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= <Sz > = ncH0 E 02 . 2
7.3.5. Poynting vector for a MPPEM wave in an absorbent dielectric such that µ is real Using the optical notation given in Section 7.2.2.1,
220
Basic electromagnetism and materials
G E G and B
G E m exp(k '' z) exp( j[k ' z Zt]) k G G n G G uuE uuE . Z c
According to Eq. (33), established for a real value of µ , we find that G G G G 1 E 0 E m exp(k '' z) exp( jk ' z) <Sz > = R(E 0 u B0 ) , where 2µ G n G G B0 u u E0 c Therefore
ª n G G º n G G G G G ªn G G º G G E 0 u B0 E 0 u « u u E 0 » E0 u « u u E0 » E0 u u u E0 c «¬ c »¼ ¬c ¼ n G G G n G E 0 .E 0 u E m ² exp(2k '' z) , from which c c G G 1 1 n' G <Sz > = R(E 0 u B0 ) E m ² exp(2k '' z) , 2µ 2µ c so that for a nonmagnetic medium (µ = µ0) cn ' H0 G <Sz > = E m ² exp(2k '' z) . 2 The power transmitted by the electromagnetic field exponentially decreases with the Z distance traversed. The extinction coefficient for this is given by 2k'' = 2n'' . c
7.4. Problem Poynting vector It is worth recalling that the physical magnitude of flux represents an amount associated with a physical magnitude which traverses a unit surface per unit time. 1.
The analogy between the charge current density and the energy current density vectors G (a) Here j denotes the charge current density vector and U the volume charge G G G G density. The flux of the vector j through a surface dȈ is dI = j dȈ . What does j alone represent? Give the equation for the conservation of charge for a rapidly varying regime in an isolated system.
Chapter 7. Electromagnetic waves in materials and the Poynting vector
221
(b) By analogy, the energy flux (which thus exhibits a power per unit surface) G G G through the surface dȈ is defined by a relationship of the type dP = S .dȈ in which G S represents the energy current density, otherwise termed the Poynting vector. What does S alone represent? By introducing the magnitude w T , which represents the volume density of the total energy, give an equation for the conservation of G G energy analogous to that for the conservation of charge ( S plays the role of j and w T that of U ).
2. The total volume density of energy w T can be considered as the sum of two terms, one being due to the density of kinetic energy ( w c ) and the other due to the energy density ( w em ), which will be detailed below. (a) The theorem for kinetic energy makes it possible to state that the variation in kinetic energy is equal to the work of the applied forces. A variation in the kinetic G G energy density (for a unit volume containing a charge U) is given by dw c F. v dt G G G G G with F q ª¬ E v u Bº¼ when charges with volume density U and velocity v are G G placed in an electromagnetic field characterized by the fields E and B . Determine from the equation for the conservation of energy the value of the expression G G G dw em . divS + E.j + dt JJG G (b) Give the Maxwell-Faraday (M-F) relationship (M-F which gives rot E ) and the G JJG B ) for a nonMaxwell-Ampere (M-A) relationship (M-A which gives rot µ0 G G B ] and magnetic medium traversed by a real current ( j ). By multiplying M-F by [ µ0 GG G M-A by [ -E ], calculate E. j . G (c) Determine the expressions for S and w em by identification between the results of 2a and 2b. Conclude. 3. (a) It is worth recalling here that the real part of a product of two complex numbers is different from the product of the real parts of two complex numbers. Show how, in a similar way, the complex amplitude of the product of two complexes is different from the complex amplitude of these two complexes. What is under consideration when dealing with the vectorial products in place of the simple products? K K (b) If E 0 and B0 are the complex amplitudes of a complex field, then the complex electric field and the complex magnetic field:
222
Basic electromagnetism and materials
notation G G G G G E 0 (r) exp(iZt) = E0 exp(iZt) and B B0 exp(iZt) , are such that: G G G G E Re(E) and B Re(B). G G G G 1 From the calculation for S [Re(E) u Re(B)] , show that the average value of S µ0 is of the form: G · §G G 1 ¨ E0 u B 0 ¸ ¢S² , Re ¸¸ 2 ¨¨ µ0 © ¹ G G where B 0 is the complex conjugated with B0 . G (c) Calculate ¢S² for a monochromatic plane progressive wave propagating in the G G direction of the unit vector u z in a medium where the wave vector k is real. G E
Answers 1. l = (v 't)'t = 1 = v (a) By definition, j represents the flux of electric charges, also termed the charge current density. This flux therefore represents the amount of electrical 6=1 charges that traverse a unit surface per unit time. Quantitatively, we can say that this quantity of charges is localized within a cylinder (or parallelepiped) with a cross-sectional unit area ¦ = 1 and length (l) given by l = (v 't) 't = 1 = v , where v is the velocity of the charges. If U = n q , is the volume charge density where n is the charge density, i.e., the number of charges per unit volume, then the quantity of charge found in the volume (V) given by V = ( ¦ ) ¦ =1 (v 't) 't = 1 = v is such that j = n q ( ¦ )¦ =1 (v 't) 't = 1 = U v . G G G The vector j is defined by j Uv . For its part, the equation for the conservation of charge is written G wU divj 0. (1) wt G (b) Similarly, if S represents the energy current density vector, S represents the quantity of energy that traverses a unit surface in a unit time. S therefore can be seen as the electrical power through a unit surface. If w T represents the total energy, the equation for the conservation of energy can be written by analogy to the equation for the conservation of charge as:
Chapter 7. Electromagnetic waves in materials and the Poynting vector
G ww T divS wt
0.
223
(2)
2. (a) Therefore, w T = w c + w em , and
dw T
dw c
dt
dt
dw em dt
. (3)
G G In addition, the theorem for the kinetic energy allows the equation dw c = F. v dt so G that by taking into account the form of F , we have: G G G G GG G G dw c G G = F. v = U[ E + v u B] v = U E v = j . E . (4) dt dw T G G dw em j. E By moving this expression into Eq. (3), we find . By dt dt introducing this equation into Eq. (2) for the conservation of energy, we arrive at G G G dw em divS j. E 0 . (5) dt (b) We have: G G JJG G wB B rot E = (equation multiplied by ) (M-F) µ wt
(M-A)
JJG rot
G B µ0
G G w (HE) j wt
G (equation multiplied by [ -E ]).
Following multiplication of the two members of each equation as indicated, we find by addition that: G G G G G B JJG G G JJG B B wB G w (HE) G G j.E . rotE - E rot =-E wt µ0 µ0 µ0 wt G G G JJG G G JJG G Using the relation div a u b =b rot a - a rot b , the above equation becomes G §G B · w § E² B² · G G H div ¨ E u =¸ ¨ ¸ j.E which can also be written as ¨ wt ¨© 2 2µ0 ¸¹ µ0 ¸¹ © G §G B · GG w § E² B² · (6) div ¨ E u ¸¸ +j.E + ¨¨ H ¸ 0. ¨ wt © 2 2µ0 ¸¹ µ0 ¹ ©
G (c) By identification with Eq. (5), it can be determined that S the Poynting vector, and that
G G B Eu which is µ0
224
w em
Basic electromagnetism and materials
H
E² 2
B² 2µ0
is the density of electromagnetic energy in the form:
w e w m , with w e
H
E²
and w m
B²
. 2 2µ0 These two expressions are for the electrical and magnetic energy densities under a stationary regime and they prove that they are still valid for variable regimes. w em
3. (a) The use of complex numbers needs to take into account (with z n = a n + i b n ): x if the real part of the sum of two complex numbers is actually equal to the sum of the real parts of the two complex numbers, as in R(z1 + z 2 ) = R(z1 ) + R(z 2 ) = a1 +a 2 ; and x or if the real part of the two complex numbers is not equal to the product of the two real parts of the two complex numbers, as in R(z1z 2 ) = R([a1 + i b1 ] [a 2 + i b 2 ]) = R (a1b1 - a 2 b 2 +i[a1b2 + a 2 b1 ]) = a1b1 - a 2 b 2 z a1b1 = R(z1 )R(z 2 ). Similarly, if the product (P) of the complexes is P = A.B , then its complex amplitude is such that (with A = A 0 eiZt and B = B0 eiZt ): P = A 0 eiZt B0 eiZt = A 0 .B0 ei2Zt = P 0 eiZt , from which P0 = A 0 .B0 eiwt z A 0 .B0 . The complex amplitude of the product of two complexes appears different from the product of the complex amplitude of two complexes. Similarly, the vectorial product, which brings in the components of the products, differs from the simple products, for example: JG G G G G G G 1 R(E u B) . R(E u B) z R(E) u R(B) , so that here R(S) z µ0 G G G G (b) However, if E E0 exp(iZt) and B = B0 exp(iȦt) , we have: G · §G G 1 ¨ E0 u B 0 ¸ ¢S² . Re ¸¸ 2 ¨¨ µ0 © ¹ G In effect, S G With E
G G G G · 1 G G 1 § 1 E u B= R(E u B) ¸ . ¨¨ ¬ª R(E) u R(B) ¼º z ¸ µ0 µ0 © µ0 ¹
JG E 0 exp(iZt)= A + iB , we have:
Chapter 7. Electromagnetic waves in materials and the Poynting vector
225
G G* G G* G E 0 eiZt E eiZt EE , from which R(E)= A = 2 2 G G G G G 1 S [E 0 exp(iZt) E 0 exp(iZt)] u [B0 exp(iZt) B 0 exp(iZt)] 4µ0 G G G G G G G 1 G [E 0 u B0 ] + [E 0 u B0 ]+[E 0 u B0 exp(2iZt)]+[E 0 u B 0 exp(2iZt)] . 4µ0
^
`
^
`
The first two terms are conjugated complexes and their sum is equal to twice their real parts [as z1 + z1* = (a1 + i b1 ) + (a1 - i b1 ) = 2 a1 ) ]. The third and fourth terms each vary as a function of 2Z so that they have average values over a half period, and therefore also a full period, equal to zero. Therefore: G G G 1 S R(E 0 u B0 ) . 2µ0
(c) If the plane wave progresses along Oz, then the electrical field can be supposed, for example, to be moving with respect to Ox. Therefore, GG GG G notation G G G E0 E 0 (r) E m eik.r E m eik.r u x . Similarly, for a magnetic field moving through Oy, we have GG GG G notation G G G B0 B0 (r) Bm eik.r Bm e ik.r u y . Therefore: G G G 1 S R(E 0 u B0 ) 2µ0 Given that E m G S
1
1
1 G G G R(E m .Bm* u x u u y ) = R(E m .Bm* u z ) . 2µ0 2µ0
vM Bm in a material (or E m
n G 2G R(E m .E m* u z ) = Em uz 2µ0 vI 2µ0 c
c Bm if in a vacuum), we have: H0 2
2G nc E m u z .
If the MPPEM wave is polarized rectilinearly, so that E m = E 0 is real, then G H0 G S nc E 02 u z . 2
Chapter 8
Waves in Plasmas and Dielectric, Metallic, and Magnetic Materials
8.1. Interactions between Electromagnetic Wave and Materials 8.1.1. Parameters under consideration So that geometrical considerations do not become a problem, this chapter will look at the interactions of electromagnetic (EM) waves with materials that have infinite dimensions. Materials with limited forms will be looked at, most notably, in Chapters 11 and 12. The propagation of EM waves can be studied by considering: x a description of the EM wave in the material with Maxwell's equations; and x a representation of the material as a collection of electronic and ionic charges that interact with the electromagnetic field through the Lorentzian force, which can be written as: G G dv G G m qE qv u B . dt In terms of moduli, the ratio of magnetic and electronic contributions are given by Fm qvB vB . Fe qE E For a plane EM wave, the E = vM B and the ratio
Fm
v
Fe
vM
. As v << vM | c , the
magnetic force (Fm) is negligible with respect to the electrical force (Fe). With the K magnetic force being weak, generally it is stated that the magnetization intensity ( I ) JJGG G G G is approximately equal to zero, so that B | µ0 H and rotI J A | 0 .
228 Basic electromagnetism and materials Different types of forces can be involved, depending on the response of the material under the deformation caused by the incident EM wave.
8.1.2. The various forces involved in conventionally studied materials 8.1.2.1. Dielectric materials The various movements and polarizations which can come about depends on whether it is electrons, ions, or permanent dipoles that are submitted to Fe. 8.1.2.1.1. Electronic polarization Much like a spring, valence electrons displaced by Fe are returned to the equilibrium position about their respective atoms by a force (fr) given by f r = - k r = - mZ0e ² r . In addition, if the electrons move within a very dense medium, they can be thought of as being subject to a friction force (ft ) which gives rise to a Joule effect and has G mG an intensity proportional to their velocity, so that f t v where W is the W relaxation time of the system and has the dimension of time (so that the equation is dimensionally correct). 8.1.2.1.2. Ionic polarization The movement of ions under an electric field approximately resembles that of electrons. Their returning force is fr = -Mnucleus Z0i2 r where, due to the greater mass of ions and their relative inertia, their ionic pulsations ( Z0i ) are much smaller than the equivalent electronic movements (Z0e). This is detailed further in Chapter 3 of Volume 2. 8.1.2.1.3. Polarization and dipole orientation It is assumed that dipoles subject to a varying electrical force are predominantly subject to a friction force associated with their rotational movement. Given that they are relatively large due to the chemical association of atoms, they present a high degree of inertia toward excitation by an electric field. The dipoles can only follow—with a dephasing—relatively low frequencies, but tend to contribute considerably to dielectric absorption at such frequencies (see also Chapters 1 and 3 of Volume 2).
8.1.2.2. In plasma In a plasma, which has a low density and is electrically neutral, it is assumed that electrons make up an electronic gas. Any displacement of the electrons from their equilibrium position by a perturbation resulting in a compression or dilation of the electron gas can be considered the result of the application of a sinusoidal electric field, which has an plasma oscillation pulse (Zp) due to the returning force of ions in the medium that are assumed to be immobile. Given the assumption that the electrons move freely, we can ignore friction and mechanical returning forces so that Zp is automatically stabilized by the longitudinal electric field associated with the
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
229
permanent returning force (thus
directly related to this electric field). The Zp corresponding frequency is therefore Q p = , and as detailed further in this 2S chapter, can appear as a breaking or “drag” frequency for EM waves.
8.1.2.3. Metals The electrons that interact most easily with EM waves are those external to atomic or molecular orbitals, in other words conduction electrons. These electrons are in effect free, or semifree, depending on the degree of approximation, and belong to no specific atom, so that any returning mechanical forces are equal to zero. The medium, however, is sufficiently dense to make friction forces nonzero and it is these forces that result in Joule effects in metals, more specifically due to collisions between electrons. The exact nature of these collisions generally is only considered in solid physics along with collisions in a network (phonons) and with impurities. 8.2. Interactions of EM Waves with Linear, Homogeneous and Isotropic (lhi) Dielectric Materials: Electronic Polarization, Dispersion and Resonance Absorption Electronic polarizations gives rise to resonance frequencies ( Z0e , which is more succinctly denoted Z0 below) in an absorbing and dispersing material, and this section is limited to studying the plots of these interactions. These interactions can be understood using the question-based tutorial below, which has answers detailed in Sections 8.2.1 through to 8.2.8. The polarization of a linear, homogeneous, and isotropic (lhi) material is the result of a total number (N) of electrons per unit volume (ne) (N = ne) each having a charge (–q) being subject to the following forces: x a Coulombic force induced by the alternating field which can be described in the G G complex form by E E 0 exp(iZt) ; G G x a returning force given by Fr mZ02 r , where me is the mass of a electron; G G G dr m dr x frictional forces given by: Ft m* = . dt W dt 1. Under a forced regime, find the expression for the displacement of electrons G G G based on the complex form r r 0 exp(iZt) . Determine r 0 .
230 Basic electromagnetism and materials
2. Give the complex expression for the electronic polarization. From this determine the real and imaginary parts of the dielectric susceptibility ( Fe ' and Fe '' , respectively) given in electrical notation by Fe = Fe ' - iFe '' . The result should be expressed in terms of Z² p = (
n e q² mH0
).
3. Now the study turns to the functions Fe’ and Fe’’ in terms of Z at two points: when Z # Z0 , and then when Z z Z0 . Parts a, b, and c of this question consider the former Z # Z0 . (a) give the sign associated with Fe ' when Z Z0 or when Z ! Z0 ; (b) for which value of Z, Fe’’ is a maximum; (c) the width at half-peak height of the function Fe’’(Z) corresponds to angular F '' frequencies Z1 and Z2, which are such that Fe ''(Z1 ) = Fe ''(Z2 ) = e max . Determine 2 Z1 and Z2 and show that for these values Fe’(Z) is at an extreme point. Determine the values for Fe’(Z1) and for Fe’(Z2); and (d) give the limiting values for Fe’ and Fe’’ (Zo f). Plot Fe’(Z) et Fe’’(Z). 4. This question concerns a regime far off the resonance position Z z Z0 ; indeed, it is now the hypothesis Z²* ² << ( Z² 0 - Z²)² that becomes relevant. Give the corresponding representations for Fe’(Z) and Fe’’(Z). 5. Staying with the hypothesis presented in question 4 [Z z Z0 and Z²* ² << ( Z² 0 - Z²)² ], show that there is a value ZA (to be determined) for Z which is such that the real part of the relative dielectric permittivity (Hr’) is zero at this pulsation. Show that for the value of Z given by Z ZA the magnetic field is zero and that the electric field is longitudinal. 6. Plane, progressive, and traversing EM waves that can be written in the form GG G G E E m exp[i(Zt kr)] and angular frequencies between but not including Z0 and ZA (outside of the resonance and ZA ) are considered in this question. From the general equation for the dispersion of waves, give the form (real, complex, or purely imaginary) of the wave vector in this domain of angular frequencies while also determining the type of wave involved. 7. Outside of the absorption and the interval [Z0, ZA ] : (a) give the explicit relation between Z and j using the expression for the relative permittivity (equation involving term expressed as Z4); (b) from the real solutions to this equation, give the relations between Z and k; and
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
231
(c) With respect to the limiting case, give the general shape of the curves representing Z = f(k) for dispersion due to electronic polarization.
8. Find for Z << Z0 the equation (for dispersion) that ties the optical index to the wavelength on a vacuum O 0 =
2Sc Z
.
8.2.1. The Drude Lorentz model: a representation of a dielectric using an electron gas and a study of the electron movement 8.2.1.1. The Drude-Lorentz model In the Drude-Lorentz model, a dielectric material is represented by a group of atoms (Na) per unit volume distributed in a vacuum. The atoms are surrounded by a field of electrons that are themselves assumed to move in a vacuum around the atoms to which they are attached. When subjected to a sinusoidal electric field, the electrons move from their positions and then tend to return to their original position through the influence of two forces : G G G dr m dr , which takes into account the x a frictional force given by Ft m* = dt W dt viscosity of the medium and results in a dephasing between the excitation and the electronic response. In this case, W is the relaxation time that represents the time required to establish the electronic polarization following application of the field. In the Drude-Lorentz model, which results in an expression for conductivity and Ohm's law, W represents the average time in between two successive collisions; and G G x an elastic force, given by Fr mZ02 r , due to the recall of electrons to their particular bonds. Physically, this force resembles that of a returning force exerted by a spring tying together an electron and its nucleus. For each type of electron, be they internal or external layer electrons, the elastic force varies along with the corresponding value for Z0. It is worth noting that for metals, which have electronic properties determined by free conduction electrons, the returning or recall force is not brought into account because the energy required (thermal energy) to remove an electron from its original atom is extremely small. As detailed in Section 8.1.2.3, only Ft is assumed to be exerted.
8.2.1.2. Equation for the movement of electrons G Here the charge of an electron is represented by –e. With respect to the direction r , G parallel to the applied field ( E ), the fundamental equation for the dynamic of the displacement of electrons is:
232 Basic electromagnetism and materials
m
G d²r dt²
G
G
G
G
¦ Fapplied = Fr Ft FCoulomb = mZ02 r G d²r dt²
G m dr
W dt
G 1 dr
e G G Z02 r = E. W dt m
G Given a cosinusoidal field of the form E
G - eE , so that:
(1)
G E 0 eiZt , the required solution for the
forced regime subject to an alternating field, particular to the equation with a second G G term, is in the form r r 0 exp(iZt) . GD With r
DGD G iZ r and r
G Z²r , the solution given in Eq. (1) gives, following
simplification of the two terms using eiZt , ZG e G G G (1’), from which - Z²r 0 i r 0 Z02 r 0 = E0 W m G e G r0 E0 . Zº ª m « Z02 Z2 i » W¼ ¬
(1’’)
8.2.2. The form of the polarization, the susceptibility, and the dielectric permittivity 8.2.2.1. Expression for electronic polarization G G G Moving a charge (q) by r is the same as applying a moment dipole P= qr , and likewise, moving an electronic charge ( q = -e ) is the same as applying a dipole G G moment P = - er (see Chapter 2, section 2.2.2). The polarization for N = n e electrons per unit volume therefore is given by: G G n e e² G G G (2) P n e P = -e n e r = -e n e r 0 eiZt = E 0 eiZt . Zº ª 2 2 m « Z0 Z i » W¼ ¬
8.2.2.2. Expressions for susceptibility and dielectric permittivity (electronic component) G G G G Given that P = H0 F E H0 F E 0 eiZt = H0 (H r - 1) E 0 eiZt (3), e
e
from Eqs. (2) and (3) it is possible to directly determine that: F = e
n e q²
1
H0
Zº ª m « Z02 Z2 i » W¼ ¬
.
(4)
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
Given that Z2p =
n e q²
1
, the result is F = Z2p e
H0 m
.
Zº ª 2 2 « Z0 Z i » W¼ ¬
233
(4’)
From this can be determined that
F = e
Z2p
Z Z i ZW = F ' - i F '' . Z² Z Z W² 2 0
2
2 0
2
e
2
(5)
e
By identification of the real and imaginary parts, F = (H r - 1) = (H r ' - 1 - i H r '') , it is possible to determine that:
and
with
e
Fe ' = (H r ' - 1) =
Z2p
Z
2 0
Z2
Z02 Z2
2
Zp2
Z² W²
Z
2 0
Z2
Z02 Z2
2
(6)
Z²* ²
Z Fe '' = H r '' = Z2p
W
Z02 Z2
2
Z*
=Z2p
Z²
Z02 Z2
=
1 n e q²
W²
Eq. (6) can be used to define that Fe '(0) =
Z2p Z02
2
. Z²* ² (8).
Z02 H0 m
Following this, the introduction of Fe’(0) into Eqs. (4), (6), and (7) yields: F = Fe '(0) e
Z02 Zº ª 2 2 « Z0 Z i » W¼ ¬
.
