.0328 0.14

� o.ot

1.16

0.0128 0.08

326

8. Y3 Al501 2 : Nd3 + Laser Crystals TABLE 8 . 1 2. Excited-state absorption from the 4F3;2 level of YAG : Nd3 + ( Ref. 20) . Terminal state

4G111 2G9;2 2P 1/2 1Ds;z 4D4/2

SJJ' (cm2 )

A. (.urn)

1 .90 x 2.19 x 3.oo x 4.oo x 2.10 x

1 .35 1 .06 0.88 0.80 0.532

+

+

w- 21 w- 20 w - 23 w - 23 w- 21

I

).nm)

(A)

025 �0.15 � 0.1

(B) FIGURE 8.8. Room-temperature ESA spectra for Nd : YAG ( Ref. 22).

This type of loss mechanism can be important in designing diode-pumped YAG : Nd3+ lasers. Examples of ESA spectra are shown in Fig. 8.8 for two spectral regions.22 The y axis in the plots is ln(Iu /lp) = (aasA - O"£sA)N*l, where lu and lp are the intensities of the probe beam transmitted through the sample of length I under unpumped and pumped conditions, and N* is the population of the excited state under pumped conditions. These spectra re­ solve the transitions terminating on different Stark components of the levels involved. Multiphoton absorption processes have also been investigated23 - 25 in YAG : Nd3 + . For excitation wavelengths in the visible spectral region, this allows the study of transitions to high-lying Nd3 + energy levels in the ultra­ violet region of the spectrum where the host material is opaque to normal single-photon absorption processes. This is the spectral region involving transitions to levels of the 4f2 5d 1 electron configuration. Broad bands were identified in the ultraviolet spectral region that may be associated with 5d levels. After multiphoton excitation, fluorescence originating on several dif­ ferent excited metastable states occurs throughout the visible region of the spectrum. For sequential, stepwise two-photon processes, the absorption cross sections for the second step in the transition were found to be of the same order of magnitude as ground-state absorption cross sections for the case where the final state of the transition is part of the 4f3 configuration, but were found to be over two orders of magnitude larger for transitions terminating on 5d levels. Configuration relaxation from 4f2 5d 1 to 4f3 was found to have a characteristic time of 2 ns. Two-photon absorption tran­ sitions with a virtual intermediate state can occur at the 064-nm laser going between the ground state and the 4G7;2 level. However, this process is too weak to be a significant loss mechanism for laser operation. Under high-resolution conditions at low temperatures with a narrow­ band laser excitation source, structure can be observed in the fluorescence spectrum of YAG : Nd3 + that cannot be explained through simple crystal­ field analysis. This can be seen in Fig. 8.7 where only four lines of the 4F3 ;2 -- 419;2 should be seen. Instead, each of the transitions appears as at least two lines. This structure is associated with ions in slightly different crystal-field sites, and the widths, positions, and intensities of the lines change with small changes in the excitation wavelength, as seen in the fig­ ure. This shows that even though the 4f electrons are shielded, they are still sensitive to changes in their local environment. These changes are caused by a variety of structural and chemical defects in the crystal that result in local strain fields. Two specific types of that defects that affect the Nd3 + spectra in YAG are hydroxyl impurities and striae from growth dynamics. 26 There is some evidence that color-center defects are produced in YAG crystals by the addition of Nd3 + ions. The defect properties of the material can be greatly improved through advanced crystal-growth techniques and treatments such as annealing and reactive atmospheric processing. These lattice defects can reduce the quantum efficiency of the material and the internal strains

1

328

8. Y3 Als012 : Nd3 + Laser Crystals

FIGURE 8.9. Fluorescence spectra of the 4F3;2 -- %; 2 transitions at 14 K for different host crystal compositions [from Ref. 14(b)].

11.

0 < 1 otm·1 )

11. 5 11 4

11.3

11.2

11.1

adversely affect laser operation. However, it has also been suggested that increasing the inhomogeneous linewidth of spectral transitions through the introduction of defects can improve the energy storage capabilities of laser materials. One example14 of doing this is substituting a fraction x of gallium ions for aluminum ions to form the mixed crystal Y3 (Al, xGax) s O J 2 . Figure 8.9 shows the variation of the fluorescence spectrum of Nd3 + near 800 nm as a function of composition for this host material. The peaks of the transitions shift to shorter wavelengths as x increases and the lines are broadest for the x = 0.50 composition. The shape of the spectral lines is consistent with a statistically random distribution of Ga ions replacing Al ions in the ten lattice sites surrounding neodymium ions in a garnet structure. The direc­ tions and amount of shift in the peak positions are different for different spectral transitions as shown in Fig. 8 . 1 0. This demonstrates the compli­ cated interactions between the Nd3 + ions and local lattice defects. Under strong pumping conditions, the optical properties of a material can change. The ESA processes discussed above is one example of this. Another is the change in the refractive index of the material. This can be associated either with thermal effects or with the difference in polarizability of the ac­ tive ion when it is in the metastable state versus the ground state. The ther­ mal effects are associated with the heat that is generated through radiation­ less relaxation processes both during pumping and lasing. The former is due to the fact that the pumping transitions terminate on levels above the meta­ stable state followed by decay down to the 4F3 ; 2 level, while the latter is due to the fact that the lasing transition terminates on the 4/1 1 ; 2 level followed by

YGc:G:Nd T RANSITIONS

575

580 585

500

595

). (nm )

870

YAG:Nd

880 BOO T RANSITIONS

FIGURE 8 . 1 0. Peak positions of absorption and fluorescence transitions of Nd3 + in garnet hosts with different compositions [from Ref. 1 4(b)].

decay to the ground state. The heating due to pump relaxation processes can be minimized through laser pumping into levels with the smallest possible quantum defect. The local heating associated with these processes changes the refractive index and this change remains until the heat has time to dif­ fuse away from the region surrounding the excited Nd3 + ion. The properties of YAG crystals controlling these processes are dnjdT 8.9 x w 6 K - 1 , the heat capacity Cp 0.59 1/g K, and the density p 4. 1 gj cm3 . The polarizability change of optically pumped Nd3 + has been charac­ terized by using degenerate four-wave-mixing techniques to establish and probe excited-state "population gratings" at described in Sec. 3.4. 2 7 The magnitude of the observed change in refractive index indicates that the off resonance 4/ -. 5d transitions make the dominant contribution the polar­ izability of the Nd3 + ions. Judd-Ofelt theory was used to calculate the po­ larizability change for free ions and the difference in the value for ions in solids was attributed to the change in the (4/ I r l 5 d) radial matrix element and in the position of the 5d energy level. The change in the refractive index induced by this polarizability change was found to be about 0.59 times the thermally induced change. This varies from host to host as discussed in the following chapter. The changes in the refractive index described above are important be­ cause no pumping configuration is spatially uniform. Thus pumping creates a nonuniform distribution of the refractive index in the laser resonator that results in beam bending, self-focusing, and wave-front distortions that are detrimental to laser operation. Both thermal lensing and population lensing can become problems at high pump levels. Electron-phonon interactions have a significant affect on the optical spectra of Nd3 + ions in Y AG. Factor group analysis of YAG crystal pre­ dicts 98 vibrational modes for k 0 phonons. Infrared and Raman spectra show vibrational transitions ranging between about 120 and 920 cm- 1 . This gives the energy range of phonons available for thermal line broadening, line shifting, and radiationless decay processes.

1

330

8. Y3 Al501 2 : Nd3 + Laser Crystals 1.0 0.8

� ·c:

::

-e 0

]

Q,.

0.6

0.4

•

25

0.2 0

100

300 400 zoo Phonon frequency [cm-1 1

500

-

600

FIGURE 8. 1 1 . Effective phonon density of states for Y3 Ais01 2 obtained from the vibronic sideband of Yb3 + ions. ( Reprinted from Ref. 28 by permission of Allerton Press.)

No vibronic sidebands have been observed for Nd3 + ions in YAG. How­ ever, vibronic spectra have been reported for both Yb3 + and Cr3 + ions in YAG. Since these sidebands can be used for effective phonon density of states, the results for Yb3 + are shown in Fig. 8 . 1 1 . 28 The widths and positions of the YAG : Nd3 + spectral lines vary with tem­ perature as predicted by the theories discussed in Chap. 5. Figure 8 . 1 2 shows examples o f the measured and theoretical fits for the widths and positions of a number of the absorption and emission lines. 29 The notation R, Y, and Z refers to 4F3;2 , 4 lu ;2, and 4/9;2 multiplets, respectively. Both two-phonon scattering terms and direct process terms make significant con­ tributions to the thermal characteristics of these lines. The line broadening is dominated by Raman scattering processes with an effective Debye temper­ ature of 500 K. Most of the lines shift to longer wavelengths as temperature is increased. However, several lines, such as the R 1 -- Z5 transition, shift to shorter wavelengths due to direct phonon decay processes affecting the terminal level of the transition. Direct phonon processes between the two crystal-field split components of the 4F5;2 level do not make significant con­ tributions to the thermal characteristics of the spectral lines because of the small magnitude of the splitting in the YAG host. The thermal shift of the main laser line is 0.04 cm- 1 /K near room temperature. It changes from about 1063.7 nm at 1 70 K to about 1 067.0 nm at 900 K. Since YAG : Nd3 + is a four-level laser system with the terminal level of the laser transition lying higher in energy than kT above the ground state, the pump power threshold is directly proportional to the linewidth of the transition. The high Debye temperature and weak electron-phonon coupling in this system results in a small linewidth as desired.

SHI FT

100

T (k)

!500

[•2cm'1 -zcm-•

100

0

200

300

T (K)

500

(B)

(A)

FIGURE 8 . 1 2. Temperature dependences of the widths and positions of spectral lines of Nd3 + ions in YAG ( Ref. 29).

There are three types of radiationless relaxation processes of great im­ portance to four-level laser applications: nonradiative decay during pump­ ing, nonradiative decay from the metastable state, and nonradiative decay from the terminal laser transition. The values of the decay rates predicted by the energy gap law for multiphonon decay processes discussed in Sec. 5.3 are given in Table 8 . 1 3 for some of the important transitions in Nd-YAG lasers. Using the energy gap law expressions given in Sec. 4.2, W C exp( and e WP / wr' , the data obtained on Nd-YAG gives c 9.7 X w-7 s- 1 ' !Y. = 3 . 1 X 10 3 em, and e = 0.045. The effective phonon energy for this material is found to be 700 cm- 1 • Multiphonon nonradiative decay of rare-earth ions is discussed further in the next chapter. The first type of radiationless relaxation process is the decay of energy to the metastable state after the initial excitation into higher-lying absorption bands. This involves energy gaps of the order of 1 000-1200 em - 1 . In hard crystals such as YAG with high-energy phonons the energy cascades down radiationlessly with almost 1 00% efficiency and no radiative leakage as long as the exci\tation is below the high-energy metastable states such as 2P3;2 and 2P 1 ;2 • This is not always true in other hosts. Measurements have been

- rxAE)

-

TABLE

8. 1 3 . Nonradiative decay rates for transitions in Nd-YAG predicted by the energy gap law for multiphonon relaxation ( Ref. 30) . ( These calculations used free-ion wave functions and the vibronic sideband of Yb3 + for the effective phonon density of states.)

Transition

4 G112 -- 2G112 4/1 1/2 -- 419/2 4/1 5 /2 -- 4/13/2 4/13/2 -- 4/1 1/2 2P3/2 -- 2D5 /2 4F3/2 -- 4/1 5 /2

�E (cm- 1 ) 1 148 1 145 1260 1400 2145 4693

w., ( s - 1 )

2 X 10 1 0 5 X 109 3 X 109 4 X 109 5 X 105 � 10- 2

made of the delay time between pumping into the 4G7 ;2 manifold and the fluorescence or laser emission from the 4F3 ;2 level. These resulted in a deter­ mining a relaxation rate of approximately 2 x 1 06 s- 1 . The largest energy gap in this cascade process is between the 4G7;2 level and the manifold made up of the 2G7;2 + 4G5; 2 levels and thus this is the bottleneck giving the longest contribution to the relaxation time. Using the simple multiphonon relaxa­ tion theory given by Eq. (4.2.24), WTIP W0 (n + l )P, where n is the phonon occupation number and p is the number of phonons participating in the re­ laxation process provides a good explanation for the temperature depen­ dencies of the radiationless processes for Nd-YAG. Fitting this model to the experimental results gives the average energy of the phonons participating in the relaxation process to be about 700 cm- 1 which indicates that the higher­ energy phonons in YAG are not strongly coupled to the impurity ions. In addition, Wo is found to obey the well-known exponential energy gap law and has a higher value in softer crystals that in Y AG, consistent with the results of vibronic, linewidth, and line-shift studies. The second type of important radiationless relaxation process is the decay from the 41" ; 2 manifold to the upper level of the %; 2 manifold since this determines how fast the terminal state of the major laser transition is de­ pleted. The energy gap for these processes ranges between about 1 1 50 and 1 300 cm- 1 • The decay rates are predicted by the energy gap law to be be­ tween 0.2 and 1 .6 x w s s- 1 for Nd-YAG, which is consistent with ex­ perimental observations at room temperature. 3 1 The third type of radiationless decay process is relaxation from the 4F3;2 metastable state. This decreases the quantum efficiency of the emission and represents a direct loss mechanism for laser operation. The determination of the quantum efficiency of the metastable state has proven to be quite dif­ ficult. For samples of YAG with low concentrations of Nd3 + so no ion-ion cross-relaxation interactions occur, the large energy gap (of the order of 4700 cm- 1 ) predicts a nonradiative decay rate several orders of magnitude =

8.4. Example: Y3 Al501 2 : Nd3 +

333

smaller than the radiative decay rate. This predicts a quantum efficiency close to 1 .0, which is consistent with the predictions of Judd-Ofelt analysis that yield a value of 259 ps for the radiative lifetime of the 4F3 ;2 level com­ pared to the measured value of the fluorescence lifetime of 250 ps. However, measurements of YJ usually produce values much smaller than this, indepen­ dent of which pump band between 350 and 900 nm is used for excitation. The decreased quantum efficiency has been shown to be sample dependent and is thought to be due to energy transfer to defects in the crystals. There are several types of ion-ion interaction processes that can be im­ portant for Nd-YAG lasers. One is sensitized pumping. Because the ab­ sorption transitions of Nd3 + in YAG crystals are sharp spectral lines, they are not very efficient in absorbing broadband flashlamp pump energy. One method of increasing the efficient use of pump light is to dope the material with Cr3 + ions to act as a sensitizer. The broad chromium absorption bands efficiently absorb the pump light and the question then is how efficiently is the energy transferred to the neodymium ions. The Cr3 + ions substitute for Al3 + ions in the garnet lattice without the need for charge compensation and without creating a significant amount of lattice distortion. At low concen­ trations they preferentially occupy the octahedral site. The excitation spec­ trum for the 4F3 ;2 -- 4/1 1 ; 2 emission of YAG : Cr; Nd is shown in Fig. 8. 1 3 along with the relevant energy levels of the two ions. 3 2 The broad absorp­ tion bands associated with transitions to the 4 T1 g and 4T2g levels of the chromium ions enhances the efficiency of pump light absorption. However, this is a strong field site for Cr3 + and thus there is fast radiationless decay to the 2Eg level which acts as a metastable state. The fact that chromium ab-

Nd3 +

•r,,

FIGURE 8. 1 3. Excitation spectrum for YAG : Cr;Nd. (After Ref. 32).

334

8. Y3 Als01 2 : Nd3 + Laser Crystals l50

0

I

I

0.1

1.0

I I I I I I I I I r-

0.8 0.6 0.4

I

0.5

! 2 � c:

..

0.2 1.0

C (at.Ofo)

2.0

5.0

0.0 10

FIGURE 8 . 14. Concentration quenching of the fluorescence intensity and lifetime of Nd3 + ions in Y AG. ( Reprinted from Ref. 1 7 with the permission of Springer­ Verlag).

sorption bands appear in the neodymium excitation spectrum indicates that energy transfer in occurring from Cr3 + to Nd3 + ions. The presence of energy transfer is confirmed by the fact that the neodymium emission in the double doped sample exhibits a double exponential decay curve with the fast com­ ponent having the decay time of Nd3 + in single doped samples and the long component having a 3.5-ms decay time associated with the Cr3+ emission. The mechanism causing the transfer has not been determined. The overall efficiency gained from chromium sensitization is estimated to be a factor of 2. However, this is not usually achieved because the quality of the double doped samples is generally much poorer than that of YAG : Nd3 + crystals. Energy transfer can also occur between Nd3 + ions and from neodymium ions to lattice defects. The dependence of the fluorescence lifetime and the fluorescence intensity on neodymium concentration1 7 is shown in Fig. 8. 14. The intrinsic lifetimes is about 250 f1 S and this is the measured lifetime for low Nd3 + concentrations. The fluorescence intensity increases monotoni­ cally with Nd3 + concentration in this region. However above a doping level of about 1 0 at. % neodymium, the fluorescence lifetime decreases signifi­ cantly and there is a similar decrease in the fluorescence intensity. The op­ timum concentration for laser operation is about 1 . 1 . at. %. This concen­ tration quenching is associated with the cross-relaxation ion-ion interaction shown in Fig. 8. 1 5( B) . In this process on ion is initially in the metastable state and the other in the ground state, while after the transfer both ions are in the 4/15;2 level and lose their energy radiationlessly as they relax to the ground state. In addition, it has been shown that energy migration among Nd3 + ions through the energy-transfer transition shown in Fig. 8 . 1 5(A) en­ hances the efficiency of the quenching transfer to defect sites in the lattice.

8 .4. Example: Y 3 Al501 2 : Nd3 +

335

1[:''":[ '

'

'

'

�

4r151 '

'

'

419 / 2 (A)

FIGURE 8 . 1 5. Energy-transfer interactions between Nd3 + ions in YAG.

This is a resonant interaction where the excitation energy moves form the

4F3 ;2 level of one ion to the 4F3;2 level of another ion without loss of energy.

The migrating energy can find defect centers where efficient nonradiative re­ laxation processes occur. The electric dipole-dipole energy transfer rate given by Eq. (5.2. 19) can be rewritten in the convenient form

_ 2nQ0"2 A.1 N2

W:

(8.4. 1 ) 3 (2nn) 4 r! ' where N is the concentration of Nd ions, r/ is the radiative decay time, 0"2 is cr

the peak cross section of the absorption transition, A J is the wavelength of the emission transition, and Q represents the spectral overlap of the two transitions given by Q [1 + (JvjAv)r 1 . Here Av is the average half-width of the two lines and Jv is their separation. Each pair of resonant crystal-field transitions contributes to the total cross-relaxation rate depending on its values of Q, 0"2 , r/, and A. 1 . The quenching rate for Nd-YAG is found to bel s Wcr

=

N2 1 .0 X 10 42 - . r!

This expression can be used to explain the concentration quenching of the fluorescence lifetime. Converting from energy transfer rate to transfer time, the concentration dependent fluorescence decay time is given by

r(N)

r!

. 1 + 1 .0 X 10 42 N2

(8.4.2)

Equation (8.4.2) gives a good fit to the data shown in Fig. 8. 14, which shows that cross relaxation is the rate-limiting step in the quenching process. At concentrations smaller than 1% the fluorescence quenching rate varies linearly with Nd concentration instead of quadratically as predicted by the cross-relaxation mechanism described above. This indicates that the migra-

336

8. Y3 Al501 2 : Nd3 + Laser Crystals

tion of energy to defect quenching sites is the rate limiting step in the quenching process. As discussed in Chap. 6, this process can be described as a random walk of a localized exciton on the lattice of Nd ions with each step caused by an electric dipole-dipole interaction. This exciton dif­ fusion process has been characterized using high resolution, time-resolved site-selection spectroscopy techniques. Narrow-line tunable laser excitation is used to selectively excite Nd ions in a specific subset of crystal-field sites and the spectral evolution with time is monitored as the energy is transferred to Nd ions in different types of sites. The results are consistent with a multi­ step migration process in which both the migration and trapping steps are assisted by two-phonon processes. 1 5•26•33 The temperature dependence of the transfer rate increases exponentially with an activation energy equal to the splitting between the ground state and the first excited state of the ground-state manifold. This indicates that the phonon assisted processes in­ volve resonant transitions between these two levels. The transfer rate in­ creases by about two orders of magnitude as 50% of the AI ions are replaced by Ga ions. This can be attributed to an increase in the Nd transition matrix elements. The exciton diffusion coefficient in pure YAG was found to be 3.5 x 10- 1 0 cm2 s- 1 , which is similar to diffusion coefficients for excitation energy in other rare-earth-doped crystals. This type of energy diffusion among the optically active ions has a direct effect on lasing characteristics since it smooths out the spatial distribution of the excitation energy. This decreases the effects of spatial hole burning and thus increases the efficiency of single frequency operation. Estimates of the energy diffusion coefficient from laser operation parameters has been esti­ mated to be 5 x 1 o- 7 cm2 s 1 , which is significantly greater than that mea­ sured directly from site-selective studies of energy transfer. 34 This may imply that there is some resonant spatial migration of energy taking place as well as the spectral diffusion. However, attempts to measure resonant spatial migration through four-wave-mixing techniques have not been successful. After the Nd excitation energy has migrated to a trapping site, the energy can be dissipated radiationlessly through energy transfer to the trap. There are several different types of trapping centers that can be effective in quenching the Nd luminescence, and it is generally difficult to identify the dominant type of quenching center in a specific sample. One example of a quenching center is another optically active ion such as Yb3 +, which can efficiently accept energy transfer from excited Nd3 + ions. Another type of quenching center for Nd excitation is a lattice defect such as hydroxyl ions. 35 A third type of quenching center is a Nd ion in a crystal-field site that is perturbed by local strains in such a way that cross-relaxation quenching processes are enhanced. Photon-echo experiments have shown that phonon-induced relaxation properties of Nd3 + ions in crystals such as YAG are very sensitive to the type of nearest-neighbor environment. 36 Even very small concentrations of these three types of defect centers can be effec­ tive in quenching the Nd luminescence if there is efficient energy migration.

References

337

The general spectral characteristics of Nd3 + are the same in other host crystals and glasses. This is because the outer-shell electrons shield the inner­ shell 4f3 electrons that make the major contributions to the observed optical properties. However, the symmetry and strength of the crystal field plus other host properties cause differences in some of the details of the spectra. Chapter 9 discusses some of the properties of Nd3 + ions in several different laser materials and how they lead to very different lasing properties. References

1 . G.H. Dieke, Spectra and Energy Levels of Rare Earth Ions in Crystals ( Wiley­ Interscience, New York, 1 968). 2. S. Hiifner, Optical Spectra of Transparent Rare Earth Compounds (Academic, New York, 1 978). 3. B.G. Wybourne, Spectroscopic Properties of Rare Earths ( Wiley-Interscience, New York, 1 965). 4. G.H. Dieke and H.M. Crosswhite, Appl. Opt. 2, 675 ( 1 963). 5. B.R. Judd, Phys. Rev. 127, 750 ( 1 962). 6. W.T. Carnall, P.R. Fields, and K. Rajnak, J. Chern. Phys. 49, 441 2, 4424, 4443, 4447, and 4450 ( 1 968). 7. M. Rotenberg, R. Bivins, N. Metropolis, and J.K. Wooten, Jr., The 3-j and 6-j Symbols ( Technology, Cambridge, MA, 1 969) . 8. C.W. Nielson and G.F. Koster, Spectroscopic Coefficients for pn, dn, and r Configurations (MIT Press, Cambridge, MA, 1 964) . 9. G . S . Ofelt, J. Chern. Phys. 37, 5 1 1 ( 1 962) . 10. T.S. Lomheim and L.G. DeShazer, Opt. Commun. 24, 89 ( 1 978); J. Opt. Soc. Am. 68, 1 575 ( 1 978). 1 1 . W.F. Krupke, Phys. Rev. A 145 , 325 ( 1 966) . 12. N.P. Barnes, D.J. Gettemy, L. Esterowitz, and R.E. Allen, IEEE J. Quant. Elect. QE-23, 1434 ( 1 987). 1 3 . S.C. Abrahams and S. Geller, Acta Cryst. 11, 437 ( 1 958). 14. (a) L.D. Merkle and R.C. Powell, Phys. Rev. B 20, 75 ( 1 979); (b) M. Zokai, R.C. Powell, G.F. Imbusch, and B. DiBartolo, J. Appl. Phys. 50, 5930 ( 1 979). 1 5 . H.G. Danielmeyer, in Lasers, edited by A.K. Levine and A.J. DeMaria ( Dekker, New York, 1 976), Vol. 4, p. 1 . 1 6. A.A. Kaminskii, Laser Crystals (Springer-Verlag, New York, 1 98 1 ) . 1 7 . H.G. Danielmeyer, M . Blatte, and P . Balmer, Appl. Phys. 1, 269 ( 1 973). 1 8 . C.A. Morrison, D.E. Wortman, and N. Karayianis, J. Phys. C 9, L191 ( 1 976). 19. W.F. Krupke, IEEE J. Quantum Electron. QE-7, 1 53 ( 1 97 1 ) . 20. G . Zeidler, IEEE J. Quantum Electron. QE-4, 1 0 1 6 ( 1 968). 2 1 . M.L. Kliewer and R.C. Powell, IEEE J. Quantum Electron. QE-25, 1 850 ( 1 989) . 22. Y. Guyot, H. Manaa, R. Moncorge, N. Garnier, E. Descroix, and P. Laporte, J. Phys. ( Paris) 4, Colloq. C4, 529 ( 1 994) . 23. M.A. Kramer and R.W. Boyd, Phys. Rev. B 23, 986 ( 1 98 1 ) . 24. G.E. Venikouas, G.J. Quarles, J.P. King, and R.C. Powell, Phys. Rev. B 30, 2401 ( 1 984) . 25. L.L. Chase and S.A. Payne, Phys. Rev. B 34, 8883 ( 1 986) .

338

8. Y 3 Als01 2 : Nd3 + Laser Crystals

26. D.P. Devor and L.G. DeShazer, Opt. Commun. 46, 97 ( 1 983); D.P. Devor, R.C. Pastor, and L.G. DeShazer, J. Chern. Phys. 81, 4 1 04 ( 1 984). 27. R.C. Powell, S.A. Payne, L.L. Chase, and G.D. Wilke, Opt. Lett. 14, 1 204 ( 1 989); Phys. Rev. B 41, 8593 ( 1 990) . 28. M.G. Blazha, D.N. Vylegzhanin, A.A. Kaminskii, S.I. Klokishner, and Yu. E. Perlin, Izv. Akad. Nauk. SSSR, Ser. Fiz. 40, 1 8 5 1 ( 1 976) [Bull. Acad. Sci. USSR, Phys. Ser. 40, 69 ( 1 976)]. 29. T. Kushida, Phys. Rev. 185, 500 ( 1 969) . 30. Yu. Perlin, A.A. Kaminskii, M.G. Blazha, and V.N. Enakii, Phys. Status. Solidi B 1 12, K125 ( 1 982). 3 1 . Yu. Perlin, A.A. Kaminskii, V.N. Enakii, and D.N. Vylegzhanin, Phys. Status Solidi B 92, 403 ( 1 979); A.A. Kaminskii and D.N. Vylegzhanin, Dokl. Akad. Nauk SSSR 195, 827 ( 1 970) . 32. A.J. Kiss and R.C. Duncan, Appl. Phys. Lett. 5, 200 ( 1 964) . 33. V. Lupei, A. Lupei, S. Georgescu, and C. Ionescu, Opt. Commun. 60, 59 ( 1 989); A. Lupei, V. Lupei, S. Georgescu, C. Ionescu, and W.M. Yen, J. Lumin. 39, 35 ( 1 987); V. Lupei, A. Lupei, S. Georgescu, and W.M. Yen, J. Appl. Phys. 66, 3792 ( 1 989) . 34. H.G. Danielmeyer, J. Appl. Phys. 42, 3 1 25 ( 1 97 1 ) . 3 5 . N . Strovola and D.L. Dexter, Phys. Rev. B 20 , 1 986 ( 1 979) . 36. N. Takeuchi, S. Chandra, Y.C. Chen, and S.R. Hartmann, Phys. Lett. 46A, 97 ( 1 973).