(9)
In turn, this gives: Fe ' = Fe '(0)
Z02
Z
2 0
Z2
Z02 Z2
Fe '' = Hr '' = Fe '(0)Z02
2
(6’) and
Z²* ²
Z*
Z02 Z2
2
. Z²* ²
8.2.2.3. Limiting cases 8.2.2.2.1. Low frequencies where Z << Z0 (Z o 0) According to Eqs. (6) and (7),
(7’)
(7)
234 Basic electromagnetism and materials
Fe ' o Fe '(0) =
Z2p Z02
and H r ' o H r '(0) = 1 +
Z2p Z02
(10), and
Fe '' o Fe ''(0) = 0 and H r '' o Hr ''(0) = 0 . The upshot is that in this frequency domain, H r | Hr '(0) . With F being real and e G G positive, according to Eq. (3), there is no dephasing between P and E . 8.2.2.2.2. At frequencies where Z = Z0 Here, Fe '(Z0 ) = 0 , so that H r '(Z0 ) = 1 Fe ''(Z0 ) =
F (Z0 ) = -i Fe ''(Z0 ) = -i e
Z2p Z0 *
Z2p Z0 * i
. As - i = e
S 2
G G , P and E are in quadrature.
8.2.2.2.3. High frequencies for which Z Z0 (Z o f) According to Eq. (6) and (7), Fe '(f) o 0 and Fe ''(f) o 0 . More precisely, with the help of Eq. (9), by neglecting the Z term but not the Z² term (as Z o f ) from the G G Z2 denominator, F | - Fe '(0) 0 (11). As - 1 = eiS , P and E have opposing e Z2 phases.
8.2.2.4. Comment concerning the Lorentz correction G The Lorentz correction replaces in Eq. (1) the field ( E ) by the locally effective field G G G G P G E al E given by . With P= -e n e r Eq. (1') now becomes 3H0 ZG n q² e G G G G - Z² r0 i r0 +Z02 r0 - e r0 = E 0 . This results in a change in the expressions W 3mH0 m for Fe and H of the term Z02 to Z '02 = Z02 -
n e q² 3mH0
.
8.2.3. Study of the curves Fe’(Z) and Fe’’(Z) for Z | Z0 (of the order of the absorption zone) In this section, the approximations Z0 ² - Z² = (Z0 - Z)(Z0 + Z) | 2 Z0 (Z0 - Z) and *Z | *Z0 are made.
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
235
8.2.3.1. Expression for Fe’ From Eq. (6), for Fe’: Fe '(Z|Z0) | Zp2
2Z0 Z0 Z 4Z02
Z
Z
0
2
, so that dividing above and below by
Z02 * ²
4 Z02 gives: Fe '(Z|Z0) |
Z2p 2Z0
Z0 Z
Z
0
Z
2
§*· ¨ ¸ ©2¹
2
=
Z0 Fe '(0)
Z0 Z
2
2
Z
0
Z
§*· ¨ ¸ ©2¹
2
.
(12)
It is noteworthy that Fe’ changes sign with (Z0 - Z). Fe’ is out of step with respect to (Z0 -Z) and Fe '(Z|Z0) 0 . Also, it is possible to state that the function Fe’ is: x positive on the left hand side of Z0. Here Z- = Z0 - H (where H >0) , from which Z0 - Z- = H > 0 and Fe '(Z- ) > 0 ; and x negative on the right hand side of Z0 so that Z+ = Z0 +H , from which Z0 -Z+ =-H < 0 , and Fe '(Z+ ) 0 . 8.2.3.2. Expression for Fe’’ Similarly to that in the previous section, from Eq. (7), Fe’’ can be found in: * 2 Z Z * p 0 2 . (13) = Fe ''(Z|Z0 ) | Z2p 2 2 2 2 2 Z 4Z0 Z0 Z Z0 * ² 0 2 §*· Z0 Z ¨ ¸ ©2¹ For its part, Fe’’ is at a maximum when the dominator is at a minimum, that is to say when Z - Z0 = 0 , so that Z = Z0 . Thus: Fe ''(Z0 ) = Fe ''max =
Z2p 1 Z0 *
.
(14)
By substituting this expression in Eq. (13):
Fe ''(Z|Z0 ) | Fe ''(Z0 )
§*· ¨ ¸ ©2¹
Z
0
Z
2
2
§*· ¨ ¸ ©2¹
2
1
=Fe ''(Z0 ) 1
4 *2
Z0 Z
2
.
(13’)
236 Basic electromagnetism and materials Fe’’(Z
| Z0)
is a function paired with (Z0 -Z) , and therefore is symmetrical with
respect to Z = Z0 . This type of curve is termed a Lorentzian curve, and as Z is separated from Z0, (Z0 -Z)2 increases and Fe’’(Z) o 0. 8.2.3.3. Evolution of Fe’(Z) and Fe’’(Z) when Z | Z0 The functions Fe’(Z) and Fe’’(Z) when Z | Z0 evolve as shown in Figures 8.1 and 8.2. The halfpeak width (hpw) of the curve representing Fe’’(Z) gives the pulsations Z1 and Z2 such that Fe ''(Z1 ) = Fe ''(Z2 ) = Fe ''(Z0 )/2 . From Eq. (13’), pulsations Zi (where i = 1 or 2) should accord to: 1
Fe ''(Zi ) = Fe ''(Z0 ) 1
4
Z0 Zi
1
2
2
Fe ''(Z0 ) .
*2 From which the following equation can de determined: * *² (Z0 -Zi )2 , so that Zi = Z0 ± . 2 4 These equations finally give: * Z1 = Z0 2 * Z2 = Z0 + 2 The hpw thus is equal to 'Z = Z2 - Z1 = * = 1/W (15). In addition, it is possible to see that the angular frequencies Zi (Z1 and Z2) give rise to the extreme limiting values of Fe’(Z). To verify this, it suffices that ª dFe ' º « » = 0 , a relatively simple calculation. ¬ dZ ¼ Zi According to Eq. (12), the corresponding values for Fe’(Zi) are: * r Z0 Fe '(0) Z F '(0) Z W 2 Fe '(Zi ) = =r 0 e = ± 0 Fe '(0) . 2 2 2 2* 2 §*· §*· ¨ ¸ ¨ ¸ ©2¹ ©2¹ Therefore: Z Z W Fe '(Z1 ) = 0 Fe '(0)= 0 Fe '(0) 2* 2 (16) Z0 Z W Fe '(Z2 ) = Fe '(0)= - 0 Fe '(0). 2* 2
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
Fe’= Hr’-1
Z0W 2
F e, (0)
'Z =
1
W
Z0 Z1
0
-
Z0W 2
F e, (0)
Z2
Z
Z0
Figure 8.1. Plot of Fe’= f(Z) when Z | Z0 .
Fe’’
Fe’’(Z0) =
Z 2p Z0 *
Z0WF e, (0)
F e'' (Z0 ) 2
Z1
Z0
Z2
Z 'Z =1/W
Figure 8.2. Plot of Fe’’= f(Z) when Z | Z0 .
237
238 Basic electromagnetism and materials
8.2.4. Study of the curves of Fe’(Z) and Fe’’(Z) when Z z Z0 (well outside of an absorption zone) Far from an absorption zone, the oscillation can be assumed to undergo only a very weak dragging force. This small force is such that it can be assumed that * ²Z² << (Z0 ² - Z²)² . 8.2.4.1. Form of Fe’(Z) Following Eq. (6'), Fe '(Z) | Fe '(0)
Z02
Z02 Z2
.
(17)
The limiting values can be given accordingly: x when Z << Z0 , Fe '(Z)
Fe '(0)
Z02
Z
0
Z
Z
0
Z
| Fe '(0)
Z0
Z
Z
0
o F'e (0) ;
Zo 0
(18)
x when Z Z0 :
Fe '(Z)
Fe '(0)
Z02 Z2
(8)
-
Z2p Z2
o 0
(19); and
Zof
Fe '(f) | 0- (when Z o f, Fe '(Z) |
1 f
o 0- ).
8.2.4.2. Form of Fe’’(Z) According to Eq. (7), Fe ''(Z) | Z2p
Z*
Z02 Z2
2
| 0 | H r '' .
A consequence of this equation is that there is no absorption in this zone, and F and G G G G H are real such that, for example, P = H0 F E or D H E . There is no dephasing between the response and the excitation, and it is only the real parts, H(Z) or F(Z), which vary with Z. The medium gives rise to dispersion and is nonabsorbent, which in effect means that it is also transparent.
8.2.4.3. Graphical representation Figure 8.3 gives a graphical representation of Fe’(Z) far from the resonance frequency (Z0). This representation gives the electrical component, as at low frequencies it is an ionic or dipolar relaxation contribution that appears.
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
239
H'r (Z) = 1 + Fe '(Z) . As Fe '(Z o f) | 0 , then In general terms, H r '(Z o f) | 1 , and the curve representing Hr’(Z) takes on the shape shown in Figure 8.4. The analytical form for H’r(Z) at points far from the resonance value is directly obtained from Eq. (17): Z2p H'r (Z) = 1 + . (17’) Z02 Z2
Fe’(Z)
Fe’(0) |
Z 2p Z02
Z0
0
Z
Figure 8.3. Curve of F’e(Z) when Z z Z0 .
Hr’(Z)
Hr’(0) = 1 +Fe’(0) 1 0
Z0
ZA
Z
Figure 8.4. Curve of H’r(Z) when Z z Z0 .
240 Basic electromagnetism and materials
8.2.5. The (zero) pole of the dielectric function, and longitudinal electric waves An angular frequency ( ZA ), which corresponds to the (zero) pole of a dielectric function, is such that ZA is different to Z0 , must be such that the following equation is true: Z2p H'r (ZA ) = 1 + 0 , Z02 ZA2 the solution of which gives rise to: ZA2
Z2p Z02 .
This also shows that ZA > Z0 (see Figure 8.4), which by consequences means that ZA z Z0 , and justifies the use of Eq. (17') to determine ZA . In order to show that the angular frequency ZA is due to a wave that has a longitudinal structure, the following may be considered. As Z z Z0 , then H r '' | 0 , G G and H is real ( H = H ). Gauss's equation, which is written div D0 = H div E 0 0 , has in reality two solutions, which are either: GG G x div E 0 0 giving for a progressive planar wave, ik.E 0 = 0 , in other terms G G k A E 0 so that the wave presents a transversal structure; or x H (Z) = 0 which corresponds to an angular frequency value such that H (ZA ) = 0 and the solution corresponds to the (zero) pole of the dielectric G function, which is thus such that div E z 0 . In this case, it remains true that: G div B0 0 JG JJG G JJG G G wD Z G rotB µ0 iZµ0 H0 H r E , so that rotB0 = i Hr E 0 = 0 when Z ZA c² wt GG G G ik.r with B0 = Bm e (form given for a progressive sinusoidal wave), these two
equations give rise to, respectively: GG i k.B0 0 G G B0 = 0 and therefore the wave is completely electric. ikuB 0 0
The Maxwell-Faraday equation makes it possible to state that: G JJG G w B JJG G JJG G G rotE 0 , so that rotE 0 jZB0 0 . With B0 = 0 , then rotE 0 0 , and wt G G G G i k u E 0 0 . This in turn means that E 0 // k . G The wave is said to have a longitudinal structure as E is directed along the wave
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
241
G G G vector k (while for the transversal wave, E 0 A k ).
8.2.6. Behavior of a transverse plane progressive EM wave which is sinusoidal and has a pulsation between Z0 and Z A sufficiently far from Z0 so that
H’’ | 0 A monochromatic planar progressive electromagnetic (MPPEM) wave that is sinusoidal can be written as: GG G G E E m exp[i(Zt k.r)] . When Z0 < Z < ZA , Figure 8.4 indicates how H r '(Z) < 0 , while H r ''(Z) = 0 at points far from Z0. Therefore H r = H r '(Z) - j H r ''(Z) = H r '(Z) which is a real negative, and thus H r = H'r < 0 . The equation for the dispersion of transverse MPPEM waves is given by k² =
Z2
Hr =
Z2
H'r < 0 , which in turn imposes that k = - ik'' = - i
Z
n '' where k is c c c a purely imaginary number. The MPPEM wave thus takes on the form given by G G E E m exp[k '' r] exp[iZt] . In effect, there is no wave propagation; the wave is simply attenuated without undergoing absorption (as Z z Z0 ). There is no propagation for the wave in the medium (because k' = 0 ), and the wave is termed evanescent. When Z º¼Z0 , ZA ª¬ , the material can be considered a perfect reflector. 2
2
8.2.7. Study of an MPPEM wave both outside the absorption zone (Z z Z0) and the range [Z0, Z A ] 8.2.7.1. Relationship between Z and k The sections above considered an MPPEM was within the range Z º¼Z0 , ZA ª¬ . This section will look at a wave outside this range with Z << Z0 and Z > ZA . In these two domains, H r ''(Z) = 0 , and H r = H'r > 0 (see Figure 8.4, noting that when Z < Z0 , H'r > 1 and when Z > ZA , 0 < H'r < 1 ). Thus, the dispersion equation Z2
Hr =
Z2
H'r > 0 , from which k = k' = k . c2 c2 With H’r(Z) given when Z z Z0 by Eq. (17’), then
takes on the form k² =
k² =
Z2p º Z2 ª «1 ». c2 «¬ Z02 Z2 »¼
(20)
242 Basic electromagnetism and materials By developing this further, it is possible to obtain: Z4 - Z²(k²c² + Z0 ² + Zp ²) + k²c²Z02
0 .
8.2.7.2. The relation Z = f(k) The preceding equation is a “bi-squared” equation of the type x 4 + b x² + c = 0 , which has solutions in the form x² = -
Zr2
k²c² Z02 Z2p 2
b 2
r
b² 4c 2
ª§ 2 2 « k²c² Z0 Zp r «¨ ¨ 2 «¬©
, so that it can be stated that: 1/ 2
2 º · 2 ¸ k²c²Z0 » » ¸ »¼ ¹
.
(21)
8.2.7.3. Shape of the dispersion curve When Z << Z0 and Z > ZA , then k² = where H'r (Z) = 1 +
Z2 c2
H'r ,
Z2p
Z02 Z2 It is worth considering the two limiting values of k: x k | 0 according to the equation for dispersion, there are two possible solutions. These are either: o Z | 0 , which corresponds to the solution Z = Z- , [which also can be directly checked by introducing k = 0 into Eq. (21), giving a solution for Z- as Z- = 0 ]; or
o H'r (Z) | 0 , in which case when Z | ZA , such that ZA2
Z2p Z02 , the
solution for Z+ is Z+ = ZA , [which again can be found by direct substitution of k = 0 into Eq. (21)]. x
k o f where once again two solutions are possible: Z Z o f and k | o for a solution to Z+. The introduction into Eq. (21) of c c k o f so that k²c² >> Z0 ² or k²c² >> Zp ² , gives Z+ = ck ; or
o H'r (Z) o f , from which (1 +
Z2p Z02 Z2
that Z o Z0 , hence the solution to Z-
) o f , and in other words means
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
243
A representation of Z = f(k) therefore shows two branched curves associated with the transversal EM waves (T branches) corresponding to the two solutions Z+ and Z- The curves are separated by a zone, or band, into which propagation of the EM waves cannot occur. The value at which Z = ZA shown in Figure 8.5 corresponds for its part to the longitudinal electric wave (L).
Z T
Z = ck
Z+ L
ZA
zone unavailable to propagation: reflection zone
Z0 T
ZO
k Figure 8.5. Dispersion curves indicated by T.
The scheme presented below gives a résumé of the different behaviors of a dielectric undergoing electric polarizations at different frequencies (and pulsations). Absorption
Hr’ > 1
0 < Hr’ < 1
Hr’ < 0 Z0
propagation with dispersion
ZA
no propagation, evanescent wave
Z
propagation, and when Z o f, Hr’ o 1, Behavior analogous to that in a vacuum, the e- cannot follow these high frequencies
8.2.8. Equation for dispersion n = f(O0) when Z << Z0 8.2.8.1. Maxwell's equation for the visible region When Z < Z0 , then H r = H r ' > 1 , and the relation for dispersion k² = for a real Hr, k = k =
Z
Z² c²
Hr gives,
Hr . This region is where Z < Z0 is in the visible range and c therefore is nonabsorbing and transparent. More precisely, the conditions on Z can
244 Basic electromagnetism and materials be written Zi
H r (Z) . This is the Maxwell equation valid for the visible region.
written that n =
It can be simplified further by stating that with a real value for Hr, the index n defined by n² = Hr is also real, so that n = n = Hr . As here H r > 1 (region Z < Z0e ), n = the result is that, with k 0 =
Z c
2Sc
Z
c
Hr
:
O= where O 0 =
H r (Z) > 1 , and vM =
2S
2S
O0
k
k0n
n
< O0 ,
and is the wavelength associated with Z in a vacuum.
8.2.8.2. Dispersion equation for Z < Z0e and Cauchy's formula In the region Z < Z0e , H r (Z) = Hr '(Z) is given by Eq. (17’), so that with Z0 = Z0e the resulting equation is: H r (Z) = 1 +
Z2p 2 Z0e
2
Z
=1+
Z2p 2 Z0e
1
1 = 1 +Fe (0) Z² 2 Z0e
1
1 Z² 2 Z0e
ª º Z² = 1 +Fe (0) «1 ...» 2 «¬ Z0e »¼ With n² = Hr (Z) and Z
2Sc
O0
n² | 1 +Fe (0)+
it is possible to say that:
Fe (0) 2 Z0e
Z2 = 1 +Fe (0)+
Fe (0) 2 Z0e
By making A = 1 +Fe (0) = H r (0) = n 02 , and B = 2Sc
2
2Sc 2
Fe (0) 2 Z0e
1
O 02
.
, then n² = A +
B
O 02
.
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials 1/ 2
From this can be determined that n so that by making C =
B 2A1/ 2
12
A
§ B · ¨1 ¸ ¨ AO 02 ¸¹ ©
245
· § B | A1 2 ¨1 ... ¸ , 2 ¸ ¨ 2AO 0 ¹ ©
, it can be stated that
n | n0 +
C
O 02
.
(22)
Equation (22), called Cauchy's equation, shows that the index varies with the wavelength.
8.2.8.3. The Rayleigh relation and group and phase velocities dZ dQ dQ = - O² The group velocity is given by vg = . In addition, dk d(1/ O ) dO O0 =
2Sc
c
Z
Q
and O =
O0
c
n
nQ
, so therefore Q =
c nO
. The result is:
ª § 1 ·º d¨ ¸» « d(1/ nO ) 1 § 1 · 1 © n ¹» c § 1 dn · vg = - cO ² = - cO ² « ¨ ¸ = cO ¨ ¸ . « » dO n © O ² ¹ O dO n © n ² dO ¹ « » «¬ »¼ c With vM = , it is possible to obtain: n O dn · § (23) vg = vM ¨1 ¸ . n dO ¹ ©
8.2.8.4. Comment: normal and abnormal dispersions In Sections 7.2.1.3 and 7.2.1.4 normal and abnormal dispersions were defined. In dielectrics, as shown in Figure 8.4 and as a general rule, Hr’(Z) increases with Z. However, as indicated in Figure 8.1, when Z | Z0 , the function Hr’(Z) decreases as Z increases, describing a behavior in the abnormal dispersion zone. 8.3. Propagation of a MPPEM Wave in a Plasma (or the Dielectric Response of an Electronic Gas) 8.3.1. Plasma oscillations and pulsations 8.3.1.1. Definition of a plasma Overall a plasma is neutral and is made up of ions, assumed to be in fixed positions (due to their high mass which accords them considerable inertia), and by electrons,
246 Basic electromagnetism and materials assumed to be highly mobile. All of this is in a vacuum. At equilibrium (rest state), the ion volume densities (n0) and the electron volume densities (ne) are identical. However, following a Coulombic interaction, if the perturbation obliges an electron to move from its equilibrium position, then it will have a tendency to return to original position due to a returning electrical force. Below, electron and ion charges are denoted by –e and +e, respectively.
8.3.1.2. Study of the displacement of electrons in a plasma subject to a mechanical perturbation
S
u(x,t)
x
u(x+dx,t)
x
x + dx
Figure 8.6. Moving a “slice” of plasma.
Supposing that under the influence of a mechanical perturbation, for example, an acoustic effect, the electrons shown in Figure 8.6 are moved together along the Ox axis by a small distance u(x,t). The thermal agitation and weights involved are negligible, much as the frictional or mechanical recall forces, as the electrons are not tied to any specific atom as is the case for dielectrics. To start there is a “slice” of electron “fluid” which has a cross-sectional area S and is placed between x and x + dx at its rest state. The initial corresponding volume of the slice is V0 = S dx . Under the effect of a perturbation, the deformed fluid moves in such a way that its limits move to x + u(x,t) and x + dx + u(x + dx, t) . Therefore its volume also changes to : Vloss = S [dx + u(x +dx, t) - u(x, t)] . With
u(x +dx, t) = u(x) + dx
wu
, then
wx variation in volume can be considered as :
Vloss = S [dx + dx
GV = Vpert - V0 = S dx
wu
wu wx
] , and the final
. wx By denoting the concentration of the charge following perturbation by ne + Gne, the conservation of charge in the occupied volume before and after the perturbation can be described by: n e V0 = (n e + Gn e ) Vpert , or in other words, n e S dx = (n e +Gn e ) S [dx + dx
wu wx
] , from which comes
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
247
wu · wu § + Gn e ¨ 1 << 1 , we obtain: ¸ = 0 . With wx wx ¹ wx © wu Gne n e . wx With the ions assumed to be fixed, their density (n0) is unchanged and the total charge density therefore is given by U = n 0 e - (n e + Gn e )e . ne
wu
With n 0 = n e , then U= - Gn e e = n e e
G Poisson's equation ( div E G limiting condition that E
U H0
wu wx
.
) makes it possible to write
wE
n e e wu
wx
H0 wx
. With the
G 0 when u 0 , the equation yields G nee G E(x, t) u(x, t) . H0 G G This relation ties together the displacement ( u(x,t) ) of the electrons to the field ( E ) generated by the same displacement. This field is a longitudinal one, as it is parallel to the displacement x. On applying the dynamic fundamental relation to the electrons (electrons localized in the slice of the fluid) we have: G G G d²u n e e² G d²u u 0. m = - e E(x, t ) , so that dt² mH0 dt² The introduction of a pulsation to the plasma, defined by Zp =
n e e² mH0
, gives
iZ t
u=Ae p . The latter relationship indicates that the electrons undergo an oscillatory movement at the “plasma pulsation” Zp.
Numerical application x n e = 106 cm-3 = 1012 m-3 (ionosphere) Ȟp = 8.97 MHz (decimeter waves). x n e = 1015 cm 3 = 1021 m-3 (dense gas discharge) Ȟp = 1.75 x 1012 Hz (millimeter waves) x ne
1029 cm-3 (metal) Q P = 1016 Hz (UV visible)
To sum, in a plasma the mechanical perturbation that moves electrons forms a longitudinal field (which has a direction depending on the induced displacement). The electric field in turn produces an electronic returning force and therefore a
248 Basic electromagnetism and materials displacement. This can be verified by studying the behavior of a plasma subject to an electric field of a form given by E = E 0 eiZt .
8.3.2. The dielectric response of an electronic gas As before, the medium is assumed to be neutral overall and consisting of fixed ions and mobile electrons all in a vacuum. In the rest state the ion volume density (n0) and the electron volume density (ne) are identical i.e. n 0 = n e . As above, electron charges are denoted by –e. G G An electric field, acting as the recalling force, ( E E 0 eiZt ) is applied to the medium along Ox. It is assumed that there are no frictional forces involved. Now, this study treats the problem by dealing with the following questions: 1. Looking for the abscissa x, which defines the position of an electron and is a solution to the permanent regime of form x
x 0 eiZt , gives x 0 with the help of a
fundamental dynamics equation. 2. Give the expression for the polarization due to the displacement of the electrons. 3. By determining the expression for the relative dielectric permittivity [Hr(Z)], also Z2p , detail what Zp is. called the dielectric function, given in the form H r (Z) = 1 Z2 G 4. The electric field ( E ) now under consideration is that of a plane sinusoidal EM wave with pulsation denoted by Z and rectilinearly transverse polarized along Ox. It propagates toward increasing values of z and is such that GG GG G G G G G E E m ei(Zt k.r ) E m ei(Zt k.r ) u x E m ei(Zt k.z) u x = E 0 eiZt .
(a) What is the dispersion equation for transversal plane EM waves? Give this relation with the notations and results of the question. (b) When Z < Zp , state the exact form and type of the corresponding waves. (c) When Z > Zp , detail the form (progressive or not) of the waves associated with the angular frequency and plot the dispersion curve [ Z = f(ck) where k is the wave vector module and c the speed of light] for these conditions. (d) In the light of the results conclude about the plasma transparency. 2Sc 5. The value of Ȝp for alkali metals is around 300 nm ( O p = ). If it can be Zp assumed that they can be represented by the electronic gas model, are they transparent to UV light?
6. Study the (zero) pole of the dielectric function which corresponds to an angular frequency given by ZL (which is such that H(ZL ) = 0 ). Is it possible to demonstrate that the associated waves are longitudinal?
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
249
Answers d²x
eE , into which dt² eE 0 eE substituting x = x 0 eiZt gives i²Z²m x = - e E , so that x = and x 0 = . mZ² mZ²
1. According to the dynamic fundamental equation, m
A field moving along Ox given by E = E 0 eiZt results in an oscillating movement eE 0
eiZt . mZ² 2. The movement of a charge (-e) by x is the same as generating a dipole moment by p = -e x. The corresponding polarization (dipole moment per unit volume) is n e² therefore P = n e p = - n e e x = - e E . mZ² 3. The dielectric function Hr(Z) is such that D(Z) = H0 H r (Z) E(Z) = H0 E(Z) + P(Z)
given by x =
H r (Z) = 1 +
P(Z) H0 E(Z)
so that H r (Z) = 1 -
n e e² H0 mZ²
H r (Z) = 1 -
= H r (Z) . With Zp ² =
Z2p Z²
n e e² H0 m
, then:
.
4. (a) The equation for the dispersion of progressive plane EM waves with transverse G structures, as here with k following Oz, can be written as k²c² = Z²Hr (Z) . In this case, Hr is real. By taking the previous expression for Hr(Z) into the equation for dispersion, we find: Z2 · Z² § ¨1 p ¸ , k² = Z² ¸ c² ¨ © ¹ which is the equation for the dispersion of EM transversal waves in a plasma. Comment: The above relationship can be obtained more directly from the Maxwell equations, detailed as Eqs. (10), (11), and (13) in Chapter 7, as in: GG GG G G G k.E 0 0 (1) k.B0 0 (2) k u E 0 ZB0 . (3) The equations show by themselves that the EM wave is obligatorily transversal. For its part, the Maxwell-Ampere equation is written in the form: G JJJG G G G wE G rot B µ0 ( jA H0 ) , with, in this case, jA Uv being the current associated wt with the displacement of electrons influenced by the oscillating field. As U = -nee
250 Basic electromagnetism and materials
and v
dx dt
, so with x =
eE 0 mZ²
eiZt it is possible to state that v
ie
E 0 eiZt , from
mZ n e e² G i E . We can remark also this same current can be mZ G G wP , and thus with seen as the polarization current jP wt G G n e² G n e² G n e² G G P = - e E - e E 0 eiZt , we found again : jP i e E { jA . mZ² mZ² mZ JJJGG § n e² ·G Finally the Maxwell-Ampere relation gives rot B µ0 ¨ i e iH0 Z ¸ E , © mZ ¹ G G § n e² ·G from which can be determined that i(k u B0 )=µ0 ¨ i e iH0 Z ¸ E 0 . © mZ ¹ G B0 from Eq. (3) into the last equation: By substituting
G which can be obtained jA
i
k² G E0 Z
n e² § n e² ·G µ0 ¨ i e iH0 Z ¸ E 0 , from which k² = µ0 H0 Z² - µ0 H0 e , and then: mH0 © mZ ¹ k² =
Z2 · Z² § ¨1 p ¸ . Z² ¸ c² ¨ © ¹ Z2p
> 1 , and H r (Z) < 0 just as k² < 0 . k is Z2 therefore purely imaginary. Using the electrokinetic notation where k = k' - i k'' , G G G then k = -ik'' ; the wave can be written in the form E E m ei(Ȧt-kz) =E m e-k''z eiȦt . The upshot is that the wave is no longer propagating but is stationary with an angular frequency (Z). The wave resembles a evanescent wave that has an amplitude that decrease exponentially with x. Z2p (c) When Z > Zp , then 1 > , and H r (Z) > 0 and k² > 0 . k is therefore real, so Z2 G G G k = k' , and the wave has the form E=E m ei(Ȧt-kz) E m ei(Ȧt-k'z) . The wave thus is
(b)
When Z < Zp , then in turn
progressive and does not exhibit a term for absorption ( k'' = 0 ). The dispersion equation thus can be written, with the more simple k' = k , as Z =
Z2p c²k² .
The curve due to this equation is shown in Figure 8.7. The slope of the dispersion dZ curve, vg = , gives vg which is the group velocity. Note that the slope is less dk than the slope of the plot Z = ck and therefore less than the speed of light.
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
251
Z Z=
Z=ck
Z2p c²k²
Hr > 0 propagation
Z > Zp
Zp
Hr < 0 : evanescent wave
Z < Zp
ck
Figure 8.7. Dispersion curve for a plasma.
(d) To conclude, the plasma (or electron gas) acts much as a high band filter toward incident waves. As indicated in Figure 8.8, the gas is only transparent when H r (Z) > 0 ; that is to say when Z > Zp . However, when Z < Zp , a component due to attenuation by e-k’’z without a term for propagation, appears.
5. Hr(Z)
1
0 O
0.5
'drag' region Z < Zp vacuum
plasma Z>Z O < Op
1
1.5
region for propagation Z > Zp vacuum
Op Z
Z/Zp
plasma
O(µm)
Zp Z > Zp
Figure 8.8. High band filtering characteristics of a plasma.
252 Basic electromagnetism and materials Given that Z ! Zp corresponds to when O < O p , waves with lengths less than Op can propagate with k'' = 0 , that is without attenuation or in other terms absorption, through the medium under consideration. Thus the alkali metals should be transparent to radiation with wavelengths below 300 nm, and this includes UV radiation.
6. The condition H(Z = ZL ) = 0 makes it possible to determine ZL, which in turn should be such that H(ZL ) = 1 -
Z2p
= 0 and that ZL = Zp . The (zeros) poles of the Z2L dielectric function therefore are equal to the plasma frequency, that is to say the frequency at which the electron gas undergoes a longitudinal oscillation. The cutoff frequency (Z = Zp) of the transversal EM waves corresponds to the longitudinal oscillation mode of the electron gas. The electrons are subject to displacement pulsations, in a direction along that of the associated field ( E = E 0 eiZt ), that is to say in the longitudinal direction of the field and along the line of electron displacement.
8.4. Propagation of an EM Wave in a Metallic Material (Frictional Forces) A study of the complex conductivity and the dispersion of waves within a metal can be carried out by resolving the following question, which considers the velocity (v) of electrons subject to a Coulombic force generated by an applied alternating field G G given by E E 0 exp(iZt) . The question given in Sections 8.2 and 8.3 is revived here except of course the electrons are now considered to be in a metallic environment so that the volume density for electron charge is given by UA = - n e e and the electrons are subject to a m
v , where Ȟ and W are the conduction W velocity and relaxation time, respectively. It is assumed that these conduction electrons are not subject to returning forces and that their movement can be studied in one dimension.
frictional force which is in the form f t = -
1. Give the equation for the movement of each electron and from this determine that the velocity when v = v0 eiZt .
2. The material is thought of as a vacuum in which the characteristic conduction electrons are spread. In the following order, calculate:
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
253
(a) the conduction current density jA = UA v = VA E and express VA as a function of Zp ² =
n e e²
, and also as a function of the conductivity (V0) for a stationary H0 m regime, i.e., when Z = 0 ; (b) the current density associated with movement through vacuum (jD0); and (c) from (b) derive the total current density (jT) and the conductivity (V), which appears in its complex form.
3.
This
question
concerns
ZW << 1 .
Q < Q c | 100 GHz , we find that ZW d 10 (a) What form does VA take on? j
(b) Calculate the ratio
D0
j
-2
In
copper,
W | 10-14 s ,
when
<< 1 .
for copper where V0 | 6 107 :-1m-1 . What
A
conclusion can be drawn from the result?
4. The metallic medium is now considered in its entirety, and therefore the characteristic used is that of its complex relative dielectric permittivity (Hr). Give the form of the displacement current in this medium with Hr. By giving the equality between two forms of complex conductivities, each obtained by the two representations found in questions 2(c) and 4, determine Hr, which is defined within the terms of this question by the relation D = H0 Hr E . 5. Here the field under consideration is that of a monochromatic planar progressive electromagnetic (MPPEM) wave which propagates in the same sense as increasing values of z. (a) Give the relation for the dispersion of these waves. (b) For ZW >> 1 , give the expression for Hr comparable to that obtained in question 3 above. From the result, draw a conclusion about the nature of the MPPEM wave. (c) Here, ZW 1 . Calculate Hr and indicate the form of the associated wave.
Answers 1. The fundamental dynamic equation along the direction of the velocity is: dv m F=m ¦ f eE v , from which the complex terms derived are: dt W dv m m v eE dt W
254 Basic electromagnetism and materials With E in the form E = E 0 eiZt , a solution for v in the form v = v0 eiZt is required. dv By substitution into the differential equation, and with iZv , it is possible to dt m v eE , so that following division by eiZt , the equation obtain: imZv + W m v0 eE 0 . becomes imZv0 + W From this can be derived: eE 0 W v0 = . m(1 iZW)
2. (a) Ohm's law gives the conduction current associated with the volume density of electrons (ne), as in: n e e² W n e e² W jA = UA v = - n e e v = E 0 eiZt E= VA E , so that m(1 iZW) m(1 iZW) VA = By introducing Zp ² =
n e e² H0 m
n e e² W m(1 iZW)
into VA , then VA =
. H0 Z2p W 1 iZW
.
When Z = 0 (i.e., under a stationary regime), VA (Z = 0) = V0 = H0 Z2p W , so that it is possible to also state that: 1 V A =V 0 . 1 iZW
(b) The displacement current associated with a vacuum, given that in a vacuum D = H0 E 0 eiZt , is jD0 =
wD wt
= i ZH0 E .
(c) The total current density is therefore: jT = jA + jD0 = (
H0 Z2p W
§ Z2 W · p + i ZH0 )E = V E , from which V = H0 ¨ iZ ¸ . ¨ 1 iZW ¸ 1 iZW © ¹
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
255
3. (a) With VA = V0
1 1 iZW
j
(b) We have
D0
j
A
|
and with ZW << 1 , we have VA | V0 .
ZH0 E
ZH0
V0 E
V0
, so that for copper and when ZW << 1 (and for
Q < Q c = 100 GHz ), the calculation gives: jD0 ZH0 W H 1011 | | 10-5 . << 0 | 7 14 V0 W V0 W 6.10 10 jA As a consequence, as long as the frequencies are not too high, i.e., Ȟ < Ȟ c = 100 GHz , the displacement current in a vacuum is negligible with respect to the internal metal conduction current. In this frequency range, we therefore find that 1 V | V A = V0 | V0 . 1 iZW
4.
With D = H0 H r E , it is possible to directly find jD = iZH0 H r E , so here
V= iZH0 H r . Equalizing the two expressions obtained for V (in 2(c) and just above in this answer) gives: Z2p W Z2p W Z2p W(ZW i) Hr = 1 + =1=1. iZ(1 iZW) Z(ZW i) Z(1 Z² W²)
5. (a) For a transverse MPPEM wave, the dispersion equation can be written with permittivity in its complex form (Hr) as k²c² = Z² Hr (Z). (b) When ZW >> 1 , the expression for Hr (obtained in answer 4) becomes Hr = 1 -
Z2p
= H r (in its real form). Once again we find the expression previously Z2 given in answer 3 for a problem treated in Section 8.3.2 concerning a plasma wherein the electrons are assumed to undergo negligible frictional forces. The MPPEM waves therefore have the same form as those written above such that when Z > Zp they are progressive and when Z Zp they are evanescent.
256 Basic electromagnetism and materials When Z W << 1, then from the answer to question 4, Hr is given by Z2p W Hr = 1 - i . Z The permittivity is complex and the dispersion equation, k²c² = Z² Hr (Z), which also Z² H r (Z) , shows that k is complex in that k = k' - i k'' . The can be written as k² = c² wave therefore can be given in the form:
(c)
\ v ei(Zt kz) = e k '' z ei(Zt k ' z) term for attenuation
propagation term
8.5. Uncharged Magnetic Media 8.5.1. Dispersion equation in conducting magnetic media For the medium under consideration, H represents the dielectric permittivity, its notation
magnetic permeability µ z µ0 , and its conductivity is such that VA
V . With
the material being electrically neutral in its natural state, then UA = 0 and Maxwell's equations from Section 5.3 can be written as:
G div E JG div B
0
0
G JJGG wB rotE + wt
(1)
(2)
JJG G rot B
G ªG wE º µ « jA İ » wt »¼ «¬
0
(3)
G ª G wE º µ «ıE İ » wt ¼» «¬
(4)
GG G G For MPPEM waves with the form E = E m exp[i(Ȧt - kr)] , Eqs. (1), (2), and (3) respectively give: GG GG G G G k.E 0 (1’) k.B 0 (2’) k u E ȦB (3’). These relationships were derived from calculations shown in Section 7.1.4 from which the results were equally applicable to amplitudes (with index m) as complex vectors, as all that was necessary was to multiply, or divide, the two terms by GG exp[i(Ȧt - kr)] . Once again, the structure of the MPPEM wave is the same as if in a vacuum (transverse wave). Equation (4) details this more specifically, in that
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
257
G G G ªG wE º ª - i k u Bº µ « jA İ » , so that with Eq. (3’), it is possible to state that ¬ ¼ wt »¼ «¬ G G G ª G k Gº « - i k u u E » µ ª¬ ıE iȦİE º¼ . Using the formula for the double vectorial Ȧ «¬ »¼ G G G G G GG G G G G product given by a×(b×c)= ac b- (ab) c , and with k A E , from eqn (1’), then: i
k² G ı ·G § E = iµİȦ ¨ 1 + ¸ E , from which comes the dispersion equation, as in: Ȧ iȦİ ¹ © ı · § k² = µİȦ² ¨ 1 - i (5) ¸. Ȧİ ¹ ©
8.5.2. Impedance characteristics (when k is real) K K When the magnetic permeability (µ) is real, B and H are in phase. Similarly, when G G G k u E K K and also are in phase. Thus k is real, B and E are related by B = Ȧ Ȧ Ȧµ Ȧµ E= B= H , so that E = Z H with Z = . k k k E Therefore, Z = , and as E is expressed in V m-1 and H in A m-1, Z has the H dimensions V A-1. The Z corresponds to an impedance that is characteristic of the medium under study and is dependent on Z. In addition, when k, µ, and H are real, according to the preceding equation, we find that k = Z İ µ , and therefore Z also is real and has a value Z= On setting Z0 =
µ0 İ0
= 377 :
µ İ
.
(6)
(the characteristic impedance of a vacuum), the
characteristic impedance of a medium is given by Z =
Z = Z0
µr İr
.
µ İ
=
µ0
µr
İ0
İr
, so that
(7)
In general terms, the complex index is defined by k = n
Ȧ c
, so k' - i k'' =
Ȧ c
(n' - in'') .
258 Basic electromagnetism and materials With a medium assumed to be nonabsorbent, then k" = 0. Therefore, k = k' is real Ȧ , and with and n'' = 0 , so that n = n' . Under these conditions, k = n c Ȧ k = Z İ µ = Z İ0 µ 0 İ r µ r İ r µ r , it is possible to derive: n = İ r µ r . c As Z = Z0
µr İr
= Z0
µr µr İ r
, the final calculation gives: Z=
µr n
Z0 .
(8)
8.6. Problems 8.6.1. The complex forms for polarization and dielectric permittivity This question concerns a dielectric that is linear, homogeneous, and isotropic (lhi) and contains N identical molecules per unit volume. Each molecule exhibits in its periphery a free electron with a charge denoted by –e. The dielectric is subject to an electric field applied along the Ox axis, and the polarization of the medium also is studied along that axis while considering that each peripheral electron is subject to a Coulombic force. This force moves the electron from its equilibrium position, to which it tends to return due to a returning force (fr) given by f r = - m Z0 ² x , where Z0 is homogeneous and of constant angular frequency, and also due to a frictional dx force (ft) which tends to brake the movement and is given by f t = - m* , where * dt is a homogenous constant that is inverse with respect to time. The movement of the ions is assumed to be negligible given that their mass is considerably greater than that of the electrons.
1. Calculations for real positions: (a) With the electric field being given by E(t) = E 0 cos Zt , write the fundamental dynamic equation. (b) From the above equation determine the differential equation that describes the displacement in x of each electron. (c) Under a forced regime, which corresponds to the particular solution obtained from the second term of the differential equation, the solution is of the form x(t) = x 0 cos(Zt - M) . Determine x0 and M. Note that in order to carry this out, it is possible to associate complex numbers to x(t) and E(t) such that x = x 0 exp(-jM) exp(jZt) = x 0 exp(jZt) , with x 0 = x 0 exp(-jM) as the complex amplitude and E(t) = E 0 exp(jZt) .
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
259
(d) Give the formula of the dipole moment induced in each individual molecule (which are assumed to not have mutual interactions). From this derive the real polarization given by P(t). (e) Is this expression an equation of instantaneous proportionality between the response of the system [P(t)] and its excitation [E(t)]? Give a conclusion concerning the dephasing between the establishment of the polarization and the application of the electric field. 2. Calculation of the complex polarization (P) and the complex induction (D): (a) With the help of the results gained from question 1, give P along with its complex amplitude P0. (b) Recall the defining relationship between the complex susceptibility from which can be given the expression for F(Z). (c) Give the relation that brings together the complex induction and the complex electric field and which permits a definition of the complex permittivity (Hr). From this, derive the expression for Hr(Z). 3. Calculation for the real electric induction (when H = H0 H r ): (a) With H = H' - jH'' give the expression for the real part of the electric induction. (b) Show how the real electric induction gives rise to a dephasing with respect to the real electric field. Express this dephasing as a function of H’ and H’’.
Answers 1. G G (a) The fundamental dynamic equation ¦ Fapplied = m J , makes it possible to state that through Ox, FCoulomb + Fr + Ff = m d2x/dt2 so that dx d²x - e E(t) - mZ0 ²x - m* =m . dt dt² (b) The differential equation that describes the movement, with E(t) = E 0 cos Zt , d²x
+*
dx
+ Z0 ²x = -
e
E 0 cos Zt . dt² dt m (c) In order to resolve a differential equation of the second order the solution can be found by adding to a general solution a second term specific to the equation with the second term. Under a permanent regime, that is when there are forcing oscillations (produced by a Coulombic force given by cos Zt) applied over a long period of time, the movement is dominated by a specific solution given by: x(t) = x 0 cos(Zt - M) . By associating with x(t) and E(t) the complex numbers, therefore is given by
260 Basic electromagnetism and materials x = x 0 exp(-jM) exp(jZt) = x 0 exp(jZt) , where x 0 = x 0 exp(-jM) is the complex dx d²x jZx, and Z²x . amplitude, so E(t) = E 0 exp(jZt)] , and hence dt dt² The differential equation then becomes - Z²x + jZ* x + Z0 ²x = -
e m
E . By dividing
the two terms by ejZt, the result is: e - Z²x 0 + jZ* x 0 + Z0 ²x 0 = E 0 , so that m e x0 = E 0 = x 0 e-jM . 2 2 m[(Z0 Z ) jZ*] c
e
E 0 , a = (Z0 ² - Z²) , and b = Z* , it is a ib m ac ibc = u - iv = Ue jT = |z0 | e-jM . Then possible to derive z 0 = a² b² c e |z 0 | = U = u²+v² = = E 0 , from which we find a² b² m[ (Z02 Z2 )² Z²* ² ]
From
z0 = - x 0 =
, where c =
that: e
x0 = m[
(Z02
Z2 )² Z²* ² ]
E0
(in which it can be algebraically verified if E0 moves according to positive values of x, and thereby if the electrons move toward negative values of x, as detailed in the figure below);
displacement toward x < 0
of
–e +
K E x
O
G E
+
displacement of towards x > 0
–e
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
and also that tanT =
u
v
261
= tan(-M) = - tan M , so that:
M = Arc tan
u v
= Arc tan
b a
= Arc tan
Z*
Z02
Z²
.
To conclude, the movement of an oscillator is cosinusoidal and undergoes the same pulsation (Z) as the excitation force, given by FC = - e E cos Zt , the same amplitude (x0) and the same dephasing (M) with respect to E. The resonance pulsation is such that x0 is at a maximum. If * = 0 , then the oscillator is unrestricted and x0 is at a maximum when Z = Z0 .
(d) Moving a charge (q) by a distance (x) is the same as applying to the system a dipole moment given by µ = qx (as detailed in Section 2.2.2.1.). Similarly, moving an electronic charge (-e) by x is the same as applying a dipole moment µ = - e x, so that: e² µ = - e x 0 cos (Zt - M) = E 0 cos (Zt - M) . 2 m[ (Z0 Z2 )² Z²* ² ] The polarization P =
¦ n i qi Gi can be written as P = N µ = - e N x , and hence i
P=
Ne²
1
m [ (Z2 Z2 )² Z²* ² ] 0
E 0 cos (Zt - M) .
(e) With the applied field being given by cos Zt, there is no proportionality between the instantaneous polarization, given by P(t) = P0 cos (Zt-M) and the instantaneous field, given by E(t) = E 0 cos Zt . The polarization is established with a phase delay (M) with respect to the applied electric field. 2. (a) The complex polarization is: P = - N e x = - N e x 0 exp(-jM) exp(jZt) = - N e x 0 exp(jZt) = P 0 exp(jZt) , which P0 = - N e x 0 exp(-jM)= - N e x 0 P0 =
Ne²
Ne² m[(Z02
1
m [ (Z2 Z2 )² Z²* ² ] 0
Z2 ) jZ*]
E 0 = P0 e-jM , with
E 0 and M= Arc tan
Z*
Z02
Z²
.
from
262 Basic electromagnetism and materials
(b) The complex susceptibility is defined by P = H0 F E , which means that P0 exp(-jZt) = H0 F E 0 exp(jZt) = H0 F E0 exp(jZt) . From this can be immediately derived that: P0 = H0 F E 0 Ne² = E0 2 m[(Z0 Z2 ) jZ*]
(c) Here D = H0 E + P =HE
H = H0 +
So H r = 1 + F = 1 + that: H r = 1 +
P E
2
Z ) jZ*]
[(Z02 Z ) jZ*]
Ne² mH0 [(Z02
Z2 ) jZ*]
= H0 + H0 F = H0 (1 + F) =H 0 H r .