9

R are-E arth-Ion Laser M aterials

The fundamental physical processes relevant to the optical pumping dy­ namics are similar for all of the trivalent rare-earth ions. However, it is through the subtle differences in the spectral properties of the same ion in different host materials or different ions in the same host that much in­ formation has been ascertained about these physical processes. The basic physics concepts were developed in Chaps. 2-5 and applied in detail to the case of Y3 Als 0 1 2 : Nd3 + in the previous chapter. In this chapter, the char­ acteristics of other host materials and other trivalent rare-earth ions are dis­ cussed and compared to those of Nd-YAG. Only laser materials based on sharp line f-to-f transitions are discussed in this chapter. Materials based on broad-band rare-earth vibronic and d-to-f transitions are discussed in Chap. 10. In the following sections, the properties of Nd3 + ions in a variety of dif­ ferent host materials are compared. Special attention is focused on the properties of energy transfer and radiationless relaxation, and on the differ­ ences between crystal and glass hosts. The characteristics of several other important rare-earth-ion laser materials are also presented with an emphasis on energy transfer and up-conversion pumping and on multiphonon relaxa­ tion processes. Figure 9. 1 shows a Dieke diagram1 which gives the relative positions of the multiplets of the trivalent lanthanide series ions and the general widths of the Stark splittings in crystals. Because of the shielding effect of the outer-shell electrons, these energy levels change only slightly from host to host. 9. 1

Nd 3 + Lasers

Although Nd-YAG is the best-known neodymium laser material, Nd3 + ions have been made to lase in many other host crystals. The differences in local environments such as site symmetry, crystal-field strength, and elec­ tron-phonon interaction strength lead to differences in the optical spectra and lasing parameters. Tables 9. 1 -9.2 list some of the important spectro33 9

9. Rare-Earth-Ion Laser Materials

340

___ u-

38 36

· -'• �·

3

__ ,,, e -- s, K ­ 2 A-- 7, 2 ..

0

· --

32

L-

o --

·­ M-

30 28

0-

· --

26

�

24

·

­

a --

' -- t,.• o -

22

p­

ML=

·--

,

__

-HG--

• --

20

16

J· -- ·

·-

' o o -- • H

S

S -- I

A � S,z

�''z

A� 4

J�S

S

" 1 t�

0 __

2

F & 1 .. 3K I

F-.!fl,.Z �--=F�

e .. z

... .

0

.. ,

14

" " 1;'1. , -s, , 0 --

12 10

p

8

'•

__ , -- ·

6

.,

--l-, -·

2

F s, 2

Ce

·-·

_,._2

-- '1

· -·

- - ·3H

4

Pr

0� �

��

�H

�=:li,,.,

41

9,2

Nd

51

--�2

v -�z

·--

6

'• --

7F � �� --'\

·

-- ·

Pm

Sm

Eu

E

z

2

\

' •

B- 5

'o -·

H,�z

� 3,2 '•

-- ·

-·

.,

� � ·,2 - ·

�'�z

v-':t,z

._,

.. ,

� 5,2

-- ·

-·

__ ,

4

F

•

-- ·

-- ·

7F o

G ._ •

- 4 v -'J,z

-- ·

Hs '2

}

o

...� · -- · w ��z .�,.,

1 __ 3,

H -w- 5

'• � -.-'' z

c -- •

c�

..

w- 7 v--1J,2

4

0

· -­

·-

·-

·-0-

� �z �\

, __

a --

c� •

c

18

2

t�l!

7F 6

Gd

Tb

·- ·

6

H1s, 2

Dy

-

·-

Ho

Er

51

8

4 11

s,.2

-

3H

6

Tm

2

F7,

2

Yb

FIGURE 9. 1 . Dieke diagram for the energy levels o f trivalent rare-earth ions (after Ref. 1 ) .

9. 1 . Nd3 + Lasers

341

TABLE 9. 1 . Comparison of spectroscopic properties of some Nd 3 + -doped laser crystals. r

Host Ca5 ( P04)J F( FAP) Sr5 (V04)J F Yz03 YLiF4 ( YLF ) CaYzMgzGe3 0 12 NdP50 1 4 ( NPP) Y3 A150 12 ( YAG) YA103 ( YALO) YA103 : Cr YV04 GdAl03 LazBez05 ( BEL) Gdz03 CaW04 LiNb03 CaMo04 Gd3 Ga50 1 2 (GOG) Gd3 SczGa3 0 12 (GSGG) LaF3 CeF3

(tLs)

R = PI.o61P u5

x = n4 ;n6

Measured

3.59

0.347 2.267 0.407 0.407 0.492 5.36 0.540 0.698 0.8 10 0.827 0.853 0.870 0.930 0.955 1 .015 1 .049 1 .093 0.728 1 .221 1 .324

230 220 260 500 305 310 250 1 80 1 80 90 100 1 55 1 20 1 80 85 120 270 28 1 600 270

3.66 3.66 3.76 3.81 4.00 4.00 4. 1 3 4. 1 5 4. 1 8 4.20 4.27 4.30 4.37 4.41 4.46 5.10 4.61 4.73

JO

161 270

259 1 57

284

Aii (cm- 1 )

4 10.6 2 12 37 40 3.6 30 65 6.9 15 27 20 15 20 12 2.3 14 25 25

TABLE 9.2. Judd Ofelt parameters, cross sections, and metastable state splittings for some Nd 3 + -doped laser crystals. nz Host (site symetry) Y3 A1 5012 ( Dz) YA103 (Cth) Yz03 (Cz) Sr5 (V04)J F (Cth ) Gd3 SczGa3 0 1 2 ( Dz) YV04 ( D 2d) LaMgAlu Ot 9 LazBe 2 05 (Ct ) YLiF4 (S4) LaF3 (Cz )

u

5.0 3.1

n6

( 1020 cm- l )

( 1 0- 1 9 cm- 1 )

6.5 1 .3 (E l l c)

�

0.2 1 .24 8.55 1 1 .9 0.35

2.7 4.68 5.25 6.8

2.35

AE (cm- 1 ) 5.0 5.85

2.89 3.0 3.23

85 129 196

65

10.7

18

0.6 1 .2 3.1

1 60 213 59 42

1 .9 0.35

2.7 2.57

5.0

2.50

342

9. Rare-Earth-Ion Laser Materials

scopic parameters and material properties for neodymium in a variety of host crystals. An example of the variation of Judd-Ofelt parameters for Nd3 + in different hosts is given in Table 9 .2. The fluorescence lifetime of the 4F3 ;2 metastable state varies from 85 to .us resulting in very different conditions for flashlamp pumping. The exact positions of the 4 F5 ;2 Stark levels and the strengths of the absorption transitions to these levels also cause significant differences in diode laser pumping. The variations in the width of the laser transition at nm demonstrates the differences in electron-phonon couling strengths. The differences in the R and X spec­ troscopic parameters show which materials are best suited for lasers operating at nm versus lasers operating at 1 357 or 885.8 nm. The polarizability change associated with Nd3 + ions pumped into the metastable state versus the ground state differs from host to host as shown2 in Fig. 9.2. Other oxide crystals are similar to YAG in this respect except for special cases such as YV04 . This is a molecular crystal with additional transitions that contribute to the polarizability resulting in a significantly higher value for 11.rxp. The fluoride crystals have signifcantly smaller values for Aap due to both higher-energy positions of the 5d levels in these hosts and smaller radial integrals. The most successful fluoride host crystal for Nd3 + ions is LiYF4 ( Nd-

600

1064

1064

oxide)

16

G)

Cl

..c u >-

= :c

"16 11 N

0 a.

YV04

•

31 47e

(inverted / phosphates) �

0.04

901 5 .

0.03 0.02 0.01

N a2Ya F1 1 • • L iYF4

-5 P-1 1 8 • N -7 5 0 LG E-1 21 ! • LH G - 5 e E-21 0 • • E-1 31

)

• K-1 066

Phosphate g lasses crystals Silicate F luorophosphate g lasses glasses

FIGURE 9.2. Measured polarizability change of optically pumped Nd3 + in different host materials (after Ref. 2) .

9. 1 . Nd3 + Lasers

343

YLF ) . This crystal has a scheelite structure and thus is birefringent. The positions of the energy levels and transitions are shifted slightly from those of Nd-YAG and the laser emission is polarized. The main laser line occurs at 1 04 7 nm in n polarization and at 1 053 nm in (J polarization. As seen from the spectroscopic R parameter in Table 9. 1 , the laser transition at 1 3 1 3 nm in (J polarization is also strong. The thermal conductivity of Nd-YLF is 0.06 Wjcm K and the thermal expansion coefficient is 1 3 x w- 6 oc - 1 along the a axis and 8 X w- 6 oc - J along the c axis. These thermal proper­ ties allow for good heat extraction during optical pumping and lasing. In addition, the phonon frequency distribution in fluorides cuts off at lower frequencies than in oxides resulting in weaker electron-phonon interactions. Thus the radiationless decay rates and the thermal contributions to the spectral linewidths and positions are all smaller for Nd-YLF than for Nd­ y AG. The metastable-state lifetime for Nd 3 + ions is much longer in YLF than in YAG. This is primarily due to the crystal-field symmetry and strength at the Nd3 + site in YLF resulting in lower radiative probabilities. The cross-relaxation quenching of this level as a function of neodymium concentration has approximately the same energy-transfer characteristics as Nd-YAG, as discussed below. For the 4 F3 ;2 --+ 4 11 1 ;2 transition, the stimu­ lated emission cross sections are similar for Nd-YLF and Nd-YAG. Since the threshold for lasing is generally inversely proportional to the lifetime-cross-section product, the threshold for Nd-YLF lasers operating on the 4 F3; 2 --+ 4 11 1 ;2 transition is somewhat smaller than that of Nd-YAG lasers. 3 For broadband lamp-pumped operation, the slope efficiencies of Nd-YLF lasers are generally slightly lower than those of Nd-YAG lasers. This is due to reduced absorption efficiency in YLF associated with the nar­ rower absorption transitions. The Q-switched operation of Nd-YLF lasers on the 4F3 ;2 --+ 4 11 1 ;2 transition has similar characteristics to Q-switched Nd-YAG lasers operating on the same transition. Nd-YLF lasers operat­ ing on the 4 F3 ;2 --+ 4 11 3 ;2 transition near 1 3 1 3 nm have a higher threshold for cw, pulsed, and Q-switched conditions compared to 1047-nm operation due to a stimulated emission cross section that is between 3 and 6 times smaller for the lasing transition. Additional losses have also been detected for 1 3 1 3-nm laser operation and attributed to ESA processes. The Q­ switched temporal pulse widths of this transition are significantly longer than those for the 4 F3 ;2 --+ 4 /1 1 ;2 transition. The different laser character­ istics for this transition of Nd-YLF compared to Nd-YAG scale approx­ imately with the differences in metastable-state lifetimes and stimulated emission cross sections. One interesting new fluoride laser crystal is KLiYF5 : Nd3 + . This crystal can be grown by either Czochralski or hydrothermal growth techniques (see Sec. 1 .3). It has been found that the host material can form in two slightly different crystalline phases. 4 The phase that is formed through hydrothermal crystal growth has only one site for Nd3 + substitution, whereas the phase formed Czochralski crystal growth has two nonequivalent sites for the neo-

344

9. Rare-Earth-Ion Laser Materials

(A)

IJ/

/

(B)

I

I

I

\

\

\

(C)

FIGURE 9.3. Mechanisms for excitation in upconversion lasers. (a) Stepwise absorp­ tion of two photons; (b) cross-relaxation energy transfer; (c) avalanche absorption.

dymium ions. The difference in number of active ion sites for the two types of crystals results in differences in spectral and lasing properties. This is another example of the importance of crystal growth in determining the properties of solid-state lasers. Up-conversion fluorescence and lasing is more efficient in fluoride hosts compared to oxides because fluorides have lower phonon cutoff frequencies resulting in lower radiationless decay rates and higher fluorescence quantum yields from high lying metastable states. This provides an alternative method to nonlinear optical crystals to obtaining visible laser emission from near infrared lasers. There are three mechanisms for obtaining up-conversion: ion-ion energy transfer between two ions in excited states, the stepwise ab­ sorption of two pump photons (from either the same laser or two different lasers) by the same ion, and avalanche absorption. These are shown sche­ matically as energy levels and transitions in Fig. 9.3. One example of a Nd3 + stepwise absorption up-conversion laser is Nd : LaF 3 pumped by two lasers. 5 The first one, at 788 nm, excites an ion to the 4 F5 ;2 level from which it quickly relaxes radiationlessly to the 4F3 ;2 metastable state. A second photon at 591 nm is then absorbed by the excited ion taking it to the 4 D 1 ;2 • After fast radiationless relaxation occurs to the 4 D3 ;2 metastable state, violet fluorescence and laser emission occurs at 380. 1 nm due to the 4 D 3 ;2 --> 4 /1 1 ;2 transition. Lasing has been obtained only at temperatures of 90 K or less. The power of cw laser emission at 20 K has been measured to be as high as 1 2 mW in the violet for incident pump powers of 1 10 and 300 mW at the infrared and yellow wavelengths, respectively. An example of a second type of Nd3 + up-conversion laser is avalanche pumped Nd : YLF. 6 In this case absorption of a 603.6-nm pump photon originates from a thermally excited 4 /15 ;2 level. After radiationless relaxation to the 4F3 ;2 metastable state and cross relaxation with a neightboring ion, there are two ions in the 4 /15; 2 level. This avalanche procedure continues to increase the excited-state absorption at the pump wavelength until a significant number of ions have been excited

9. 1 . Nd3 + Lasers

345

to the 2 P3; 2 metastable state. Fluorescence and lasing then occurs at 4 1 3 nm due to the 2 P3 ;2 -- 4 11 1 ;2 transition. Again, lasing has been obtained only at low temperatures. One host crystal of special interest is gadolinium scandium gallium garnet, Gd3 Sc2 Ga3 0 1 2 , known as GSGG. Since they are both oxide garnet crystals, GSGG and YAG have many properties that are similar. As seen in Table 9 . 2, the laser transition cross section for Nd3 + in GSGG is about twice that in YAG. This implies that about twice as much energy density can be stored in the metastable state, which is advantageous for Q-switched performance. The metastable-state fluorescence lifetime and nonradiative decay properties are similar for the two hosts. However, the differences in spectral properties results in the energy transfer from Cr3 + ions to Nd3 + ions being much more efficient in GSGG than in YAG. Thus Cr; Nd-GSGG has the advantage of sensitized pumping with broad-band lamp excitation, resulting in approx­ imately a factor of 2 lower threshold and higher efficiency for laser opera­ tion than Nd-YAG. The differences in the properties of the Cr-Nd energy transfer between YAG and GSGG hosts is due to the difference in crystal-field strengths. The gallium garnet has larger crystal lattice parameters than the aluminum gar­ net and thus a smaller crystal-field strength. As described in Chap. 6, the fluorescence emission properties of Cr3 + ions are highly sensitive to the local environment. The strong crystal-field environment in YAG results in the 2 E level of Cr3 + lying well below the 4 T2 level and the fluorescence spectrum of YAG : Cr3+ is dominated by sharp, spin-forbidden R-line emission. As dis­ cussed in the last chapter, there is very poor spectral overlap between this emission line and Nd3 + absorption lines, leading to a low efficiency of en­ ergy transfer. For the weaker crystal-field environment in GSGG, the split­ ting between the 2 E and the 4T2 levels is much smaller and the Stokes shift of the latter causes the emission spectrum to be dominated by the broad, intense band associated with the spin-allowed 4 T2 -- 4 A 2 transition. This has excellent overlap with Nd3 + absorption transitions as shown in Fig. 9.4, and thus the energy transfer is very efficient. 7 The kinetics of the energy transfer from chromium to neodymium ions in GSGG have been described8 by a simplified version of the expressions for the time evolutions of the sensitizer and activator excited-state populations derived in Chap. 5 t (9 . 1 . 1 ) Ncr ( t) Ncr ( O) exp y - y yfi - Wt and

(

rcr

)

(9 . 1 . 2) where

9. Rare-Earth-Ion Laser Materials

346

0.7

j.. ,'v Emission

0.6

I I I I I I I I

0.5

.. CJ 0.4 1: ..

.c

�

.c oct

0.3

0.2

\ \ \ \ \ \ \ \

0.1 7 00

Wavelength (nm)

600

900

1 000

FIGURE 9.4. Fluorescence of Cr3 + and absorption spectrum of Cr;Nd GSGG (after Ref. 7).

r (t) and

{

[erf (B0 +

[1 - exp( -B2t - yVt)] }

J (1 (9.1.3)

1 + W - -. 1 B= (9.1.4) rcr TNd Here y is the Cr-Nd interaction parameter and Wis the migration enhanced energy-transfer parameter. This is essentially an expansion of Eq. (5.4.23) for diffusion-enhanced single-step energy transfer. It has been found8 that a reasonably good fit between these equations and the observed fluorescence decay kinetics after pulsed excitation can be obtained assuming electric dipole-dipole interaction for both the Cr-Nd transfer step and the Cr-Cr energy migration. However, an improved fit can be obtained if it is assumed that transfer from Cr ions to Nd ions in the first- and second-nearest­ neighbor positions takes place at a greater rate than predicted by electric dipole-dipole interaction. For a crystal with PNd Per 1020 cm-3; the values of the energy-transfer parameters that are obtained from theoretical2 fits to experimental data are W(Cr-Nd)8 2000s-1, y(Cr-Nd) l l o s-1 1 , W(Nd-Nd) 124 s-1, y(Nd-Nd) .5 s-112, 'Nd 280 ps, and rcr 120 ps. The effective energy-transfer efficiency from chromium ions to neo-

9. 1 . NdH Lasers

347

dymium ion in GSGG can reach levels of greater than 80%, which makes sensitized pumping an important process for this neodymium laser material. Another type of neodymium laser crystal of particular interest is neo­ dymium pentaphosphate, NdP5 0 1 4 · This is representative of a special class of laser crystals known as stoichiometric laser materials that can be useful in low-threshold, high-gain minilaser applications. In general, these materials are the high-concentration end of a compositional series NdxA I -xP5 0 1 4 where A La3 + or y3 + . The distinguishing feature about this class of materials is that there is much less concentration quenching of the emission compared to normal host crystals, and thus strong fluorescence and laser operation occurs in crystals with 1 00% Nd3 + concentrations. This is due to the fact that the positions of the energy levels in NdP5 0 1 4 are shifted in such a way that the 4 F3 ;4 -- 4 /15 ;2 transition occurs with less energy than the 4 19;2 -- 4 /15 ;2 transition. Thus the usual mechanism for concentration quenching through an ion-ion cross-relaxation process involving a resonant interaction through these two transitions is not very efficient. However, with the close spacing of the Nd3 + ions, there is strong resonant transfer of excitation energy in the metastable state. NdxA I - xP5 0 1 4 crystals exhibit spectral energy transfer associated with the short-range interaction between Nd ions in nonequivalent types of crystal­ field sites. The characteristics of this process are similar to those of Nd3 + ions in YAG and other host materials. In addition, long-range spatial mi­ gration of energy among the Nd3 + ions takes place in NdP5 0 1 4 and mixed crystal systems of this class. This process has been investigated using laser­ induced transient grating techniques. 9 The excitation diffusion coefficient for NdP5 0 1 4 was found to be in a range between x w- 6 and 2.5 x 1 2 w-4 cm s- , depending on the type of excitation (resonant or vibronic) and temperature. The large value of D was found for resonant excitation and the signal decay dynamics in this case exhibited nonexponential, oscillatory behavior. The ion-ion interaction causing the energy migration was found to be consistent with an electric dipole-dipole mechanism. The strong inter­ action due to the high concentration of Nd3 + ions results in long mean free paths for the migration process. The diffusion coefficients decreased with decreasing Nd concentration in mixed crystals and were about an order of magnitude for concentrations of 20%. Another type of process that can be important in some types of laser crys­ tals is host-sensitized energy transfer. Molecular crystals such as tungstates or vanadates are examples of host materials where this can take place. In YV04 : Nd3 + , the (V04 ) 3 - vanadate molecule has absorption transitions in the near-ultraviolet spectral region and emission bands in the visible spectral region of this material. When these molecular ions are excited, the energy can migration from molecule to molecule in the crystal like a Frenklel exciton. As the exciton gets near to a Nd3 + it becomes trapped and the energy is transferred to Nd3 + ion. This process has been investigated using

5.4

348

9. Rare-Earth-Ion Laser Materials

time-resolved spectroscopy techniques, 1 0 and the results are consistent with both the energy migration and trapping mechanisms being due to electric dipole-dipole interaction. At low temperatures, the energy transfer is due to a single-step process while at high temperatures the transfer is enhanced by multistep migration. The latter mechanism is a thermally activated hopping process with a diffusion coefficient of the order of 5.4 x 10- 7 cm2 s 1 . So far the mechanism of host-sensitized energy transfer has not been exploited in commercial laser systems. This is due mainly to the strong interest in direct diode laser pumping of the Nd3 + ions. YV04 : Nd3 + is an excellent material for a diode laser-pumped laser because it has strong, single-line transitions with cross sections of the order of five times greater than those of Nd : YAG. Due to the cross-relaxation interaction shown in Fig. 8 . 1 5(A), concentra­ tion quenching is an important limiting process Nd-doped laser materials. Figure 9.5 shows the quenching of the fluorescence of the fluorescence life­ time of Nd3 + in several host materials as a function of neodymium concen­ tration. An empirical expression that gives a good fit to these data is 1 1 ro (9. 1 .5) - = - + Wer :: rf = + W , 1 ro er rf ro where rf is the fluorescence decay rate, r0 is the decay rate with no cross­ relaxation present, and Wcr is the rate of cross relaxation. Assuming electric dipole-dipole interaction, the cross-relaxation rate can be described using the Forster-Dexter theory outlined in Chap. 5. For this application a quenching parameter Q can be defined as Q=

p

(9. 1 .6)

where p is the neodymium concentration. Substituting this into Eq. (9. 1 .5) gives the fits to the data shown in Fig. 9.5. The Q parameters obtained from these fits are listed in Table 9.3. For cases where the Q parameter has been calculated using the energy-transfer expression given in Eq. (5.4. 1 5) and the spectral overlap of the 4 F3 /4 --> 4 1, 5 ; 2 emission transition with the 4 19;2 --> 4 /15 ;2 absorption transition, a good agreement has generally been found between measured and predicted values. The exceptions to this are materials such as NdP5 0 1 4 where quenching is enhanced by migration to "killer sites." Another interesting class of host materials are nonlinear optical crystals. For some applications, specific laser wavelengths are required in spectral regions where no primary laser exists with the appropriate characteristics. For these situations it is possible to use a nonlinear optical crystal to change the wavelength of a primary laser through processes such as harmonic gen­ eration or parametric mixing. Usually the nonlinear optical crystal is a sep­ arate component from the laser crystal, either external or internal to the laser cavity. However, there have been several demonstrations of doping

9. 1 . Nd3 + Lasers

349

Dexter's Theory 't f 40

0

2

4

6

8

Nd CONCENTRATION (1020 cm·3 )

10

(A)

..

:I.

... E

600

·� >

B .. ..

"t)

> ·;: u

.. ;:

w

Nd

1 concentration, 020 ion/cm 3

(B)

FIGURE 9.5. Concentration quenching of the fluorescence lifetime of Nd3 + ions in several hosts. (A) Crystals ( Ref. I I ) . ( B) Glasses [Ref. 12 (a)].