Ne² mH0 [(Z02 Z2p 2
F=
Ne²
, and by making Zp ² =
mH0
, we find
.
3. (a) With H = H' - j H'' , D = H E = (H' - jH'') E and E = E 0 exp(jZt) = E 0 cosZt + j E 0 sinZt D = H'(E 0 cosZt + j E 0 sinZt) - j H'' E 0 cosZt + H'' E 0 sinZt , and D = R(D) = (H'cos Zt + H''sin Zt) E 0 , or: D = Acos Zt + Bsin Zt , with A = İ'E0 and B = İ"E0 .
(b) D therefore is given by D = A cos Zt + B sin Zt , which can be changed to D = C cos (Zt - M) . In effect, if we make: A = C cos M B = C sin M
C = A² B² and tg M =
B A
, then
D = A cos Zt + B sin Zt = C cos M cos Zt + C sin M sin Zt = C cos (Zt - M) . Consequently, the real induction is D = C cos (Zt - M) , with C = A² B² = E 0 H ' ² H '' ² , tg M =
B
H ''
A
H'
Finally, this can be written as: D = C cos (Zt - M) = E 0 H ' ² H '' ² cos (Zt - M) , with M = Arc tan
H '' H'
.
.
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
263
8.6.2. A study of the electrical properties of a metal using the displacement law of electrons by x and the form of the induced electronic polarization (based on Section 8.4) This exercise concerns the polarization, conductivity, and optical properties of a metal assumed to consist of a collection of N fixed ions (per unit volume) with N (N = ne) free electrons, there being one free electron for each atom, of mass and charge denoted, respectively, by m and –e. All are assumed to be in a vacuum with a permittivity denoted by H0. The electronic gas is subjected to the following forces: x a Coulombic force induced by an alternating field (applied along Ox) that is given G G in its complex form as E E 0 exp(iZt) ; and G G m dx x a frictional force given by Ft . W dt The metal of choice for this problem is copper, which has a relaxation time (W) of 10-14 sec. Its conductivity is denoted by V(0) and is approximately equal to 6 x 107 ȍ-1 m-1. The plasma angular frequency (Zp) is defined by the relationship Nq² Zp 2 and typically is of the order of 1016 rad sec-1. mH0 1. Under a forcing regime, we are looking for the expression for displacement in the G G G form x x 0 exp(iZt) (complex expression). Determine x 0 . G 1 2. Give the expression for the complex polarization ( P ) as a function of . 1 iZW G G G wD 3. The complex conductivity (ı) is defined in the general equation: j VE . wt G (a) By giving D as a function notably of polarization, give the equation for V that will then be used to give the static conductivity [V(0)] which describes the conductivity when the frequency is zero and can be given as a function of N, q, W, and m. (b) Give the physical significance of the two terms that appear in the equation for ı. (c) Express V(0) and then İ r as a function of Zp².
Answers G 1. The dynamic fundamental solution given for ¦ Fapplied gives
m dx
W dt
eE
m
d²x dt²
, so that
G mJ moving along Ox
264 Basic electromagnetism and materials d²x
1 dx
e
E. dt² W dt m The solution for this differential equation, under a forcing regime where E = E 0 exp(iZt) , is required in the form x = x 0 exp(iZt) . With x iZx and
x
Z² x substituted into the differential equation, it is possible to obtain Z e - Z²x 0 + i x 0 = E 0 , following division of the two terms by exp(iZt), and m W hence : e eW x0 = E0 = E0 . Zº ª imZ[1 iZW] m « Z² i » W¼ ¬
2. The electronic polarization is such that for each individual electron (of charge –e) is displaced by x by the electric field and generates a dipole moment given by µ = - ex. Denoting the electron density by ne, the corresponding polarization is given n e² W 1 E 0 e jZW . It is interesting to note that by by P = - n e e x = e imZ [1 iZW] n e e² , it is possible to write that: introducing Zp ² = mH0 P=
Z2p H0 W iZ[1 iZW]
E 0 e jZW .
3. (a) With D = H0 E + P , it is possible to state that n e² W 1 ª º = iZ «H0 E e E » . It is possible to imZ 1 iZW ¼ wt wt wt ¬ immediately derive the relationship n e² W 1 V = iZH0 + e = V(Z) . m [1 iZW] n e² W , and finally, From this, when Z = 0 , V = V(0) = e m V(0) V = iZH0 + . [1 iZW] wE . This term can be considered (b) The first term given as iZH0 corresponds to H0 wt as being tied to the displacement current through a vacuum, as in V vide = iZH0 . The j = VE =
wD
= H0
wE
+
wP
Chapter 8. Waves in plasmas and dielectric, metallic, and magnetic materials
V(0)
wP
, meaning the current due to the volume [1 iZW] wt polarization of the material (that is to say inside the material). This conductivity is an internal conductivity V(0) Vinternal = . [1 iZW]
second term,
, corresponds to
265
(c) With D = H0 Hr E , we have j = VE =
wD wt
= H0 H r
wE wt
= iZH0 H r E , from which
V = iZH0 H r . By equalizing the preceding expression for conductivity, V = iZH0 + possible to obtain H r = 1 +
(d) With V(0) = Zp ² =
V(0)
1
iZH0 [1 iZW]
=1-
i V(0) ZH0 [1 iZW]
.
n e e² W
n e e²
m
mH0
V(0) = Zp ² H0 W , and H r = 1 - i
Z2p W Z[1 iZW]
(= 1 -
Z2p W[ZW i] Z[1 Z² W²]
).
V(0) [1 iZW]
, it is
Chapter 9
Electromagnetic Field Sources, Dipolar Radiation, and Antennae
9.1. Introduction Until now the properties of electromagnetic (EM) waves have been covered without considering the mechanisms for their production or destruction. In fact, the laws of electromagnetism and classic mechanics applied to quasipoint charges such as electrons gives rise to a theory for wave emission, the principal result of which is that when a particle undergoes an acceleration, an EM wave may be emitted. In this chapter, it will be shown how the sinusoidal oscillation of an electric dipole can yield an EM wave. Electric dipolar radiation can be obtained by either sinusoidally oscillating the distance between dipolar charges or the actual dipolar charges themselves, as long as the dipolar moment is in the form p = p0 e jZt . This configuration is used in antennae, the principal types of which are detailed below with particular attention being paid to the half-wave antenna. G A preliminary determination of V and A potentials, associated with the dipoles, is required to calculate the EM radiation field. In addition, this calculation requires the solutions to Poisson's equations, which guide the propagation of potentials, generally called “retarded potentials”. In this chapter we will limit ourselves to the study of radiation in a vacuum, while in next chapters (Chapter 10 among others) we will look at interactions between EM waves and materials. Complex notation: It is worth noting that if a signal is in the form g = G m cos(Zt - M) , the complex notation is g = R(G m e jZt e-jM ) , and that by
convention we can write more simply g = G m e jZt e-jM . In this case, the complex amplitude is G = G m e- jM , which makes it possible to state that g = G e jZt . Strictly speaking, and as mentioned in Chapter 6, it is best to write g with an underline so that the equation becomes g = G e jZt ; however, the simplified if criticized notation G G also is used. If g and Gm are the vectors g and G m , G becomes a complex vector
268
Basic electromagnetism and materials
G denoted by G , which generally has the components Gx, Gy, and Gz. We thus have for the differentiation or integration operations of g = G ejZt with respect to time, the following: dg = jȦ G e jȦt = jȦg , dt dg dG jȦt dG R = e , so by identification = jȦG (= jȦ G m e-jM ); and dt dt dt
³ gdt =
1 jȦ
1
G e jȦt
jȦ
g,
R ³ gdt=( ³ Gdt)e jȦt , and ³ Gdt =
1 jȦ
G.
9.2. The Lorentz Gauge and Retarded Potentials 9.2.1. Lorentz's gauge 9.2.1.1. Poisson's equations for potentials within an approximation of quasistationary states In a homogeneous dielectric media, and under a regime of an approximation of quasistationary states (AQSS), we have: G JJJJG G JJG G G wA (1) B rot A (2) E grad V wt G The V and A potentials are expressed as a function of the deliberately applied and acting charge ( UA ) and current ( jA ) densities, as in: G G P jA dW 1 UA dW (3) (4) V= A ³³³ ³³³ 4SH r 4S r Under an AQSS regime,
wUA wt
0 , and the equation for the conservation of charge
G G gives div jA = 0 , from which can be deduced that div A = 0 (see Section 1.4.5). In addition, the potentials follow the Poisson equations, so that: JJJGG G U (5) 'A µjA 0 (6) 'V + A 0 H
269
Chapter 9. Sources, dipolar radiation, and antennae
9.2.1.2. Under rapidly changing regimes For systems subject to rapid variations, the displacement current must be considered G G G G wD by adding in jD to the conduction current jA VE , so that he current density wt which now intervenes is: G G G j jA jD . G The V and A potentials are instantaneous potentials calculated in Eqs. (3) and (4) for a time t at a certain point (r) from charges and currents at the same t. Propagation does not intervene in these equations for the potentials. The G instantaneous potentials V and A thus appear as intermediates in the calculation of G G E and B , having brought in a term for the displacement currents rather than the G G G G G propagation. In effect, in Eq. (4), where j jA jD now takes the place of jA , A G G G wD is bound to the derivative of D (in terms of jD ) and is not an intermediate in wt G G G the simple calculation of E in Eq. (1) and D , because knowledge of A supposes G G that the magnitude of D , or rather the derivative of D , is known. In order to resolve the problem more simply, the displacement currents are ignored and other potentials are used, such as those defined and denoted below G as Vr and A r , to represent the propagation. 9.2.1.3. Lorentz's gauge
G In Maxwell's equations, div B G ( A r ), such that:
JJGG The equation rotE
G wB
wt
G 0 so that B is still derived from a potential vector G B
JJG G rotA r .
(7)
G JJG § G wA r becomes rot ¨ E ¨ wt ©
· ¸¸ ¹
0 , which indicates that
G G wA r E is derived from a scalar potential (Vr) such that: wt G JJJJG G wA r E gradVr . (8) wt G In a homogeneous media, possibly charged, the Poisson equation, div E
be written as:
UA H
, can
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Basic electromagnetism and materials
- 'Vr -
w
G (divA r )
UA
. (9) H G JJG G §G wE · In addition, Ampere's theorem given by rotB µ ¨ jA H ¸ , brings us to: ¨ wt ¹¸ © G JJG G JJGJJG G § wE · (10) rotB rotrotA r µ ¨ jA H ¸. ¨ wt ¹¸ © wt
JJG JJG With rot rot
JJJJG G grad div ' , and by bringing Eq. (8) into Eq. (10), we have: G JJJJG JJJJG wVr G GG §G w ²A r · (11) grad divA r 'A r = µ ¨ jA H grad H ¸. ¨ wt wt² ¸¹ ©
Equations (9) and (10) can be rewritten in the following form to give Poisson's equations for a rapidly varying regime: G w ²Vr UA w wV (9’) 'Vr - HP (divA r Hµ r ) wt² H wt wt G G JJJJG G GG w ²A r wV 'A r - Hµ + µ jA =grad(divA r Hµ r ) wt² wt
(11’)
G G JJG G In addition, Eq. (7), B rotA r , can define only A r to the closest gradient, as JJG G JJG G JJJJG G G rotA r rotA 'r when A 'r A r gradf . JJJJG G G wf For its part, Vr can be determined only to within , as A r = A 'r gradf wr substituted into Eq. (8) gives: G JJJJG G wA 'r JJJJG wf E grad Vr grad wt wt G JJJJG § wf · wA 'r grad ¨ Vr ¸ wt ¹ wt © G' JJJJG wA r wf = gradVr' , with Vr' Vr . wt wt G G The change of ( Vr , A r ) to ( Vr' , A 'r ) is called a gauge transformation, as in: Vr o Vr'
Vr
wf wt
Chapter 9. Sources, dipolar radiation, and antennae
271
G G G JJJJG A r o A 'r = A r +gradf .
The determination of the potentials is carried out by imposing a measuring G condition so as to find Vr and A r ; f is an arbitrary function called a gauge G G function or simply a gauge. The invariance of E and B is termed the EM field gauge invariance. The most convenient method to determine the potentials therefore is to G impose on A r and Vr a condition that simplifies Eqs. (9) and (11) by removing in both cases the terms in parentheses. This condition, G wV divA r Hµ r 0 , (12) wt wV 0 it takes the place of the is called Lorentz's gauge, noting when in addition wt G condition div A 0 for stationary regimes. Equations (9’) and (10’) therefore yield: U A 0 (13) wt² H G JJJGG G w ²A r 'A r - H µ + µ jA = 0 (14) wt² 'Vr - HP
w ²Vr
G In these equations, Vr and A r are decoupled, in contrast to Eqs. (9’) and (11’), and follow Eqs. (13) and (14) if f satisfies the wave equations discussed in Comment 3 below. Equations (13) and (14) generalize Eqs. (5) and (6) to rapidly varying regimes.
Comment
1:
We
have
H µ = H 0 µ0 H r µ r =
1 c²
H r µr ,
where
H r µr = n²
(with µr = 1 , we again find Maxwell's equation, H r = n² ), so that H µ = Equations (13) and (14) can also be written as: 1 w ²Vr UA 'Vr H v² wt² G JJJGG 1 w ²A G r 'A r + µ jA = 0 v² wt²
0
n²
1
c²
v²
.
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Basic electromagnetism and materials
Comment 2: In a vacuum without charge or current, Eqs. (13) and (14) reduce to: G JJJGG 1 w ²A 1 w ²Vr r = 0. 'Vr 0 and 'A r c² wt² c² wt² By convention, the following operators are noted thus: 1 w² '{ , called the d'Alembertian, Vr = 0 c² wt² G 1 w² '{ c² wt²
, called the d'Alembertian vector
G Ar = 0 .
Comment 3: A Lorentz condition is imposed on the gauge function, in which case, G wV we still find that divA r Hµ r = 0 . wt G ' G JJJJG wf With A r = A r +gradf and Vr' Vr we also have: wt JJJJG G G div A 'r = div A r + divgradf (a); and Hµ
wVr' wt
Hµ
wVr wt
Hµ
w ²f wt²
(b).
By adding each successive member in (a) and (b), we obtain JJJJG G G wV ' wV w ²f 0 div A 'r +Hµ r = div A r +Hµ r + div gradf Hµ wt wt wt² = 0 as the Lorentzian gauge JJJJG w ²f =0 divgradf Hµ wt² 1 w ²f R 'f =0 v² wt²
Finally, Lorentz's gauge condition requires that f equation.
satisfies a wave
Comment 4: This additional remark concerns Coulomb's gauge. If there are no charges or currents, then Eq. (9) becomes: G w 'Vr + (divA r ) 0 wt
Chapter 9. Sources, dipolar radiation, and antennae
If Vr is chosen so that 'Vr G div A r = 0 as in the stationary regime.
0
273
(Coulomb's gauge), we than have
9.2.2. Equation for the propagation of potentials and retarded potentials
r M x dW q(t)
Figure 9.1. Potential generated by a charge [q(t)] throughout dW.
Equations (13) and (14) are the equations for propagation and they can be integrated by hypothesizing that at a given instant (t), a charge [q(t)] is in an elementary volume (dW), which is assumed to be spherical given the symmetry caused by the isotopic propagation and is situated at a point (M) as described in Figure 9.1. The action of the charge depends only on the distance (r) from which its effect is studied. Within the spherical symmetry of the problem, the Laplacian of Vr is in the form: 1 w ² rVr 'Vr = , r wr² And Eq. (13) gives: 1 w ² rVr 1 w ² Vr UA 0 (13’) r wr² v² wt² H Outside of the element dW, the singularity of which can be considered to be a quasipoint, we have U = 0, and therefore can write for the outside that: w ² rVr 1 w ² rVr 1 w ² rVr 1 w ² Vr 0 =0. (15) r wr² v² wt² wr² v² wt² Equation (15) is for propagation, and the function rVr at t and r has a solution in the form: r r rVr (r, t) G(t - ) F (t ) . v v As the source is at M and it is only waves emitted toward r > 0 that have physical r solutions which are in the form G( t - ) , we therefore find: v r G(t ) v . (16) Vr (r, t) r
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Basic electromagnetism and materials
In order to determine G, a limiting scenario might be used where r o 0 and in 1 1 w ² rVr , which is such that: Eq. (13) we find that becomes very large, just as r r wr² 1 w ² rVr 1 w ² Vr >> . r wr² v² wt² 1 w ² rVr UA + 0 , so that : Equation (13') thus tends toward r wr² H U 'Vr + A = 0 . (13’’) H Equation (13') therefore tends toward the more usual Poisson equation, which has a solution that is for r o0 (point charge at M): q(t) q(t) Vr (r,t) = , so that also, [r Vr (r,t)]r o 0 = . 4SHr 4SH From this can be determined that ª § r ·º q(t) ½ [r Vr (r, t)]r o 0 = q(t) ° «G ¨ t ¸ » . v ¹¼ r o0 4SH ¾ G(t) = ¬ © 4SH ° = G(t) ¿ And therefore, from Eq. (16), we find: r q(t ) v . Vr (r,t) = 4SHr
(17)
If in a volume (V) there is a distribution of charges q(t), then there is an accumulation of potentials from the elementary charges such that q(t) = U(t) dW and the general expression for the potential at t is: r· § U¨ t ¸ 1 v © ¹ dW . (18) Vr (r,t) = ³³³ 4SH r In addition, Eq. (14) yields potential vector components similar to those of Eq. (13), with the condition that 1/H is replaced by µ. The components of the potential vector thus have solutions of the type given by Eq. (18) in which 1/H must be replaced by µ. These solutions can be brought together in a single equation: G § r· j ¨t ¸ G µ v¹ © (19) A r (r,t) = dW . ³³³ 4S r
Chapter 9. Sources, dipolar radiation, and antennae
275
For a distribution of rectilinear currents, we have: r· § I¨ t ¸ G µ v © ¹ dlG . (19’) A r (r,t) = ³ 4S r To conclude, we thus obtain as a function of the active charges and currents at an G r r instant (t - ) the potentials Vr and A r for t and r with a delay of , which takes v v G into account the duration of the propagation. The potentials Vr and A r are delayed G G potentials and substituted into Eqs. (7) and (8) yield E and B . 9.3. Dipole Field at a Great Distance G 9.3.1. Expression for the potential vector A generated by a domain D 9.3.1.1. General formula dWi
ri
Mi
qi O
P r
G u
domain D JJJG
Figure 9.2. Distribution of charges (qi) placed around O by OM i
G Gi .
Given an element, with a volume (dWi) and an associated charge (qi) such that qi = U dWi , which is within a domain (D) that has a distribution of qi charges placed JJJJG around O by OMi , we can associate with D a nonzero dipole moment given by JJJJG G G G p ¦ qi Gi z 0 where Gi OMi . i
G Of interest is the form of the potential vector A at a point (P) located with respect to the origin (O) at the heart of D and the distribution of charges, by JJJG G G OP r ru . Given Eq. (19) and that the system is in a vacuum, with ri = Mi P , its possible to write that: G § r · ji ¨ t i ¸ G G µ0 c¹ © A r (r,t) =A(P,t) = dWi . ³³³ 4S ri
G It is possible to state that if vi (t) is the velocity of qi associated with dWi in D, G G G G G G with ji Uvi , we have ji dWi Uvi dWi vi UdWi vi qi , and can then write:
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Basic electromagnetism and materials
MP G qi vi (t i ) G µ0 c , A(P,t) = ¦ 4S i Mi P
(20)
where the summation is overall the charges within D. In order to calculate the electromagnetic field that comes from this potential vector, the space can be divided into three: x a zone defined by r << O , inside which the field can be thought of as quasistationary; x an intermediate zone where r | O ; and x an external zone for which r >> O and where the radiation is the concern of this study. G 9.3.1.2. The form of A in the radiation zone In the outer zone, it can be assumed that ri = MiP | r, as all the points in D are, in practical terms, a distance r from P. From this, Eq. (20) can be rewritten so that G µ r G A(P,t) = 0 ¦ qi vi (t ) . (21) 4Sr i c JJJJJG dOMi G G G G With vi , we have ¦ qi vi (t) p(t) , where p(t) is the derivative with dt i respect to time of the resultant dipole moment due to all charges in D. This finally gives: r G ) p(t G µ0 c A(P,t) = . (21’) 4S r G It is interesting to note that the form of A therefore corresponds to a spherical wave, which at a great distance from D would be considered to be a source generating a plane wave (see Section 6.2.2.3). Indeed, at a great distance from the source, the surface of such a sphere can be associated with a tangential plane wave. With r dW W = t - , and as 1 , it can also be noted that: c dt G dp(W) d G dW d G = p(W) p(W) . dt dW dt dW
The derivative therefore can be taken indifferently in respect of t or W.
Chapter 9. Sources, dipolar radiation, and antennae
277
9.3.2. Expression for the electromagnetic field in the radiation zone (r >>O) G JJG G r G G G ) = p( W) = p W (which in effect With B rotA , and placing by notation, p(t c means placing into lower magnitudes with respect to the index W): r º ª G p(t ) » JJG G JJG pG W G µ0 JJG « c » µ0 rot B(P, t) rot A(P, t)= rot « . 4S r r « » 4S «¬ »¼ JJG G JJG G JJJJG G By using rot(aA) arotA grada u A , it is possible to write: G µ0 JJGG µ JJJJG 1 G B rotp W + 0 grad u p W , the second term of which brings in 4Sr 4S r G G JJJJG 1 1 r u grad , a term that varies with respect to and at a great distance 3 r r² r² r 1 becomes negligible with respect to the first term which only varies as . Therefore: r G µ0 JJGG B| rotp W . (22) 4Sr G JJGG u G w² K G p W , where A quite involved calculation then gives rotp W =- u p p(W) . In c wW² order to get this result, one can verify that: x on the one hand, we have JJG d wp (W) d ª wp y (W) º ª rotpG W º = ª« z º» ¬ ¼ x dy ¬ wW ¼ dz «« wW »» ¬ ¼ G JJJJG ²p w w ²p z wW w ²p(W) º ª y wW «gradW u » wW² wy wW² wz ¬ wW² ¼ x so that by reproducing the calculation in three dimensions, we find: G JJGG JJJJG w ²p( W) rotp W gradW u ; and wW² G G JJJJG r u x on the other hand, gradW . This result can be obtained by calculating, cr c JJJJG wW 1 w x for example, that [gradW]x = (x² y² z²)1/ 2 , from which we wx c wx cr have: G JJJJG G G 1 G r gradW [xi yj zk] . cr cr
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Basic electromagnetism and materials
JJGG On taking the result rotp W
G B
u c
G p W into Eq. (22), we finally reach: u
G G µ0 p(W) u u 4Sc
r
.