350 TABLE

9. Rare-Earth-Ion Laser Materials 9.3. Concentration quenching parameters for some Nd-laser materials.

Crystal

Y3 A1 s Ou Gd 3 Sc 2 Ga 3 0 1 2 YV04 YLiF4

Q ( 1020 cm- 3 )

3.8 5.0 3.3 3.7

Glass Ed 2 Silicate LHG 7 Phosphate LG 8 12 Fluorophosphate

Q ( 10 20 cm- 3 )

3.89 6.88 3.99

rare-earth ions in nonlinear optical crystal to obtain a "self-doubling" laser. One important example of this is LiNb0 3 : Nd3 + . The normal 4 F3 ; 2 -- 4 /11 12 near-infrared laser transition is converted to second-harmonic green emis­ sion for this case. 1 3 Because of the phase-matching requirements for the primary- and second-harmonic wave in a nonlinear optical crystal, efficient self-doubling laser crystals are very sensitive to crystal orientation, align­ ment, optical path length, and temperature. The crystal and cavity design parameters required to maximize laser performance are generally not the same as the design requirements for optimizing nonlinear optical conversion efficiency. Lasers based on nonlinear optical materials are discussed further in Chap. 1 0. Thus far systems with separate laser and frequency conversion crystals have achieved better performance characteristics than self-doubling systems. However, with new host materials such as NdxY 1 -xA1 3 (B0 3 ) 4 ( NYAB) being developed, the properties of self-doubling lasers may be significantly enhanced in the future. This material has been operated as a free-running and a Q-switched system with flashlamp and with diode laser pumping and with Cr3 + activation. Both the 1 .34- and the 1 .06-flm emission lines have been self-doubled. NYAB has low concentration quenching rates allowing for high concentrations of Nd ions. The for a sample with a Nd con­ centration of x 0. 1 , the metastable-state lifetime is 60 flS and the 1 .06-flm laser line has an emission cross section of 2 x 10- 19 cm2 and a linewidth of 30 em- 1 . The luminescence quantum efficiency is about 1 5%. The nonlinear optical coefficient of the host material is almost four times that of KH2 P04 crystals. One of the major problems with self-doubling in NYAB is the strong absorption at the wavelength of the doubled light. Another problem is the temperature-dependent change in the phase-matching conditions that occurs due to the heat generated in the optical pumping. A new material with a nonlinear optical host that may prove to be inter­ esting is neodymium-doped potassium gadolinium tungstate KGd(W04 h ( Nd : KGW ) . This has diode laser-pumped laser properties similar to Nd : YAG. 14 In addition, KGW has very strong Raman gain. Thus the stimulated Raman properties of the host could be used to convert the laser emission to a different frequency. Glass materials are also excellent laser hosts for Nd3 + ions. Along with the comparative ease of synthesizing the material compared to many types

9. 1 . Nd3 + Lasers ::

'

N�

20

� c0

�

15

�3

c

351

3

10

�

.0 <(

0

200

300

400

°300

400

600

500

700

600

90

1 000

Wavelength (nm)

(A)

1

0

0.8

0.6

0.4

0.2

0

1 000

1 02 0

1 040

1 080

1 060

Wavelength (nm)

1 1 00

1 1 20

1 1 40

(B)

FIGURE 9.6. (A) Absorption and (B) fluorescence spectra of Nd3 + in ED-2 silicate glass [from Ref. 1 2 (b)].

of crystals, glass can be fabricated into special laser configurations such as slabs and fibers as well as standard rod designs. The major difference be­ tween the optical spectra of ions in glass and crystal hosts is that the former exhibit much strong inhomogeneous broadening due to the lack of long­ range order of the environment, Typical absorption and fluorescence spectra of Nd3 + in a glass host is shown in Fig. 9.6. The general features of these spectra are similar to those seen in Figs. 8.5 and 8.7 but much less structure

352

9. Rare-Earth-Ion Laser Materials

is visible. Beacause of the inhomogeneous linewidths, it is not possible to resolve the individual Stark components of the multiplets as is usually pos­ sible in crystalline spectra. There are many different types of glasses and different compositions within each type that have been investigated as laser hosts. 1 2 Major classes include silicates, phosphates, and fluoride glasses. These have very different structural properties. 1 5 For example, silicates form continuous random network structures while phosphates form chain struc­ tures. Fluoride glasses are transparent further into the infrared spectral region and have lower phonon energies than oxide glasses. The rare-earth dopant ions generally occupy interstitial positions in glass hosts and distort the local environment to give a high corrdination number of ligands, result­ ing in low local site symmetry. The ability to induce this local distortion depends on the rigidity of the glass structure, which in turn depends on the type and ocncentration of network modifier ions that are present. Thus it is possible to fine-tune the spectral and lasing properties of Nd-glass lasers by changing the composition of the host glass. Judd-Ofelt theory can be applied to rare-earth ions in glass hosts in the same way it is used for crys­ tals. Through compositional studies it has been found that flz is propor­ tional to the covalency of the glass while n4 and n6 are inversely propor­ tional to the rigidity of the material. 1 6 Examples of some of the properties of Nd-glass laser materials1 2• 1 7 are given in Table 9.4. The exact composi­ tions of the samples listed in the table can be found in Refs. 1 2 and 17. There is a general trend in silicate glasses that the radiative lifetime increases and the transition cross section decreases as the alkali modifier ion is varied from Li to Na to K. The trend is opposite in phosphate glasses. 1 2 The low thermal conductivity of glasses can cause problems with thermal lensing during high-average-power laser operation. Thus Nd-glass lasers generally operate in a low-repetition-rate pulsed mode instead of a cw or high-repetition-rate mode. The data in Fig. 9.2 show that the polarizability change induced by pumping Nd3 + ions into the metastable state follows the same trend with respect to host type as found for crystals. The peak transi­ tion cross sections for glass hosts are generally smaller than those for crystal hosts and the transition linewidths are broader. Thus Nd-glass lasers gen­ erally have higher thresholds and lower gain than crystalline lasers. The properties of glass lasers are generally favorable for Q-switched operation. Spectral saturation and hole burning can be a problem because of the in­ homogeneous broadening of the lasing transition. The strong inhomogeneous broadening of the spectral transitions of Nd3 + ions in glass hosts indicates that there is a continuous distribution of differ­ ent types of local crystal-field sites occupied by these ions. Narrow-linewidth laser beams can be used to selectively excite subsets of ions in specific types of sites. By scanning the pump lase racross the inhomogeneous bandwidth, this type of "site-selection spectroscopy" can be used to probe the differ­ ences in spectroscopic properties of ions in idfferent types of sites. Examples of line-narrowed absorption and wavelength-dependent branching ratios for

ZBAN

Lithium silicate ( ED-2) Potassium silicate ( L21 7) Lithium phosphate ( P-107) Potassium phosphate ( P-1 14) Germanate (5012) Fluorophosphate ( E-122)

Host

3.2 4.5 3.4 4.0 5.9 2.4 3. 1

Q2

4.6 2.2 4.8 5.1 3.3 4.6 3.7 4.8 2.0 5.4 5.9 2.9 5.4 5.7

Q6 !4 ( l o -20 cm2 ) 34.4 36.6 26.2 23.4 36.4 30.4 30.5

__, 4 lt lj2 )

AA.:rr (nm)

(4 £3/2 359 946 356 347 458 391 419

rrad

( J.Ls) 2.7 1 .0 3.8 4.5 1 .7 3.1 3.2

-->

u ( 1 0 - 20 cm2 ) 4 ( F3/2 4 lt t/2 ) 0.421 0.439 0.406 0.402 0.447 0.401 0.067

p9/2

0.484 0.471 0.496 0.499 0.466 0.500 0.494

Pt t/2

0.091 0.085 0.093 0.095 0.083 0.095 0.360

pl3/2

0.004 0.004 0.005 0.005 0.004 0.005 0.079

Pts;2

TABLE 9.4. Properties of Nd 3 + ions in selected glass host materials. [Data for the first six samples from Ref. 1 2(b); data for the fluoride glass from Ref. 1 7(a) .]

w Vt w

"" "' (I) .. "'

t"

+

w

p.

..

z

�

354

9. Rare-Earth-Ion Laser Materials

z 0 ;=

� "'

::!

0 0: <.>

• 3

� �0: � "' <

424

425

426

427

428

429

430

WAV E L E N G T H (nm)

433

431

435

434

(A) f LUOR OPHOSPHATE

2.0

� 0 ;:

S I L IC A T E

1 .6

� 1.4

BORATE

Cl z

:;:

� 12 " a: "'

1 .0

426

427

428

429

431

E X C I T A T I ON WAV E L ENGTH (nml

432

433

(B) FIGURE 9.7. (A) Line-narrowed absorption and (B) site-dependent branching ratios for Nd3 + in different types of glass hosts (from Ref. 1 8) .

different types of glasses are shown in Fig. 9.7. One of the standard laser glasses that has been investigated in detail using site-selection spectroscopy techniques1 8 is ED-2, which is a silicate glass with the composition 60% Si02 , 27.5% Li2 0; 10% CaO; 2.5% Ah03 with about 1 .0% Nd2 0 3 added as a dopant. A pulsed dye laser was used to excite at different places in the 4 /9; 2 __ 2 P 1 ;2 absorption band and the line-narrowed fluorescence was for

9. 1 . Nd3 + Lasers

355

433 EXCITATION WAV E L E NGTH tnml

WAV E L E NGTH (pm)

FIGURE 9.8. Line-narrowed fluorescence of Nd3 + in ED-2 silicate glass. The spectra were recorded at liquid-helium temperature 100 JlS after the excitation pulse. The fluo­ rescence intensities are adjusted for equal numbers of excitation photons absorbed (from Ref. 1 8) .

transitions from the 4 F3 ;2 metastable state to both the 4 19; 2 and 4 In;2 multi­ plets. Examples of the spectra from Ref. 1 8 are shown in Fig. 9.8. Even in the line-narrowed fluorescence it is not possible to resolve individual Stark components of the transitions. Spectroscopic properties such as radiative and nonradiative decay rates and branching ratios are different for each subset of Nd3 + ions. This can be seen as a continual variation of these properties as the excitation wavelength is scanned across the absorption band. Examples of the results obtained in Ref. 1 8 are shown in Fig. 9.9. As the excitation wavelength increases, the radiative decay rate increases while the nonradiative decay rate decreases resulting in a significant increase inrelative quantum efficiency. Similar site­ selective variations in spectral properties are observed for other types of glass hosts. Concentration quenching through cross-relaxation energy transfer and migration to "killer sites" is similar in glass and crystal hosts. For example, the primary quenching mechanism for the fluorescence in Nd-doped glasses is the electric dipole-dipole interaction through the coupled 4F3 ;4 -- 4 /15 ;2, 4 19;2 -- 4 /15;2 transitions. There are special cases where these transitions are not in resonance so this mechanism is not very effective. For these materials the quenching is enhanced through energy migration to perturbed Nd sites or to impurities that can dissipate the energy radiationlessly. Some lithium phosphate glasses fall in this category and have quenching properties similar to NdP5 0 1 4 crystals. 1 9 Nd-Nd quenching interactions along with Nd-OH energy transfer have been identified for this glass. These are enhanced

0.90

300

0.70

429

0

430

4 3 '1

432

(b)

E XC I TA T I ON WAV E L E N G T H

0

0

(a)

(nm )

a

w

<( a: ><( u

� 1-

I�

-

1 000

2000

3000

429

•

430

431

432

E X C I TA T I O N WAV E L E N G T H

t 433

(nm)

NON-RADIATIVE

(c)

0

�

1><(

�

tf) U.J

1

4000

2000

1 200

1 000

800

700

600

500

4 50

400

350

300

FIGURE 9.9. Excitation wavelength dependence of several spectral properties of Nd3 + in ED-2 silicate glass at liquid helium temperature (from Ref. 1 8) .

5 �

;:

U.J

>

0 0.80

::;) 12 <( ::;)

�

LL.

LL.

!2

w

>­ u 2

w 0

(.)

><(

::;: ;:

3 w

400

e:. "'

:: .

�

P>

�

..

P> "' (1)

t"

0 ;:l

I -

::r'

.. ..

P>

tn

@

:0 ;;o P>

w V1 0'1

9. 1 . Nd3 + Lasers 5

357

00.0

400.0

.. .3 ..

�::J

3 00.0

2

200.0

1 00.0

o o .

o o

15

0 0

TEIAPERATURE

2o

o

(K)

FIGURE 9. 1 0. Temperature dependence of the fluorescence lifetime for Nd : ZBAN glass. [Reprinted from Ref. 1 7(a) with permission of iEEE, © 1 991 IEEE.]

through both hopping transport and long-range resonant energy migration. The strengths of each of these ion-ion interaction processes are similar to those found in crystal hosts. The data for several glasses are given in Fig. 9.5( B) and Table 9.4. A typical example of a fluoride glass laser is Nd : ZBAN, which has the composition 53.33 ZrF4 ; 19.84 BaFz; 3. 14 AlF3 ; 1 8 .70 NaF; S.O NdF 3 in mole percent. This has been successfully operated as a laser in both bulk and fiber configurations. The temperature dependence of the fluorescence life­ time is shown in Fig. 9 . 1 0 and the absorption and fluorescence spectra are shown1 7 in Fig. 9. 1 1 . The linewidths of the transitions are somewhat smaller than in oxide glasses but the individual Stark components still can not be resolved. The theoretical fit to the lifetime data is given by 1 . 1 = r£ 1 + r exp (9. 1 .7) 1

[ (��) r

As described in Chap. 4, the form of this equation describes the coupled decay time of two energy levels that are in thermal equilibrium due to direct phonon absorption and emission processes across the level splitting In this case, is the splitting between the two Stark components of the 4 F3 ;2 metastable state. Various spectroscopic properties of neodymium in ZBAN are given in Table 9.4. One interesting result of the investigation of this material is that excited state absorption of pump photons can be an im­ portant process for laser pumping into some of the absorption bands in the 750-nm spectral region.

flE

flE.

358

9. Rare-Earth-Ion Laser Materials

"';"' E

CJ

C') 0

.. -

> CJ a: w z w

-

(A)

·;: "

�

� :e

�

6.0

4.0

�

z

I! �

2.0

0 "0 750.0

WAVELENGTH

(nm)

1 1 50.0

(B) FIGURE 9 . 1 1 . (A) Absorption and (B) fluorescence spectra for Nd : ZBAN glass. [Reprinted from Ref. 1 7(a) with permission of IEEE, © 1 99 1 IEEE.]

9.2. Other Trivalent Lanthanide Lasers 9.2

359

Other Trivalent Lanthanide Lasers

Although no lasing ion has been as successful as Nd3 + , all of the other trivalent lanthanide ions have been made into lasers. The majority of these laser transitions are 4f-4f transitions with properties similar to that of neodymium. The same fundamental physical processes apply to these types of lasers. Concentration quenching is associated with ion-ion cross-relaxa­ tion interactions and this can be enhanced through energy migration pro­ cesses. Sensitization by Cr3 + is effective for several trivalent rare-earth ions and there are a wide variety of energy-transfer processes between different types of rare earth ions. Multiphonon radiationless relaxation is one type of physical process that has been studied through measurements of different types of ions in the same host material. The nonradative decay rate for a specific level is gen­ erally determined by measuring the fluorescence decay rate for the level and subtracting the radiative decay rate from this value. The radiative decay rate is usually calculated from the transition cross sections measured in absorp­ tion spectra or from Judd-Ofelt theory. If the initial level of the transition of interest does not fluoresce, the nonradiative decay rate can sometimes be determined from the delayed from the delayed rise time of the fluorescence from the terminal level of the transition through an appropriate kinetic rate equation model. Examples of results obtained on a variety of different types of rare-earth­ doped host materials20 - 24 , are given in Table 9.5 and Figs. 9.12 and 9. 1 3. The interpretation of these results is based on the weak electron-phonon TABLE 9.5. Parameters for multiphonon radiationless decay processes in some rare-earth-doped laser materials ( From Refs. 20 24). Host LaBr3 LaCl 3 LaF3 LiYF4 Y2 03 YA10 3 Y3Als 0 12 Fluoride Germanate Silicate Phosphate

(cm-1 )

hwerr

(s- 1 )

175 260 350 400 550 600 700

Crystals 1 .2 X 10 1 0 1 . 5 x 10 1 0 6.6 X 1 08 3.5 X 1 07 2.7 X 1 08 5.0 X 1 09 9.7 X 107

1 .9 X J.3 X 5.6 X 3.8 X 3.8 X 9.6 X 3.1 X

10- 2 10- 2 10- 3 10- 3 10- 3 10- 3 10- 3

0.037 0.037 0. 14 0.22 0. 1 2 0.063 0.045

500 900 l iOO 1 200

Glasses 1 .6 X 10 1 0 3.4 X 1 0 1 0 1 .4 X 1 0 1 2 5.4 X 1 0 12

5.2 X 4.9 X 4.7 X 4.7 X

10- 3 10- 3 10- 3 1 0- 3

O.o75 0.014 0.0057 0.0036

c

IX

(em)

e

�

w

·= ·e.. 5

10

1 02

0 o,

I

9/ 2

"'

77° K

s

2H

5/2

2

s0

2P 3/2

5F 5

0 2

2

"'

55

4s 3/2

50 3

4F 9t2

( 3 PI , 30 3 )

5r

6

• Thu l i u m

.o. Erb i u m

• Ho l m i u m

0 Euro p i u m

o Neodym i u m

(A)

E n ergy gop to next- lower level (cm· I J

YAI 0 3

40

2G 91

5r 1

:; ::

-�

c 0 c 0 �

"'

"'

c 0 · u;

2! e

"'

1 0° 0

10 1

1 02

103

104

105

i 06

1 07

1000

LaBr 3 (175cm - l )

(B)

Energy gap ( cm- 1 )

2000

LaCI 3 (260 cm-1) 3000

4000

FIGURE 9 . 1 2. Energy-gap dependences of multiphonon nonradiative decay rates for different host crystals. ( Reprinted from Ref. 22 with permission of Elsevier Science-NL, Sara Burgerhartstraat 25, 1 055 KV Amsterdam, The Netherlands.)

::E

:;

c 0 .c a.

3 10

.. 0 .. 104 5

7 10

9.2. Other Trivalent Lanthanide Lasers

361

FIGURE 9.13. Energy-gap dependences of multiphonon nonradiative decay rates for different host glasses (from Ref. 2 1 ) . !l �

5' 105

-1: g

c

� 104

coupling, energy gap law model discussed in Chap. 4. The parameters in the table come from the expressions

(9.2. 1 ) where WKr is the nonradiative decay rate for a process involving p phonons, C and rx depend on the host but not on the specific transition, and hwerr is the energy of the effective phonon involved in the transition. e is the ratio of the ratio of the decay rates of the p- and ( p 1 ) -phonon processes as described in Chap. 4. Recalling that the phonon occupation number is given by n(hwerr ) = [exp(hwerr/kBT) 1r 1 , the temperature dep�ndence of the nonradiative decay rate for a p-phonon process is given by

W�r( T)

=

W�r(O)

(

exp -- - 1

kBT

)P

,

(9 . 2.2)

where the temperature-independent factor is given by Eq. (9.2. 1 ) to be WKr(O) = ce ai!.E. In most cases the effective phonons involved in non­ radiative relaxation processes are the highest-energy phonons available since this minimizes the number of phonons required to conserve energy for a given energy gap and thus results in the lowest possible order of the decay process. However, this is not necessarily always true since the coupling strength or density of states of lower-energy phonons may be greater than these quantities for high-energy phonons. The order of the process and thus the value of the effective phonon energy can be determined from the tem­ perature dependence of the decay rate. An example of the variation of the nonradiative decay rate with temperature is shown in Fig. 9. 14.

362

9. Rare-Earth-Ion Laser Materials

- 10 ..

' u·

=

Q

!!

.. cr 0: 0 ·;; 0: ..

�

0: 0

6

..: .. ::

:IE

La �

Ho

••

F (1F1 l - E I' F. .1S1.l 6 E • 1 800cm.-• •

8 6

4

100

200 T( ° K )

300

FIGURE 9 . 14. Temperature dependence of the p 6 multiphonon nonradiative decay rate for the transition 5F3 -- 5F4 , 5S2 in Ho : LaF3 showing the experimental points and theoretical fit. ( Reprinted from Ref. 22 with permission of Elsevier Science NL, Sara Burgerhartstraat 25, 1 055 KV Amsterdam, The Netherlands.) =

The results of extensive investigations of nonradiative decay processes of rare-earth ions in crystal and glass hosts shows that the simple phenomeno­ logical energy gap law fits the observed results for energy gaps spanning over six orders of magnitude. Some of the noted exceptions to the pre­ dictions of the energy-gap model can be readily explained through selection rules applicable to specifications or phonon coupling properties that can be verified independently through the shape of vibronic sidebands. Also, for very small energy gaps that can be bridged by one or two phonons, the basic ssumption of the multiphonon transition model break down and the one- or two-phonon theories discussed in Chap. must be used. The variation of the multiphonon decay rates from host to host as shown in Figs. 9. 12 and 9. 13 is associated with differences in the phonon spectrum of the host materials and differences in the electron-phonon coupling strengths. The former is demonstrated by the value of the effective phonon energy and the latter is reflected by the magnitude of the parameter e. Despite the general success of the energy-gap law for describing the char­ acteristics of multiphonon relaxation rates of rare-earth ions in crystal and glass hosts, there are some deviations from theoretical predictions. For

4

9.2. Other Trivalent Lanthanide Lasers

363

example, 2 5 for a YLF crystal host, the energy gap law fits the nonradiative relaxation rates for one set of values of C and IX for A.E :;: 2000 cm- 1 and for a very different set of values of C and IX for transitions with A.E > 2000 cm- 1 . This has been attributed to the weakness of using a point-charge model for the ion-ligand interaction. By adding terms in the interaction Hamiltonian that account for the spatial distribution of the electron orbits including, exchange effects and the dipole moments of the ligands, the dif­ ferences in the energy-gap law parameters for transitions with small and large gaps can be explained. By investigating the temperature dependencies of spectral linewidths in the series of trivalent lanthanide ions in the same host crystal ( YLF ), it has beefi possible to determine the variation of the electron-phonon coupling strength from ion to ion. 26 Transitions were chosen that are well resolved in the spectrum and intense enough to be studied at high temperatures. The results were interpreted in terms of the theoretical expression given in Eq. (4.4.8) with the Raman scattering of phonons found to be the dominant line-broadening processes and the effective Debye temperature for YLF chosen to be 250 K. The results show that the electron-phonon coupling strength is strong in the beginning (Ce 3 + ) and end (Yb3 + ) of the lanthanide series and weak in the middle (Gd3 + ) . The explanation of this trend is com­ plicated by the contributions of several effects. As the atomic number of the ions in this series increases there is a contraction of the electron wave func­ tions, which will decrease the electron-phonon coupling strength. However, for ions beyond Gd3 + , the 5s and 5p orbitals contract more than the 4f orbi­ tals thus decreasing the screening of the electrons in the 4f orbitals and therefore enhancing the electron-phonon coupling of the optically active electrons. In addition, the relative position of the 5d levels With respect to the 4f levels can affect the coupling strength. There has been a significant amount of interest in lasers operating in the 2-3-,um region to be used for a variety of applications. The ions of major interest for these aplications are Er3 +, Tm3 +, and Ho 3 + . These can be used in several different combinations along with Cr3 +, depending on the specific pumping and lasing characteristics desired. There are relatively large energy gaps between the multiplets of Ho 3 + ions, resulting in small radiationless decay rates leading to several metastable states. Thus holmium ions lase on several different transitions in a variety of oxide and fluoride hosts. The transition of greatest interest if 5 h --+ 5 !8 which produces laser emission at about 2. 1 ,urn. This wavelength has strong laser-tissue of interest in medical applications and occurs in an "eyes safe" spectral region with good atmo­ spheric transition for lidar applications. Since the terminal level of this transition is one of the upper Stark components of the ground-state multip­ let that has significant thermal occupation at room temperature, it is difficult to operate as a four-level laser with holmium and the pump thresholds are relatively high. To minimize ground-state absorption of laser light, low con­ centrations of Ho 3 + are generally used. Since this also decreases pumping

364

9. Rare-Earth-Ion Laser Materials

5

Cross

5

I 1 1

1 1 1

1

Tm

3+

3+

5

5

Transfer

5

U p c o nversion

IJ

(8 -+I I I I

17

--

-'Tm

Energy

1

:

(8

1 1 1 I

7

6

Laser

Transition

Ho

3+

5

FIGURE 9 . 1 5 . Energy levels and optical pumping transitions in Y 3 Al501 2 : Tm3 + , Ho3 + [from Ref. 27 (a)].

efficiency, it is common to use sensitizer ions such as Cr3 + or Tm3 + or both. One important example of this type of laser material is Y 3 Al5 0 1 2 : Tm3 +, Ho 3 + . The spectroscopic and lasing properties of this material have been investigated in great detail for laser-pumped systems and flashlamp-pumped operation where Cr3 + is then included as an initial sensitizer ion. The laser­ pumped system without Cr3 + ions will be used for this example. 2 7 The energy levels and pertinent transitions for the optical pumping dynamics are shown in Fig. 9. 1 5. Laser pumping excites the 3 H4 level of Tm3 + . The excitation energy can then stay on the same ion and relax down to the 3 F4 metastable state where both radiative and nonradiative emission occurs. However, for the high concentrations of thulium that are used to make this an affective sensitizer ion, there is a strong probably for ion-ion interaction to occur between Tm3+ ions. The dominant interaction mechanism for these pumping conditions is cross relaxation involving the excited ion undergoing the transition 3 H4 __ 3 F4 , while an unexcited ion simultaneously undergoes the transition 3 H6 __ 3 F4 . Since this process leaves two Tm3 + ions in the 3 F4 metastable state for every ion originally excited, the quantum efficiency of any process involving emission from this metastable state has twice the quantum efficiency as achieved through normal linear pumping processes. In addition to this cross-relaxation process, the ion-ion interaction can cause energy migration among Tm3 + ions. This can occur in the 3 H4 level before relaxation or in the 3 F4 level after relaxation. The latter is generally the dominant energy-transfer process. The migration among Tm3 + ions in the 3 F4 level can occur of long distances which makes the final transfer to Ho 3 + ions very efficient. This takes place through the coupled pair of transitions 3 F4 __ 3 H6 ( Tm3 + ) , 5/s __ sh ( Ho 3+ ) . The energy-transfer processes for this system have been characterized in detail. 2 7 Rewriting the critical interaction distance for electric dipole-dipole

9.2. Other Trivalent Lanthanide Lasers

365

energy transfer in Eq. (5.2.2 1 ) in terms of both transition oscillator strengths and the integrated absorption cross section (9.2.3) the measured absorption and emission spectra can be used to characterize the strength of the ion-ion interaction. Here the functions fsm and Jt rep­ resent the normalized emission spectrum of the sensitizer and absorption spectrum of the activator, respectively. For example, at room temperature the value of Ro for a step in the Tm-Tm migration is calculated to be 1 6.0 A while the value of R0 for the Tm-Ho transfer step is calculated to be 1 7 .3 A. As temperature is lowered to 12 K the value of Ro for Tm-Tm interaction decreases to 1 1 .5 A. This is due to the decrease in the population of upper Stark components, which decreases the transitions available for spectral overlap. For this ion-ion interaction strength and the high concentrations of Tm3 + ions used for sensitization, the average energy-transfer time between pairs of thulium ions is of the order of microseconds while the metastable­ state lifetime is of the order of milliseconds. Thus multistep long-range energy migration occurs among the Tm3 + ions. The long-range energy migration in the 3 F4 level was investigated using laser-induced grating spectroscopy techniques. The signal decay was inter­ preted with the Kenkre model28 using Eq. (5.5. 12),

where V is the nearest-neighbor interaction rate sausing the energy to migrate, rx is the excitation scattering rate, a is the nearest-neighbor distance, A is the grating spacing, and r is the excitation lifetime. The value of r was measured, A was determined from the wavelength and crossing angle of the laser beams, and a was calculated with the assumption that the Tm dopant ions were distributed randomly in the sample. The theoretical expression was then fitted to the experimental results treating V and rx as adjustable parameters. The results of these measurements are shown as functions of temperature and Tm concentration in Figs. 9. 1 6-9. 19. The results shown in Figs. 9. 1 6 and 9. 1 7 demonstrate that the ion-ion interaction rate required to produce the measured long-range thulium energy migration in the metastable state is consistent with the predictions of an electric dipole-dipole mechanism. There are several types of scattering mechanisms that can limit the mean free path of migrating excitions. If the dominant mechanism is scattering by acoustic phonons, the scattering rate should exhibit a temperature dependence that varies as T3 12 . The solid line in Fig.9. 1 8 reflects this type of temperature dependence and the consis-

366

9. Rare-Earth-Ion Laser Materials 2!50xl03 ..

1

u ..

200

� .!l

" II:

1 !10

c 0 ., u " .. ..

1 00

£ E

.. I

!10

E

..

5

0

Tm 3+

10

Concentration

20 x 1 0

15

(em -3)

20

FIGURE 9. 1 6. Concentration dependence of the ion-ion interaction rate of Tm3 + ions for the 3 F4 energy migration process at room temperature. The circles are the experimental points and the line is the theoretical prediction [from Ref. 27(a)].

..

!

1

u

.!l

0 II:

§

., u

e .!l .5

200 1 50 1 00 50

0

50

1 00

1 50

(K)

200

Temperature

250

300

FIGURE 9. 1 7. Temperature dependence of the ion ion interaction rate of Tm3 + ions for the 3 F4 energy migration process for a sample with 14. 1 x 1 020 cm 3 thulium ions. The open circles are the experimental points and the solid circles are theoretical predictions [from Ref. 27(a)].

9.2. Other Trivalent Lanthanide Lasers 6x10

ju

..