(23)
G In the plane wave approximation, for an electric field, E G E
µ0 1 G G G ª p(W) u u º u u . ¬ ¼ 4S r
G G cB u u , so that :
(24)
G To conclude, if p(W)= 0 , the Eqs. (23) and (24) show that the radiated magnetic field is zero. By consequence, only accelerated charges radiate, as G dv G G p ¦ qi i and the wave corresponding to p z 0 is termed the “acceleration dt i wave”.
9.3.3. Power radiated by a dipole G In vacuum, the Poynting vector is defined as : S
G G EuB µ0
(Chapter 7, Section 7.3.). G G G cB u u , S can be written
G Staying with the planar wave approximation, so that E as : G cB² G S u. µ0
By taking Eq. (23) and substituting it into this equation, for an angle (T) between G G p(W) and u as shown in Figure 9.3(a), we find: G S
µ0
G p 2 (W)
16S²c
r²
sin ²T
G u.
(25)
If T = 0 , then in other words if the radiation intensity is observed in the direction G of p it would be found to be zero. However, the radiation maximum is found when T =
S
2
.
Chapter 9. Sources, dipolar radiation, and antennae
G p
T=0
P r
T
G u
T=-
(a)
279
G S
S
T=+
2
(b)
S 2
Figure 9.3. Definition of T (a) and the variation of S with T (b).
The variation in S as a function of T is represented in Figure 9.3.b, and it G S can be seen that |S| is at a maximum when T = ± , and S = 0 when T = 0 . 2 G S G d6 n O
G
Figure 9.4. Calculation of the flux ( S ) through a sphere (6) with center O.
G The power radiated through a surface can be calculated from the flux of S through the total surface (see Section 7.3.1.2). For the spherical surface under G consideration, the total radiation is given by a calculation of the flux S across the sphere (6) around a center O of radius r, as detailed in Figure 9.4. G G As S // n.d²6 ), we have P = w ³³ S.d²6 . With d²6 = r²sinT dT dM , so that also 6
d6 = 2S r² sinT dT , we find that T S
P = ³ S 2Sr² sin T dT= T 0
T S
With: ³ sin 3 T dT T 0
cos T 1
³
sin 2 T d(cos T)
cos T 1
P=
µ0 G 2 p (W) 8Sc
µ0 G 2 p (W) 6Sc
4 3
T S T 0
, we obtain the Larmor equation:
2 G 2 p (W) . 4SH0 3c3 1
3 ³ sin T dT .
(26)
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Basic electromagnetism and materials
9.4. Antennas 9.4.1. Principle: a short antenna where A << O 9.4.1.1. Oscillating charge (or current) and oscillating dipole equivalency: Hertz's dipole G G er { u
z P G r
N
G eT
T O
x
y M
Figure 9.5. Positions taken up by an oscillating charge.
A charge (Qm) is moved from a point (M) toward N, as shown in Figure 9.5. This movement also means that in going from its initial position, through the middle point (O), there is a dipole moment represented by: JJJJG G G p m = Q m MN Qm sm (- Qm at M and Qm at N). If we now impose upon Qm an oscillating elongation that is harmonic and linear G G G represented by s = sm cosȦt = sm exp(jȦt) (by convention the notation Re is omitted in front of the complex term), then in a manner similar to the generation of a dipole with an instantaneous moment, it can be considered that: G G G G p(t) = Qm s = Qm sm cosȦt = Q m sm exp(jȦt) , (27) G G G G where p(t) = Qm sm cosȦt = p m cosȦt = p m exp(jȦt) is the sinusoidal electric dipole moment created by charge oscillation. This dipole is called Hertz's dipole, and given Eq. (27) describing its moment, it can also be seen as a two oscillating JJJJG G charges, placed at M and N such that MN = sm , and with an instantaneous value given by: Q Q m cos Ȧt = Q m exp(jȦt) . It also is possible therefore to think that between the two charges at M and N there is a alternating current, as shown in Figure 9.6, and of a value given by: dQ (28) I= = jȦQm exp(jȦt) = I m exp(jȦt) , dt where Im = jȦ Qm , and with j = exp[jS/2] the dephasing of S/2 is brought in.
Chapter 9. Sources, dipolar radiation, and antennae
281
Q Qm N
t
Q
I
I t
sm Q
M
-Q
t - Qm
Figure 9.6. Variations in charges and current with time (t).
Dividing Eq. (28) by (27), it is possible to form a relation between p(t) and I(t) , as in s js p(t) = m I(t) = - m I(t) . (29) jȦ Ȧ 9.4.1.2. Practical elements of a Hertzian dipole antenna The left-hand side of Figure 9.6 means that in practical terms an antenna can be A sm represented by two wires of length connected by a coaxial cable, as in 2 2 Figure 9.7, so that the resultant current is always zero. The antenna is traversed by a sinusoidal current given by I = Im exp(jZt). Q A /2 = sm/2
I Oscillator
A
A /2 = sm/2
I -Q
Figure 9.7. Hertzian dipole.
282
Basic electromagnetism and materials
9.4.1.3. Radiation field due to a Hertzian dipole Given that the two charges are separated by a small distance ( A ), seen from P in the radiation zone, they can be assumed equivalent to a point charge. Evidently, Figure 9.5 is not to scale as in reality MN= A is much greater than OP = r. The two charges are in effect the same distance (r) from P and the wave is pretty much a plane wave at P. These assumptions were used in Section 9.3 to perform the calculations, G leading in particular to an expression for E in Eq. (24). G G G From Figures 9.5 and 9.7, p(t) = Q m sm exp(jȦt) = Qm A exp(jȦt) ez , so we have G G G p(W) = - Z²p(W) = - Z²p(W) ez . (30) JJJG OP G G In terms of spherical coordinates, and with u { er = OP 9.5), the double vectorial product given in Eq. (24) is such that:
§ cos T · § 1 · ½ § 1 · °¨ ¸ ¨ ¸° ¨ ¸ ®¨ sin T ¸ u ¨ 0 ¸ ¾ u ¨ 0 ¸ °¨ 0 ¸ ¨ 0 ¸° ¨ 0 ¸ ¹ © ¹¿ © ¹ ¯©
G G G > ez u er @ u er
§ 0 · §1 · ¨ ¸ ¨ ¸ ¨ 0 ¸ u ¨ 0¸ ¨ sin T ¸ ¨ 0 ¸ © ¹ © ¹
G r
r
§0 · ¨ ¸ ¨ sin T ¸ ¨0 ¸ © ¹
(from Figure
G sin T eT
From this can be deduced that: G E
µ0 1 G G G ª p(W) u u º u u ¬ ¼ 4S r
so that in addition, with Z² G E
k² 4SH0 r
k²c²
k²
H0µ0
G p(W) sin T eT
µ0 Z² 4Sr
G p(W)sin T eT ,
, and k
S H0 O ²r
2S O
(31)
, we find:
G p(W) sin T eT .
(32)
It is possible to see that a field radiated by a Hertzian dipole has only one G component with respect to eș , is inversely proportional to r, but also depends on r and t through the term p(W). 9.4.1.4. Radiation power of a Hertzian dipole when A = sm << O In supposing that A = sm << O, for any given moment the intensity (I) can be assumed constant over the entire length ( A ) of the dipole, which is itself considered in practical terms much as a single point from the external position (P) situated
Chapter 9. Sources, dipolar radiation, and antennae
283
outside the radiation zone, so that r >> O. Larmor's formula given in Eq. (26), obtained by removing p from the integral, therefore can be applied. Thus we find that with Q = Qm cosZt , and that I =
dQ dt
= - Z Qm sinZt = Im sinZt, then
p = Qm A cosȦt , and p= - Q m AȦ2 cos Zt , from which p²= Q 2m A 2 Ȧ4 cos 2 Zt and we now have: P =
µ0 6ʌc
p 2 (IJ)
µ0 Q2m A 2 Ȧ4
cos 2 Zt .
6ʌc
By taking I²m = Z²Q²m and given that
µ0 A 2 Ȧ 2 12Sc
1
I 2m
2
R ray I2m .
(33)
This relationship gives a definition of Rray, the so-called radiation resistance, which has resistance as a dimension and is directly obtained from: µ Ȧ2 R ray = 0 A 2 6Sc
With µ0 c = µ0
1 İ 0µ0
=
R ray =
µ0 İ0
2
2ʌµ0 c § A · ¨ ¸ . 3 ©O¹
= Z0 = 378 : , we can also write that:
2
2
§A· §A· Z0 ¨ ¸ | 780 ¨ ¸ . 3 ©O¹ ©O¹
2ʌ
(34)
(35)
9.4.2. General remarks on various antennae: half-wave and “whip” antennae Antennae can be seen everywhere in modern day life. They can be found in portable telephones, televisions, cars, and automatic doors, just to give a few examples. According to Eq. (35), short antennae radiate poorly, and it is for this reason that antennae with sizes of the order of the wavelength are preferred. The most basic versions are still made up of a simple dipole, as shown in Figure 9.8a. When the antenna is emitting, it is powered by an oscillator and parallel wire cables as detailed above (Section 9.4.1.2). If the antenna has a length O/2 and establishes a stationary current so that there are nodes at the two ends and an antiO node at the origin in the middle ( z r ), then the current has the form: 4 I(t) = Im cos kz sin Zt.
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Basic electromagnetism and materials
Each part along z with a length dz thus radiates as if it were a small dipole. This concept is used in a problem at the end of the chapter. When the antenna is in the receptive mode, the electric field around it induces a current (see more precisely Section 12.3.3.3), which propagates through the cable up to the receiver.
(a)
O O
(b)
4
2
Figure 9.8. (a) Half-wave antenna; and (b) quarter wave or “whip” antenna.
One of the more common types of antennae are the “whip” antennae often seen on cars. They are electrical monopoles, as shown in Figure 7.8a, and form an electrical image in the plane of the conductor so that a positive charge that moves along the antenna toward the top produces a “reflected” image negative charge that moves symmetrically toward the bottom. Typically, the length of an antenna is O/4, but its actual length needs to be determined quite precisely before determining its exact radiation diagram. When its length is equal to O/4, the quarter-wave length antenna acts in effect as a half-wave antenna as it can generate its symmetrically opposite image in the plane of the conductor. There are many other types of antennae. To cite just a few: x “disk” antennae formed from microstrips that are applied for example onto airplane wings; x frame antennae, also called a magnetic dipole antenna, based on a ferrite iron core, which concentrates the magnetic field; x horn antennae (used with wave guides); and x parabolic antennae that concentrate waves at a focal point where a horn antenna is placed.
Chapter 9. Sources, dipolar radiation, and antennae
285
9.5. Problem Radiation from a half-wave antenna For a half-wave antenna, that is with a length ( A ) such that A = O/2, and ignoring any energy losses such as the radiation, it can be assumed that it carries a current (I) which can be written as: 2S I Im cos kz exp jZt Im cos z exp jZt . O Assuming that the radiation is isotropic, it is independent of the angular coordinate (M). This problem concerns the radiation zone where the radius or distance from the antenna (r) is such that r >> O. 1. Determine the expression for the electric field at a point (P), which is such that P { P(r,ș) , i.e., has spherical coordinates (r,T) and M = 0 in the plane of the paper. K K 2. Assuming that when r >> O, the vectors E and H are orthogonal as in a plane wave and are related in a vacuum by
E
c . Given that
B
G calculate H . 3. Calculate the average Poynting vector.
E
µ0
H
H0
Z0 | 378 : ,
Answers 1. z A
P(r,T) r’
O/4 z
M
T’ dz T
I(z)
O
Im cos
I A
O/4
r 2S
O
z
O 2
B It can be assumed that T’ | T at P in the radiation zone where r >> O | A . The actual dipole can be considered to be a resultant of many small dipoles of a length dz centered on a point M such that MP = r’ = r – z cos T, as schematized in the diagram above.
Basic electromagnetism and materials
286
An elementary Hertzian dipole with a moment denoted by dp radiates, as G detailed in Section 9.4.1.3, an elementary electric field dE that follows the form: G µ Ȧ² G dE = - 0 dp(W) sinș eș , where r' = MP, [see Eq. (31)]. 4ʌ r' From Eq. (29), we can also write that: G µ0 jȦ I(IJ) G dE = dz sinș eș , where sm = dz, so that in this problem, 4ʌ r' § ª r 'º· § r' · I(W) = I ¨ t- ¸ Im cos kz exp ¨ jZ « t » ¸ . c ¼¹ © c¹ © ¬ As r >> O, and r >> A , the denominator r' can be replaced by r. This cannot be said to be true for the numerator as the exponential phase varies rapidly with r'. So, integrating over the length of the antenna and using r’ = r – z cos T to clean up, we end up with: O / 4 G § ª µ0 r º· jZz cos T G E jZIm exp ¨ jZ « t » ¸ sin T ³ cos kz exp dz eT . 4Sr c c ¼¹ © ¬ O / 4 By using cos kz
G E
µ 0 jZI m 8Sr
eikz eikz 2
and
Z
2S
c
O
k , we obtain:
§ ª r º· exp ¨ jZ « t » ¸ sin T c¼¹ © ¬ O / 4
u ³
O / 4
^exp jkz >cos T 1@ exp jkz >cos T 1@ ` dz eGT
The integral gives: O / 4
³
O / 4
^exp jkz >cos T 1@ exp jkz >cos T 1@ ` dz
2S O 2S O 2S O j 2 S O > cos T1@ j j j >cos T1@ >cos T1@ >cos T1@ ½° 1 °° e O 4 e O 4 e O 4 e O 4 ° = ® ¾ jk ° cos T 1 cos T 1 cos T 1 cos T 1 ° °¯ °¿ S ½ S ½½ ° 2 jsin ® > cos T 1@¾ 2 jsin ® > cos T 1@¾ ° 1 ° ¯2 ¿ ¯2 ¿° ® ¾ jk ° cos T 1 cos T 1 ° °¯ °¿
Chapter 9. Sources, dipolar radiation, and antennae
With
µ0 jZIm 2 8ʌr
k
=
µ0 jZI m c Ȧ
4ʌr
=
jIm 4ʌİ 0 cr
287
, we obtain:
S ½ S ½½ ° sin ® > cos T 1@¾ sin ® > cos T 1@¾ ° ° ¯2 2 ¿ ¯ ¿ ° eG sin T exp jZW ® ¾ T 4ʌİ 0 cr cos T 1 cos T 1 ° ° °¯ ¿°
G E
jI m
As: S ½ sin ® > cos T 1@¾ ¯2 ¿ S ½ sin ® > cos T 1@¾ 2 ¯ ¿ G we have E
§S · cos ¨ cos T ¸ ©2 ¹ S § · cos ¨ cos T ¸ 2 © ¹
° 2 §S · ½° G sin T exp jZW ® cos ¨ cos T ¸ ¾ eT so that finally: 4ʌİ 0 cr °¯ sin ²T ©2 ¹ ¿° jI m
§S · cos ¨ cos T ¸ j ©2 ¹ I exp jZW eG . T m 2ʌİ 0 cr sin T
G E
It is worth noting that this expression is indeterminate when sin T = 0, i.e., when T = 0 or T = S. 2. Assuming that at great distance the wave still assumes a plane wave structure, the G G G G G G vector H will be orthogonal to E . So with the trihedral E, H, er where er indicates G G the direction and sense of propagation, H is collinear with eM . Therefore with H
E H0 / µ0 we obtain
G H
§S · cos ¨ cos T ¸ ©2 ¹ I exp jZW eG . M m 2ʌr sin T j
3. By applying Eq. (33) of Section 7.3.3, we can write that
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Basic electromagnetism and materials
G S
1 2
G G Re E u H* where B = µ0 H,
from which can be obtained
G S
§S · cos 2 ¨ cos T ¸ 2 2 © ¹ Im eG u eG T M 2 2 2 2Sr H0 c sin T §S · cos 2 ¨ cos T ¸ 2 1 1 ©2 ¹ Ieff eG = r 2 4SH0 Sc sin T r2 1
§S · cos 2 ¨ cos T ¸ 2 © ¹ 9.5 sin 2 T
2 Ieff G e (S.I.), that is in W/m². 2 r r
Integration over the sphere of radius r gives the total radiated power, much as the same method used to obtain the Larmor relation given in Section 9.3.3. The 2 result that can be obtained from the ensuing numerical calculation is P | 73 Ieff and gives a final radiation resistance for the half-wave antenna of the order of 73 :.
Chapter 10
Interactions between Materials and Electromagnetic Waves, and Diffusion and Absorption Processes
10.1. Introduction Having looked at the way in which electromagnetic (EM) waves are emitted by a particle accelerating in a vacuum in Chapter 9, this chapter is concerned with how EM waves interact with materials, and in particular: x the form of wave emitted by a material subject to an incident EM wave (Rayleigh diffusion); x how EM waves are formed by charged particles interacting with a material as the so-called Rutherford or Bremsstrahlung (German for “braking radiation”) radiation; and x the absorption and emission phenomena of EM waves interacting with excited (more exactly “perturbed” as detailed below by the intervention of Hamiltonian perturbation) materials. The description used will be semiclassical, or possibly semiquantic, depending on whether the glass is half full or half empty with regards to a knowledge of quantum mechanics! That is to say that EM radiation can be written in a classic form while the material can be understood in quantic terms generally known by readers. The quantic terms used however, generally are for students following their first university degree course, and although Section 10.4 details in the simplest terms this semiquantic theory, it can be skipped over during an initial read. The essential results have been set out in Section 10.5 and these can be compared in a second read with the results given in chapter 3 of the volume, “Applied Electromagnetism and Materials” (notably the last Figure of that chapter). 10 .2. Diffusion Mechanisms 10.2.1. Rayleigh diffusion: radiation diffused by charged particles An electron with a charge denoted by q is subject to a monochromatic planar polarized electromagnetic (MPPEM) wave which is polarized along Ox (complex
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Basic electromagnetism and materials
JJJG G amplitude of the incident wave given by Einc //Ox ) and propagates along Oz, as G G shown in Figure 10.1. The electric wave can be described by E Einc exp(jZt) .
x G E inc
P
T
z
q
G Binc
M y
Figure 10.1. MPPEM wave incident on an electron with charge denoted by q.
If the electron is within a dense material, following its displacement along Ox by the electric field of the incident wave, it can be assumed that it will be subject to a returning force in the direction of its equilibrium position and in the form f r = - k x while simultaneously being subject to frictional forces of the form f t = - f
dx
. dt Rigorously speaking, the interactive force between the EM wave and the velocity G G G G G ( v ) of the electron is given by Fem q(E v u B) E
E
< E assuming that v c , so that the c c electromagnetic force can be reduced simply to Coulomb's force, as in :
With B
, we have vB
v
G G Fem | FC G The fundamental dynamic equation, ¦ F qEinc e jZt - k x - f
G qE . G mJ , given in terms of Ox thus gives: dx dt
=m
d²x dt²
The solution for a steady state can be looked for in the form x = x 0 e jZt , and the placing of this into the preceding fundamental equation gives: qEinc x e jZt x 0 e jZt . k mZ² jZf
The result is that q displaced by x generates a dipole moment:
Chapter 10. Diffusion and absorption processes
291
p(t) = q x(t) = q x 0 e jZt and radiates according to the result of Section 9.3.2 concerning an EM field such G G G G G p(W)= q x(W)= q J , so that that E and B are expressed as a function of
p(W)= -Z²q x = - Z²q x 0 e jZW . The power radiated and given by Eq. (26) of Section 1.3.3 is proportional to G2 p (W) , and thus p(W)= - Z²q x 0 e jZW . Two simple examples can now be considered.
10.2.1.1.
Diffusion by bound electrons (valence electrons of the atmospheric molecules O2 and N2) The electrons are well bound to the molecules, and at the level of the forces involved, the constants introduced are such that k >> mZ² and k >> Zf . The result is that the dynamic fundamental equation, as in qEinc x= e jZt , k mZ² jZf qEinc jZt qE e , so that x 0 = inc , where x 0 thus appears is reduced to x | k k p(W)= - Z²q x 0 e jZW , we have: independent of Z . With G p 2 (W) v Z4 . G p 2 (W) and is in the form P v Z4 , that is The radiated power is proportional to the power radiates to the fourth power of the angular frequency (Z).
sun at zenith average yellow radiation
(a) maximum diffusion is blue sky (b) radiation with reduced blue appears more orange Earth
Figure 10.2. Observations made when the sun is at its zenith.
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Basic electromagnetism and materials
Given that it is radiation with a high angular frequency that dominates: x When we look at the sky, but not at the sun, the sky appears blue as shown in Figure 10.2 a. The power radiated, or rather diffused, by electrons in the molecules that make up the atmosphere is at a maximum for the highest angular frequencies, i.e., those that correspond to the color blue in the incident optical radiation from the sun. x When we look at the sun when at its zenith (an observation worth avoiding due to possible eye damage!) as shown in Figure 10.2 b, the sun appears on earth more orange than the yellow observed by a satellite outside of earth's atmosphere. This is because the transmitted light is equal to the incident light minus light lost to diffusion during its passage through the atmosphere. In effect, the light we see has an “impoverished” blue region, especially if the sky is cloudy, resulting in an apparent color shift toward red. x When we see the setting sun, as shown in Figure 10.3, the observed waves have had to travel through a long distance in the atmosphere, much longer than that when the sun is at its zenith, with the result that there is an greater impoverishment of blue light diffused all along the light's pathway. This makes the light look red as all other light has been diffused.
setting sun long pathway resulting in considerable diffusion of blue light sky
radiation in red due to loss of blue earth
Figure 10.3. Observation of the setting sun.
10.2.1.2. Diffusion by free electrons This section looks at the study of free electrons using, for example, Laue type diffusions in crystalline solids. Free electrons, which are little or even not at all bound to atoms, are not subject to returning forces toward a given atomic nucleus. Thus, k = 0 and the frictional forces are also negligible, so that f = 0 . qEinc e jZt thus is The solution to the differential equation, x = k mZ² jZf
Chapter 10. Diffusion and absorption processes
reduced to x =
qEinc mZ²
e jZt ; p(W)= - Z²q x 0 e jZW then gives q²
p(W)
m
Einc e jZW .
In addition, according to Eq. (24) of Section 9.3.2, we have:
G E
µ0 1 G G G ª p(W) u u º u u ¬ ¼ 4S r
G G and as in Figure 10.4, we have p // ex .
x
M G G G G G puu
puu uu G eT T G p
G u
z y Figure 10.4. Orientation of the vectors used in the text.
G E
G For a point (M) located by the direction of a vector ( u ), it can be written that G EeT , so that in terms of moduli, E = E T .
S G G G G ª º ª º ¬ p(W) u u ¼ sin 2 ¬ p(W) u u ¼ µ µ0 q² we have E T = 0 p sin T Einc sin T e jZW . 4Sr 4Sr m r With W = t - , and F = m J = q Einc , so that J c As
G G G ª º ¬ p(W) u u ¼ u u
ET =
qJ
sin T
4SH0 c²
r
jZ
e
p sin T
qEinc m
, finally we have
r c e jZt
= E 0 T e j Zt .
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Basic electromagnetism and materials
294
The wave appears to be well proportional to the acceleration denoted by J ; hence the term “accelerating wave”.
10.2.2. Radiation due to Rutherford diffusion 10.2.2.1. A general description of Rutherford diffusion Here the incident particle is an electron denoted –e with a mass given by m that is a great distance from a proton that is assumed to be fixed and at a reference origin (O). The trajectory of the electron is considered rectilinear with a velocity vf . If there are no forces acting on the incident electron, then it will have a trajectory given by the straight line (D), as shown in Figure 10.5, which passes at a minimum distance (b) from the target (b is called the impact parameter).
D vf
b
R
T
U
O
Figure 10.5. Trajectory and the Rutherford diffusion.
In reality, when an electron nears a proton, the Coulombic interactive force is no G longer negligible. The electron is subject to a Coulombic force ( f ) that is directed G along the line between the proton and electron distance (R). If V denotes the G G electron's kinetic moment, the theory of kinetic moment is written (with f and R being collinear) as G dV G G Ruf 0. dt G G G The result is that V R u mv is a first integral of the movement where: G x the direction of V is fixed and the movement is thus plane; and G x the modulus (ı) of the vector V is a constant.
Chapter 10. Diffusion and absorption processes
295
With the electron being located in the plane of its movement by the polar angle T, it is possible to state that: V
mRv
mR²
dT
, dt from which the surface law can be deduced, as in: dT dS V R² 2 C . dt dt µ By denoting the potential energy of an electron by Ep(R) and its total energy by E, the application of the law of energy conservation results in the equation: 2 V² ª E E p (R) º mR² ¬ ¼ dT dR V m m ²R² r r dR d T V 2 V² ª E E p (R) º mr² ¼ m²R² m¬ dR
changes sign and R abruptly changes from a decreasing to an dT increasing region, or the inverse, when the numerator of the last equation cancels out so that for a value U for R is such that: V² E p (U) E . 2mU² The derivative
In addition, by taking the origin of the potential energies at infinity, it also is possible to write for the two constants for the movement that the kinetic moment (ı) and the energy (E) that 1 2 V mbvf and E , mvf 2 from which can be deduced for E p (U) that 2 § mvf b² · ¨1 ¸¸ . ¨ 2 © U² ¹ Here the interaction potential [ E p (U) ] is negative and the equation shows that U < b.
E p (U)
It is equally feasible to introduce a parameter (G) that corresponds to a distance between the two particles at which the Coulombic energy of the electron, given by e² e² , is equal to its mass energy as in mc2, so that G . 4SH0 R 4SH0 mc
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Basic electromagnetism and materials
10.2.2.2. Rutherford diffusion and radiation During the deviation caused by electrostatic interactions, the electron – proton pair G G constitutes a dipole that has a dipole moment given by p e( R) , which varies G continuously, as R varies just as well in terms of direction as in terms of modulus. G G The sign changes [in (R) ] due to the fact that R is orientated with respect to the origin (the proton) toward the electron, while the dipole moment goes from the negative (the electron) to the positive (the proton) charge. The consequence of this is that the dipole radiates, which is assumed, to a first approximation, to not modify the movement. It thus is possible to calculate the electromagnetic field radiated at a point (P), which is in the radiation zone. Also, OP >> O , where the origin of the dipole also is at O (by supposing that R/2 << OP = r). G The field ( B ) thus is given simply by Eq. (23) in Chapter 9, as in G G G µ0 p(W) u u B , 4Sc r G G G G into which, here, p e( R) , so that p e( R) . G The field ( E ) is given, for its part, by Eq. (24) in Chapter 9, by supposing that at a great distance from the radiation zone the wave is plane and is such that: JJJG G G G G OP E cB u u , where u . r Across a sphere with a given radius (r) the radiation power is given by Eq. (26) in Chapter 9, i.e., 1 2 G 2 r P(r,t) = p (t ) . 3 4SH0 3c c G With R given by the fundamental dynamic relationship: e² G e3 G G G G p e R eR , mR eR , we have 2 4SH0 R² 4SH0 mR
from which, it can deduced that:
P(r,t) =
2 G 2 r p (t ) 3 4SH0 3c c
so that with R
e6
1
2m 3
4SH0 c
4 m 3R 3
r R(t ) , we have c P(r, t)
2 mc3G3 3 R4
.
Chapter 10. Diffusion and absorption processes
297
To conclude, the electron and the associated dipole radiate essentially when the G G electron passes with the neighborhood of the nucleus, at which point R and thus p vary the most both in terms of modulus and of direction. The expression for the radiated power shows that as the electron distances itself from the proton (R increases), the power of the radiation decreases very rapidly, as it is proportional to R 4 .
10.3. Radiation Produced by Accelerating Charges: Synchrotron Radiation and Bremsstrahlung 10.3.1. Synchrotron radiation Spectroscopy is an essential method in “divining” materials when the most precise structural representations are required. Sensitivity in these methods therefore is of primary importance and can be addressed properly by increasing the signal to noise ratio. This can be done by increasing the power of the radiation source, which also permits a reduction in sampling time. Synchrotron radiation displays exactly this advantage. It was first observed at the end of the 1940s and exhibited the additional benefit of being easily controlled. A considerable number of synchrotrons since have been constructed, and they remain in heavy demand. An electron synchrotron is made up of a large vacuum chamber, in the form of an annular, into which are pulsed fast electrons that have undergone an acceleration in an electric field. Magnetic fields, resulting from magnets placed around the ring, oblige electrons to follow the ring's curve. At speeds close to light, the accelerating electrons emit, at a tangent to their trajectory, a fine beam of radiation. The modification of the acceleration field permits a wide variation in this so-called synchrotron radiation. This controllability and sensitivity of the source has allowed a large number of experimental difficulties to be overcome. Experimental techniques such as extended X-ray absorption fine structure (EXAFS) spectroscopy were developed using synchrotron sources. 10.3.2. Bremsstrahlung: electromagnetic stopping radiation When electrons or ions penetrate a material, their movement is stopped by their interaction with the constituent electrons or ions of the target material. The depth of penetration is governed by electronic and nuclear-stopping strengths. The deceleration of the particles results in an emission of radiation called Bremsstrahlung radiation, given in German to mean “braking radiation” and signifying stopping radiation. The loss in kinetic energy occurs progressively and is fractionated by the order of successive interactions. The resulting electromagnetic radiation is polychromatic. With total energy being conserved, the highest frequency radiation corresponds to a particle that has lost its energy in a single and unique interaction. An application of this type of radiation is the production of polychromatic X-rays via X-ray tubes.
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10. 4. Process of Absorption or Emission of Electromagnetic Radiation by Atoms or Molecules (to Approach as Part of a Second Reading) 10.4.1. The problem The study here considers the interaction of an atom or a molecule, in terms of a quantic description that takes on board the discreet energy levels associated with the positions of electrons in their orbitals, with an incident wave (here a luminous wave) assumed to be periodic and described in classical terms by a periodic vector potential. By using a Coulomb gauge (see comment 4 from Section 9.2.1.3), the vector G potential accords with the gauge condition, that is divA r = 0 , and the associated scalar potential (U(r,t)) can be taken as zero, as in U(r,t) = 0. In order to deal with the problem, we will consider the effect of an incident wave as a perturbation to the state of an atom or molecule. This perturbation can be characterized by a Hamiltonian that must first be evaluated to then show that it typically can be reduced to a electric dipole Hamiltonian. Finally, by applying the theory of time-dependent perturbations, we will be brought to studying the electric dipole transitions generated inside an atom that interact with the incident radiation. 10.4.2. Form of the interaction Hamiltonian 10.4.2.1. Hamiltonian in analytical mechanics: form for a particle interacting with a wave In analytical mechanics (see also, for example, “Elements of analytical mechanics” by J. W. Leech, Dunod 1961), it is simple enough to show that the Lagrangian associated with the movement of a particle, with a charge denoted by q and a mass denoted by m placed in an electromagnetic field derived from a scalar potential (V) where V is the only potential (VCoul) generated by the atom, and the vector potential G ( A ) associated with the incident wave has the form: 1 G G L = mv² + q ª¬ v.A - V º¼ . (1) 2 The limiting conditions of the integral I obtained from Eq. (2):
wI
³ L dt give the movement equation
G G G G G F = q ª¬ E + v×Bº¼ = mȖ .
(2)
This result can be obtained by using the fact that there is an equivalence between 0 with Euler's equation: d § wL · wL 0 , (3) ¨ ¸ dt ¨© wq i ¸¹ wqi
299
Chapter 10. Diffusion and absorption processes
which gives rise to, for example, for the variable qi = x [Eq. (4) identical to Eq. (2) depending on values of x]:
mx
ª dy § wA y wA x wV º ª wA q « x » q « ¨¨ wx ¼ wy ¬ wt ¬« dt © wx wL
With pi
· dz § wA x wA z ¸¸ dt ©¨ wz wx ¹
·º ¸» . ¹ ¼»
(4)
(conjugate moment), we obtain with L given by Eq. (1) and in a
wq i
mx qA x , so that in terms of vectors G G G p = mv + qA . (5)
Cartesian frame p x
Calculating then the Hamiltonian given by:
¦ pi q i L
H
(6)
i
where L is given by Eq. (1), an expression in which according to Eq. (5): G
G p - qA 2m
1
1 G G mv² = mv 2 2 2m
1
2
.
G G 1 G From this can be deduced that v = p - qA , so that Eq. (1) can be written: m
G G q G p - qA A - q VCoul m G G G G p - qA G = p - qA + 2qA - q VCoul 2m
L=
G
G p - qA 2m 1
2
+
so that finally: G
G
G G p - qA p + qA L= -qV
G G p² - q²A²
Coul
2m
2m
- q VCoul .
In additional terms, from Eq. (5) we have: pi = m
dqi dt
+ qAi
dqi dt
= q i
1 m
pi - qAi ,
(7)
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Basic electromagnetism and materials
which moved with Eq. (7) into Eq. (6) gives: G 1 G G H = p p - qA - L m G G 1 G G G = p - qA 2p - p - qA + qVCoul 2m
so that: H=
G
G p - qA 2m 1
2
+ qVCoul .
(8)
10.4.2.2. Hamiltonian operator and the perturbation Hamiltonian G G G In terms of operators, p corresponds to the operator p { j= , so that on development, we have as an operator associated with Eq. (8): G G p² q G G G G q²A² l H= pA + Ap + + qVCoul 2m 2m 2m
(9)
G f (r) º by applying the function \ (r) we have: To calculate the commutator ª p, ¬ ¼ G G ª p, f (r) º \ (r) = j= ª¬, f (r) º¼ \ (r) ¬ ¼
G G j= f (r)\ (r) f (r)\ (r)
from which G ª p, f (r) º \ (r) ¬ ¼
G G G j= f (r) . \ (r) + f (r)\ (r) f (r)\ (r) G j=f (r) . \ (r).
So definitely: G ª p, f (r) º ¬ ¼
G j=f (r) ,
G and by making A take on the role of f (r) , we have: G G º ª p, A ¬ ¼
G G j=.A
G j= div A ,
(10)
Chapter 10. Diffusion and absorption processes
G so that with Coulomb's gauge, for which div A G G º ª p, A ¬ ¼
0.
301
0: (11)
By using the result of Eq. (11) in Eq. (9), we have: G G G l = p² q pG A + q²A² + qV H Coul , 2m m 2m which we can write in the form:
l=H l0 + H l (1) H with
l 0 = p² + qV H Coul 2m G (1) q G G q²A² l H = pA+ . m 2m
(12)
0
l ) is the atomic Hamiltonian that describes the particle (the The first term ( H electron undergoing an interaction with the wave) in the presence of a Coulombic potential in the atom. l (1) represents the interaction of the electron with the wave For its part, H G characterized by the vector potential denoted as A . In an approximation of the scale of the wavelengths (optical waves with wavelengths of the order of 600 nm), we can assume that the wave is practically uniform over all of the atom (which has a dimension of the order of Bohr's radius, G 0.053 nm) or indeed of the molecule. Under such conditions, A depends only on the G position ( ra ) of the atom or molecule and the second term of the Hamiltonian l (1) ) is a scalar, for which the atomic states between the two different perturbation ( H states of the atom are zero. The result of this is that the second term cannot undergo a transition and consequently is ignored in the following calculations. The perturbation Hamiltonian thus is reduced to (with q = -e as the charge of the interacting electron)
Basic electromagnetism and materials
302
G G l (1) = q pG A = e pG A . H m m
(13)
GG This Hamiltonian sometimes is called the “ A.p Hamiltonian” in which the vector potential of the Coulomb gauge intervenes, evaluated for the atom's position.
10.4.3. Transition rules 10.4.3.1. Preliminary introduction to quantum mechanics (see also, for example, C. Cohen-Tannoudji, Quantum Mechanics, Chapter XIII, Hermann, Paris, 1973) By looking at the course notes of most courses in quantum mechanics, we can see the expression for the probability of obtaining a state (m) from a state (n) after an instant (t), where 't = t – t0 and t0 = 0, for a particle subject to a time-dependent l (1) ) . Thus found, the probability is in the form: perturbation ( H Pnm
a*m (t)a m (t)
(14)
where a m (t)
1
t
(1)
³ H mn (t ')dt '
(15)
j= 0
and equation in which:
H (1) mn (t ')
l (1)
< 0m (t ') | H
| < 0n (t ') =
\ 0m e
j
E 0m =
t'
l (1)
|H
| \ 0n e
j
E 0n =
t'
.
10.4.3.2. Application to a wave perturbation In the approximation of an electric dipole, the size of the atom is smaller than the G G 2S wavelength of the luminous perturbation ( k.r | r is low, whereas the value of O O is high) and the vector potential, which has the general form GG G G G G A(r, t) A 0 cos(Zt k.r) , can be reduced to A(t) A 0 cos Zt . l (1) given by Eq. (13) thus takes on the form H
G l (1) = e pG A cos Zt H 0 m
j=
e G G A 0 cos Zt . m
(16)
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Chapter 10. Diffusion and absorption processes
By making j=
:
e G G A0 m
(17)
definitively, we find that:
l (1) H
By making : mn
:eiZt :e iZt
: cos Zt
2
\ 0m | : | \ 0n
.
(18)
notation
m | : | n and by remarking that :
is Hermitian and time independent, we have according to Eq. (15):
a m (t)
j 0 0 : mn t = E m E n t ' ª jZt ' e jZt º dt ' . e e ³ ¬ ¼ 2 j= 0
(19)
With: Z0
E 0n E 0m
(20)
=
and following integration, Eq. (19) gives:
a m (t)
j Z Z t j Z Z t : mn ª e 0 1 e 0 1 º « ». 2 j= « Z0 Z Z0 Z » ¬ ¼
(21)
This result indicates that a m (t) is large only when the denominator tends toward zero, that is when Z rZ0 . If Z
Z0 , the first term is large with respect to the second, and Eq. (21) gives:
a m (t)
2
a*m (t)a m (t)
so that in addition,
: mn
e
2
=² Z0 Z
2
j> Z0 Z@t
*
e
1
4
j> Z0 Z@t
1
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Basic electromagnetism and materials
a m (t)
2
2
: mn
¬ª1 cos(Z0 Z)t ¼º .
(22)
Z0 Z 2
2= ²
The term shown in Eq. (22) represents the probability of finding a particle in the state m at an instant t, and the probability of the transition Pn o m is defined by the 2
variation as a function of time of the probability a m (t) of finding the particle in the final state m, as in: sin Z0 Z t 1 d 2 2 Pn o m a n (t) m|:|n . dt Z0 Z 2=² If we suppose that the time t is large with respect to the period of the electromagnetic field, which is f the order of 10-15 sec in the optical domain, and if we make x = Z0 - Z, we obtain: Pn o m
1 2= ²
m|:|n
2
lim
sin xt
t of
x
.
Dirac's function is defined by Gx
1
lim
S t of
sin xt x
and the probability of the transition thus is written as Pn o m
S 2= ²
m|:|n
2
G Z0 Z .
(23)
This indicates that for the transition probability from one state n to another state m is zero, then G Z0 Z z 0 Z = Z0 . Therefore, the angular frequency (Z) of the incident electromagnetic wave must be such that:
=Z
=Z0
E 0n E 0m .
(24)
This expression confirms in some way the principle of the conservation of energy and can be considered as a “resonance condition”.
Chapter 10. Diffusion and absorption processes
305
With Z Z0 , we have E 0n E 0m + =Z . There is an absorption of energy ( =Z ) by the atom. If we had taken as the more important term in Eq. (21) that which corresponded to Z Z0 , we would have obtained E 0n
E 0m - =Z . Therefore, there is an
emission of an energy =Z induced by the perturbation.
10.4.3.3. Determination of the transition rule, and dipole transition moment The probability of transition between the state n and the state m is given by Eq. (23), in which the resonance condition intervenes through the intermediary of Dirac's function ( G Z0 Z ). With the resonance condition being fulfilled ( Z Z0 ), the transition probability thus brings in the term P0
S 2= ²
m|:|n
2
, which
evaluated with the help of Eq. (17) is of the form:
P0
S 2= ²
m|:|n
2
2 JJJG G 2 S § =e · 2 ¨ ¸ A0 m | | n . 2= ² © m ¹
In order to evaluate the term
(25)
G m | | n the following mathematical term
(described in the problems at the end of this Chapter) can be used: G =² G \m | | \n \m | r | \n En Em . (26) m Under these condition, Eq. (25) which can be written as:
P0
G G º ª§ = ²e · S G 2 ª§ =²e · A m | | n » «¨ ¸ m||n 4 0 «¨ m ¸ 2= ¹ ¬© ¼ ¬© m ¹
G brings in the terms in the form (where µ G § = ²e · ¨ ¸ m||n © m ¹
G \ m | er | \ n
º » ¼
*
G er , the dipole moment)
En Em
G µmn E n E m .
To conclude, the probability of a transition is proportional to the transition dipole moment defined by: µmn
\m | µ | \n
* ³ \ m µ \ n dW
(27)
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Basic electromagnetism and materials
Once the resonance condition is established, it is the result of the calculation for this integral that finally indicates the permitted transitions between the various possible m and n atomic states.
10.5. Conclusion: Introduction to Atomic and Molecular Spectroscopy In this last part, which gives the most important results from Section 10.4 on the semiclassic calculations for dipolar approximations, we will underline the importance of dipolar radiation and apply it to the description of the possible different internal transitions (for absorption or emission) in atoms and molecules. 10.5.1. Result concerning the dipole approximation In very general terms, the dipolar approximation, which states that the electromagnetic field is practically constant at the level of an atom or a molecule, means assuming that there is only a weak coupling between an electromagnetic field and an atom or a molecule. In effect, it is inefficient because its length is way below that of the length of the electromagnetic waves. We thus can treat the effect of an electromagnetic field as that of a perturbation. By carrying out a development of the interaction energy of the electromagnetic field in terms of various contributions in an order of decreasing importance (analogous to that of a distribution of charges at multipoles, with successive terms due to total charge, dipole moment, quadripolar moment, etc.) we find that the term for the electric dipole interaction is greater than the term for the magnetic dipole interaction, which in turn is greater than the term for the quadripolar electric interaction and so on. In physical terms, the luminous perturbation, characterized by its electric field G GG ( E ) moves electrons through a coupling energy in the form W µ.E , where G G G wA E ZA 0 sin Zt in the Coulombic gauge, where the scalar potential wt associated with the luminous wave field is zero (we can see that this coupling energy intervenes in Eq. (25) taking Eq. (26) into account).
If the following electronic redistribution is symmetric, there is no overall variation in the electric dipole and the transition is forbidden. Inversely, if the redistribution is asymmetric, then the transition is possible even if the atom (or molecule) showed no initial permanent dipole moment.
10.5.2. Different transitions possible in an electromagnetic spectrum Various types of transitions (for emissions or absorptions) can be written, depending on the wavelength of the electromagnetic field.
Chapter 10. Diffusion and absorption processes
307
10.5.2.1. Rotational transitions A molecule with a permanent magnetic dipole can appear as a carrier of a variable dipole moment when touring on its own axis (figure 10.6). Inversely, if the molecule is symmetric (without a permanent moment), there is no apparent variation in the dipole moment with rotation. Only molecules that possess a permanent electric dipole can emit or absorb radiation (in the far infrared or microwave region) by making a transition between rotational states.
+
+
Figure 10.6. Variation of the dipole moment of a rotating polar molecule.
Figure 10.7a shows a rotating dipolar molecule, such as HCl, being influenced by an alternating electric field (E) along Ox at an initial time t = 0. Figure 10.7b represents in the plane Oxy the rotation of the dipole at the various instant when t = 0, t1, t2, t3 etc., and Figure 10.7c shows the variation of a component of the dipole in the direction Ox with respect to time. As shown in Figure 10.7b, the dipole periodically changes position with the electric field due to the variation of the orientation of E and the variation in the GG coupling energy, which is of the form W µE and when t = 0, Wmin µ x E as E is directed along Ox at the initial time. As a consequence, the component of the dipole in a given direction (shown for Ox in Figure 10.7c) fluctuates regularly, and thus the component µx exhibits a fluctuation with a form resembling that of the electric field from the electromagnetic wave. (In emission, the radiation would be of the same frequency as long as the rotation is not slowed by the interaction of the dipole with neighboring molecules in the material.)
Basic electromagnetism and materials
308
Ex
(a) 0 z
t4
t3
t
y
(b)
t2
t1
+
µx
x
+
+
+
+
+
+ y
x
µx
(c) 0
t1
t2
t3
t4
t
Figure 10.7. Variation of (a) a component (Ex) of a varying electric field; (b) the position of a permanent dipole moment (µ) in the plane Oxy; and (c) the value of µx with time.
10.2.2. Vibrational transitions R
(a)
(b)
+
R
+
Figure 10.8. A vibration results in a stretching of the bond, and for (a) the nonpolar molecule shows no variation in dipole moment, but (b) a polar molecule exhibits a change in dipole moment due to the oscillation.
Chapter 10. Diffusion and absorption processes
309
These transitions appear when there is a vibration associated with, for example, a stretching or a torsion of the bonds between the atoms of a molecule, resulting in a variation in the dipole moment of the molecule. Only those vibrations that are associated with a modification of the dipole moment of a molecule are accompanied with an emission of absorption of electromagnetic radiation. Given the energies brought into play, the mechanism is apparent in the infrared.