.! .. .. 0 0:

01 c: "I:

�0

u

en

367

3

5 4 3

•

•

2

50

1 00

1 50

200

•

250

Temperature (K)

300

350

FIGURE 9. 1 8. Temperature dependence of the excitation scattering rate for the 3 F4 energy migration process of Tm3 + ions in a sample with 14. 1 x 1 020 cm 3 thulium ions. The circles are the experimental points and the line is the theoretical prediction [from Ref. 27(a)].

.. .. .. ..

E �

·u

.. c: .. -=

u

0 u

c: 0

·;;

:I

rs

l.a 2.5 2.0 1 .5 1 .0 0.5 0.0

0

2

Concentration (em-1 4

8

a

10

12

14

1 8x 1 0

20

FIGURE 9 . 1 9 . Concentration dependence of the excitation diffusion coefficient for the 3 F4 energy migration process of Tm3 + ions at room temperature. The circles are the experimental points and the line is the theoretical prediction [from Ref. 27(a)].

368

9. Rare-Earth-Ion Laser Materials

tency between the experimental results and theoretical predictions implies that acoustic phonon scattering is major scattering mechanism. Using the primary energy-transfer parameters V and a determined by experimental measurements, the secondary parameters given in Eq. (5.5. 14) can be determined: V2 Diffusion coefficent: D = 2a2 ; Mean free path:

Lm =

v

IY.

v'la - ;

Diffusion length: LD = ..; . vtstte . . d per scattenng event: Ns = Lm . S ttes a For this material, the mean free path for the exciton migration is of the order of 1 0 6 em, while the diffusion length is of the order of 10- 5 em. The num­ ber of sites an exciton visits between scattering everts is of the order of 50. These values show that for the experimental conditions used, the properties of energy migration for this physical system are a < V; a < Lm « A. This is consistent with a long mean-free-path type of random walk. If energy­ transfer models based on the assumption of scattering at each step in the random walk are used for this system, the predicted values of the diffusion coefficient are more than two orders of magnitude different from those measured experimentally. The theoretical curve shown in Fig. 9. 1 8 that gives a good fit to the experimental data is based on the dependence D oc n41 3 ( Tm). This is predicted by all of the theories developed to describe a random-walk type of excitation migration (see Chap. 5). The overall energy-transfer efficiency from Tm to Ho ions in co-doped samples can be determined from a simple statistical interpretation of the energy transfer parameters described above assuming uniform distributions of the two types of ions. In this approach, the Ho3 + ions are treated as trapping sites for the migrating Tm3 + excitons. The transfer efficiency can also be determined by measuring the risetime of the fluorescence of the Ho ions as described in Chap. 5. In this case, the two methods are consistent within a factor of 2. They give an overall energy-transfer rate of the order of 104 s- 1 . Since the Ro for the Tm-Ho trapping step is greater than the Ro for the Tm-Tm migration step, the energy transfer is a "diffusion-limited" pro­ cesses. The Ho-Tm backtransfer process is not negligible in this system since thermal equilibrium of the excited-state populations of the two types of ions is measured to occur in about 200 Jl.S. In the energy migration model described here, this type of backtransfer is treated as a reduced trapping probability. Due to the significantly lower concentration of Ho 3 + ions com­ pared to Tm3 + ions, energy migration among the holmium ions is negligible. There are a variety of different upconversion processes that can occur on both Tm3 + and Ho 3 + ions resulting in luminescence from higher-energy IY.

.

(9.2.5)

-

9.2. Other Trivalent Lanthanide Lasers

369

metastable states. The significance of these processes depends critically on the concentrations of the two types of ions, the wavelengty of the pump laser, and the pump intensity. The extensive amount of spectroscopic information that has been obtained on Tm, Ho : YAG can be used as input to a rate equation model that describes the optical pumping dynamics of this system. This can be used for computer simulations of the laser-pumped laser operation of this system and the results compared to experimental observations. Using the energy levels and transitions shown in Fig. 9. 1 5, the rate equations describing the time evolutions of the populations of the eight energy levels labeled in the figure n ; plus the photon density in the cavity at the output wavelength np are27 dt

dn1

dt

dn2

dt

dn3

dt

dn4

dt

dns

dt

dn6

dt

dn7

dt

dns

dt

dnp

- U14 + fJ-41 - k42n4n1 - k62n6n1 + k26nsn2 + n2rz 1 + n4r4 1 P41 + n3r] 1 P3 1 + k468n4n6 + k261n2n6 - n 7r7 1 ,

(9.2.6)

2k42n4n1 + k62n6n1 - k26nsn2 - n2rz I + n4r4 I P42 + n3r] 1 P32 - k261n2n6, n4r4 l p43 + n 3r3 1 + n 7r7-1 ,

(9.2.7)

U14 - fJ-41 - k42n4n1 - n4r4 1 - k468n4n6,

(9.2.9)

W.

65 -

(9.2.8)

+ k62n6nl - k 26nsn2 + n6r6 1 + n 7r7- 1 + n s r8 1 , (9.2. 1 0)

1 Ws 6 - Wtis - Wtis - k62n6n1 + k26nsn2 - n6r6 - k468n4n6 - k261n2n6, _ k 261n2n6 - n7r7 1 ,

(9.2. 1 1 )

1 Wtis + k468n4n6 - n s r8 ,

(9.2. 1 3 )

_1 Wtis - Ws 6 + n6Wel - nprc .

(9.2. 14)

(9.2. 12)

Here the Wii parameters represent the rates of stimulated transitions between levels i and j, r; is the fluorescence lifetime of the ith level, kii is the rate of energy transfer from level i to level j, kiik is the rate of energy transfer from levels i and j to level k, pii represents the branching ratio for a transition between levels i and j, rc is the cavity lifetime for photons at the laser output wavelength, and Wei a6s cl/lc. The latter expression describes the stimu­ lated emission due to one photonjcm3 and is used to seed the cavity equa­ tion. In this expression, a65 is the stimulated emission cross section for the laser transition, I is the sample length, and lc is the cavity length. The vari­ ous energy-transfer transitions appearing in the rate equation are obvious

370

9. Rare-Earth-Ion Laser Materials

except the one originating on level 7. Any population of the 4 /5 level of Ho3 + (level 7) undergoes rapid radiationless relaxation to the 5 h level fol­ lowed by energy transfer to the 3 H5 multiplet of Tm3 + (level 3). This series of events is characterized only by the decay rate r:y 1 for simplicity. The stimulated transition rates must be determined from the sum over transition cross sections between individual Stark components taking into account the Boltzmann population distribution of the components of the initial level and the degeneracies of the components of the final level. The stimulated tran­ sition rates at the laser frequency are directly dependent on the concen­ tration laser photons in the cavity. For the pumping conditions and sample consentrations of interest here, the up-conversion processes have very little effect on the spectroscopic and lasing properties. However, when high con­ centration samples and the appropriate pumping conditions are used, up­ conversion can have a significant effect on the pumping dynamics of Tm3 + and Tm 3 + -Ho 3 + laser systems. Sequential up-conversion and relaxation processes can cause the populations of the metastable states to cycle in and out of inversion. From the extensive spectroscopic investigations of Tm, Ho : YAG, the values of all of the transition-rate parameters, branching ratios, cross sec­ tions, and lifetimes apearing in the rate equations are known. The laser cavity parameters are determined by experimental conditions. These equa­ tions can thus be used as a model for describing the optical pumping dynamics of Tm, Ho : YAG lasers with no adjustable parameters. This model does not account for the spatial distribution of excitation energy within the sample or specific cavity modes. Despite these simplifications, this model can be used to predict the temporal behavior of laser emission from this system for different excitation conditions. A computer simulation was performed with Eqs. (9.2.6-9.2. 14) using a fourth-order Runge-Kutta routine to predict the density of laser photons as function of time after the excitation pulse. 2 7 To simplify the model, the up-conversion loss mecha­ nisms were neglected. Examples of the results are shown in Fig. 9.20. Near threshold the numerical modeling predicts a single laser output spike about 200 f.1S after the pump pulse. At higher pump energies the time between excitation and lasing decreasing and the predicted laser emission appears as a series of relaxation oscillations. Figure 9.20 also shows the results of a laser-pumped Tm, Ho : YAG laser experiment. There is excellent agreement between the experimental results and the computer simulations in terms of the temporal characteristics of the laser operation. The threshold energy predicted by the numerical model is significantly lower than that found ex­ perimentally. This may be due to additional active and passive loss mecha­ nisms not take into account in the model. The results shown in Fig. 9.20 demonstrate the use of spectroscopic data and theoretical modeling to pro­ vide a better understanding of laser operational characteristics. In this case the results are especially useful in determining the optimum concentrations of Tm and Ho ions since these concentrations control the energy-transfer

9.2. Other Trivalent Lanthanide 15 10

Lasers

371

7.89 mJ

5 n I

�

:':

E u

0 -

._:

.:;-

ii c "' 0

c 0

0

.J: CL

0 2000

1 50 0

0

200

1 4. 2 mJ

' "' [] 0

500

3000 2000

0

1 8 . 2 mJ

200

400

400

'·::" [] 0

0

0

200

1 000 0

20

Ti m e

(A)

0

200

(11-s)

200

400

400

A L EXANDRITE

10

50

1 000

0

400

0

Ui .. z :: a:i a:

::5

)..

i.i5

z UJ ..

is:

11.4

mJ

:I

j

li

2

0

,

I

0

TIME

( B)

200

I;.<S)

400

: -1

FIGURE 9.20. Comparison of (A) numerical modeling and ( B) experimental results for a Tm, Ho : YAG laser pumped by an alexandrite laser. [Reprinted from Ref. 27(b) with permission of Elsevier Science NL, Sara Burgerhartstraat 25, 1 055 KV Amsterdam, The Netherlands.]

rates and thus are important in determining the temporal characteristics of laser operation. Similar laser measurements and computer simulations performed on the fluoride glass Tm, Ho : ZBAN give qualitatively similar results but with reduced time between the pump pulse and laser emission. 29 Thulium lasers are interesting because they provide the opportunity for continuous tuning of the emission over a broad wavelength range. 3° Figure 9.21 shows the fluorescence spectrum from the 3 F4 level in YAG : Tm3+ . At room temperature, the crystal-field Stark levels are significantly broad­ ened by phonon processes. There are 1 1 7 possible transitions between spe­ cific Stark levels. Each of these transitions has a width of approximately 1 0 nm. The combination of line broadening and the large number of closely spaced spectral lines allows the laser emission based on the 3 F4 -- 3 H6 tran­ sition to be tuned from 1 .87 to 2. 1 6 �tm. Similar tunability is observed for Tm3 + in other host crystals such as YSGG and YLF. 30 A smaller range of

372

9. Rare-Earth-Ion Laser Materials

"§

"iO.I

-e o.e

�

� 0.4 . en

�

0.2

FIGURE 9.2 1 . Room-temperature fluorescence spectrum from the 3 F4 level in YAG : Tm3 + (from Ref. 30).

pl2

N

1

azst

I

/

4 rl l /2

I

N3

a-3 +

( A)

Il S / 2

I

I

/ F

7

/

/2

FIGURE 9.22. Up-conversion pumping dynamics of Yb; Er : YAG lasers.

tunability in the 1 .04-,um spectral region can be obtained from Yb3 + in YAG and other host crystals. 3 1 This is associated with the 2 F5;2 -- 2 F7 ;2 transition and is due the same situation of phonon-troadened, closely spaced Stark levels as discussed for Tm3 + . The energy-level structures of rare-earth ions involving significant numbers of excited levels, some of which have long fluorescence lifetimes, are well suited for up-conversion lasers. These offer an alternative to pumping non­ linear optical crystals for frequence conversion and have the advantage of no phase-matching alignment problems and reduced problems with laser damage. Yb; Er : YAG is an example of a system in which the optical dynamics of up-conversion through sensitized energy transfer have been investigated in detail. 3 2 Figure 9.22 shows the relevant energy levels and transitions involved in the excitation dynamics of the green laser emission in this system. The rate equations describing the time evolution of the level

9.2. Other Trivalent Lanthanide Lasers

373

populations for this system are32

dt

dnt

dt

- P12nt + W2sn2n3 - Ws2nsnt + rx2sn2ns + A 2n2,

dn2 P12nt + Ws2nsnt - W2sn2n3 - rx2sn2ns - A 2n2, dn 3 dt = Ws2nsnt - W2sn2n3 + rx44n42 + rxssn 25 + As 3 ns + A 73n7 + A63 n6 + A s3ns + A 4 n4 , dn4 2 dt = 2rx44 n4 + A 74n7 + A64 n6 + As4 ns A 4n4 , dns 2 dt W2sn2n 3 - Ws2nsnt - 2rxssn 5 - rx2sn2ns + A6sn6 Asns, dt

dn6

dt

rx44n42 + A67n7 A6n6,

dn7 As - A7n7, 1 ns dns 2 dt rxssn s + rx2sn2ns - Asns,

(9.2. 1 5) (9.2. 1 6)

(9.2. 1 7) (9.2. 18) (9.2. 1 9) (9.2.20) (9.2.2 1 ) (9.2.22)

where n; is the population density of the ith level, PiJ is the external pumping rate from level i to level j, A; is the fluorescence decay rate for level i and A iJ is the total transition from level i to a specific level j, wij is the energy­ transfer coefficient from an ion in level i to an ion in level j, and aiJ is the up-conversion coefficient for two ions initially in levels i and j. All of the rate parameters in this model can be determined by spectro­ scopic measurements interpreted with the theories for radiative and non­ radiative transitions and energy transfer discussed previously. 33 The radia­ tive absorption and emission rates were determined through measured spedtral line strengths and Judd-Ofelt analysis. The nonradiative decay rates were determined by measuring the temperature dependencies of the fluorescence lifetimes and applying the energy-gap law using the parameters C 9.7 x 1 07 s- 1 , rx 3 . 1 x w 3 em, and an effective phonon energy of 700 cm- 1 determined previously for YAG. The energy transfer was found to involve multistep migration among the ytterbium ions before transfer to erbium. At average ion-ion separations determined with the assumption of random distributions. the dominant energy-transfer mechanism is electric dipole-dipole for both Yb-Yb and Yb-Er transfer. However, for the ions pairs that are spaced less than 1 0 A apart, higher-order multiple interaction may be important. The strength of the interaction between two Yb ions is about the same as the interaction strength between Yb and Er ions. How­ ever, the relative populations of the initial levels involved in the energy­ transfer transitions results in the Yb-Yb migration step rate being about an

374

9. Rare-Earth-Ion Laser Materials TABLE 9.6. Parameters for transitions in Yb; Er : YAG (after Ref. 33). Nonradiative decay rates Initial level

J .3 X 104 3 .9 X 104

Er: 4 S3;2 4 F7j2 4 /9/2 4 /1 1/2 4 /13/2 Yb: 2 F512

2.8 X 105 2.9 X 103 0.9 1 .4 X 10- 5

Energy transfer parameters Transition

Comment

Yb Yb migration step Yb Er up conversion Er Er up conversion

2.5 X 104 2.1

3.5

X

X

103 102

10 A separation 4 /1 1;2 --> 4 F7;2 at a 10 A separation 4 /1 112 -->4 F112 at a 10 A separation

For 6.5% Yb and 1 .0% Er: Total Yb Er transfer efficiency = 0.66 Csa (EDD) = 5.5 X 1 0-40 cm6 s- 1 Css (EDD) 1 .0 x 10- 39 cm6 s- 1 =

order of magnitude greater than the Yb-Er tansfer rate for the same ion­ ion separation. The backtransfer process from Er to Yb differs from forward transfer only in terms of spectral overlaps for the transitions and for this case the forward transfer is only 50% larger than the backtransfer proba­ bility. For the up-conversion processes, the Yb-Er transition was found to be stronger than the Er-Er process. This is due to the greater strength of the Yb transitions compared to Er transitions which offsets the larger spectral overlap of the Er-Er process. The results of these measurements are sum­ marized in Table 9.6. Figure 9.23 shows the time evolution of the up-conversion fluorescence from the 4S3 ;2 level at 550 nm after a 40 ns pump pulse at 940 nm. 33 The delay tetween the pump pulse and the maximum of the fluorescence is asso­ ciated with the temporal dynamics involving ion-ion energy transfer and is consistent with the predictions of the rate-equation model described above. The Yb; Er : YAG system described above has not been operated as an up­ conversion laser although the pumping dynamics leading to up-conversion fluorescence are similar to the excitation processes in systems that have been made to lase. Along with the characteristics of Nd3 + discussed previously, Er3 +, Pr3 +, Ho 3 +, and Tm3 + ions have been used for up-conversion lasers in both crystalline and glass hosts. The latter have been in fiber laser config­ urations, which can generally operate at room temperature. Fibers have the advantage for stepwise two-photon absorption processes of confining the

9.2. Other Trivalent Lanthanide Lasers 550 n m 50000

.. "'

40000

·�: "'

.30000

·;;

20000

..0

}; 0

:£

c:
375

U pconv e rs io n 0

Q 'l

0

1 7.Er 6.57.Yb 0. 1 7.Er 6.57.Yb 1 7.Er 6.57.Yb 27.Er

1 0000

FIGURE 9.23. Time evolution of the up-conversion fluorescence Yb; Er : YAG. The emission occurs at 550 nm from the 4 S3 ;2 level after excitation by a 40-ns pulse at 940 nm (from Ref. 33).

pump light over a long region of gain. This allows for the use of low doping levels and minimizes thermal effects. Some of the interesting up-conversion lasers are listed in Table 9.7. 34 As mentioned above, there are three exci­ tation mechanisms for up-conversion pumping, and the low phonon cutoff energies generally favor fluorides over oxides for both crystal and glass fiber hosts. Trivalent erbium is one of the most successful ions for up-conversion lasing, and the optical pumping dynamics of some Er3 + systems are discussed here as an example of this type of laser. Trivalent erbium has several important laser transitions including the 4 /1 1 ;2 -+ 4 /1 3 ;2 transition at 3 ,urn, the 4 /1 3 ;2 -+ 4 /1 5/2 transition at 1 .6 ,urn, and the 4 S1 3 ;2 -+ 4 /1 5 ;2 transition at 551 mn. The green laser emission near 551 nm can be obtained from up-conversion pumping of Er3 + ions in YLiF4 crystals. 3 5 As shown in Fig. 9.24(A), the terminal level for this transition is one of the higher-lying Stark components of the ground-state manifold. Pumping can be achieved through transfer by initially building up a signifi­ cant population of ions in the 411 1 ;2 metastable state. This level has a fluo­ rescence lifetime of 8 ms and can be excited directly by 970-nm radiation. Pumping the 4 /9; 2 level by 800-nm light is followed by efficient radiationless relaxation to the 4 11 1 ;2 and thus offers an alternate excitation wavelength. This can be advantageous for diode laser pumping. Cross-relaxation energy transfer can then occur between two erbium ions in the metastable state. In this process, one ion loses its energy decaying to the ground state

376

9. Rare-Earth-Ion Laser Materials

TABLE 9. 7. Examples of rare-earth up-conversion lasers. ( Data taken from Ref. 34 where original references are listed.) Material Ion Nd3 + : Ho3 + : Pr3 + : Er3 + :

Host

Laser wavelength (nm)

Temperature ( K)

Pumping mechanism"

LaF3 YLiF4 Ba(Y, Yb)Fs ZBLAN fiber ZBLAN fiber

380. 1 413.0 551.5 540 553 491 520 549.6 551 . 1 560.6 469.7 551 551 .7 470.3 561 546 486.2 453 483.0 450.2 455 510 480 455

90 20 77 300 300 300 77 90 35 40 300 30 30 300 300 30 300 1 60 1 60 90 200 300 77

STP AVA ET pulsed STP STP STP STP STP ET ET STP pulsed ET ET STP pulsed STP STP STP pulsed AVA STP ET ET STP STP

YAI03 YLiF4

BaY2 Fs

Tm3 + :

KYF4 ZBLAN fiber Y3 Als 0 1 2 YLiF4 Ba(Y, Yb)Fs ZBLAN fiber

"Pumping mechanisms: STP = stepwise two photon; ET = energy transfer; AVA = avalanche. All cw lasers except those marked pulsed.

(4111 ;2 -- 4 /15; 2 ), while the other ion accepts this energy and is raised to the 4 F7;2 level ( 4 11 1 ;2 -- 4 F7;2). The excited ion then decays nonradiatively to the 4 S3 ;2 metastable state from which the fluorescence and lasing occurs. Another mechanism for initiating the 4 /1 1 ;2 -- 4 F7;2 transition is through excited-state absorption of a pump photon if 970-nm pumping is used. As excitation of the system contimues, population builds up in other energy levels giving rise to a variety of possible energ-transfer proceses. One interesting combination of three cross-relaxation processes leads to blue laser emission at 470 nm. 33 The initial pumping and up-conversion step is the same as described above. In addition to populating the 4 S3 ;2 and 411 1 ;2 metastable states, some population will build up in the 4 /1 3 ;2 level due to radiative and nonradiative decay processes from higher excited levels. Then cross-relaxation energy transfer from .one ion undergoing the transition ' 4 [1 1/ -- 4 4 [1 3 /2 -- 4 [15;2 to another wn ' und ergomg the transition as 2 shown in Fig. 9.24( B). The third energy-transfer step is a cross relaxation between one ion undergoing the deexcitation transition 4 S � -- 4/15 ;2 while .

References

--

1

1

I

I

377

470 nm

I

(B)

FIGURE 9.24. Up-conversion pumping dynamics of Er : YLF lasers.

the other ion is excited through the transition 4F9;2 -- 2 K1 3 ;2• The excited ion then decays radiationlessly to the 2 P3;2 metastable state which is the initial state of the laser transition to the 4/1 1 ;2 level. The terminal level is an upper Stark component of the multiplet as shown in Fig. 9.24( B). The longest-wavelength rare-earth laser36 provides output at 7.24 pm. This was achieved by doping Pr3 + in a LaCh crystal. Chloride crystals have very low phonon energies, which minimizes radiationless relaxation processes between energy levels with small energy gaps. This allows more energy levels to have long enough lifetimes to become metastable states and exhibit stimulated emission. The lasing transition for this case is 3 F3 -- 3 F2. Since lasing was achieved only at low temperatures, and since LaCh is a hygroscopic material, this laser is not convenient for practical applications. However, the demonstration of lasing in the mid-infrared spectral region is an important result. References

1 . G.H. Dieke and H.M. Crosswhite, Appl. Opt. 2, 675 ( 1 963). 2. R.C. Powell, S.A. Payne, L.L. Chase, and G.D. Wilke, Phys. Rev. B 41, 8593 ( 1 990) .

378

9. Rare-Earth-Ion Laser Materials

3. N.P. Barnes, D.J. Gettemy, L. Esterowitz, and R.E. Allen, IEEE J. Quant. Elect. QE-23, 1434 ( 1 987) . 4. H. Weidner, W.A. McClintic, Jr., M. McKaig, B.H.T. Chai, R.E. Peale, J.F.H. Nicholls, K.M. Beck, and N.M. Khaidukov, in Proceedings of the Advanced Solid State Laser Conference, 1 995, edited by B.H.T. Chai and S.A. Payne (OSA, Washington, DC, 1 995), p. 545. 5. R.M. Macfarlane, F.Tong, A.J. Silversmith, and W. Lenth, Appl. Phys. Lett. 52, 1 300 ( 1 988). 6. R.M. Macfarlane, A.J. Silversmith, F.Tong, and W.Lenth, in Proceedings of the International Confernce on Laser Science and Laser Materials, edited by Z.J. Wang and Z.M. Zhang ( World Scientific, Singapore, 1 989), p. 24. 7. W.F. Krupke, M.D. Shinn, J.E. Marion, J.A. Caird, and S.E. Stokowski, J. Opt. Soc. Am. B 3, 1 02 ( 1 986) . 8. A. Beimowski, G. Huber, D. Pruss, V.V. Lapterv, I.A. Shcherbakov, and E.V. Zharikov, Appl. Phys. B 28, 234 ( 1 982); D. Pruss, G. Huber, A. Beimowski, V.V. Laptev, I.A. Shcherbakov, and E.V. Zharikov, Appl. Phys. B 28, 234 ( 1 982); V.G. Ostroumov, Yu.S. Privis, V.A. Smimov, and LA Shcherbakov, J. Opt. Soc. Am. B 3, 8 1 ( 1 986). 9. C.M. Lawson, R.C. Powell, and W.K. Zwicker, Phys. Ref Lett. 46, 1 020 ( 1 98 1 ); C.M. Lawson, R.C. Powell, and W.K. Zwicker, Phys, Rev. B 26, 4836 ( 1 982); J.K. Tyminski, R.C. Powell, and W.K. Zwicker, Phys. Rev. B 29, 6074 ( 1 984); R.C. Powell, J.K. Tyminski, A.M. Ghazzawi, and C.M. Lawson, IEEE J. Quantum Electron. QE-22, 1 355 ( 1 986). 10. D. Sardar and R.C. Powell, J. Appl. Phys. 51, 2829 ( 1 980). 1 1 . L.G. DeShazer and R.J. St. Pierre, presented at the Conference on Lasers and Eletro-Optics (CLEO 1 992), Anaheim, CA, May 1 992. 1 2. (a) S.E. Stokowski, in The CRC Handbook of Laser Science and Technology edited by M.J. Weber (CRC, Boca Raton, FL, 1 982) Vol. 1 , p. 2 1 5; (b) 1 2.S.E. Stokowski, R.A. Saroyan, and M.J. Weber, Nd-Doped Laser Glass, Spectro­ scopic and Physical Properties ( Lawrance Livermore National Laboratory, Livermore, CA, 1 98 1 ) . 1 3 . N.F. Evlanova, A.S. Kovalev, V.A. Koptsik, L.S. Komienko, A.M. Prokhorov, and L.N. Rashkovich, JETP Lett. 5, 29 1 ( 1 967) . 1 4 . T . Braf and J.E. Balmer, Opt. Eng. 34, 2349 ( 1 995). 1 5. T. lzumitani, Optical Glass ( Kyoritsu Shuppan, Tokyo, 1 984). 1 6. R. Reisfeld and C.K. Jorgensen, in Handbook on the Physics and Chemistry of Rare Earths, edited by K.A. Gschneidner, Jr., and L. Eyring ( Elsevier, Amster­ dam, 1 987), Chap. 58, p. 1 . 1 7. (a) R.R. Petrin, M.L. Kliewer, J.T. Beasley, R.C. Powell, I.D. Aggarwal, and R.C. Ginther, IEEE J. Quantum Elect. QE-27, 1 030 ( 1 99 1 ) ; (b) R.R. Petrin, R.J. Reeves, M.L. Kliewer, R.C. Powell, I.D. Aggarwal, and R.C. Ginther, in Ad­ vanced Solid State Lasers, edited by H.P. Jenssen and G. Dube (Optical Society of America, Washington, DC, 1 99 1 ), Vol. 6, p. 236. 1 8 . C. Brecher, L.A. Riseberg, and M.J. Weber, Phys. Rev. B 18, 5799 ( 1 978). 1 9. A.G. Avanesov, T.T. Basiev, Yu.K. Voron'ko. B.I. Denker, A.YA. Karasik, G.V. Maksimova, V.V. Osiko, V.F. Pisarenko, and A.M. Prokhorov, Sov. Phys. JETP 50, 886 ( 1 979) . 20. L.A. Riseberg and H.W. Moos, Phys. Rev. 174 , 429 ( 1 968). 2 1 . C.B. Layne, W.H. Lowdermilk, and M.J. Weber, Phys. Rev. 16, 1 0 ( 1 977) .

References

379

22. L.A. Riseberg and M.J. Weber, in Progress in Optics edited by E. Wolf ( North Holland, Amsterdam, 1977), Vol. 14, p. 89. 23. Yu. E. Perlin and A.A. Kaminskii, Phys. Status Solid B 132, 1 1 ( 1 985). 24. R. Reisfeld and C.K. Jorgensen, in Handbook on the Physics and Chemistry of Rare Earths, edited by K.A. Geschneidner, Jr., and L. Eyring ( Elsevier, Am­ sterdam, 1 987), p. I . 25. Yu.V. Orlovskii, R.J. Reeves, R.C. Powell, T.T. Basiev, and K.K. Pukhov, Phys. Rev. B 49, 3821 ( 1 994); Yu.V. Orlovskii, K.K. Pukhov, T.T. Basiev, and T. Tsuboi, Opt. Mater 4, 583 ( 1 995). 26. A. Ellens, H. Andres, M.L.H. ter Heerdt, R.T. Wegh, A. Meijerink, and G. Blasse, presented at the Conference on Dynamical Processes of Ions and Mole­ cules in the Excited States of Solids, Cairns, Australia, August 1 995, to be pub­ lished in the Journal of Luminescence. 27. (a) V.A. French, R.R. Petrin, R.C. Powell, and M. Kokta, Phys. Rev. B 46, 801 8 ( 1 992); (b) R.R. Petrin, M.G. Jani, R.C. Powell, and M . Kokta, Opt. Mater. 1, 1 1 1 ( 1 992); (c) M.G. Jani. R.J. Reeves, R.C. Powell, G.J. Quarles, and L. Esterowitz, J. Opt. Soc. Am. B 8, 741 ( 1 99 1 ) . 28. V.M. Kenkre and D. Schmid, Phys. Rev. B 3 1 , 2430 ( 1 985); V.M. Kenkre, Phys. Rev. B 18, 4064 ( 1 978). 29. R.R. Petrin, R.C. Powell, M.G. Jani, M. Kokta, and I.D. Aggarwal, in OSA Proceedings on Advanced Solid State Lasers, edited by L.L Chase and A.A. Pinto (Optical Society of America, Washington, DC, 1 992), Vol. 1 3, p. 1 39. 30. R.C. Stoneman and L.Esterowitz, Opt. Lett. 15, 486 ( 1 990); J.F. Pinto, L. Esterowitz, and G.H. Rosenblatt, Opt. Lett. 19, 883 ( 1 994) . 3 1 . L.D. DeLoach, S.A. Payne, L.L. Chase, L.K. Smith, W.L. Kway, and W.F. Krupke, IEEE J. Quantum Electron. 29, 1 1 79 ( 1 993). 32. P. Lacovara, in Advances in Solid State Lasers, edited by A. Pinto (OSA, Washington, DC, 1 992), Vol. 1 3 , p. 296. 33. P. Lacovara, Ph.D. thesis, Boston College, Chestnut Hill, MA ( 1 992) . 34. R.M. Macfarlane, J. Phys. ( Paris) 4 Colloq. C4-289 ( 1 994). 35. W. Lenth, A.J. Silversmith, and R.M. Macfarlane, in Advances in Laser Science III, edited by A.C. Tam, J.L. Gole, and W.C. Stwalley, Proc. AlP 172, 8 ( 1 987); R.A. McFarlane, Appl. Phys. Lett. 54, 2301 ( 1 989); R.A. McFarlane, Opt. Lett. 16. 1 397 ( 1 99 1 ); T. Hebert, R. Wannemacher, W. Lenth, and R.M. Macfarlane, Appl. Phys. Lett. 57, 1 727 ( 1 990). 36. S.R. Bowman, L.B. Shaw, B.J. Feldman and J. Ganem, in Proceedings of the Advanced Solid State Laser Conference, 1995, edited by 1 995 B.H.T. Chai and S.A. Payne (OSA, Washington, DC, 1 995), p. 1 1 6.

10

Miscell aneous Laser M aterials

As described in the previous four chapters, the most common solid-state laser systems are based on first row transition-metal ions and trivalent lan­ thanide ions. These are shown in their position in the periodic table of the elements in Fig. 10. 1 along with the spectral region covered by each class of laser ion. In addition to these laser ions, solid-state lasers have been demon­ strated based on divalent lanthanide ions, actinide ions, color centers, and various molecular centers. Some of these are discussed in the following sec­ tions. It is obvious that there are many other ions that might be developed into lasers in the future, and this is currently an active area of research. However, most of the optically active centers that are obviously useful for laser materials, according to the theoretical considerations discussed in this book, have been investigated, and the development of any significant new systems will require some novel aspect such as new types of hosts, new TRANSITION METAL ION LASER RANGE: 0.61 to 3.5 11m \ / \ / / \

H

Li

Be

Na

Mg

K

Ca

Sc

Rb

Sr

y

Cs

Ba La

Fr

Ra

Ti v

Cr

Zr

Mo Tc

Ru

w

Os

Nb

Hf Ta

Ac '

'

'

Ce 3+

Th ..

Pr

Pa ..

..

Mn

Fe

Re

Nd 3+ u

Co

Rh Ir

Ni

B

c

N

0

AI

Si

p

s

CI

Ar

Ne

Cu

Zn

Ga

Ge

As

Se

Br

Kr

Pd Ag

Cd

In

Sn

Sb

Te

I

Xe

Pb

Bi

Po

At

Rn

Pt

Pm

Sm

Eu

Np

Pu

Am

..

He F

Hg

Au

Gd

Cm

11

Tb Bk

Er

Cf

Es

RARE EARTH ION LASER RANGE: 0.17

to

7.2 !liD

Fm

.,

..

Lu Md No ..

..

..

Lr

FIGURE 1 0. 1 . Common lasing ions in the periodic table and their spectral coverage. 380

1 0. 1 . Other Rare-Earth-Ion Lasers

381

pumping schemes, or new cavity configurations. Still the laser systems cur­ rently available have a limited choice of operational characteristics, and the search for new solid-state lasers is driven by applications that require lasers with specific characteristics that are not currently available. One of the most important areas of research and development is intracavity nonlinear optical processes. The typical way of utilizing nonlinear optical processes in solid­ state laser systems is to pump frequency-shifting crystals outside the stan­ dard laser cavity. Recent research has demonstrated that having the non­ linear crystal inside the laser cavity can be advantageous in improving conversion efficiency as well as utilizing the nonlinear optical transmission to improve the beam quality of the laser output. An example of this type of laser system is discussed below. 1 0. 1

Other Rare-Earth-Ion Lasers

The major rare-earth-ion lasers that were discussed previously in Chaps. 8 and 9 all involve f-to-f transitions of trivalent ions. However, laser action can be obtained from d-to-f transitions of both trivalent and divalent rare­ earth ions, and lasers can operate on /-to-transitions of divalent rare-earth ions. Figure 10.2 shows the energy-level diagrams and major lasing tran­ sitions for some of these ions. For most trivalent lanthanide ions, the manifold of 5d electron energy levels is at high enough energy that it is not important for laser operation. The absorption bands associated with f-to-d transitions are so far into the ultraviolet spectral region that they are masked by the band edge of the host material and out of the normal spectral range of pump lamps. Usually there are some high-energy levels of the 4f configuration that allow for fast non­ radiative decay of excitation energy from the 5d to the 4f energy levels. Trivalent cerium is an exception to this situation. The energy level for Ce3 + is shown in Fig. 1 0.2. The relevant electron configuration for the ground state of this ion is 4d 1 leading to only one term, 2 F. Spin-orbit coupling splits this into J = � and � multiplets. The first excited state above the ground-state term is a state of the 5d configuration. Since there are no high­ energy levels of the 4f configuration to enhance nonradiative decay pro­ cesses, the 5d level fluoresces with a high quantum efficiency in several host materials such as LiYF4 , LaF 3 , and LiSAF. The exact position of the 5d energy levels and their splitting due to local crystal fields varies significantly from host to host. The fluorescence transitions originate on the lowest 5d level and terminate on the two multiplet components of the ground state. Since the absorption and emission transitions are between states of different electron configurations having opposite parity, they are highly allowed tran­ sitions. This results in strong, broad absorption bands and very fast fluore­ scence lifetimes (of the order of w- s s). The interesting aspect of Ce3 + lasers is that they produce tunable emission in the near-ultraviolet spectral region.

10. Miscellaneous Laser Materials

382

";"Ei u "'

g .. "' � "" z ""

25

� 5o

15 10

I

I '"' __It;

5 0

o

7 p1 sm2+

(CaF2)

sm2+

(SrF2l

I: o y2+

I

-411 5/2 -411 3/2

r

�l l/2 2F7/2 2F5/2 _ 4r9/2 2F7/2 ce 3+ Tm2+ u 3+

FIGURE 1 0.2. Energy-level diagrams for several laser ions.

For example, 1 LiYF4 : Ce 3 + vibronic laser emission can be tuned between 306 and 3 1 5 run, and between 323 and 328 nm. There are problems with photoionization of cerium in fluoride hosts, and excited state absorption losses prevents Ce3 + from lasing in oxide crystals such as YAG. One important difference of divalent lanthanide ions compared to tri­ valent ions is that the energy levels of their 4r- ' sd configurations generally occur at lower energies. Thus in some host materials the 4/ .. 5d transitions occur in the visible region of the spectrum and can be useful for laser oper­ ation. Since these are allowed electric dipole transitions, they are much stronger than f-to-f transitions and have oscillator strengths near unity. Also, the crystal-field perturbation strength is stronger for electrons in the 5d level than for those in the 4/ level. This results in broad transition bands with large Stokes shifts and a higher probability for phonon-assisted tran­ sitions. As seen in Fig. 1 0.2, it is common for divalent rare-earth ions in fluoride host crystals to have energy levels of the 4f configuration high enough in energy that radiationless decay processes occur efficiently from 5d to the 4f levels. For these materials the f-to-d transitions can provide effi­ cient pump bands, but the laser emission is associated withf-to-f transitions as with standard trivalent rare-earth ions. Laser materials of this type emit in the red and near-infrared spectral regions. One exception to this situation

10. 1 . Other Rare-Earth-Ion Lasers

383

is Sm2+ in CaF2 crystals. For this case the levels of the 4f5 5d configuration lie below the excited state multiplets of the 4f6 configuration. Thus the laser transition is a d-to-f transition that occurs near 708 nm with some tunability due to the vibronic coupling of the d level. However, poor crystal quality has limited the usefulness CaF2 : Sm2+ for laser applications. Although all of the lanthanide series ions have been made to lase, the only actinide ion that has lased is trivalent uranium. The crystal field strengths for the sr actinide ions is about twice as large as it is for the 4fn lanthanide ions. This results in a larger splitting of the Stark levels and stronger f-f transitions for the actinide ions compared to the lanthanide ions. The energy-level diagram for U 3 + is shown in Fig. 10.2. The energy levels of actinide ions are based on 5fn configurations, and spin-orbit interaction is generally stronger than that found for the 4r configurations of lanthanide ions. U 3 + has three f electrons, and thus its ground-state term of 4 I similar to that of Nd3 + . The large multiplet splitting reduces the nonradiative decay rates between the ground-state term multiplets and allows for a laser tran­ sition to occur between the 4 1 1 1 ;2 level and the upper crystal-field split com­ ponent of the 4 19 ;2 multiplet of u3 + in CaF2 and similar crystals. 2 The laser emission occurs in the near-infrared spectral region at wavelengths ranging from 2.23 to 2.83 J.Lf, depending on the host material. Typical host crystals for trivalent uranium are fluorides2•3 such as CaF2 and LiYF4 . The absorp­ tion and emission spectra of LiYF4 : U3 + are shown in Fig. 10.3. The emis­ sion spectrum is composed of 30 unresolved transitions between the 6 Stark components of the 4 1 1 1 ;2 metastable state and 5 Stark components of the 4 19; 2 ground-state manifold. Because of the strength of the crystal field, J is no longer a good quantum number, so spectral properties such as tran­ sition oscillator strengths and branching ratios must be calculated using the crystal-field energy levels. The room-temperature lasing transition3 has a

PUMPING RANGE

�

� 0. 3

.. ..,

Q.

0

3.0

(A)

0.1

0.6

08

A ( )l m l

10

(B)

FIGURE 10.3. (A) Absorption and ( B) emission spectra of LiYF4 : U3 + [taken from Ref. 3(a)].

384

10. Miscellaneous Laser Materials

cross section of 3 x 10- 20 cm- 2 and a lifetime of 1 92 /ls (for small uH con­ centrations where there is no ion-ion interaction). The lifetime lengthens at low temperature, implying a quantum efficiency of about 40%. The potential for more efficient optical pumping and broader tunability of the emission make actinide ions an attractive alternative for solid-state laser materials. So far, however, little work has been done to exploit these properties for prac­ tical solid-state laser systems.

1 0.2

Nonlinear Optical Lasers

There are still many specific spectral regions where no solid-state lasers have emission lines or the operational characteristics of lasers operating at these wavelengths are not satisfactory. One technique for expanding the spectral coverage of solid-state lasers is through nonlinear optical effects. The prop­ agation of intense electromagnetic waves through a nonlinear optical mate­ rial produces coherent radiation at new wavelengths through processes such as harmonic frequency generation, sum and difference frequency mixing, parametric generation, and stimulated Raman scattering. Generally fre­ quency shifters are devices used external to primary laser systems, and therefore the detailed theoretical treatment of nonlinear optical processes, devices, and materials is not covered here. 4 However, in some cases the fre­ quency shifting occurs inside the laser resonator and thus becomes a fun­ damental property of the laser system. One good example of this is self­ doubling laser crystals discussed in Sec. 9.3. Therefore, a general outline of the theory of nonlinear optical processes in solids relevant to frequency­ shifting devices is given here. The electrons bound to atoms in a solid respond to an incident electro­ magnetic field as an ensemble of dipole oscillators, as shown schematically in Fig. 1 0.4. Under normal conditions of weak driving fields, the response of the oscillators is linear and they reemit electromagnetic waves with the same frequency as that of the incident field. However, for strong driving fields the INCIDENT ELECTROMAGNETIC WAVE

NCNLINEAR DIELECTRIC MJITERIAL

TRANSMITTED ELECTROMAGNETIC WAVE

NONLINEAR GENERArED WAVE

1 0.4. Polarization response of atomic dipoles to the driving force of an elec­ tromagnetic wave.

FIGURE

385

1 0.2. Nonlinear Optical Lasers

restoring force of an oscillator may be too weak to maintain a linear re­ sponse. In the presence of strong interaction with the force of the external field and restrictions of the local geometry and bonding of the lattice struc­ ture, response frequencies different from that of the driving force can be generated. This nonlinear response causes the oscillators to produce electro­ magnetic waves at frequencies different from that of the incident electro­ magnetic field. The atomic system of interest for nonlinear optics can be modeled as an ensemble of driven harmonic oscillators. For each oscillator, the displace­ ment of electron from the ion in the ith direction, is r; and this is described by the equation ( 1 0.2. 1 ) r;.. + Y;r. ; + OJ;2 r; eE; where y is a damping factor, OJ; is an oscillation frequency of the dipole, and E; is the driving field. For a linearly polarized electromagnetic wave, ( 10.2.2) The displacement can be expressed as a Fourier series for different fre­ quency components, 00

r; = L r;( OJn) eiwnt . n= oo

( 10.2.3)

Substituting Eqs. (10.2.2) and ( 1 0.2.3) into Eq. ( 10.2. 1 ) gives the component of the displacement at the driving frequency OJrx as (10.2.4) For N oscillators per unit volume with polarization p ; (OJrx) polarization of the material is given by

eo er; (OJct ) , the

Ne2 E;( OJrx) . 2 OJ; - OJ2rx + lY;OJrx x) i l (OJrx)E; (OJrx),

P;( OJrx ) Np; (OJrx )

where the linear susceptibility i s defined as (I) X; (OJrx)

Ne2 . OJ;2 OJ2rx + lY;OJrx

( 1 0.2.6)

The nonlinear response of the system can be treated by expanding the potential for the electron-atom interaction as

V Vo Vo

o

o L:(a ) o

r;r1 + r;rJ rkJ + r; + i r, i,J r, rJ i,J,k r, rJ rk + L P; r; + L PiJ rirJ + L PiJk r;rJ rkJ + ( 10.2.7) i,j iJ,k +

·

·

·

.

·

·

·

10. Miscellaneous Laser Materials

386

Many of the terms in the sums are equivalent under the permutation of the subscripts. It is common to reduce the terms by defining tensors such as

Vijk = Pijk + Pikj + Pjik·

Then the equation of motion for a perturbed system is

3 ( 10.2.8) L Vijkrjrk eE; . j, k=l If the driving field has two driving frequencies Wa and wp , Eq. ( 10.2.8) can r;

+ y;h + wfr; +

be solved as described above and the displacement at each frequency com­ ponent in the Fourier spectrum listed separately,

eE; eE; (wa ) r; ( Wa ) = 2 2(w+a ). (Wa ) W; - Wrx lY;Wa eE; (wp) eE; (wp) r,· ( wp ) . A; (Wp ) W;2 - Wp2 + lY;Wp

ri ( Wa + Wp ) =

'

'

+ wp ) Aj(wa ) Ak (wp )

A; ( Wa - wp) Aj (wp) Ak(wa ) · ( 10.2.9) The last two expressions in Eq. ( 10.2.9) produce polarizations of the media at sum and difference frequencies,

P;(wa + wp)

L X�� (wa + wp; Wa , wp) Ej(wa )Ek(wp ) jk

+

(2) (wa + wp; Wa , wp) Ej(wp )Ek(wa ), Xijk

jk P;( Wa - wp) L x�� (wa + wp; wa, wp) Ej(wa )Ek(wp ) jk + L xW (wa - wp; Wa , wp) Ej(wa )E'k (wp) . jk

( 1 0.2. 10)

where

- ViJkeoe3 N Xij(2d) wa + Wp •, Wa, wp ) _ A · ( + ) A · ( ) Ak ( ) , Wa Wp Wa Wp ViJkeo e3 N ) - wp,. wa , wp) _ · XiJ(2dwa A (Wa Wp ) A · (Wa ) Ak* ( Wp ) ' z

1

�

_

( 1 0.2. 1 1 )

�

This perturbation approach can be extended to higher orders and com­ bined into a single driving polarization,

10.2. Nonlinear Optical Lasers

387

( 1 0.2. 12) This polarization can be used in conjunction with the normal wave equation derived from Maxwell's equations in describing the propagation of an elec­ tric field in a dielectric medium. 5 Assuming propagation in the z direction so it is possible to work with a one-dimensional equation, and making the normal slowly varying amplitude approximation for an optical field, the wave equation is

oE( w) + aE( + ! oE( w) if.lCW ( (10.2. 1 3 ) w) v ot oz 2n P w) where k nw I c, f.lB 1 I v2 , and !Y. f.lCJC12. Substituting the nonlinear po­

larization into this expression yields a set of equations that can be solved to give the expressions for the fields associated with each frequency component present in the Fourier spectrum. Harmonic generation. One important example of nonlinear optical pro­ cesses for shifting the output wavelength of a laser is harmonic generation through frequency doubling, tripling, or quadrupling. As an example, con­ sider second-harmonic generation (SHG). In this case the general driving polarizations and second-order susceptibilities given in Eqs. ( 1 0.2. 10) and ( 1 0.2. 1 1 ) can be written specifically with Wrx wp w . Two photons with the pump frequency w combine to give one photon with twice that fre­ quency as shown in Fig. 10.5(A). It is general convention to write the second-order susceptibility in terms of the piezoelectric tensor d, 6

xW ( w3 ; w, , w2) dijk ( w3 ; w, , wz).

The expression for the driving polarization at twice the incident wave fre­ quency is

( 1 0.2. 14) where

( 1 0.2. 1 5)

k

(A)

2k

k

(B)

FIGURE 10.5. Feynman diagrams of three-photon nonlinear optical processes. (A) Second-harmonic generation. ( B) Optical parametric generation.

388

10. Miscellaneous Laser Materials

is the phase mismatch vector, and the effective second-order nonlinear co­ efficient is ( 1 0.2. 16) Here the a;'s are the direction cosines of the polarization and the electric fields. Substituting the driving polarization expression from Eq. ( 10.2. 14) into the wave equation in Eq. ( 1 0.2. 1 3), an expression for the SHG field E2w (r) can be derived. A similar expression for the field can also be found. For a lossless media, in the steady-state limit, the coupled set of nonlinear field equations are

Ew (r)

E

d zw (z) dz where

=

. Ew (z )Ew (z) lKzw

e

illkz ,

( 1 0 . 2. 1 7)

( 1 0.2. 18) The exact solutions to this set of equations 7 are complicated mathe­ matical expressions involving Jacobi elliptic integrals. The expressions com­ monly used for describing SHG are found by making several simplifying assumptions, including the initial condition of zero field at the second har­ monic frequency, no wave-vector mismatch, and no depletion of the funda­ mental beam. Under these conditions, the expressions for the intensities of the waves at the fundamental and SHG frequencies as they travel a distance L in the nonlinear optical material are lw (L) = lw (O)sech2 (rL) , ( 1 0.2. 19) fzw (L) = lw (O)tanh2 (rL) , where r Kw I £1 ( 0) I · The limiting form of the Jacobi elliptic integral func­ tion when pump depletion is not considered gives the following expression for the SHG intensity: !l.kL . (10.2.20) l2w (L) lw (O) (rL) 2 smc2

=

=

•

( )

It should be noted that this treatment assumes plane waves. If the treatment is expanded to include generalized wave fronts, analytic solutions are not possible. The expression in Eq. ( 10.2.20) demonstrates that the most efficient SHG occurs when there is no phase mismatch. If !l.k is not zero, there is an inter­ ference between SHG waves originating at different z planes in the material. Figure 10.6 shows how the SHG intensity varies as a function of the phase

1 0.2. Nonlinear Optical Lasers 1 .00

FIGURE 1 0.6. SHG intensity output pat­ tern versus phase mismatch ( Ref. 12) for a gain coefficient of G rL 1 .

0.75

=

� 0.50

.:I'

389

=

0.25

0.00 -4- 3-2- 1 0 1 2 3 �kL

4

mismatch-interaction length product. The interference effects cause peaks in the SHG intensity at different positions. The distance between adjacent peaks is called the coherence length and is given by 2n 2n ( 1 0.2.2 1 ) lc where kw wnw/ c and A. i s the wavelength of the fundamental wave in free space. The coherence length is the maximum useful length in the material for producing SHG. It is typically of the order of only 100 11m, which demonstrates why it is necessary to have exact phase matched conditions in order to have efficient SHG. One method for achieving phase matching is to take advantage of the bi­ refringence and dispersion properties of an anisotropic crystal. As an exam­ ple of this, consider three-wave interactions in a biaxial crystal. For waves of frequency w; and wave vector k;, conservation of energy and momentum requires that ( 10.2.22) where phase matching occurs when Ilk 0. Expressing the wave vector as k; (n;w;jc)l;, where i; represents a unit vector in the k; direction, and as­ suming that all three waves propagate in the same direction, the two-phase matching conditions become 1

W J n 1 + w 2 n2 . W3 n 3 W 3 n3

( 1 0.2.23)

The first of these expressions shows that the ratios of the frequencies

w J /w3 and w2 /w3 must both be less than one. Using this information in

the second expression leads to three possible conditions for the refractive indices: ( 10.2.24)

10. Miscellaneous Laser Materials

390

or (10.2.25) or >

1,

<

1

=>

n1

>

n 3 , n2

<

(10.2.26)

n3 .