10.5.2.3. Electronic transitions Electronic transitions are apparent if, following an electromagnetic excitation or emission, the resulting redistribution of electrons changes the dipole moment of the molecule. For this to happen, the redistribution must be asymmetric; if the redistribution is symmetric, then the corresponding transition is forbidden. The energies brought into play are in the optical region. 10.5.3. Conclusion This chapter has described different phenomena of absorption and emission of EM waves by materials. Absorption phenomena will be detailed further in Chapter 3 of the second volume called “Applied Electromagnetism and Materials” using a more phenomenological approach based simply on the interaction of electromagnetic waves with molecules. The deformation that the molecules undergo will be described as a function of the region in which the frequency of the EM wave falls and as part of a more classic mechanical treatment, as detailed in the last figure of Chapter 3 of Volume 2. It is worth adding that if the energies are of the order of radio frequencies, then it is the magnetic spin moment that interacts with the electromagnetic wave. The reversing magnetic spin dipole thus is at the origin of electronic paramagnetic resonance (EPR) that follows the reversing electron spin. If the effect involves the proton spin, then phenomena associated with nuclear magnetic resonance (NMR) are then observed. 10.5. Problems 10.5.1. Diffusion due to bound electrons This problem concerns a bound electron with a charge denoted by –q, which is initially situated at the origin of a trihedral defined by Oxyz of an atom or molecule subject to: x a monochromatic plane polarized (along Ox) electromagnetic wave incident along G G Oz and with an electric field in the form E E 0 exp(iZt) u x ; and G G x and a returning force toward its equilibrium position of the form f -mZ0 ² x u x . It is assumed that the angular frequency (Z) of the incident wave is well below that of Z0 . The atom or molecule thus is an oscillating dipole.
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Basic electromagnetism and materials
1. It is also assumed that the velocity acquired by the electron is negligible with respect to that of the speed of light. (a) Give the fundamental dynamic equation. (b) From this deduce the solution for the permanent steady state along the x abscissa of the electron. (c) What is the value of the dipole moment (p(t)) of the system? 2. The equation for an electromagnetic field radiated at a great distance by a dipole JJJJG G at a point (M) such that OM r is: G G G G µ0 1 G µ0 p(W) u r W) uG ) uG . B and E (p( r r 4S c r 4S r
G ur G uMG uT
x M
G p T M
m
G uM
z
G G G What does W correspond to? Write the equation for p(W) . Thus calculate B and E . G G 3. If E W and BW are the complex amplitudes of the electric field and the magnetic G G G G field, respectively, given by E E w exp(iZt) and B Bw exp(iZt) , in general G terms, the average value of the Poynting vector ( S ) is of the form G ¢S²
G G § E B w w Re ¨ 2 ¨© µ0
1
· G G ¸ , where B w is the conjugate complex of Bw . ¸ ¹
Determine the average Poynting vector for the problem.
4. Determine from this the total power radiated by the electron and from the result makes a conclusion.
Chapter 10. Diffusion and absorption processes
311
Answers 1. (a) The displacement with respect to Ox can be found by considering that for this axis: G G G Fem Frappel mJ , so that G G G q E v u B Freturn
With B =
E c
, we have v B =
G mJ vE c
<< E if
v c
1 .
With respect to Ox, we therefore have qE 0 eiZt mZ02 x
m
d²x dt²
.
(b) Under a permanent steady state, we are looking for a solution of the form: x
x 0 eiZt , that with
qE 0 mZ02 x 0 x0
qE 0 mZ02
d²x dt²
Z²x and by simplifying with eiZt , then
mZ²x 0 x 0
qE 0 m ª Z02 Z2 º ¬ ¼
, so that with Z0 >> Z , we find
, and finally
x
qE 0 iZt e mZ02
In terms of vectors, G x
G xu x
G x 0 eiZt u x
G where u x is the unit vector along Ox.
(c) Moving the charge –q is equivalent to applying a dipole moment of p = - q x, for q²E 0 iZt which p(t) qx e , so that in terms of vectors, mZ02
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Basic electromagnetism and materials
q²E 0 iZt G e ux . mZ02
G p(t)
G 2. We have B(M)
G G µ0 p(W) u r
4S c
r
DD G JJJJ with the propagation) . With p(W)
, where W
t
r
(change in variable associated
c
r
§ · q²E 0 iZ¨© t c ¸¹ G e u x , we find mZ02
G Z²p(W) and p(W)
G that for B(M) G B(M)
r
G G Similarly, with u x u u r results in: G E(M)
µ0 q² Z²E 0 4ScrmZ02
§ · G iZ¨© t c ¸¹ sin T u M e
G Bw eiZt . G G sin T u T , the equation for E
G G G sin T u M and sin T u M u u r
r
µ0 q²Z²E 0 4SrmZ02
§ · G iZ¨© t c ¸¹ sin T u T e
G G 3. The average Poynting vector thus is (where u T u u M G ¢S²
G G § E B w w Re ¨ 2 ¨© µ0
1
· ¸ ¸ ¹
G E w eiZt . G u r ) given by
µ0 q 4 Z2 E 02 32S²r²m²Z04 c
G sin 2 T u r .
4. The total power radiated is given by: P=
G JJJG
w ³³ S d²6 . 6
JJJG G With d²6 = r²sinT dT dM u r , we have:
P
µ0 q 4 Z2 E 02 32S²r²m²Z04 c
r² ³³ sin 3TdTdM T,M
µ0 q 4 Z2 E 02 32S²m²Z04 c
S
2S
3 ³ sin TdT .
T 0
Chapter 10. Diffusion and absorption processes
With
S
4
T 0
3
3 ³ sin TdT
, we finally obtain:
µ0 q 4 E 02 § Z4 ¨ 12Sm²c ¨© Z04
P
2Sc
Comment: With Z
313
O
2Sc
and Z0
O0
, we also can write that:
µ0 q 4 E 02 § O 04 ¨ 12Sm²c ¨© O 4
P
· ¸ ¸ ¹
· ¸. ¸ ¹
As detailed in the course, the smaller Ȝ (0.4 µm in the visible spectrum, that is blue light), the greater the power radiated (hence the blue color diffused by electrons in molecules present in the sky).
10.5.2. Demonstration of the relationship between matrix elements [see Eq. (26) of Section 10.4.3.3 as in G =² G \m | | \n \m | r | \n En Em ] m For this, we will establish the following equation: d A
1
dt
i=
> A, H @
which is a relative equation for the physical magnitude A, which is not explicitly l ) is Hermitian (proper real values) time dependent, so that the associated operator ( A
l|\ . \|A
and is such that A
Answers d A
d\ l l | d\ . |A|\ \|A dt dt dt dt d\ d \ l \ j= , for which However, according to Schrödinger, H dt dt Calculate
d
l|\ \|A
1 l H\ . j=
Basic electromagnetism and materials
314
We thus obtain:
d A
1 l l|\ \|A l| 1 H l\ H\ | A j= j=
dt =-
1 l l|\ 1 \|A l|H l\ . H\ | A j= j=
H and A are Hermitian (being equal to their associated terms and are such that, for example, \ | H | M H\ | M ), so we also find that: d A dt 1
=
1 ª l l | \ \ | AH ll | \ º \ | HA »¼ j= «¬
l H lº | \ \ | ª A, ¬ ¼ j=
1 j=
> A, H @
j =
> H, A @
.
l { mrG , in terms of operators we also can write that (see also, for With A example, E. Durand, Quantum Mechanics, Masson, p.69, 1970):
m As m
G dr dt
G dr dt
jm ª l G º H, r . ¼ = ¬
G corresponds to an operator of a quantity of movement, j= , we also
have: G j=
jm ª l G º =² G H, r , from which ¼ m = ¬
H lº . ª Gr, ¬ ¼
Multiplication of the left side of the last expression by \*m and the right side by \ n , and then carrying out with a scalar product gives: =² m
G \m | | \n
Gl l G | \ . \ m | r H | \ n \ m | Hr n
l is Hermitian (just as is Gr ), so that: H l G | \ \ m | Hr n
l | Gr\ \m | H n
G l|\ * r\ n | H m
G
l\ dr . ³ r\ n H m *
Chapter 10. Diffusion and absorption processes
Similarly, we obtain: Gl \ m | r H | \n
G l \ m | r | H \n
l\ | Gr\ * H n m
G *
l\ dr . ³ r\ m H n
The result is that: =² m
l\* As H m
G \m | | \n
l\ E m \*m and H n =² m
G \m | | \n
G
\* H l l * ³ r[ m \ n \ n H\ m ]dr .
E n \ n , we have: G G E m ³ \*m r\ n dr E n ³ \*m r\ n dr ,
and hence, the looked-for equation: =² m
G \m | | \n
G \ m | r | \ n
En Em .
315
Chapter 11
Reflection and Refraction of Electromagnetic Waves in Absorbent Materials of Finite Dimensions
11.1. Introduction In practical terms, electromagnetic (EM) waves in materials of limited dimensions correspond to those in systems such as coaxial cables, optical fibers, and wave guides. It is by this process that signals, and therefore information, are transmitted with as low a degree of attenuation and parasitic phenomena as possible. This chapter will look at the influence of discontinuity, at the interface between two materials, on the propagation of EM waves. Typically, the two materials are: x linear, homogeneous, and isotropic (lhi) dielectrics and not magnetic so that their G G G G G magnetization intensity can be written as I = 0 , so that B=µ0 H+I=µH , µ = µ0 G with I = 0 ; G x uncharged, as in UA 0 , and not traversed by real currents, as in jA 0 ; and x described by their complex dielectric permittivity as H = H' - jH'' (electrokinetic notation), or as H = H' + jH'' (optical notation). In this chapter, and then in Chapter 12, which mostly looks at applications using the optical region, the notation used is that classically used elsewhere in optics as in sinusoidal planar progressive EM waves are written in the form (see also Section 6.4.2): GG G G E=E m exp j[kr-Ȧt] . (1)
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Basic electromagnetism and materials
G ki
medium (1)
11.2. Law of Reflection and Refraction at an Interface between Two Materials 11.2.1. Representation of the system
x
G G n12 = n z incident plane
z
medium (2)
y
O
Figure 11.1. Interface between media (1) and (2).
As a general system, the volume under consideration is made up of two media, denoted (1) and (2) which have permittivities H1 and H 2 , respectively, and are separated by an interface on the plane Oxy. The incident wave has an angular GG G G frequency denoted by Zi, is written in the form Ei =Eim exp j[k i r-Ȧi t] , and thus by G G notation for the incident wave E m = Eim . In addition, by choosing the origin of G G G phases on this wave such that Eim = Ei0 exp(jMi ) = Ei0 when Mi 0 , then it is possible to state: GG G G Ei =Ei0 exp j[k i r-Ȧi t] . (2) G The incident plan is defined by the plane that contains the incident wavevector ( k i ) G G as well as the normal to the interface ( n12 = n z ). In order to simplify the representations, the incident plane can be taken as Oxy in which consequently, is the G wavevector k i . Also, we will assume that the first material, medium (1), is G G nonabsorbent so that H r1 = n12 so that the modulus of k i is k i = k1 = k 0 n1 .
At the reflection on the plane of the interface Oxy, the incident wave will observe a modification in one part of its amplitude (which now will be denoted G E r0 ), while the other part will be subject to a dephasing by a quantity denoted as Mr. The reflected wave then can be written in a general form where Zr designates the angular frequency following reflection:
Chapter 11. Reflection and refraction in absorbent finite materials
G G G G E r =E 0r exp(jMr )exp j[k r r-Ȧr t]
G G G0 E r exp j[k r r-Ȧr t] .
319
(3)
Similarly, for the transmitted wave, self-evident notations are used: G G G G G G G0 E t =E 0t exp(jMt )exp j[k t r-Ȧ t t] E t exp j[k t r-Ȧt t] . (4) G G In these equations, k r and k t are the wavevector relating to the reflected and G transmitted wave, respectively. As detailed below, k t can be a complex magnitude G (thus denoted k t ).
11.2.2. Conservation of angular frequency on reflection or transmission in linear media Two different types of reasoning can be used in this case: the first is by a physical analysis of the problem, and the second via an equation using the conditions of continuity at an interface. 11.2.2.1. Physical reasoning As shown in Chapter 10 dipoles associated with a displacement of charges inside a G dielectric that are subject to an incident wave ( Ei ) with a given angular frequency ( Zi = Z ), and within an approximation of a linear oscillator (returning force proportional to the induced displacement), emit in turn a wave with the same angular frequency as the incident wave. The result is that little by little, in media (1) and (2), the incident wave with angular frequency is reemitted with the same angular frequency, so that Zi = Zr = Zt = Z . (5) 11.2.2.2. Reasoning with the help of equations of continuity G G0 G0 By denoting ai, ar and at as the proper projections of Ei0 , E r , and E t at the plane of the interface Oxy, the continuity condition of the tangential component of the electric field gives rise to: GG G G G G a i exp j[k i r-Ȧi t] a r exp j[k r r-Ȧr t] a t exp j[k t r-Ȧ t t] . By multiplying the two GG members by exp - j[k i r-Ȧi t] , we obtain:
ai + a r e
G G G j[(k r - k i )r-(Ȧr - Ȧi )t]
G
G G
a t e j[(k t - k i )r-(Ȧt - Ȧi )t] .
(6)
This equation must be true for all instants (t), and each term of the preceding equation should have the same temporal dependence, so that 0 Zr - Zi = Zt - Zi . The same result is found as that given by Eq. (5), that is Zi = Zr = Zt = Z .
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Basic electromagnetism and materials
11.2.3. Form of the wavevectors with respect to the symmetry of the media Once again, the same two types of reasoning may be applied. 11.2.3.1. Reasoning based on physics of the symmetry of a system The EM field of the incident wavevector is given by: k 0 G ° ix k i ®k iy z 0 ° ¯k iz z 0 Taking into account the disposition of the interface, which is such that the discontinuity only occurs in the direction Oz, the propagation in the directions x and y is not perturbed at the interface. This results in perturbation being only in the direction Oz, where the wavevector thus is modified on reflection and transmission. Physically then we should have: k ix = k rx = k tx = 0 k iy = k ry = k ty
(7)
k iz z k rz z k tz
11.2.3.2. Mathematically based reasoning and the use of the continuity equation Again using the result given by Eq. (6) and also from the imposition of the condition of continuity of the tangential component of the electric field, the equation is valid G only at the level of the interface where the vector ( r ) has the components given by x z 0 G° r ®y z 0 °z = 0 ¯ G ki G
G kr Ti
Tr
G
[i
[r G
[t Tt
z
y
G kt
Figure 11.2. Representation of the projected vectors ( [ ) at the interface.
Chapter 11. Reflection and refraction in absorbent finite materials
321
G G G By denoting ȟ i , ȟ r , and ȟ t , the projected vectors at the interface Oxy of the G G G vectors k i , k r , k t , respectively, we can state that they have components given by: k G ° ix ȟ i ®k iy ° ¯0
k G ° rx ȟ r ®k ry ° ¯0
k G ° tx ȟ t ®k ty ° ¯0
G Eq. (6) for the vectors r located at the interface thus is in the form: G
G
G G
G G
a t e j[([ t - [i )r-(Ȧt - Ȧi )t] .
a i +a r e j[([r - [i )r-(Ȧr - Ȧi )t]
(6’)
G This Eq. (6') should be true for all r vectors at the interface, and we should find for each term in Eq. (6') the same spatial dependence, which can be written: G G G G G G G G G 0 = ȟ r - ȟ i r = ȟ t - ȟ i r, so that ȟi = ȟ r = ȟ t . G With k i in the plane of the wave k ix 0 , it can be determined that:
k ix = k rx = k tx = 0 and k iy = k ry = k ty .
(7’)
Following reflection and transmission of the EM wave at the level of the interface, there remains only k rz and k tz to be determined. This can be done only through a use of the equations for the propagation in each of the media.
11.2.4. Symmetry and linear properties of the media and the form of the related field 11.2.4.1. In medium (1)
A medium (1) medium (2) Figure 11.3. Representation of a wave at point A in medium 1.
A wave at any point (A) in medium (1) is in fact a superposition of the incident and reflected wave. Taking the results of Sections 11.2.2 and 11.2.3 into account, which made it possible to state that Zr = Z , that k rx = k ix = 0 and that k ry = k iy , along with that of the form of Eq. (2) for the incident wave, the resultant wave in medium (1) therefore must be of the form: G G G0 E1 (r,t) = E1 (z) exp ª j k iy y- Ȧt º . (8) ¬ ¼
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Basic electromagnetism and materials
The exponential term contains the well-determined (and unchanged) part of the wave, where the temporal variation is Zr = Z and the spatial variation with respect to x is given by k rx = k ix = 0 and with respect to y by k ry = k iy . G0 The term E1 (z) contains the behavior with respect to Oz, which remains for the moment unknown; the resolution of the propagation equation in medium (1) will G0 allow a determination of E1 (z) and krz (as a function of kiz).
11.2.4.2. In the medium (2) Once again, Zt = Z and k ty = k iy , while only ktz remains unknown. In medium (2) the wave thus is in the form: G G G0 E 2 (r,t) = E 2 (z) exp ª j k iy y- Ȧt º . (9) ¬ ¼ G0 The exponential contains the known terms, while E 2 (z) encloses that which is unknown for the moment (behaviour with respect to Oz) and it is the resolution of the propagation equation for medium (2) which will permit a determination of G0 E 2 (z) and k tz { k 2z .
11.2.5. Snell-Descartes law for the simple scenario where media (1) and (2) are nonabsorbent and k1 and k2 are real 11.2.5.1. The wavevectors associated with incident, reflected, and transmitted waves are all in the same plane (incidence plane) The result given in the subtitle can be determined immediately from the fact that we have k ix = k rx = k tx = 0 from the first expressions of Eq. (7). The result is that the G G G wavevectors k i , k r , and k t are all in the plane Oyx, which is their hypothetical plane of incidence. 11.2.5.2. Law of reflection: Ti Tr The first equivalence of the second expression of the equations given in (7) is that Ȧ Ȧ k ry = k iy . With k i = n1 and also k r = n1 , the relation k ry = k iy , which c c G G represents the equality of the projection of vectors k i and k r with respect to Oy, Ȧ n1 sin și = n1 sin ș r , so that Ti c c G G we orientate the angles with respect to the normal, then n12 = n z , and
gives in moduli (see Figure 11.2)
Ti
Ȧ
Tr .
(10)
Tr , and if
Chapter 11. Reflection and refraction in absorbent finite materials
323
11.2.5.3. Law of refraction: n1 sin și = n2 sin șt Still with respect to the second of the expressions in Eq. (7), we also have k ty = k iy . When the materials are nonabsorbing, the wavevectors are real just as the optical indices. We thus have k i =
Ȧ c
n1 and k t =
Ȧ c
n 2 and the relation
G G k ty = k iy , which brings in a projection of the vectors k i and k t with respect to Oy, Ȧ
n1 sin și =
Ȧ
n 2 sin ș t , so that : c c n1 sin și = n 2 sin ș t . (11)
gives rise to (Figure 11.2)
11.2.6. Equation for the electric field in medium (1): the law of reflection As above indicated, medium (1) is not magnetic or absorbent, n1 and G Ȧ k i = n1 = k 0 n1 are real, and the components of k i are (0, k iy z 0 and k iz z 0 ). c In medium (1), the monochromatic wave is in accordance with the wave equation [Eq. (7") in Chapter 7]: JJJGG Ȧ² 2 G ǻE1 + n1 E1 = 0 (12) c² G where E1 is the form given by Eq. (8), as in G G G0 E1 (r,t) = E1 (z) exp ª j k iy y- Ȧt º . ¬ ¼ G G G0 By making f(y)=exp ª j k iy y- Ȧt º , we have E1 (r,t)= E1 (z) f(y) , with the result ¬ ¼ that
JJJGG G G G ǻE1 = i ' E1x + j ' E1y + k ' E1z G ªG0 G G G0 G0 i ' E1x (z) f(y) º j ' ª E1y (z) f(y) º k ' ª E1z (z) f(y) º »¼ ¬« ¼» ¬« ¼» ¬« G G G G ª w² 0 º ª 2 0 0 0 = f(y) « E1 (z) » + i² k iy i E1x +j E1y +k E1z f(y)º « ¬ ¼» ¬ wz² ¼
ª w² G 0 º 2 G0 E1 (z)-k iy E1 (z) » exp ªi k iy y-Ȧt º . « ¬ ¼ w z² ¬ ¼
Following simplification of the two f(y)=exp ª j k iy y- Ȧt º , Eq. (12) becomes: ¬ ¼
latter
terms
with
the
term
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Basic electromagnetism and materials
G0 d²E1 (z) dz² 2 By making k1z =
Ȧ² c²
§ Ȧ² 2 · G0 + ¨ n12 - k iy ¸ E1 (z) = 0 . © c² ¹
(13)
2 , then Eq. (13) can be written as n12 - k iy
G0 d²E1 (z) dz²
2 G0 + k1z E1 (z) = 0 .
(13’)
It is notable that: 2 2 2 2 k i2 = k ix + k iy + k iz = k iy + k i2z
k i2
k 0 n1
2
Ȧ² c²
2 = k iz
n12
Ȧ² c²
2 2 n12 - k iy = k1z .
Where k iz = k1z > 0 so that the incident propagation can occur for a positive value G0 of Oz, the general solution for E1 (z) is given by G0 G0 G0 E1 (z)= Ei exp(i k1z z)+E r exp(- i k1z z) . (14) incident propagation
reflected propagation
G0 By substituting E1 (z) into Eq. (8), we obtain: G G G0 E1 (r,t) = E1 (z) exp ª j k iy y- Ȧt º ¬ ¼ G0 G0 Ei exp ªi k iy y + k1z z - Zt º +E r exp ªi k iy y - k1z z - Zt º ¬ ¼ ¬ ¼ G G Ei + E r . (15)
The first term details the incident wave for which the wavevector is given by: k = 0 G ° ix k i ®kiy °k = k > 0. 1z ¯ iz The second term represents the reflected wave at the level of the interface and is the G monochromatic plane progressive wave for which the vector k r has the component:
Chapter 11. Reflection and refraction in absorbent finite materials
325
k = k ix = 0 G ° rx k r ®k ry = k iy z 0 ° ¯k rz = - k1z = - k iz < 0 From the relations k rx = k ix = 0 and k ry = k iy , we find once again the laws of reflection obtained in Section 11.2.5. The incident and reflected rays are in the incident plane Oyz and Ti Tr . It is worth noting that these laws do not depend on the nature of medium (2), as no hypothesis has been formulated concerning it, and it could be nonabsorbing or absorbent and even a perfect insulator or conductor.
11.2.7. The Snell-Descartes law for reflection a system where medium (2) can be absorbent: n2 and k2 are complex Here medium (1) is nonabsorbent and n1 and k1 are real. 11.2.7.1. Form of the field in medium (2) Here medium (2) can be absorbent and is characterized by a complex dielectric permittivity, so that İ r2
n 22 .
JJJGG Ȧ² 2 G The wave equation in medium (2) is ǻE 2 + n 2 E 2 =0 . c² G Into this the substitution of the form of E 2 given by Eq. (9), G G G0 E 2 (r,t) = E 2 (z) exp ª j k iy y- Ȧt º , gives ¬ ¼ G0 d²E 2 § Ȧ² 2 2 · G0 (16) + ¨ n 2 - k iy ¸ E2 = 0 . dz² © c² ¹
By making k 22z
i.e.,
§ Ȧ² 2 2 · ¨ n 2 - k iy ¸ , so that with k iy = k 0 n1 sin și , then c² © ¹ k 22z = k 02 (n 22 - n12sin²și ) .