Thus for this type of phase matching, a crystal must be used that has differ­ ent refractive indices for different directions, and it must be cut and oriented in such a way that the three beams are traveling and polarized in directions that the relevant components of the refractive indices satisfy one of the con­ ditions in Eqs. ( 1 0.2.24)-( 1 0.2.26). For a biaxial crystal, the spatial variation of the index of refraction has the form of an ellipsoids with values in the directions of the semimajor and semiminor axes of ne1 and ne2 , respectively. The propagation of a wave traveling in an arbitrary direction in a biaxial crystal is described by the equations ( 10.2.27) where x, y, and z are the directions of the principle axes of the ellipsoid. The solutions of this quadratic equation are n = ne1 and ne2 • Thus a wave with arbitrary polarization traveling in the crystal will be resolved into two com­ ponents traveling at different speeds with mutually perpendicular polariza­ tion components in the e1 and e2 directions. Combining these results with the phase-matching conditions in Eqs. ( 10.2.24)-( 10.2.26) defines the three types of phase matching for biaxial crystal. These are listed in Table 10. 1 . In order to orient a crystal in the appropriate way to achieve one of the phase-matching conditions for waves traveling colinearly through the mate­ rial with different wavelengths and directions of polarizations, it is necessary to know the variation in the index of refraction with wavelength of light for different crystallographic directions. This dispersion relation is given by Sellemier's equation, TABLE 10. 1 . Phase matching conditions for a biaxial crystal. Type I

e2

n3

w1 : w2 : W3 :

_

e1 e1 e2

ray ray ray

W I e1 + 0J2 e1 2 W3 n WJ n i

Type lib

Type Ila

e2

n3

w1 : e1 ray cv2 : e2 ray e2 ray w3 : W I e1 + ()2 e2 2 W3 n WJ ni

ez

n3

WJ : w2 : w3 :

_

e2 e1 e2

ray ray ray

CO J ez + 0J2 e1 2 WJ ni w3 n

10.2. Nonlinear Optical Lasers

n21 (A.) 1 = A , + 2 A 2 2 + A 4 A.2 + A. A 3 -

·

·

·

391

( 10.2.28)

where the ni are the values of the refractive indices along the three principal axes of the index ellipsoid. Equation ( 10.2.24) is an empirical expression that is fit to experimental data points treating the values of the A i coeffi­ cients as adjustable parameters. This equation is an example of one of several different empirical expressions that thave been used to describe the variation n with the wavelength of light. In order to obtain an accurate dis­ persion curve, the data used for obtaining the theoretical fit must be dis­ tributed over a large spectral range. Using spherical coordinates in a biaxial crystal, the expressions for the induced second-order nonlinear polarizabilities with type I and type II phase matching are

�� e 1 (w3) eoa�2 dijki(w3 ; w2 , w, )a? a�1 Ee 1 (w2) Ee1 (w, ) = d;ff( fJ , l/J)Ee 1 (w2)Ee 1 (w i ), type I P=� e1 (w3) = eoa?dijki(W3i w2, w! )a?a�1 Ee2 (w2)Ee 1 (w1 ) d;� ( O, ¢J)Ee2 (w2) g1 (w, ) , type II

( 10.2.29)

where

delff (0 , eoaei 2 dijki ( W3 ., w2 , w, ) aje1 ake 1 , d�� ( O , f/J) eoa�2 difki(w3 ; w2, w! )a?a�1 -

( 10.2.30)

are the effective second-order nonlinear coefficients for type I and type II phase matching, respectively. As long as the material is transparent in the spectral range covering all three frequencies of interest, the nonlinear co­ efficients are independent of frequency. It is common practice to simplify calculations involving the nonlinear co­ efficient dijk by noting that the expressions in Eqs. ( 10.2.29) and ( 10.2.30) are invariant under the interchange of subscripts j and k . Therefore the non­ linear coefficient tensor can be written as a 3 x 6 matrix with elements dim, where the subscript m represents the independent jk elements with jk = 1 1 =? m = 1 ; jk 22 =? m = 2; jk 33 =? m = 3 ; jk = 23 or 32 =? m = 4; jk = 13 or 3 1 =? m = 5; jk = 1 2 or 21 =? m = 6. Using this reduction allows the expressions in Eq. ( 10.2.29) to be written in the form

G:)

d" d!2 d!3 d, 4 d, s d,6 o ' � d2! d22 d23 d24 d25 d26 d3! d32 d33 d34 d3 5 d36

E2I E22 E32 2E2E3 2E,E3 2E,E2

( 10.2.3 1 )

392

1 0. Miscellaneous Laser Materials

Symmetry arguments can be used to reduce the number of nonzero, inde­ pendent matrix elements of the nonlinear coefficient. Kleinman's symmetry condition requires the dijk elements to be independent of permutation of i , j, and k. In addition, the third rank tensor for dijk must be invariant under the symmetry operations of the crystal symmetry group.9 Using the product function dxyz xyz as the basis function, the symmetry operations of the crystallographic point group can be applied to all of the matrix elements to determine which ones remain invariant. For example, using the five sym­ metry operations of the C2v point group show that only five dim matrix ele­ ments are invariant (and therefore nonzero) . Kleinman's symmetry con­ dition shows that only three of these are independent matrix elements. The final nonlinear coefficient matrix for C2v symmetry has the form 0 0 0 0 d1 s 0 ( 10.2.32) 0 0 0 d24 0 0 0 0 0 Equations ( 10.2.29) and ( 10.2.30) can now be expanded in terms of the nonzero dim coefficients and the direction cosines expressed in terms of the crystallographic directions in spherical coordinates. The expressions for derr are maximized with respect to the angles 0 and rp. This gives the maximum coupling between the fields, and the angles found in this way are called the phase-matching angles Om and fJm · As an example of phase-matched SHG, consider a crystal of KTiOP04 (designated KTP) pumped by a YAG : Nd3 + laser at 1 064 nm. The fre­ quency doubled emission occurs at 532 nm. The point-group symmetry for KTP is C2v so the nonlinear coefficient tensor is given by Eq. ( 10.2.32). The Sellemier coefficients and dim coefficients have been measured. 1 0 • 1 1 This information can be combined to give the plot of derr versus the crystallo­ graphic angles 0 and rp shown12• 1 3 in Fig. 1 0.7. These results show that the maximum conversion efficiency for SHG will occur for type II phase matching with Om 90° and fJm 25.2° . Since phase matching depends critically on the refractive index of the material and n ( T , A., Om , rpm ) changes with temperature, wavelength of light, and crystallographic direction represented by the angles Om and fJm , there are "acceptance parameters" for phase-matching conditions. The physical condition of an experimental setup must be within the acceptable ranges of these parameters for efficient optical conversion to occur. This allows the ability to have angular and temperature tuning of nonlinear frequency con­ version. However, it also can result in instability of the nonlinear optical process. Another important performance limiting side effect of birefringe­ ment phase is "walk-off" of the interacting beams caused by double refrac­ tion. When Om 90° the angular acceptance bandwidth is at its maximum and the walk-off angles are zero. This condition is referred to as "noncritical phase matching." Some of the important nonlinear optical materials for second-harmonic generation are listed in Table 10.2. As discussed in Chap. 9, some of these

1 0.2. Nonlinear Optical Lasers 20

..

40 9

''

'

50

''

'

''

70 80

.. · ' ,

0

,

10

·

., .. .. ·

, • ..

20

3.5 3.0

60

90

4.0

• W a x occurs at 8 = 9 0 ° , t = 2 5 . 2

30

2.5 ',

II ' ''

'

.. - ..� .. _ _ .. .. .,. .. ..

,.

,,

.. .. ., ··

30

393

2.0 1 .5

>

.. e

� I

0

.. '-'

1 .0

"C

G

0.5 40

50

60

70

80

00 0 90

FIGURE 10.7. SHG phase-matching curves for KTP pumped at 1 064 12).

nm

(from Ref.

TABLE 1 0.2. Examples of nonlinear optical materials ( Data taken from Ref. 4.) . Material Formula KHzP04 KDzP04 NH4HzP04 LiNb03 (MGO) KNb0 3 KTiOP04 BazNa(Nb03 )5 P BaBz04 LiB3 0s KTiOAs04 AgGaS2 AgGaSe2 ZnGeP

Transparency range Common name KDP D KDP ADP

KTP Banana BBO LBO KTA

(run)

176.5 200 1 84 330 400 350 370 198 1 60 350 500 710 740

1 700 2000 1 500 5500 4500 4500 5000 2600 2600 4000 1 3000 1 8000 12000

l dii l

(lo-1 3 m/V) 4.35 4.02 5.28 54.4 2.05 1 37 1 82 22.2 1 1 .7 6.96 1 34 333 754

materials can act as host crystals for active dopant ions such as Nd3+ to self­ doubling lasers. However, as seen from the discussion in this section it is difficult to design a system to simultaneously optimize laser performance and SHG efficiency. Thermal loading, due to the finite pumping quantum defect, and small optical damage thresholds of doped nonlinear optical crystals are the most serious deleterious effects to overcome to make self-

394

10. Miscellaneous Laser Materials

doubling lasers practical. Cavity effects such as diffraction and walk-off can destroy phase-matching conditions. In addition, some of the active lasing ions such as Nd3 + can has absorption transitions at the wavelength of the second-harmonic light, and this acts as a loss mechanism in a monolithic system. Thus most frequency-doubled solid-state laser systems involve ex­ ternal cavity frequency doublers with separate laser materials and nonlinear optical materials. Frequency tripling and quadrupling are also generally done with nonlinear optical frequency multiplying or mixing crystals outside the primary laser cavity. Optical parametric generation. Optical parametric generation is another important second-order nonlinear optical process for solid-state lasers. Devices based on this process include optical parametric oscillators (OPOs) and optical parametric amplifiers (OPAs). OPOs are especially interesting since they provide tunable coherent output at two wavelengths simulta­ neously. Since these are external devices to the primary solid-state laser, they will not be treated in complete detail. The same coupled differential equations for the nonlinear driving polar­ izations resulting from three-wave interactions that was outlined above can be used to describe OPOs. In this case the pump wave is the high-frequency electromagnetic wave and it interacts with two lower-frequency electro­ magnetic waves called the signal and idler waves. If the nonlinear material is in a cavity with feedback, gain can occur in either the signal or the idler wave or both. The mathematical description of optical parametric gen­ eration is similar to that of SHG. There is a set of three coupled differential equations describing the pump, idler, and signal waves. The expressions for the driving polarizations for these three waves are similar to Eqs. ( 1 0.2. 1 0) and ( 1 0.2. 1 1 ), but the frequencies of the waves are W 3 Wp , w2 Ws, and WJ w;, where the subscripts p, s , i designate pump, signal, and idler, re­ spectively. In this case the pump photon decays into the signal and idler photons as shown in Fig. 10.5( B). This is a "parametric" process since the values of the signal and idler frequencies can vary with the only restriction being that their sum is the pump frequency. For a lossless material, the equations describing the colinear propagation of the three coupled fields in the media give phase-matching and conservation of energy conditions of

kp - ks - k;, Wp Ws + W; with coupling coefficients given by Eq. ( 10.2. 1 8). /1k

( 10.2.33)

Maximum gain in intensity in the signal (and/or idler) beam a s it travels through the crystal requires maximizing the coupling coefficient and main­ taining phase-matching conditions. The latter condition limits the angular, temperature, and spectral acceptance bandwidths of the OPO crystal. Diffrac­ tion and birefringence walk-off limit the effective interaction length of the material, and for strong coupling of the three waves the interaction length is limited by backtransfer of intensity from the signal (or idler) beam to the pump beam. For ultrashort pulsed systems (i.e., femtoseconds), maintaining

1 0.2. Nonlinear Optical Lasers

395

temporal overlap of the three fields is also important. This is limited by the varying group velocities of beams at different frequencies as well as dis­ persion spreading of the pulses. These problems have made it difficult to develop OPOs that have the efficiency and stability required for use in prac­ tical laser systems. However, recent advances in new nonlinear optical mate­ rials along with the development of innovative resonator designs have now made OPOs important devices for solid-state laser systems. Using angular or temperature tuning, the phase-matching conditions in Eq. ( 10.2.33) can be satisfied over a wide range of signal and idler frequencies for a given pump frequency. For example, a nonlinear optical material such as BBO can provide phase-matching conditions that allow output wavelength to be tuned between 0.4 and 2.5 pm. Some of the important properties of non­ linear optical materials useful for OPO applications are listed in Table 1 0.2. One interesting type of system is the combination of a tunable solid-state laser source with an OP0. 14 In this case the properties of the OPO remain constant, and tunable output is obtained by tuning the wavelength of the pump laser. An example of this is shown in Fig. 10.8. The theoretical and experimental tuning curves for the signal and idler beams of a KTP OPO are shown as a function of the pump wavelength of a tunable alexandrite wavelength. Stimulated Raman scattering . Another physical process that is useful in shifting the wavelength of a laser output is stimulated Raman scattering. Most of the work on Raman shifting has been done using gas cells as the nonlinear medium. This provides the required conversion efficiency through small gain over a long path length. Recent studies of Raman scattering in solids have identified new crystals that provide high gain over short path lengths. The properties of some of these materials are listed in Table 10.3. Solid-state Raman shifters are rugged, compact devices compared to gas

�

-

1 . 64 1 . 60 1 . 56

� Q)I=i a:l

1 . 52

i

1 . 44

� �

0

1 . 48

1 .40

1 . 36

700

720

740

760

780

BOO

Pump Wavelength (nm)

820

FIGURE 10.8. Output wavelengths of the signal and idler beams for a KTP OPO pumped by a tunable alexandrite laser (from Refs. 1 2 and 14).

396

10. Miscellaneous Laser Materials

TABLE 10.3. Examples of materials solid-state Raman shifting. Material (line in cm- 1 )

Gain (cm/GW) 0.21

References 18

Li3 BazGd3 (M04) 8 (768)

4.7

3.1

KGd(W04) z (901 .5) Ba(N0 3 )z ( 1 046) CaC0 3 ( 1 085) NaN03 ( 1066)

6

5.9

11

1.5

13

2.3

47

2.0

Handbook of Laser Science and Technology ,

edited by M.J. Weber (CRC, Boca Raton, FL, 1987), Vol, III, p. 293 N. R. Belashenkov, V. D. Velayev, S. V. Gagarsky, and A. N. Titov, presented at the Advanced Solid State Laser Conference, Memphis, TN, February 1995 T. T. Basiev e t al. , Sov. J. Quant. Electron. 17, 1 560 ( 1 987) T. T. Basiev e t al. , Sov. J. Quant. Electron. 17, 1 560 ( 1 987) S. N. Karpukhin and V. E. Yashin, Sov. J. Quant. Electron. 14, 1 337 ( 1984) S. N. Karpukhin and A. I. Stepanov, Sov. J. Quant. Electron. 16, 1027 ( 1 986)

cells. Their size and high-gain characteristics have made them ideal for intracavity operation. This has led to a new class of all solid-state coupled cavity Raman lasers. Raman shifting is a third-order nonlinear optical pro­ cess that does not require phase matching. Therefore, it does not suffer from stability problems associated with thermal, angular, and wavelength accep­ tance parameters as compared to SHG and OPO processes. In addition, the fur-wave mixing interaction associated with stimulated Raman scattering provides automatic beam cleanup of the output as discussed below. Thus, intracavity stimulated Raman scattering can be an attractive method for shifting a primary laser emission wavelength to a different wavelength re­ quired for a specific application. However, it should be noted that contin­ uously tunable wavelength output cannot be obtained from Raman shifting as it can with OPOs (except for pump wavelength tuning) . The physical processes of Raman scattering takes place through photon­ phonon interaction. As an electromagnetic wave travels through a dielectric media out of resonance with any absorption transition, it drives the atomic dipoles to oscillate as shown in Fig. 10.3, and this induced polarization acts as the source of the transmitted wave. If a vibrational phonon is absorbed or emitted during this process, conservation of energy requires that the trans­ mitted wave will have a frequency greater than or less than the incident wave by an amount equal to the phonon frequency. These waves are re­ ferred to as anti-Stokes and Stokes emission, respectively. The energy-level diagram and transitions associated with Stokes Raman scattering process are shown in Fig. 10.9. The dashed line represents a virtual electronic state and I v) is the vibrational level reached by the creation of the phonon. The photon coupling that produces Raman scattering takes place through

1 0.2. Nonlinear Optical Lasers --le>

397

le>

- - - - -

(rop. kp )

(rol .kl)

E8(ro8,k8)



-

1irov

lg>

(A) S pontaneous R aman Scattering

(B) Stimulated Raman Scattering

FIGURE 1 0.9. Energy levels and transitions for a Stokes Raman scattering process in which a pump photon of frequency Wp is destroyed while a phonon of frequency Wv and a Stokes photon of frequency Ws are created.

the induced polarizability of the material. This is given by the second tenn in a Taylor-series expansion of the polarizability a with respect to a vibra­ tional mode coordinate q, a

=

ao

+

()

oa q+ oq o

·

.. .

( 10.2.34)

The polarization of the medium resulting form this induced polarizability and acting to generate the scattered wave is ( 10.2.35) P = PnP = Pn aE , where Pn is the density of molecules and p is the molecular dipole moment. In a solid, the molecular vibrations contribute to form a vibrational wave that is described by a wave equation. This can be expressed in terms of a driven harmonic oscillator as given by Eq. ( 10.2.8), with a term added for the wave propagation 2 o q

oq

+

F + v2v v2 q + wv2 q = - '

( 10.2.36)

where r is the damping constant, V v is the group velocity of the vibrational wave, Wv is the fundamental oscillation frequency, n1 is the reduced mass of the molecule, and F is the driving force. The force of the electric field acting on the molecular dipole moment is F

=

� (p oq

·

E)

�

()

oa E E, oq o ·

( 1 0.2.37)

where only the first term in the expansion of Eq. ( 10.2.35) has been used.

398

1 0. Miscellaneous Laser Materials

The total field in the medium can be treated as being composed of copropagating pump wave and Stokes wave, ( 10.2.38)

Eq(z, t),

where the pump wave has a complex amplitude a propagation vec­ tor in the z direction with magnitude and a frequency Wp . The Stokes wave has the same properties with S replacing the p subscripts. is the unit vector in the x direction. The interference between the pump and Stokes waves produce a beat wave at the molecular vibration frequency = ks. This is a Raman-active = - ws and wave-vector magnitude optical phonon. This vibrational wave obeys Eq. ( 1 0.2.36) and has the gen­ eral form

kp,

ex

Wv

kv kp -

Wp

q

[qv(z, t) ei(k,z w,t) q� (z, t) e i(k,z w, t) ] ex .

( 10.2.39) + =! The polarization of the medium is now found by substituting Eqs. ( 10.2.34), ( 1 0.2.38), and ( 1 0.2.39) into Eq. ( 10.2.35) to give p

=

[�0 + G:) 0 ] (Epei(kpz wpt) + E; e i(kpz wpt) q

( 10.2.40)

Expanding this expression and keeping only the terms with the same phase factors for the pump and Stokes waves gives P�

[� Ep + ( 0qvEs] ei(kpz wpt) + c.c. � + [� E + ( 8 ) v Ep ] ei(ksz wst) + c . c , 8q 2 o

Pn

o s

o

q*

.

( 1 0.2.4 1 )

where c.c. stands for complex conjugate. This expression can be used as the source term in Eq. ( 10.2. 13) to give the set of coupled partial differential equations describing the pump and Stokes fields 1 ( 10.2.42) = +s,

8Ep 8Ep l. VpOJp K! qvE 8z Vp 8t VsOJs 8Es 1 8Es = lK! qv EP .

*

K! - f.loPnVsWs (88q�)o ' 4

(10.2.43)

where the coupling constant is given by _

( 1 0.2.44)

and vs and Vp are the phase velocities of the Stokes and pump waves, re­ spectively. Substituting Eqs. ( 10.2.38) and ( 1 0.2.39) for the wave amplitudes in Eq. ( 10.2.36) and applying the slowly varying amplitude approximation give an expression for the vibrational wave in terms of the pump and Stokes field

1 0.2. Nonlinear Optical Lasers

Kz: dqv dqv + rqv lKz. EpE* dt - Vv dz

399

amplitudes and coupling coefficient

(10. 2 .4 5)

S>

where

Kz 2m1wv (ooct.q) o . (10.2 .46) Equations (10. 2 . 42), (10. 2 . 4 3), and (10. 2 . 4 5) form a set of three coupled dif­ ferential equations describing Raman scattering in a dielectric medium. These are simplified by transforming to the retarded coordinate system and assuming V v Vp vs. The equations then become dqv + rqv lKzEp . ( t ) Es* , dt' aEp, . wp KJqvEs(t, ), (10. 2 .47 ) z ws 8Es lKJqv EP (t, ). oz' «

=

=

0

-

1

'

-

*

.

These equations must then be solved for different experimental conditions to determine the buildup of the Stokes wave in the material. For the case of steady state excitation (i.e . , when the excitation pulse width is much larger than the dephasing time of the optical phonon) the vi­ brational wave equation can be easily solved and the expression obtained for substituted into equations for the pump and Stokes fields

qv

aEp Wp KJKzi Esi 2Ep Ws r oz' 8Es KJKzi Epi 2Es r oz' _

(10.2.48)

The right-hand sides of these expressions involve the products of three elec­ tric field and thus can be written in terms of a third-order susceptibility and related to the third-order nonlinear polarization,

(10.2.49) where

XRaman (ws, -wp , Wp, ws) - - l. 4eoPmn wv (-aoct.q)2 -r1 . (3)

.

_

0

(10.2 . 50)

400

10. Miscellaneous Laser Materials

The electric fields of the pump and Stokes beams can be expressed in terms of the intensity (irradiance) of the light beams through the usual rela­ tionship I (eov/2) I E I 2 . The expressions in Eq. ( 10.2.48) then become dfp dz'

-

Wp

(10.2. 5 1 )

ps ws yl l ,

where the gain coefficient is given by

( )

a 2. Pn Ws oerwvm aq 0 a

y - 4e

(10.2.52)

Thus the intensity of the Stokes beam traveling through a material increases in direct proportional to r- 1 and (arxjaqf The expression for the rate of spontaneous Raman scattering of photons in a solid angle iln is given by1 5

= /A A

--

dWsp

Pn ).� a(J Is Wp ,

h vsLlvs an

(1 0.2.53)

where W with being the cross-sectional area of the pump beam, Llvs is the full width at half maximum of the Stokes spectral line, and a(J/an is the differential cross section for spontaneous Raman scattering. Comparing this expression with Eqs. ( 1 0.2.5 1 ) and (10.2.52) show that the can coeffi­ cient can be expressed as

Pn A� y ( v ) = h vs an S ( v) a(J

pn ).� a(J __, Ymax = h2vsLlvs an '

( 10.2.54)

where a Lorentzian line-shape function has been used in the latter expres­ siOn. Since measurements of gain or scattering cross section are generally made at a specific wavelength, it is important to develop expressions relating these parameters at different wavelengths. 1 6 Using second-order perturbation theory and Fermi's golden rule, an expression can be derived for the Raman transition probability per unit time per unit volume per unit energy. This can be multiplied by the density of radiation modes per solid angle at the Stokes frequency and by the population density of the initial state of the material, and then divided by the effective material excitation density to obtain an expression for the differential scattering cross section per solid angle,

a(J an

8n3 ns

2P

i-

Here the n; represent the photon occupation numbers at the pump and Stoke frequencies, Mfi is the matrix element for the transition, and P; is the population density of the initial state of the material. This can be used with Eq. ( 10.2.54) to obtain the following relationships between gain at two wavelengths and scattering cross sections at two wavelengths,

1 0.2. Nonlinear Optical Lasers

( a(JI an) Ap2 ( a(J1 an) Ap ]

401

( 10.2.55)

From a theoretical perspective, expressing the Raman gain coefficient in terms of the derivative of the molecular polarizability as in Eq. ( 10.2.48) can be useful. However, this quantity is difficult to measure directly, so for quantitative interpretation of experimental results it is more useful to use the expression for the gain coefficient in Eq. ( 10.2.50) where the Raman line­ width and differential cross section for spontaneous Raman scattering can be measured from conventional Raman spectroscopy. Solving Eqs. ( 1 0.2.5 1 ) gives an expression for the change in the intensity of the Stokes beam as it propagates through the medium 1 2 ' yloz -- e

ls(O) , Ip (O) ( 10.2.56) ls(z ) Io , (O) w + __ 1 ws lp (O) lp (z' ) + (wp lws)Is(z' ) is the conserved ( photon conservation) loz eY

,

where Io total field intensity. This expression shows that the intensity of the Stokes beam initially increases approximately exponentially through the medium, and then tends toward a saturation value. For a distance of travel L, the total gain G is given by ( 10.2.57) G yl0 L with y given by Eq. ( 10.2.52) or ( 10.2.54). For efficient Raman lasers, it is generally best to use a material with a high Raman gain G. Using the theory outlined above, the gain given in Eq. ( 10.2.57) can be expressed in terms of material parameters as

1

a(J

( 10.2.58)

where Pn is the density of states of the phonons involved in the scattering event, Vs is the frequency of the Stokes transition, and a(JI an is the Raman differential scattering cross section. In this expression a Lorentzian line shape has been assumed and this is proportional to the inverse of the tran­ sition linewidth Ll vs. The density of states is important in understanding the temperature dependence, but most laser systems are required to operate at room temperature. Thus for obtaining a specific laser output frequency from a given Raman-shifted pump laser, the criteria for a high-gain Raman mate­ rial are a narrow Raman linewidth and a high Raman-scattering cross sec­ tion. The material must also be transparent in the spectral region of interest. Table 10.3 lists the Raman gain and linewidth properties of several crystals that may be important for Raman laser applications. The width of a line in a Raman spectrum is determined by the lifetime of

402

10. Miscellaneous Laser Materials

the final state of the transition. This is generally dominated by the lifetime of the of the phonon that is produced in the Stokes emission process. The phonon generated through Raman scattering will decay into other phonons with lower energies as the system evolves toward thermal equilibrium. The phonon-phonon interaction mechanism causing this decay to occur involves anharmonic coupling of the vibrational modes. The most important contri­ bution to the lifetime of this optic phonon is a three-phonon anharmonic interaction in which the initial phonon scatters into two other phonons with conservation of energy and wave vector. The expression for the width of the Raman spectral line is given by1 7 2 nV Lhs dK L i ( nsK1 + l , ns' K2 + l , no O I HA i nsK p ns' K2 , no 1 ) 1 3 2 n (2n) s ,s

J

1

( 10.2.59)

with a similar term describing the self-energy that shifts the frequency of the Raman line due to interaction with the phonon field. The process described here involves the destruction of the initial phonon of frequency w0 and wave vector K simultaneously with the creation of two other phonons of frequen­ cies Ws, Ws , and wave vectors K ! , K2, with the requirements of conversation of energy and wave vector. At high temperatures other processes involving phonons in different branches may contribute to the line broadening and line shifting. The rate at which the Stokes phonon decay occurs depends on the strength of the anharmonic coupling of the vibrational modes and the con­ servation laws that are required for the transition to occur. The latter re­ quire that the sum of the energies of the two final phonons equals the energy of the initial phonon, and that the sum of the wave vectors of the two final phonon equals the energy of the initial phonon. This depends on the density of phonon states for the material. The photon-phonon interaction in the Raman scattering process creates an optic phonon with a near zero wave vector. Therefore the relaxation process for this phonon can involve any phonons with equal and opposite wave vectors. The key parameter in the Eq. ( 10.2.59) is the anharmonic interaction Hamiltonian HA . The strength of this operator depends on the anharmo­ nicity of the crystal which is characterized by the Gruneisen parameter Y a · This can be expressed in thermodynamics quantities as Ya

r:x VB

Cv

3 r:xL V

KCv .

( 10.2.60)

Here r:x is the volume expansivity while r:xL is the linear expansion coefficient, V is the volume, Cv is the specific heat, B is the bulk modulus, and K is the compressibility. These thermal properties have been tabulated for many materials, but unfortunately are not available for most of the materials of interest for Raman lasers.

10.2. Nonlinear Optical Lasers

403

Thus crystals having the smallest Raman linewidth will be those with small values of y0 and a vibrational density of states that is not compatible with conservation of energy and wave vector for the decay of the Raman phonon. 1 8 The temperature dependence of the stimulated Raman scattering Stokes spectral line in Ba(N0 3 ) 2 crystals has been investigated in detail and interpreted in terms of specific phonon decay processes. 1 9 The second parameter in Eq. ( 10.2.58) of importance for high gain in Raman laser materials is the scattering cross section. The cross section for Raman scattering depends on the strength of the photon-phonon coupling which is proportional to the modulation of the laser-induced polarizability. An accurate theoretical calculation requires knowledge of the electronic and vibrational wave functions as well as the local electron-phonon interaction. This information is not usually available. Therefore it is necessary to resort to qualitative arguments based on the lattice structure of the crystal. The vibrational modes of a crystal are classified according to the irredu­ cible representation of the symmetry group according to which they trans­ form. To satisfy conservation of wave vector and energy, only optic phonon modes near the center of the Brillouin zone will make significant contribu­ tions to Raman scattering. In some cases a crystal structure can be described in terms of molecular units. In this case the local mode molecular vibrations generally produce larger changes in polarizability than the lattice modes. Although the local crystal-field environment will alter the vibrational modes of a free molecule, the molecular vibrations are a good starting point for analyzing the vibrational modes of the crystal. The Raman scattering cross section is proportional to the square of the change in the polarizability of the vibrating group with respect to the normal mode of vibration, There are three aspects of the structural symmetry that are important in determining the cross section. 2° First, the symmetry classification of the vibrational mode should have allowed selection rules for Raman transitions and should give the maximum value of The totally symmetric breath­ ing modes of vibration classified as A 1 g have the greatest change in the polar­ izability tensor. Tightly bound molecular groups such as tungstates, molyb­ dates, nitrates, etc., generally have A 1g vibrational modes that maintain their integrity when the molecular group becomes a structural unit of a crystalline solid. Second, molecular groups such as the tungstates, molyb­ dates, nitrates, etc., mentioned above, all have allowed electronic transitions to charge transfer levels at low energies. Since the polarizability tensor has a resonant denominator that is dominated by a virtual transition to the lowest-lying energy level as the intermediate state, these molecular com­ plexes have high values of In a crystalline environment, these charge transfer levels generally shift to lower energies resulting in enhanced values of polarizabilities. With the appropriate symmetry conditions as discussed above, this results in high values of Third, the structural symmetry should result in chemical bonding characteristics that maximize that rate of change in polarizability during a vibration. Covalent bonding allows for a

(oaf oq) 2 .

oafoq.

a.

oafoq.

1 0. Miscellaneous Laser Materials

404

TABLE 1 0.4. Character table for D 3h and basis functions.

D 3h A 'I A 2' E' A "I A 2" E"

E

2 I I 2

2CJ

3 C2

(Jh

2SJ

3 av

1 I 1 I

I 0 I I 0

I 2 I I 2

I I I 1 I

I 0 1 I 0

Normal modes of AB 3

Basis functions

Rz

(x, y) z

ctxx + !Xyy, Cl.zz

VI

(ctxx - ctyy, ctxy)

VJ , V4

( Rx, Ry)

(ctxz , IXyz)

V2

greater change in polarizability than ionic bonding. Conjugated bonds enhance the polarizability change. Crystal structures such as diamond, zinc­ blende, and wurtzite are favorable for covalent bonding, and thus can have high Raman scattering cross sections. As an example of the group theory analysis required for interpreting Raman spectra, consider an AB3 molecular group such as N0 3 or C03 . The symmetry point group for this structure is D 3h with the characters of the irreducible representations given in Table 1 0.4. The basis functions listed in the table show how the components of a vector, a rotation operator, and the components of the polarizability tensor transform according to the irredu­ cible representations of the group. Using the group theory techniques de­ scribed in Sec. 2.2 along with the normal mode vibrational analysis tech­ niques described in Sec. 4. 1 , the four normal modes of vibration for the AB3 molecule are shown in Fig. 10. 10. By analyzing the symetry transformation properties of these vibrational modes, they can be associated with the irre­ ducible representations of the D 3h point group as shown in the character table. Note that v3 , V4 transform together as the bases of the doubly degen­ erate representation E'. From the character table it can be seen that V J , v3 , and v4 all transform in the same manner as some of the components of the

vl

?

v2

v3

FIGURE 1 0. 1 0. Normal modes of vibration of the AB3 molecule.

6

0

.

�

,0

0

G

0 v4

,P

10.2. Nonlinear Optical Lasers

405

TABLE 10.5. Character table for D 3h and basis functions. c3 A E

E

c3

c3z

e

e'

*

e

e

D3h Correlation

A; , Ar E ' , E"

polarizability tensor and therefore have allowed Raman transitions. On the other hand, v2 is not Raman active. In general, the strongest Raman tran­ sitions are those associated with the diagonal elements of the polarizability tensor. Thus, this analysis predicts that the A'1 ( v 1 ) normal mode vibration will produce the biggest peak in the Raman spectrum of the AB3 complex. When a molecular group is part of a crystal structure, the factor group method described in Sec. 4. 1 for local mode vibrational analysis is used to determine the change in symmetry properties and splitting of degenerate modes. For example, the NO .J molecular ions occupy a lattice site with C3 point-group symmetry in a Ba (N0 3 h crystal. The correlation of irreducible representations between the C3 and D3 h point groups is given in Table 10.5. The crystallographic point group for Ba ( N03 h is Th . Thus, the next step in the factor group analysis is to determine the correlation between the irredu­ cible representations of the C3 site symmetry and the Th crystallographic point-group symmetry. The most important representation under considera­ tion is the one according to which the VI vibrational mode of the molecular ion transforms. The correlation relationships for this are A'1 (D3h ) -A ( C3 ) -- A9, Au, T9 , Tu ( Th) . Thus the VJ vibrational mode of the free molecule splits into two nondegenerate and two triply degenerate vibra­ tional modes in the barium nitrate crystal. Components of the polarizability tensor transform according to two of these vibrational modes of the crystal, A9 and T9, and thus they have allowed Raman transition . The sum of the diagonal components 1Xxx + !Xyy + IXzz transforms according to the totally symmetric mode A9. This VJ (A9) vibrational mode produces a strong, nar­ row line in the Raman spectra. For the free nitrate ion the Raman shift for this line is 1 050 cm- 1 , while the position of this peak in the Ba (N0 3 h crys­ tal spectrum is 1047.3 cm- 1 • Thus crystals having the highest Raman cross section will be those having molecular structures where the molecular breathing mode acts as a zone center vibrational mode of the crystal that transforms as the A 19 irreducible representation of the crystallographic point group symmetry. If molecular polarizabilities are known, those with the highest polarizability will have the highest Raman cross section. The high Raman gains of the materials listed in Table 10.3 are associated with totally symmetric vibrational modes of molecular groups such as nitrates, tungstates, and molybdates. Because of this high gain, it is difficult to use these materials in an external cavity con­ figuration for frequency shifting a laser beam as is done with Raman gas

406

10. Miscellaneous Laser Materials

cells. Nonlinear cascade and higher-order photon interaction processes cause the first Stokes emission to be depleted and emission to occur at other frequencies. 2 1 Thus the best use of solid-state Raman shiftiing is in an intracavity configuration. An intracavity Raman laser operatin at 1 .5 11m consisting of a Nd-YAG laser operating at 1 .3 J1f pumping a Ba (N0 3 h Raman laser has recently been demonstrated. 22 The coupled cavity design used for this laser was optimized to produce an optical-to-optical conversion efficiency near to the quantum limit of 85% along with an output beam having near-diffraction­ limited beam quality. The poor beam quality of the pump laser was con­ verted to a Gaussian beam output of the Raman laser through the process of stimulated Raman scattering. This Raman beam cleanup effect is attrib­ uted to the four-wave-mixing nature of this x (J) processes. No detailed mathematical description has been derived to describe the mechanism of Raman beam cleanup in a Raman laser. The general process that is taking place can be understood by examining the driving polari­ zations responsible for stimulated Raman scattering (SRS). In the slowly varying envelope approximation, the third-order nonlinear polarizations associated with SRS are given by the expressions derived above to be ( 1 0.2.6 1 ) and

2 p ( 3 \ws , r) = x�; : I Ep(r) 1 Es (r) e-iksZ , where i� ) and x�; represent the third-order tensoral nonlinearty of the medium, and Ep (r) and Es (r) represent the slowly varying envelope of p

the three-dimensional electric field vector of the pump and Stokes beams, respectively. Applying the convolution theorem in terms of the spatial fre­ quency spectrum of the pump and Stokes fields, the polarization can be written as ( 1 0.2.62) and

respectively, where k is the wave vector. In generating the stimulated Stokes field it is evident that each plane wave component of the pump field mixes with every other pump component via the complex autocorrelation Yp (k) �p (k)** �; (k) . This apparent pump spectrum is then combined with all pos­ sible plane-wave compontens of the seeding Stokes field via the convolution Yp (k)** �s (k) . Hence the stimulated Stokes field is a smoothed version of the seeding Stokes spectrum. This smoothing operation is performed twice per

10.3. Color-Center Lasers

407

round-trip in the Raman laser resonator. The central limit theorem predicts a Gaussian spectrum to result from a multiorder spectral convolution where the order is greater than three. Therefore a Gaussian transverse field distri­ bution is produced after only a few round trips in the laser cavity. Under the conditions of adequate Raman gain per pass and cavity mode matching, a TEM00 diffraction limited Raman laser output can be achieved over a very large dynamic range without the use of intracavity apertures. This technique of beam shaping enables very efficient high-power laser oscillators to be built. The role of four-wave mixing in Raman beam cleanup becomes evident when the plane-wave approximation is applied to these equations. With �p ( k) �s ( k)

= =

�ptJ ( k - kpi )

+

�p2J ( k - kp2 ) ,

�st J ( k - ks t ) ,

as the example spectra, the driving polarization for the stimulated Stokes fields is given by p

( J) (Ws2 , r )

= �; : X

+

2 2 i r [( j �pt i �Sl + � �p2 1 �s t ) e- (ks i -ks2 ) · ei(kp2-kpl +ksl -ks2 ) ·r l ei (kpl -kp2+ks i -kS2 ) ·r +

( 10.2.63) where the first term represents the normal Raman gain and the second and third terms represent secondary Stokes radiation generated through four­ wave mixing. The normal Raman gain is shown to be independent of pump direction and is hence "phase matchless." The Stokes radiation generated by four-wave mixing, however, depends strongly on the direction of the pump radiation as a result of the phase mismatch. These are the terms responsible for the ultimate smoothing of the stimulated Stokes radiation. 10.3

Color-Center Lasers

Dopant ions such as those discussed above are one type of point defect in a host material that can act as a luminescent and lasing center. Another type of optically active point defect is a color center. A typical color center con­ sists of an electron trapped at ion vacancy site in the lattice. Lasers based on color centers have been demonstrated in both fluoride and oxide crystals. 2 3 The production of color centers in crystals generally depends on thermal or radiation treatments. Color-center lasers in the visible and near infrared spectral region generally operate on electronic transitions of trapped elec­ tron defects, while far-infrared color center lasers operate on the vibrational transitions of molecular defects such as CN - . Neutral atom defects can also produce laser emission in the near-infrared spectral region. Typical absorp­ tion and emission spectra for color centers exhibit strong, broad bands as-

408

10. Miscellaneous Laser Materials TABLE 1 0.6. Examples of common color-center lasers. ( Data from Refs. 24 30.) Type of center p2+

F2 + : o 2

p2 FA ( II )

Tl0 ( 1) eN

Host crystal

Tuning range (tml)

LiF KF NaCI KBr LiF KCI : Li RbCI : Li KCI KBr

0.82 1 .05 1 .22 1 . 50 1 .42 1 .78 1 . 86 2. 1 6 1 . 1 1 .25 2.3 3 . 1 2.5 3.65 1 .4 1 .6 4.86 (at 4 K)

sociated with allowed transitions having high oscillator strengths (f � 0.2) and radiative lifetimes of the order of 100 ns. The quantum efficiencies can be close to 1 00%. These are favorable for optical pumping with lasers and result in high gain cross sections and low thresholds. The color centers are strongly coupled to the host lattice and operate as quasi-four-level systems similar to other vibronic lasers. The homogeneously broadened emission band is useful for tunable laser emission or mode-locked operation. The large Stokes shift between the emission and absorption minimizes losses due both to ground-state and excited state absorption of laser emission. Table 1 0.6 lists some of the common color-center laser. 24 - 28 The nomenclature for designating specific types of defect centers can be found in various review articles on color centers. 29 One major problem with color-center lasers is the lack of stability of color centers at room temperature. The mobility of electrons and ions in the host crystals can cause the number of color centers to decay with time or change into aggragate centers. Since color center lasers are pumped by other lasers, photobleaching effects are especially important. Techniques including the introduction of dopant ions have been developed to stabilize specific types of color centers. However, cw color center lasers still operate generally only at low temperatures and only a few pulsed systems exhibit stable operation at room temperature. Alkali halide crystals such as LiF are common host materials for color­ center lasers. Fi defect centers have been especially useful for laser opera­ tion with these crystals. 3° Figure 10. 1 1 shows a portion of the LiF lattice with an Fi defect center. This type of color center involves an electron in an interstitial position between two fluorine ion vacancies in the lattice. The energy levels for this type of center are found by modeling it as a hydrogen molecular ion. 3 1 Using molecular orbital notation, the relevant energy levels and transitions for an Fi center is shown in Fig. 10. 1 2. Note that when the electron is excited to an upper-state energy level, the spatial extent of its orbital increases, leading to a stronger interaction with neighboring ions.

10.3. Color-Center Lasers u+

0

•

0 •

0

'

0

F2+ center

I

oO Do 0

•

•

0

•

0

0

•

•

0

0

•

•

409

FIGURE 1 0. 1 1 . Section of a LiF crystal lattice with a Fi defect center.

0

This results in a lattice relaxation that changes the position of the energy levels as shown in the figure. The operation of LiF : Fi color center lasers is associated with the lsag -2pau transition shown in Fig. 1 0. 12(A) . Due to the parity change, this tran­ sition has a high oscillator strength for strong pump absorption and a high quantum efficiency for radiative emission at room temperature. The excited state lattice relaxation effect mentioned above results in a large Stokes shift, as seen from the absorption and emission spectra shown in Fig 10. 1 2( B). This allows for laser output to be tuned across the entire emission band without ground-state absorption loss. In addition, there are no energy levels in a position that gives rise to excited state absorption. The positions of the energy levels depends on the dielectric constant of the host crystal. This can change significantly for different alkali halides, and thus the peak of the laser emission band for Fi color center lasers ranges from about 0.9 to about 2.2 fJ.m depending on the host crystal. The optical properties of this type of color center are highly anisotropic with the electric dipole transition being polarized along the line joining the two vacancies that make up the defect. The ensemble of Fi defect centers are distributed in orientation along the six different [1 1 0] crystallographic directions. Only those centers orientated in the direction compatible with the polarized pump light take part in the lasing operation. At high levels of pumping, it is possible for multiphoton excitation to take place, as shown in Fig. 10. 1 2(A). If excita­ tion to a high-lying energy level takes place at room temperature, the center can reorient itself along a different crystalographic direction. This decreases the number of Fi centers aligned in the appropriate direction for laser operation and thus results in photobleaching of the material. This process can be minimized by operating at low pump irradiances or by maintaining the crystal at low temperatures where the ionic motion required for defect

410

10. Miscellaneous Laser Materials

(A) EMISSION

ABSORPTION

0.5

0.6

0.7

0.8

A(!lm)

0.9

1.0

1.1

1.2

(B)

FIGURE 1 0. 1 2. Energy levels and spectra of LiF : Fi . (A) Energy levels for Fi center. ( B) Optical spectra for LiF : Fi -

reorientation is less probable. In addition, by putting an impurity ion such as 02 - next to the Fi center, orientational bleaching is inhibited. F2 centers in LiF crystals also provide room-temperature operation. 30 The tuning range is shifted to longer wavelengths compared to Fi centers. A color-center laser based on a different class of defect is KCl : T1° ( 1 ) . In this case the neutral thallium atom substitutes for a cation on a lattice site that is perturbed by a single nearest-neighbor anion vacancy. The perturba­ tion lifts the degeneracy of the energy levels and mixes even- and odd-parity states to give transition oscillator strengths on the order of 0.008. This is a stable defect center that gives tunable laser output between 1 .4 and 1 .6 f.i.

1 0.4. Other Solid-State Lasers

41 1

One interesting problem that occurs with lasers that have a high gain and a broad homogeneous linewidth, such as color-center lasers, is spatial hole burning. This makes it difficult to obtain single longitudinal mode opera­ tion. In a standing-wave cavity with a homogeneous gain profile, it would normally be expected that when one cavity mode begins to oscillate, all other modes would be surpressed. However, for a high-gain material the peak electric field intensity saturates the gain media in the spatial regions where it is a maximum for his lowest threshold mode. This intensity has a cosine-squared pattern. The spatial regions between these maxima are not saturated and can support gain for a longitudinal mode with a different frequency. This results in the simultaneous oscillation of two longitudinal modes. If single-mode operation is desired, it is necessary to use etalon in the cavity to select one mode and surpress the other. Color center lasers have been very useful laboratory sources for tunable laser emission in the near-infrared spectral region. However, for most prac­ tical applications outside the laboratory their usefulness is limited by their lack of stability at room temperature. One important application for mode­ locked color center lasers is coupling them to fibers lasers to produce soliton lasers. 32 The sync-pumped mode-locking produces trains of pulses having temporal widths between 5 and 1 0 ps. The dispersive qualities of fibers can be used to produce pulse compression that reduces the pulse widths to the femtosecond time regime. 1 0. 4

Other Solid-State Lasers

Several other types of optically active centers have been investigated for use as lasers. These include both organic and inorganic molecules as well as different types of ions such as those with closed-shell electronic configura­ tions. Some of these novel systems are beginning to be developed to the level of commercial laser systems. Organic dye molecules such as rhodamine 6G can exhibit very strong, broad-band fluorescence emission that make them very attractive for lasing centers. Dye lasers based on these molecules in liquid solvents such as alcohol have been very successful in the visible spectral region using either flashlamp or laser pumping. Using different dye-solvent combinations, broadly tunable laser emission can be obtained in bands between about 3 1 0 and 1 50 nm. The dye abosorption and emission bands are associated with transitions between the vibrational sublevels of the electronic states of the molecule. A generic energy-level diagram for molecular transitions is shown in Fig. 10. 1 3 . The ground-state manifold is a singlet state while there are sets of both singlet and triplet excited states. The singlet-singlet transitions are spin-allowed and result in strong absorption and emission bands. The latter have fast fluorescence lifetimes (of the order of a few nanoseconds). If intersystem crossing occurs to the triplet manifold, the molecule can pro-

412

10. Miscellaneous Laser Materials

FIGURE 1 0. 1 3. Typical energy-level diagram and optical transition for dye molecules.

RADIATIONLESS_ , DECAY (Ips)

•

I

,

I

I INrERSYSTEM ' , . :- c ROSSING (IOns)

(l ns) PHOSPHORESCENCE (lJ.IS)

STATE SINGlET

duce a long-lived emission (of the order of a few microseconds) . Excited state absorption can be a significant problem in the triplet energy-level mani­ fold. Initial attempts to incorporate organic dye molecules in solid host materials were unsuccessful as lasers because of rapid photodegradation under optical pumping. These were generally based on the same types of organic molecules used for liquid dye lasers and standard glass or polymer host materials. Recent developments of new types of dye molecules and host materials have significantly decreased the photodegradation problem. 3 3 One class of these new materials includes pyrromethene-BF2 complex dyes doped in acrylic plastic or xerogel hosts. 3 3 • 34 A typical dye molecule, optical spectra, and laser emission band are shown in Fig. 10. 14. Note that for this particular example, the tuning range is limited on the high-energy side by the ground-state absorption spectrum. These types of solid-state sys­ tems have the potential to replace liquid dye lasers in the near future. How­ ever, the average power handling capability of the host ultimately limits their utility. One spectral region that is not easily covered by primary solid-state lasers is the near ultraviolet. Since lasers in this region have important applications in areas such as medical procedures and material processing, there is signifi­ cant interest in developing solid-state sources that operate in this region. Currently it is possible to have solid-state systems operating between 190 and 400 nm by using frequency multiplier nonlinear optical crystals in con­ junction with visible and near-infrared lasers. Because of the complicated nature of these systems, it would be preferable to generate the uv light di­ rectly from the laser material. There are several types of materials that have the potential for accomplishing this task; however, it should be noted that pump sources are a limiting technology. Long-lived uv lamps are not avail­ able and pumping with uv gas lasers defeats the purpose of having a solid­ state source. One possible solution may be surface discharge sources, but their use for laser pumping devices has not yet been thoroughly explored.

1 0.4. Other Solid-State Lasers

FIGURE 1 0. 14. Spectral properties of PM-HMC solid-state dye laser (after Ref. 34) . (A) PM­ HMC molecular complex. ( B) Optical spectra of PM-HMC in xylene. (C) Broadband laser emission of PM-HMC.

(A) :r

s

>-

1.2

� ·a "

� � BCl

E:�

0.9

j

0.0 400 500

!:

>

t.tl

0 Cl � <

0

�

413

t.tl

60 700 800 wAVELENGTH (run)

� 0

�

(B)

WAVELENGTH (nm)

(C)

One class of uv systems that are being investigated are solid-state excimer lasers. Gas lasers based on rare-gas halide excimers such as xenon, krypton, or argon as fluorides or chlorides are important lasers with emission lines at various wavelengths from 1 93 to 351 nm. Recently it has been shown that excimer molecules such as XeF can be incorporated as dopants in rare-gas host crystals such as Ar and Ne. 3 5 These laser systems are currently being pumped with gas excimer lasers. They have very high stimulated emission cross sections and large gain coefficients. The major emission lines reported so fare are at 286, 41 1 , and 540 nm. There are several other types of ions that have exhibited gain but have not yet been made into lasers. As discussed in Chap. 1 , single-pass gain is asso­ ciated with generating a significant amount of stimulated emission. This is a necessary process for laser action but not sufficient to ensure that a material can be made into a laser. For a laser to operate, the gain per pass must ex-

414

10. Miscellaneous Laser Materials

ceed the loss per pass. Some types of losses (such as those associated with scattering centers) can be reduced through techniques to enhance the quality of the material. Other types of losses such as excited-state absorption may not be able to be overcome. Thus the potential exists for several new classes of solid-state laser materials, but it is uncertain which of these will reach any degree of importance in the field. One type of ion that has exhibited gain 36 is Rh2+ . This is an example of a transition-metal ion with a 4dn electronic configuration. Ions of both the 4dn and 5dn series have unfilled d-electron shells similar to the 3dn configu­ rations. One of the rapidly developing areas of solid-state laser technology in­ volves gain confinement through material configurations other than bulk rods or slabs. The most important configurations are guided wave geom­ etries, either fibers or channel waveguides. Microchip lasers are also being developed in which a small piece of the active laser material is pumped with a semiconductor diode laser. These new configurations result in laser opera­ tional parameters that are different than those obtained from the same material used in a rod or slab configuration. They also allow for the use of new materials and new fabrication techniques that are not useful for bulk lasers. One of the important motivations for developing these new laser configurations is that their compactness allows them to be easily integrated into optoelectronic devices and systems. Both single crystals and glasses doped with both transition-metal and rare-earth ions have been used in these configurations. Waveguide configurations involve optically active regions with small cross­ section dimensions and can involve long interaction lengths of the pump and laser light with the gain media. A critical parameter in determining the threshold power for cw laser operation is the product of the pump intensity and the interaction length. For end pumping, a laser with length I in the waveguide configuration can have a cross-sectional area of a few wave­ lengths of light squared over its entire length, while in the bulk configuration with a focused pump beam the cross-section area will be the length times the wavelength. This will result in the laser threshold for the waveguide laser being less than that for the bulk laser by a factor of several wavelengths of light divided by the length. However, this factor must be multiplied by the ratio of the losses in the waveguide versus the losses in the bulk material. Generally the losses in the bulk material are less than those in the wave­ guide, but this factor will improve as waveguide fabrication techniques improve. The long-gain region with the a low density of optically active ions mini­ mizes the effects of ion-ion interaction and makes optimum use of the pump photons. By restricting the numerical aperture of a guided wave laser, single-mode operation can be obtained. Instead of using mirrors to provide the feedback in the active material, distributed feedback can be obtained by fabricating Bragg reflection gratings in the material. This can lead to

References

415

low-threshold, single-frequency, highly efficient, three- or four-level laser operation. Up-conversion lasers using sequential two-photon absorption pumping transitions have been especially successful for rare-earth ions in fluoride fiber (ZBLAN ) configurations where nonradiative decay processes are minimized. Guided-wave structures are well suited for diode laser pumping. They can be configured in a oscillator-amplifier configuration to generate high powers, and can be mode-locked to produce ultrafast pulses. With the appropriate host materials, nonlinear optical effects such as second­ harmonic generation can be incorporated into the laser material. There are some important differences between the two major guided wave configurations, fiber and planar. Planar waveguides generally have higher losses per interaction length than fibers. Also, planar waveguides are much shorter than fibers so the advantages of long interaction length with lower concentration of active ions is lost and the processes involved with ion-ion interaction may be greater. However, planar waveguides make maximum use of new fabrication techniques and can utilize a much wider variety of crystal and glass materials compared to fibers. They are also easier to inte­ grate into optoelectronic devices, and they offer easy access to the active region for applying electrodes and utilizing active and passive modulation techniques. Bulk lasers still have advantages for obtaining high power emission, espe­ cially for Q-switched pulsed operation. This is associated with the fact that the high gain in fibers results in amplified stimulated emission that limits the amount of energy that can be stored and extracted through Q-switching. Beam divergence of the laser output can also be a problem with guided­ wave structures compared to bulk lasers. References

1 . D.J. Ehrlich, P.F. Moulton, and R.M. Osgood, Jr., Opt. Lett. 4, 1 84 ( 1 978). 2. S.P.S. Porto and A. Yariv, J. Appl. Phys. 33, 1 620 ( 1 962). 3 . (a) M. Louis, E. Simoni, S. Hubert, F. Auzel, D. Meichenin, and J.Y. Gesland, in OSA Proceedings on Advanced Solid State Lasers, 1 995, edited by B.H.T. Chai and S.A. Payne (OSA, Washington, 1 995), Vol, 24, p. 1 4 1 ; (b) D. Mei­ chenin, F. Auzel, S. Hubert, E. Simoni, M. Louis, and J.Y. Gesland, Electron. Lett. 30, 1 257 ( 1 994). 4. V.G. Dmitriev, G.G. Gurzadyan, and D.N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer, Berlin, 1 99 1 ) . 5. J.D. Jackson, Classical Electrodynamics ( Wiley, New York, 1 975). 6. N. Bloembergen, Nonlinear Optics ( Benjamin, New York, 1 965). 7. J.A. Armstrong, N. Bloembergen, J. Ducuing, and P.S. Pershan, Phys. Rev. 127, 1 9 1 8 ( 1 962) . 8 . M. Born and E. Wolf, Principles of Optics ( Pergamon, Oxford, 1 965) . 9. J.F. Nye, Physical Properties of Crystals (Oxford Univ., Oxford, 1 957). 10. K. Kato, IEEE J. Quant. Elec. QE-24, 3 ( 1 988). 1 1 . H. Vanherzeele and J.D. Bierlein, Opt. Lett. 17, 982 ( 1 992).

416

1 0. Miscellaneous Laser Materials

12. J.T. Murray, M.S. thesis, Oklahoma State University, Department of Physics, 1 992. 1 3 . T.Y. Fan, C.E. Huang, B.Q. Hu, R.C. Eckardt, Y.X. Fan, R.L. Byer, and R.S. Feigelson, Appl. Opt. 26 , 2390, ( 1 987) . 14. M.G. Jani, R.C. Powell, B. Jassemnejad, and R. Stolzenberger, Appl. Opt. 31, 1 998 ( 1 992). 1 5 . M.D. Levenson, Introduction to Nonlinear Laser Spectroscopy (Academic, New York, 1 982) . 16. J.T. Murray, Ph.D. thesis, University of Arizona, Optical Sciences, 1 996. 17. R. Loudon, Proc. R. Soc. London A 275, 2 1 8 ( 1 963). 1 8. J.T. Murray, R.C. Powell, and N. Peyghambarian, presented at the Dynamical Processts Conference, Cairns, Australia, 1 995. ( Proceedings (to be published) in J. Luminescence, 1 996) . 19. P.G. Zverev, W. Jia, H. Liu, and T.T. Basiev, Opt. Lett. 20, 2378 ( 1 995); P.G. Zverev and T.T. Basiev, in OSA Proceedings on Advanced Solid State Lasers, 1995, edited by B.H.T. Chai and S.A. Payne (Optical Society of America, Washington, DC, 1 995), Vol. 24, p. 288. 20. G. Eckhardt, IEEE J. Quant. Electron. QE-2, 1 ( 1 966) . 21 . P.G. Zverev, J.T. Murray, R.C. Powell, R.J. Reeves, and T.T. Basiev, Opt. Commun. 97, 59 ( 1 993). 22. J.T. Murray, R.C. Powell, N. Peyghambarian, D. Smith, and W. Austin, pre­ sented at the Nonlinear Optics Conference, Hawaii, July 1 994; J.T. Murray, R.C. Powell, N. Peyghambarian, D. Smith, and W. Austin, presented at the Advanced Solid State Laser Conference, Memphis, January 1 995; J.T. Murray, R.C. Powell, N. Peyghambarian, D. Smith, W. Austin, and R.A. Stolzenberger, Opt. Lett. 20, 1 0 1 7 ( 1 995). 23. L.F. Mollenauer and D.H. Olson, J. Appl. Phys. 46, 3 1 09 ( 1 975). 24. L.F. Mollenauer, N.D. Vieira, and L. Szeto, Opt. Lett. 7, 414 ( 1 982); L.F. Mol­ lenauer, D.M. Bloom, and A.M. Del Gaudio, Opt. Lett. 3, 48 ( 1 978); L.F. Mol­ lenauer and D.M. Bloom, Opt. Lett. 4, 247 ( 1 979). 25. J.F. Pinto, E. Georgiou, and C.R. Pollock, Opt. Lett. 11, 5 1 9 ( 1 986). 26. D. Wandt, W. Gellerman, and F. Luty, J. Appl. Phys. 61, 664 ( 1 978). 27. K. German J. Opt. Soc. Am. B 3, 149 ( 1 986). 28. R.W. Tkach, T.R. Gosnell, and A.J. Sievers, Opt. Lett. 9, 1 22 ( 1 984). 29. L.F. Mollenauer, in Tunable Lasers, edited by L.F. Mollenauer and J.C. White (Springer-Verlag, New York, 1 987), Chap. 6. 30. T.T. Basiev, S.B. Mirov, and V. Osiko, IEEE J, Quant. Electron. 24, 1 052 ( 1 988). 3 1 . L.F. Mollenauer, Phys. Rev. Lett. 43, 1 524 ( 1 979). 32. L.F. Mollenauer and R.H. Stolen, Opt. Lett. 29, 1 3 ( 1 984). 33. R.E. Hermes, T.H. Allik, S. Chandra, and J.A. Hutchinson, Appl. Phys. Lett. 63, 877 ( 1 993); B. Dunn, F. Nishida, R. Toda, J.I. Zink, T.H. Allik, S. Chandra, and J.A. Hutchinson, Mater. Res. Soc. Symp. Proc. 329 267 ( 1 994) . 34. T.H. Allik, R.E. Hermes, G. Sathyamoorthi, and J.H. Boyer, Proc. SPIE 21 15 , 240 ( 1 994) . 35. N. Schwenter and V.A. Apkarian, Chem. Phys. Lett. 154, 4 1 3 ( 1 989); G. Zerza, G. Sliwinski, and N. Schwentner, Appl. Phys. B 55, 3 3 1 ( 1 992); Appl. Phys. A 56 , 1 56 ( 1 993). 36. R.C. Powell, G.J. Quarles, J.J. Martin, C.A. Hunt, and W.A. Sibley, Opt. Lett. 10, 2 1 2 ( 1 985).

Index

absorption coefficient 5, 1 04 cross section 6, 1 04 104, 108, 1 1 3 1 1 5 excited state 24, 26, l l l , l l 5 photon 3 7, 25 26, 93 94, 104 transition probability 4 transition rate 4, 93 94 two photon l l 0 l l 5 accepting modes o f vibration 1 65 acceptor ion 1 82 activation energy 1 57 1 59, 1 95, 203 activator ion 1 82 active ion distribution 29 30, 1 75 1 77, 1 8 1 1 82, 1 9 1 1 92, 1 96, 205, 259, 276, 285, 343 344, 352 ADP 393 alexandrite 254 261 absorption spectrum 255 257, 262 energy levels 256 energy transfer 258 260 exchange coupled pairs 258 259 fluorescence lifetime 256 258 fluorescence spectrum 256 259, 262 N lines 258 259 quantum efficiency 260 R lines 256 site occupancy 255 stimulated emission cross section 260 vibronic laser 260 allowed transition 94 98 amplification 7

angular momentum coupling 37 49, 1 77 1 82 electron 33 49, 1 7 1 L S coupling 37 42, 96 98 quantum number 34 38 raising and lowering operators 35 Russell Saunders coupling 37 42, 96 98

annihilation operator phonon 1 20 122 photon 87 88 antiferromagnetic coupling 1 79 1 82 atomic polarizability 1 08, l l 3, 26 1 , 263 267, 328 329, 342, 352

avalanche absorption 344 back transfer 204 banana 393 basis 53 Ba3 (V04) 2 : Mn 285 287 absorption spectrum 286 fluorescence lifetime 287 fluorescence spectrum 286 stimulated emission cross section 287 BBO 393 Beer Lambert law 5, 1 04 BEL : Nd 341 Bohr magneton 42 Boltzmann distribution 1 3 1 , 1 59, 1 62, 1 65, 1 80 1 8 1

borate 26 1 Born approximation 20 1 Born Oppenheimer approximation 1 60, 1 69 1 74

Bose Einstein distribution function 1 30 branching ratio 3 1 5, 325, 341 342, 353 R parameter 341 x parameter 341 cavity dumping 22 cavity quality factor 20 Ce3 + 382 character 52 54 chromium 2 1 5 27 1 , 282 285, 290 292 chrysobery1 25 5

417

Index

418

class 5 1 53 Clausius Mossatti relationship 108 Clebsch Gordan coefficients 38 41 , 56 57 , 72 75, 227 229, 303 407 408 Co2 + 273, 287 288 coherence factor 1 94 collision broadening 103 color center lasers 407 4 1 1 colquiriite 265 concentration quenching 1 76, 202 203, 245, 334, 336, 347 350, 355 357 Condon approximation 60 62 configurational coordinate diagram 1 57 1 60, 249, 250, 280 28 1 configuration mixing 240, 3 1 0 3 19 corundum 237 Coulomb gauge 85, 88 Coulomb integral 46 48, 8 1 , 2 1 9,230 23 1 Coulomb interaction 32 33, 36, 4 1 , 1 77 crl+ 27 1 272, 290 292 Cr3 + 21 5 270 atomic polarizability 261 , 263 crystal field strength 232, 261 exchange coupled pairs 245 25 1 , 258 259 excited state absorption 261 263 free ion terms 2 1 6 224 octahedral crystal field terms 224 229 optical damage 264 265 quantum efficiency 263 Racah parameters 226, 261 saturation fluence 264 265 selection rules 233 237 spin orbit splitting 233 237 strong field material 233, 237 253 Tanabe Sugano diagram 232 weak field materials 233, 254 270 Cr4+ 27 1 , 282 285 eN-

creation operator phonon 1 20 1 22 photon 87 88 critical concentration 1 89 critical interaction distance 1 89 crystal field strength 58 59, 77 80, 1 2 1 1 22, 1 59, 1 60,232, 26 1 , 267 268, 27 1 275, 285, 301 309 operator equivalents 302, 306 309 Racah algebra 302 tensor operators 301 crystal field theory 57 59 crystal growth 27 28, 343 344

density of states, phonon 129 1 30 142 Debye cutoff frequency 1 30, 1 42 , 14 7 Debye temperature 1 42, 142 1 47, 1 56 density of states, photon 93 94 diagonal sum rule 44 Dieke diagram 340 diffusion model 1 99 203 diffusion coefficient 1 99, 202 203, 213

diffusion length 1 99, 2 1 3 mean free path 203, 2 1 3 dipole moment operator 95 direct transitions 1 27 1 32' 1 43 ' 1 47 donor ion 1 82 Doppler broadening 1 03 Dy2+ 382 dye lasers 4 1 1 4 1 2 ED 2 : Nd 3 5 1 353 effective Bohr radius 1 90 effective cross section 1 9 20 effective phonon mode 1 34 efficiency 7, 12, 1 4 1 6, 1 8 pump efficiency 1 2, 1 9 quantum efficiency 7 , 106 slope efficiency 1 6 Einstein coefficients 94, 100, 1 04 1 05, 1 88, 315

electromagnetic multipole multipole interaction 1 84 1 85 electron configuration 66 69 electron phonon coupling 1 2 1 1 22 , 1 27 , 1 54 , 1 56 1 60, 1 69 1 70, 269, 363

anharmonic electron phonon coupling 1 56 1 57

strong electron phonon coupling 1 56 1 74 weak electron phonon coupling 1 27 147 electron phonon interaction 1 2 1 1 22, 1 58 1 60, 1 69 1 7 1

emerald 261 262 emission, photon 4, I I effective cross section 1 9 20 emission cross section 6, 1 9 20, 1 04 1 05 emission intensity 1 05 107 emission lifetime (see lifetime) emission transition probability 7, 3 1 5 fluorescence emission I 05 spontaneous emission 4, 7, 26 spontaneous emission rate 93 95 stimulated emission 6, 1 9 20 stimulated emission rate 1 3, 93 94, 1 03 105

Debye model 1 30, 1 35 1 36, 142, 146 1 47, 1 94

energy gap law 1 34, 1 68, 1 95, 3 3 1 332, 359 363

Index energy migration 1 76 1 77 diffusion theory 198 203 hopping model 1 96 1 98 Kenkre model 365 368 Monte Carlo simulation 202 random walk 1 96 198 energy, pulse 21 energy transfer 25, 1 75, 333 336, 345 348, 364 37 1 coherent energy transfer 203, 213 phonon assisted energy transfer 192 195 radiative energy transfer 175, 5.31 resonant energy transfer 1 82 192 strong coupling 17 5 weak coupling 175 energy transfer rate 1 88 192 diffusion limited energy transfer rate 201 diffusion model 1 99 200 electric dipole dipole interaction 1 88 191 electric dipole quadrupole interaction 1 89 electric quadrupole quadrupole interaction 1 89 exchange interaction 1 89 190 multistep random walk 196 199 phenomenological energy transfer rate parameter 206 209 phonon assisted energy transfer 192 195 trap limited energy transfer rate 201 Er3 + lasers 340, 372 377 exchange integral 46 48, 8 1 , 219, 230 23 1 exchange interaction 1 75, 1 77 1 82, 1 84, 1 89 190, 245 250 excited state absorption (ESA) 1 1 1 , 1 1 5, 25 1 , 261 , 263, 285, 287, 289, 323, 326 327, 343, 357 exciton 1 76 177 diffusion 1 99 203, 2 1 2 21 3 Frenkel 176 localized 196 mean free-path 203, 212 migration 1 % 2 1 3 random walk 1 96 198 self trapped 203 trapping 196 Wannier 177 F2+ 408 41 1 FAP : Nd 341 Fe2+ 289 290 Fermi Dirac statistics 33 Fermi's Golden Rule 92 93, 129, 162, 1 83 ferromagnetic coupling 1 79 1 8 1 fiber lasers 414 4 1 5 Findley Clay analysis 1 5 1 6

419

fluorescence emission 105 107, 207 209 fluorescence linenarrowing 210 21 1 , 354 357 / number 6, 99 100, 105 forbidden transition 96 98 forced electric dipole transition 235, 309 3 19 F iirster Dexter energy transfer process 1 83 forsterite 282 four level system 8 17, 1 8 Frank Condon overlap 1 57 1 58, 165, 250 gain 8, 14 1 6 cavity roundtrip gain 8 saturated gain 1 3 smalJ signal gain 8 , 13 17 GFG 261 , 269 270 GGG 261 262 GGG : Nd 341 GSAG 261 GSGG 261 262 GSGG : Nd 34 1 , 345 Cr Nd energy transfer 345 347 glass : Nd 349 353 branching ratios 353 354 effective linewidth 353 emission cross section 353 fluorescence linenarrowing 354 357 Judd Ofelt parameters 353 radiative lifetime 353 group 5 1 basis 5 3 character 52 54 class 5 1 54 double valued representation 55, 62 representation 5 1 54 representations of lattice vibrations 123 127 space 1 23, 149 153 subgroup 55 56 theory selection rules 97 99 of the wavevector 149 Ham effect 1 74 Hamiltonian crystal field 58 electric dipole interaction 90 electric dipole dipole 1 85 electric quadrupole interaction 90, 185 electromagnetic field 85, 87 electron phonon interaction 120 122, 1 6 1 162, 1 70 energy transfer 1 84 186 exchange interaction 184 ion 32 33, 43, 49

420

Index

Hamiltonian (continued) ion pairs 1 77, 1 79 ion photon interaction 85, 88 9 1 lattice vibrations 1 1 7 1 20 magnetic dipole interaction 90 9 1 , 1 86 multielectron atom 33, 1 70 nonadiabatic 1 6 1 1 62 Hartree Fock approximation 36 Ho3 + lasers 340, 364 37 1 , 376 Cr sensitized 364 rate equation model :369 371 Tm sensitized 364 368 homogeneous broadening 1 0 1 1 03, 143 hopping time 1 97, 202 host sensitized energy transfer 1 82, 347 Huang Rhys factor 1 54, 1 60, 1 66, 1 68 1 69, 250, 267, 285

Hund's rules 42, 67, 2 1 7 296 Hypersensitive transition 3 1 8 3 1 9 impurity sensitized energy transfer 1 82 inhomogeneous broadening 1 0 1 , 1 03 1 04, 143, 248 250, 258 259, 352

ion pairs 1 75, 1 77 1 82, 245 250, 258 259, 276, 285

group theory representations 1 23 1 28 kinetic energy 1 1 8, 1 6 1 1 62 local modes 1 1 7, 1 23, 1 54 normal modes 1 22 1 28, 1 49 1 54, 1 60 1 65 potential energy 1 1 8 1 19, 1 6 1 1 63 promoting modes 1 65 self energy 144 symmetry coordinates 125 1 28 LBO 393 lensing 23 25, 1 09 1 1 0, 1 1 3 1 1 4 nonlinear lensing 2 3 , 1 09 1 1 0, 1 1 3 1 14 thermal lensing 23 25, 1 09 1 1 0 LGS 26 1 LiCAF 26 1 , 265 269 lifetime 4 cavity lifetime 1 0, 20 concentration quenching 1 76 fluorescence lifetime 7, 207 radiative lifetime 4, 1 00, 1 05, 1 8 8 radiative trapping 1 76 lifetime broadening 1 0 1 , 140, 143 ligand field theory 57 LiNb03 : Nd 350 lineshape, spectral 5, 1 1 , 1 00 1 04, 249 Gaussian lineshape 5, 1 1 , 1 0 1 , 1 03 1 05, 143

Lorentzian lineshape 5, 1 1 , 1 00 1 03, Jahn Teller effect 1 69 1 74, 269, 276, 280 282, 29 1

Judd Ofelt theory 1 90, 309 3 1 9 Einstein A coefficient 2 1 5 intensity parameters 3 1 3, 324, 34 1 , 352 353 linestrength parameters 3 1 4 reduced unitary matrix elements 3 1 3 314 selection rules 3 1 3 KDP 393 Kerr effect 1 09 Kerr lens mode locking 1 09 KGW : Nd 350 KLiYF 5 : Nd 343 344 Kramer's degeneracy 62, 305 KTA 393 KTP 393 LaF3 : Nd 341 342, 344, 376 laser induced gratinng spectroscopy 2 1 1 213, 258, 347, 365 37 1

Lande interval rule 42 Lande factor 43 Laport rule 96 lattice vibrations 1 1 7 1 28 accepting modes 1 65

1 43

Pekarian lineshape 1 69 Voigt lineshape 5, 1 0 1 , 143 linewidth, spectra1 140 144, 1 55, 248, 276 279, 330 33 1 , 363

collision broadening I 03 Doppler broadening I 03 homogeneous broadening 1 0 1 , 1 43 inhomogeneous broadening 1 0 1 , 143 lifetime broadening 1 0 1 , 140, 143 natural linewidth 1 0 1 , 140 radiationless decay processes 143 Raman scattering of phonons 1 4 1 143 temperature dependence of the linewidth 143

lineshift, spectral 144 147, 249, 330 3 3 1 radiationless decay processes 1 47 Raman scattering of phonons 1 44 1 47 temperature dependence of the line position 147

LiSAF 26 1 , 265 269 absorption spectrum 266 crystal field 267 268 electron phonon coupling 269 fluorescence lifetime 268 269 fluorescence spectrum 266 Huang Rhys factor 267

Index Jahn Teller effect 269 spin orbit coupling 260 LiSGF 269 LLGG 261 262 local modes of vibrations 1 1 7, 1 23, ! 54 Lorentz local field factor 1 00, 108, 3 1 0 3 1 1 losses 8 , 14 20 active 8 passive 8 LS coupling 4 1 , 97, 3 1 5 master equation 204 205, 212 material properties 23 30 Maxwell's equations 85 mean free path 203, 2 1 3 MgFz : Co2 + 287 288 absorption spectrum 287 288 excited state absorption 287 fluorescence lifetime 287 288 fluorescence spectrum 287 288 MgF2 : Ni 2+ 289 290 absorption spectrum 290 fluorescence lifetime 290 fluorescence spectrum 290 MgF2 : V2+ 289 minilasers 17 5 Mn 5 + 272, 282, 286 mode locking 22 23, 1 09, 4 1 1 Monte Carlo simulation 202 multiphonon processes 1 02 107, 1 60 1 69, 359 363 multiplet 41 natural linewidth 1 0 1 , 1 40 Nd3 + 294 360, 34 1 345 concentration 348 350 configuration mixing 309 3 1 9 electron phonon interaction 3 1 7 energy level diagram 299, 340 free ion terms 295 299 glass hosts 349 353 hypersensitive transitions 3 1 8 multiphonon decay 3 59 363 Racah parameters 298 reduced unitary matrix elements 3 14 R parameter 341 Slater parameters 296 30 1 spin orbit coupling 300 301 x parameter 341 nephelauxetic effect 306 Ni2 + 275, 289 290 nonadiabatic Hamiltonian 1 6 1 1 69 nonlinear optical crystals 348, 350, 384 407

421

coherence length 389 harmonic generation 387 394 optical parametric generation 387, 394 395 phase matching 388 394 stimulated Raman scattering 395 407 nonlinear optical processes I 07 1 1 5 nonlinear refractive coefficient 1 09 1 10 nonlinear refractive index 24 25, 1 09 1 1 4, 263 normal modes of vibration 1 1 8 1 20, 1 23 1 25, 1 49 1 54 NPP 34 1 , 347 348 concentration quenching 347 energy transfer 347 number operator, photon 87 88 NYAB 350 optical damage 1 9, 25, 264 265 Orbach processes 1 32 1 37 orbitals, electron 34 35, 70 77 oscillator strength 6 7, 99 1 00, 1 05 output coupling 8, 1 5 1 7 partition function 1 8 1 Pauli exclusion principle 3 3 , 3 6 , 67, 2 1 6, 227, 295 pentafluoride 261 perovskite 26 1 phase matching 350 photoconductivity 1 76 point group 50 56 polarizability (see atomic polarizability) population grating 212 population inversion 8 3, 20 2 1 power 1 1 1 9 Pr lasers 376 377 principal quantum number 34 promoting modes of vibration 1 65 pulse width, spectral 5 6, I I , 23 pulse width, temporal 20 23 pumping schemes 1 9 avalanche pumping 1 9 energy transfer pumping 19, 5 . 7 up conversion pumping 19, 1 1 5 pump efficiency 12, 1 9 Q switching 20 22 quantum defect 1 2 quantum efficiency 7 , 106, 250, 285, 333 quenching parameter 348 350 Racah algebra 302

422

Index

Racah parameters 47 48, 8 1 , 2 1 9, 222, 224, 232 233, 240, 26 1 , 308 radiationless decay processes 1 27 1 37, 1 62, 1 68, 250, 278 280, 33 1 332 direct transitions 1 30, 143, 147 multiphonon processes 1 32 1 37, 1 62 1 69 selection rules 1 27, 1 3 1 Raman laser 350, 395 407 Raman scattering of phonons 1 32, 1 35 1 37, 1 4 1 143 random walk 1 96 198 rate equations 206 209, 269, 273 reduced mass 32 Rh2+ 414 ruby 2 1 5, 232 233, 237 252, 26 1 absorption spectrum 239, 262 B lines 241 charge transfer bands 241 configuration coordinate diagram 249 250 crystal field strength 233, 240 energy levels 238 energy transfer 251 exchange coupled pairs 245 250 excited state absorption 25 1 fluorescence lifetime 242 245 fluorescence spectrum 242, 246, 262 Huang Rhys factors 250 N lines 245 250 photocurrent 252 quantum efficiency 250 Racah parameters 240 radiationless relaxation rates 250 R lines 240, 248 250 S lines 241 stimulated emission cross section 252 vibronic transitions 242 243 Russell Saunders coupling 4 1 , 96, 2 1 5, 295 representation 5 1 55 sapphire 237, 276 282 saturation fluence 1 8 , 264 265 second harmonic generation 1 08, 350 selection rules group theory 97 99 radiationless transitions 1 3 1 radiative transitions 95 96, 98 99 vibronic 140, 1 49 1 54, 1 65 self doubling 1 09, 350 sensitizer ion 1 82 site selection spectroscopy 2 1 0, 327 329, 336, 352, 355 357 six } symbol 304 Slater Condon parameters 46 48, 2 1 9, 222, 296 301

Slater determinant 43, 72, 23 1 slope efficiency 1 6, 292 Sm2+ 382 383 space group 1 23, 1 49 1 53 spatial hole burning 4 1 1 spectral hole burning 2 1 0 2 1 1 spectral overlap integral 1 8 8 spectroscopic notation 3 7 , 4 1 multiplet 4 1 multiplicity 41 spectroscopic term 41 spectroscopic quality parameter 3 1 5, 324, 341 spin flip transition 67 spin orbit coupling 32 34, 36, 4 1 42, 58, 1 70, 233 237, 269, 278, 280, 300 301, 304 spin spin coupling 1 78 1 80 spontaneous emission rate 93 95 SrTi03 : Cr3 + 147 1 56 fluorescence spectrum 1 48 normal modes of vibration 1 49 1 53 vibronic selection rules 149 1 53 vibronic spectrum 1 53 155 zero phonon linewidth 145 1 56 Stark components 305 Stark effect 4 1 , 57 stimulated emission rate 93 94 stoichiometric laser materials 1 75, 341 , 347 Stokes shift 1 57 1 59 subgroup 55 56 superexchange 1 8 1 susceptibility 1 07 1 08, 1 1 1 1 1 3 symmetry coordinates 1 25 128 symmetry operation 49 5 1 Tanabe Sugano diagrams 232, 272 275 tensor operators 301 3 1 9 term 4 1 thermal equilibrium 1 30, 1 62, 1 80 1 8 1 thermal lensing 25, 1 09 1 1 0, 1 1 4 thermal relaxation time 1 3 1 , 1 36 1 37 three-;j symbols 38, 303, 3 1 2 three level system 8 18, 20, I 06 1 07 threshold 8 1 7, 1 9 Ti3 + 27 1 272, 276 282 Ti sapphire 276 282 absorption spectrum 276 277 configuration coordinate model 280 28 1 energy level diagram 278 fluorescence lifetime 276 277 fluorescence spectrum 276 277 Jahn Teller effect 276, 280 282 radiationless decay rates 278 280 R lines 276 279

Index Tl 0 410 Tm2 + 382 Tm3 + 340, 365 368, 37 1 372, 376 energy migration 365 368 laser 340, 371 372, 376 transition rate, radiationless 1 30 1 34 absorption transition rate 1 30 1 34 emission transition rate 1 30 1 34 transition rate, radiative 92 95 absorption transition rate 93 95 spontaneous emission rate 93 95 stimulated emission rate 93 95 transition rate, vibronic 1 37 140 absorption transition rate 1 39 emission transition rate 1 39 transition strength, radiative (see oscillator strength; / number) trapping, exciton 1 96 202 triangle rule 36, 4 1 , 45 tungstate 261 two photon absorption 1 1 0 1 1 1 , 1 1 4 1 1 5, 329 sequential two photon absorption transition I l l , 1 14 virtual two photon absorption transition 1 1 1 , 1 14 3 U + 382 384 up conversion 19, 1 1 5, 344 345, 372 377, 415

V2+ 288 289 y2+ 272, 282 vibronic lasers 260 270, 276 282, 287 290 vibronic sideband 1 37 1 40, 1 54 1 55, 242 243, 330 waveguide lasers 4 1 4 4 1 5 Wigner coefficients 3 8 Wigner Eckart theorem 57, 303, 3 1 6 XeF laser 4 1 3 x parameter 3 1 5, 324, 34 1 YAG 3 1 9, 320, 329 YAG : Cr4+ 282 286 absorption spectrum 284 285 crystal field strength 28 5 energy levels 283

423

excited state absorption 285 fluorescence lifetime 285 fluorescence spectrum 284 285 nonradiative decay rates 285 quantum efficiency 285 stimulated emission cross section 285 YAG : Nd 299, 302, 307 308, 3 1 9 336 absorption spectrum 320 32 1 , 324 atomic polarizability 328 329, 342 concentration quenching 334 336 crystal field parameters 302, 307 308 energy gap law 3 3 1 332 energy levels 299, 323 energy transfer 333 336, 345 346 excited state absorption 323, 326 327 fluorescence lifetime 324 fluorescence spectrum 321 322, 325, 327 329 Judd Ofelt parameters 324 linewidths 330 3 3 1 Jineshifts 330 3 3 1 nonradiative decay processes 3 3 1 332 operator equivalents 307 308 quantum efficiency 333 R parameter 34 1 342 selection rules 309 site selection spectra 327 329, 336 stimulated emission cross section 3 1 9 two photon absorption 327 vibronic spectrum 330 x parameter 324, 341 342 Y3 (AII - x Gax ls0 12 : Nd 328 329 YAIO : Nd 341 YAP : Nd 341 Yb3 + lasers 372 375 YGG 26 1 YLF : Nd 341 344, 362 363, 376 YSAG 261 YSGG 261 YV04 : Nd 341 342, 347 348 host sensitized energy transfer 347 348 ZBAN : Nd 353, 357 358, 371 Zeeman effect 4 1 42 zero phonon line 1 40 147, 1 55 1 56 ZnSe : cr2+ 290 292 absorption spectrum 29 1 fluorescence lifetime 291 fluorescence spectrum 29 1 Jahn Teller effect 29 1 slope efficiency 29 1

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Physics of Solid-State Laser Materials by Richard C. Powell Tlw inve ntion of the laser has had a profound impact on tlw qualit y of our li ves. Lasers a re <'Ommonly being used in a multitude of applications ranging from price scanne rs at r he<·kout counte rs to a variety of me dical s urgery procedures. The laser has been the cata lyst for d eveloping new technologies that have revolutionized important industries. Although man y diffe re nt types of lasers arc used in latlay's applications, solid-sta le lasers a re always preferable if they are available with the d esired operating c haracteristics. This graduate-level text presents the fundam ~ntal phys ics of solid-stale lase rs, including the bas is of laser action and the optical and e lectronic prope rties of laser materials. Some knowledge of quantum mechanics and solid-stale physics is assumed, but the discussion is as self-contained as possible, making the book an excellent re fe re nce as well as useful fo r inde pendent s tud y. After an overview of the topic, the book is divided into two parts. The first section opens with a review of the quantum mechanics and solid-s tale phys ics, s peclrosc·opy, and c rys tal field theory. It the n treats the quantum theol-y of radiation, the e mission and absorption of radiation, and nonlinear optics. lt concludes with discussions of lallice vibrations and ion-ion inte ractions and of the ir e iTects on optical properties a nd laser action. The setond section treats spe_c ific solid-slate laser mate rials, the prototypical ruby and Nd-YAG systems bejng treated in ~realest de tail. It condurles with a discussion of novel anrl nons tandard mate rials .

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