(17)
Then the equation for the propagation of the wave takes on its more normal form [see for example Eq. (15) in Chapter 7]: G0 G0 d²E 2 (16’) + k 22z E 2 = 0 . dz² The general solution is thus of the form: G0 G0 G E 2 (z) = E t exp(ik 2z z) + E '0t exp(- ik 2z z) .
(18)
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Basic electromagnetism and materials
Physically, in medium (2) there is no source of light and the only wave that actually can be found is a wave transmitted and distancing itself from the interface. In this case, the second component of Eq. (18) must be equal to zero, as it represents G the effect due to a wave that nears the interface (the E '0t also must be taken as equal to zero in order that the condition is held). Only the first component of Eq. (18), which clearly represents the effect of a wave leaving from the interface, can be retained on a physical basis, so that finally we have G0 G0 E 2 (z) = E t exp(ik 2z z) . (18’) For its part, k 2z defined by Eq. (17), generally has a complex form and can be written using a notation in the form: k 2z = k tz = k 'tz + ik 'tz' .
(19)
On substituting Eq. (18’) into Eq. (9), we obtain: G G G0 G0 E 2 (r,t) = E 2 (z) exp ªi k iy y- Ȧt º E t exp ªi k iy y + k 2z - Ȧt º , ¬ ¼ ¬ ¼ so that with Eq. (19): G G G0 (21) E 2 (r,t) = E t exp( - k ''tz z) exp ªi k iy y + k 'tz z - Ȧt º . «¬ »¼
(20)
propagation term
attenuation term
11.2.7.2. Form of the wavevector in the medium (2) Equation (21) for the wave in medium (2) can be rewritten in the form: G G G G G G G0 (21’) E 2 (r,t) = E t exp( - k ''t r) exp ª i k 't r - Ȧt º . «¬ »¼ G G This relation defines the vectors k 't and k ''t for which the components are given from Eq. (21):
G k 't
k 'tx = 0 (= k ix ) k 'ty = k iy =
(22)
k 0 n1 sin și
(19) (17) k 'tz = R(k 2z ) = k 0 R ® n 22 - n12 sin²și ¯
1/2 ½
¾ ¿
Chapter 11. Reflection and refraction in absorbent finite materials
G k ''t
k ''tx = 0
327
(23)
k ''ty = 0
19 (17) k ''tz = Im(k 2z ) = k 0 Im ® n 22 - n12sin²și ¯
1/2 ½
¾ ¿
Finally, the wave can be written in the form: G G G0 ª G G º E 2 (r,t) = E t (z) exp «i §¨ k t r - Ȧt ·¸ » , ¹¼ ¬ © G G G G where k t is defined by k t = k 't + ik 't' , such that: 1/2 ½ G G' 2 G 2 ¾ ez °k t =k iy e y + k 0 R ® n 2 - n1 sin²și G ° ¯ ¿ kt ® ° kG '' = k Im n 2 - n 2 sin²ș 1/2 ½ eG . ® 2 1 ¾ z 0 i ° t ¯ ¿ ¯
G G k 'ty e y + k 'tz ez
(24)
Consequences: inhomogeneous and homogeneous waves Generally speaking, the wavevector for the transmitted wave can be written in accord with Eq. (24) in the form: G G G G k t =k 'ty e y + k 'tz ez + i (k ''tz ez ) . (25)
According to Eq. (21), the propagation in medium (2), which is determined by G the real component of k t , is along Oy and Oz, while the attenuation associated with G the imaginary component of k ''t is only in Oz. The directions of propagation and attenuation thus are different and the wave is termed inhomogeneous. In the specific case where k 'ty = 0, so that when k 'ty = k1 sin ș1 we have T1 (normal incidence), we have:
0
G G G k t = k 'tz ez + i (k ''tz ez )
and the propagation and attenuation are in the same direction (Oz), and thus the wave is termed homogeneous.
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Basic electromagnetism and materials
11.2.7.3. First law of refraction
G G Here we have k 'tx = 0 and k ''tx = 0 , so that ª k t º = 0 . The k t has no component «¬ »¼ x along Ox and the transmitted wave is propagated in the plane of incidence, i.e., Oyz. 11.2.7.4. Second law of refraction
When k 'tz z 0 , the transmitted wave effectively propagates in medium (2); G G supposing T2 =T t =(n12 ,k 't ) , so that k 'ty = k 't sin ș 2 (see Figure 11.2). In addition, it also is possible to state, from Eq. (22), that k 'ty = k iy = k 0 n1 sin și which can written with k1 = k 0 n1 and by making T1
Ti by notation, that k 'ty = k1 sin ș1 . As
a consequence, we have the equation: k1 sin ș1
k 't sin ș 2 ,
(26)
which can be rewritten using the notations introduced, as k i sin și
k 't sin ș t .
(26’)
G G 11.2.7.4.1. Case where k t is real R k ''t = 0
Here, according to Eq. (23), the term n 22 - n12sin²și
must have a positive and real
magnitude: if it were negative, its square root would be purely imaginary and k ''t would be nonzero. Looking at the following conditions:
x first condition, n 22 - n12sin²și that n 22
n 22
has a real magnitude, imposes that n
2 2
is real, so
H r 2 where Hr2 has a real magnitude. Medium (2) thus is intrinsically
[that is to say by itself without medium (1)] nonabsorbent as k 22 =
Ȧ² c²
is real if H r2 = H r2 is real (and positive)
n 22 - n12sin²ș1 = n 22 - n12sin²ș1 ; x second condition, the term n 22 - n12sin²ș1 has a positive magnitude if
n 22 > n12sin²ș1 .
(27)
n 22 =
Ȧ² c²
İ r2
Chapter 11. Reflection and refraction in absorbent finite materials
329
This is the supplementary condition to n 22 being real [medium (2) nonabsorbent and G G Hr2 real] so that k t can be real (R k ''t = 0) when medium (2) is in the presence of medium (1). 0 , we have k t = k 't , which is such that according to Eq. (22):
When k ''t
k 2t = k '2t = k '2ty + k '2tz = (k 0 n1 sin ș1 ) 2 +(k 02 [n 22 - n12 sin² ș1 ]) = k 02 n 22 = Also, when
k ''t
0 , Eq. (26) is written as k1 sin ș1 n1 sin ș1
n 2 sin ș 2 .
Ȧ² c²
n 22 = k 22 .
k 2 sin ș 2 , so that in turn: (28)
G The conditions required so that n 22 > n12sin²ș1 (R k ''t = 0) are: x by simple evidence, that n 2 > n1 (see also Figure 11.4); x or n 2 < n1 and T1 TA where TA is defined by n1 sin TA = n 2 .
(29)
In effect, so that n 2 n1 we have T2 ! T1 (Figure 11.5). The T2 cannot exceed T2A
S
2
and TA also cannot exceed a limiting value given by T1
such that n1 sinTA = n 2 sin
TA and
S
. The limiting angle ( TA ) above which there can be 2 no refracted radiation is associated with the total reflection and thus is defined by the n equation sinTA = 2 . n1 The condition T1 TA yields sinT1 < sinTA =
n2 n1
, which otherwise can be
G stated as n1 sinT1 < n 2 and corresponds well to Eq. (27), for which k t is real G (R k ''t = 0) .
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Basic electromagnetism and materials
G ki
n2 > n1, T2 < T1. if T1= T’1 = S/2, T2 = T’2 such that sin T’2 = n1/n2
T1
n1
n2
T1 TA
n1
T1 = S/2
n2 T2
n2 < n1, T2 > T1. if T2 = T2A = S/2, T1= TA such that sin TA = n2/n1
G kt
T2l= S/2 G kt T2
T’ 2
n2 > n1 Figure 11.4. Refraction when n2 > n1 .
n2 < n1 Figure 11.5. Refraction when n2 < n1 .
G G 11.2.7.4.2. When k t is complex: R k ''t z 0 G When k t is complex, then k ''tz z 0 must be true. The attenuation occurs in the Oz
direction. According to the third relation given in Eq. (23), the following must be true: x either n 22 = İ r2 is imaginary and medium (2) is absorbent; or x n 22 = İ r2 = İ r2 is real [medium (2) is not absorbent] and n 2 n1 and T1 ! TA
(which correspond to n1 sinT1 ! n1 sinTA = n 2 ). We thus have n 22 - n12 sin²ș1 0 , so that n 22 - n12sin²ș1 i 2 ( n12sin²ș1 - n 22 ) and, according to Eqs. (22) and (23), we can deduce that, respectively: ' °k tz ° ® ° '' °k tz ¯
1/ 2 ½ k 0 R ® ª i²(n12sin²ș1 - n 22 ) º ¾ 0 ¬ ¼ ¯ ¿ 1/ 2 ½ k 0 Im ® ª i²(n12sin²ș1 - n 22 ) º ¾ = r k 0 n12 sin²ș1 - n 22 ¬ ¼ ¯ ¿
1/ 2
(30) z 0.
So that the wave undergoes an exponential “braking”, Eq. (21) must have k ''tz z
! 0 . As z ! 0 in medium (2), the positive solution from Eq. (30) for k ''tz is retained, and in terms of vectors, we have [from Eq. (24)]:
Chapter 11. Reflection and refraction in absorbent finite materials
G k ' =k eG =k n sin ș eG iy y 0 1 1 y ° t ®G 1/2 G ° k ''t = k 0 n12 sin²ș1 - n 22 ez ¯
331
(30’)
the wave is inhomogeneous. It is as shown in Figure 11.6. T1 > TA
G ki
TA
n1
k 't = k 0 n1 sinș1
T1
y n2
G ki
k ''t
z n 2 < n1 Figure 11.6. Inhomogeneous wave where k ''t z 0 , n2 < n1 and T1 > TA .
11.3. Coefficients for Reflection and Transmission of a Monochromatic Plane Progressive EM Wave at the Interface between Two Nonabsorbent lhi Dielectrics (n1 and n2 are real), and the Fresnel Equations 11.3.1. Hypothesis and aim of the study We thus can suppose that n12 = İ r1 = İ r1 are real, just as n 22 = İ r2 = İ r2 . Once again it is assumed that the wave source imposes an angular frequency (Z) and that the state of the polarization of the incident wave, which is always a combination of two orthogonal polarization states, is either such that: G G x Ei0 is perpendicular to the plane of incidence (Ei0A ) in which case it is termed a transverse electric (TE) polarization, as the electric field is orthogonal to the plane of incidence (it can also be stated that the wave is in an orthogonal polarization as in Figure 11.7 a); or G G x Ei0 is parallel to the plane of incidence ( Ei0& ) and in which case the term is one of transverse magnetic (TM) polarization as the magnetic field thus is orthogonal to the plane of incidence. Another term used is that of parallel polarization, as shown in Figure 11.7b.
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Basic electromagnetism and materials
G Ei0
G0 G 0 Bi E i
Ti ki
Ti
ki
n1 n2
y
G
x n12
Tt
n1 y
G
x n12
n2
kt
Tt
kt
z
z Fig. 11.7(a). TE polarization.
Fig. 11.7(b) TM polarization.
G G Given the incident wave, as in Ei = Ei0 exp(j[k i sinși y + k i cosși z - Ȧt]) , it should G0 G0 be possible to determine either the vectors E r and E t for the reflected and
transmitted waves, respectively, or the coefficients for the amplitudes of reflection (r) and transmission (t) defined by: r=
E 0r Ei0
and t=
E 0t Ei0
.
(31)
The r and t are a priori complex magnitudes that can take into account any possible dephasing between the reflection and the transmission. We thus have two unknowns to determine, which can be done with the help of two equations established from the limiting conditions. With the media assumed to be nonmagnetic, the following two relations for continuity at the interface are used: E1t = E 2t and B1t = B2t (as B1t = µ0 H1t = µ0 H 2t = B2t in nonmagnetic media where µ = µ0 ). The indices “1” and “2” denote the media with refractive indices n1 and n2, respectively. Given that the study concerns monochromatic planar G G progressive electromagnetic (MPPEM) waves, between E and B there now is the G G G kuE . well-used equation B = Z
Chapter 11. Reflection and refraction in absorbent finite materials
333
11.3.2. Fresnel equations for perpendicular polarizations (TE) 11.3.2.1. When kt is real (n2 ! n1 or n2 > n1 and T1 TA )
G Ei0
n1
G Bi0 Ti
G Ti ki
Tr Tr
G B0r
G kr G0
E r
y
G
x n12
Tt
G0 Et
n2
Tt
G B0t
G kt
z Figure 11.8 In TE mode where n2 > n1 or n2 < n1 and ș1 < șA .
By symmetry, the reflection and transmission fields conserve the same polarization directions as the incident wave. It is supposed that the fields thus are placed as indicated in Figure 11.8. The eventual dephasing with respect to the reflection or the transmission will be determined through an argument around the coefficients for reflection or transmission. For the configuration then it is possible to write equations for the continuity at the interface: x with respect to Ox: Ei0 + E 0r = E 0t , so that on dividing the two members by Ei0 ,
we have: 1 + rA = tA
(32);
x with respect to Oy: Bi0 cosși - B0r cosși = B0t cosș t , so that also
ki Ȧ
Ei0 cosși -
kr Ȧ
E 0r cosși =
kt Ȧ
E 0t cosș t from which with
k i = k r = k 0 n1 and k t = k 0 n 2 and division of two members by Ei0 : (33) (1 - r1 ) n1 cos T1 = t A n 2 cos T2 .
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Basic electromagnetism and materials
By substituting
rA = tA - 1
from Eq. (32) into Eq. (33), and with
și = ș r = ș1 and ș t = ș 2 , it can be immediately deduced that: tA =
2n1cosș1
and r A =
n1cosș1 +n 2 cosș 2
n1 cosș1 - n 2 cosș 2 n1 cosș1 + n 2 cosș 2
.
(34)
With the help of Eq. (28) ( n1 sin ș1 n 2 sin ș 2 ), we can eliminate n1 and n2 from Eq. (34), and then by multiplying the top and bottom of the preceding sin ș 2 yields: equations by n1 tA =
2 sinș 2 cosș1 sin(ș1 +ș 2 )
and r A =
sin(ș 2 -ș1 ) sin(ș1 +ș 2 )
The angles ș1 and ș 2 vary at most between 0 and
.
(34’)
S
, and sin T2 and cos ș1 2 are always positive just as is sin(ș1 +ș 2 ) as (ș1 +ș 2 ) also has a variation limited to between 0 and S. The tA is in fact always real and positive here, and the transmitted wave does not exhibit a dephasing with respect to the incident wave. 10.3.2.1.1. When n 2 > n1 With kt being real, in agreement with the relation n1 sin ș1 n 2 sin ș 2 , we have ș 2 < ș1 , and rA is always real and negative as the maximum variation in (ș 2 - ș1 ) is given by -
ʌ
< (ș 2 -ș1 ) < 0 . We can conclude therefore that a TE wave reflected by 2 the more refractive material undergoes at the reflection a dephasing ( MA ) such that exp(jMA ) = -1 , so that: MA = S (as exp(jS) = cos S + j sin S = - 1). In the limiting case, Eq. (34) shows that: n -n x for a normal incidence, where ș1 = 0, rA = 1 2 0 , so that in terms of moduli: n1 +n 2 rA = x for a glancing incidence, where T1
n 2 -n1 n1 +n 2 S
(35)
then rA 1 . 2 As T1 increases, the modulus of rA continuously increases from a value given by Eq. (35) up to unity. Finally, we obtain the representation given in Figure 4.9 for when n 2 > n1 .
Chapter 11. Reflection and refraction in absorbent finite materials
1
335
S
n2 > n1
n2 > n1
|rA|
MA
0
90°
T1
0
90°
T1
Figure 11.9. Modulus and phase angle of the reflection coefficient as a function of the angle of incidence (T1) for a TE wave with n2 > n1 .
11.3.2.1.2. When n 2 n1 and T1 TA (kt real) According to Eq. (28), T2 ! T1 and 0 (T2 T1 )
S
so that sin(T2 T1 ) ! 0 , 2 and according to Eq. (34'), rA is real and positive, so that when medium (2) is less refractive than medium (1) and when T1 TA , the reflected wave rests in phase with the incident wave. According to Eq. (34), we can see that as T1 increases, the
modulus of rA steadily increases from a value given by rA =
n1 -n 2 n1 +n 2
! 0 , obtained
for a normal incident wave (ș1 = 0) , up to the unit value given for the limit of incidence ( ș1 = șA ), as at this point ș 2 takes on the value representation of this zone. 1
S
2
. Figure 11.10 gives a
0
n2 < n1
n2 < n1
|rA|
MA
0
Tl
T1
90°
- S0
Tl
T1
90°
Figure 11.10. Modulus and phase angle of the reflection coefficient as a function of the angle of incidence ( T1 ) for a TE wave and with n2 n1 .
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Basic electromagnetism and materials
11.3.2.2. When kt is complex (n2 n1 and T1 ! TA ) , the wave is inhomogeneous and is as shown in Figure 11.6 The relations for the continuity at the interface (Figure 11.8) give: x with respect to Ox, we again find Eq. (32); and x with respect to Oy, we have: G G0 ·G ki 0 k 1 G G0 G 1§ G Ei cosși - r E 0r cosși = k t ×E t e y ¨ ¬ª k t ¼º y ¬ª k t ¼º z ×E t ¸ e y . Ȧ Ȧ Ȧ Ȧ© ¹ G0 As E t is directed along Ox (TE wave), the term on the right-hand side is reduced to
^
`
G G G G Ei0 '' G ª k t º ×E 0t e y . With ª k t º ik e and by simplifying with the two tz z ¬ ¼z Z Ȧ ¬ ¼z members of the continuity equation for Oy we have:
1
(1 - r A ) k1 cosT1 = i k ''tz t A ,
with according Eq. (30’) : k ''tz = k 0 n12 sin²ș1 - n 22
1/ 2
. We thus obtain
k cosș1 - ik ''tz and r A = 1 . (36) k1cosș1 +ik ''tz k1 cosș1 + ik ''tz For r A , the numerator is the conjugate complex of the denominator, for which r A =1 . The angle for dephasing at the reflection thus is given by: tA =
2k1cosș1
MA = Arg (r A ) tel que r A = |r A | exp(iMA ) = exp(iMA ). We
therefore
z = k1 cosș1 + ik ''tz r A = exp(iMA ) =
§ M · tan ¨ - A ¸ © 2 ¹
have
z
exp(iMA )
rA
z
,
and
then
by
making
U exp(i D ) , we also find that
z
ȡ exp(- iĮ)
z
ȡ exp(iĮ)
tan D
k ''tz k1 cosș1
= exp(- i 2Į) , so that
.
By using n 2 = n1 sinT1 (which makes it possible to replace n 22 in Eq. (30’) by n12 sin 2 TA ), we find that (in the shaded zone of Figure 11.10):
§M · tan ¨ A ¸ © 2 ¹ If T1
TA , MA
0 , and if T1
shaded part of Figure 11.10).
=-
S 2
sin²ș1 - sin²șA 1/2 cosș1 ,
MA 2
S 2
.
, thus MA
(37)
S (see the plot on the
Chapter 11. Reflection and refraction in absorbent finite materials
337
10.3.3. Fresnel's equations for parallel magnetic field polarizations 10.3.3.1. When kt is real (n 2 ! n1 or n 2 n1 and T1 TA ) . G G0 G0 kr Bi Br
G0 Ti Tr G0 G T T r i Ei Er ki y
G
x n12 Tt G0 Et
G Tt
B0t G kt
z Figure 11.11. TM wave with n2 > n1 or n2 < n1 , and ș1 < șA .
As detailed in Section 10.3.2, the fields retain the same polarization directions at reflection or transmission through symmetry. We thus can assume that the fields, once reflected and transmitted, are laid out as described in Figure 11.11. For this configuration, the equations for continuity at the interface (still with |Ti | = |Tr | = T1 and Tt T2 ) give:
x with respect to Ox: Bi0 + B0r = B0t , so in addition, k 0 n1 Ei0 Ȧ
k 0 n1 E 0r Ȧ
k 0 n 2 E 0t from which can be deduced that: Ȧ (38); and n1 (1 + r|| ) = n 2 t||
- with respect to Oy: - Ei0 cosș1 + E 0r cosș1 = - E 0t cosș 2 , from which: cosT1 (1 - r|| ) = t|| cosT2 . (39) From this can be deduced that: n 2 cosș1 - n1 cosș 2 r& and t& n 2 cosș1 + n1 cosș 2
2n1 cosș1 n 2 cosș1 + n1 cosș 2
.
(40)
By again using Eq. (28), it is possible to eliminate n1 and n2 from the above equations, so that they become:
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Basic electromagnetism and materials
r&
sinș1 cosș1 - sinș 2 cosș 2 sinș1 cosș1 + sinș 2 cosș 2
from which r&
tan(ș1 - ș 2 ) tan(ș1 + ș 2 )
Similarly, we obtain t &
.
sin2ș1 - sin2ș 2
2 cos(ș1 +ș 2 ) sin(ș1 -ș 2 )
sin2ș1 + sin2ș 2
2 sin(ș1 +ș 2 ) cos(ș1 -ș 2 )
,
(40’)
2 cosș1 sinș 2 . sin(ș1 + ș 2 ) cos(ș 2 - ș1 )
(40’’)
Using Eq. (40), it is immediately evident that t & is always real and positive as cosș1 and cosș 2 are both always positive ( ș1 and ș 2 being between 0 and
S
). 2 Hence the dephasing between the incident wave and the transmitted wave is always zero. With regard to the reflected wave, according to Eq. (40), it is possible to state that r& r& is a real magnitude and the dephasing of the reflected wave is dependent on the sign of r|| , which in turn is such that: sgn r|| = {sgn > tan(ș1 - ș 2 ) @}x{sgn > tan(ș1 + ș 2 ) @} . 11.3.3.1.1. When n 2 ! n1 : kt is real and ș 2 ș1 For a normal incidence (ș1 r&
0D ) , according to Eq. (40) r&
n 2 - n1 ! 0 so that n 2 + n1
r& is a magnitude that is both real and positive, and the dephasing M|| thus is
zero. In general terms though, ș 2 <ș1 and ș1 > 0, S / 2@ , (ș1 - ș 2 ) > 0, S / 2@ , and sgn > tan(ș1 - ș 2 ) @
always
is
positive. The result is such that {sgn r|| } changes sign with {sgn > tan(ș1 + ș 2 ) @} and r& goes from being positive to negative when ș1 ș B so that: (ș B + ș 2 ) =
S
. (41) 2 ș B , is called Brewster's angle and is such that
The angle of incidence, ș1 S tan(ș B + ș 2 ) = tan f , with r|| = 0 . 2 transmitted.
The
wave
therefore
is
entirely
Chapter 11. Reflection and refraction in absorbent finite materials
When ș1
S
ș B , so that T2 =
2
339
- TB , we have
S n1 sin TB = n 2 sin T2 = n 2 sin ( - TB ) = n 2 cosTB , from which: 2 n tan TB = 2 . (42) n1
At an air/water interface, we have TB = Arc tan 1.5 | 57 ° . With ș1 = 90°, we have r& =-1 and |r|| | = 1 and M& =S. So, to display the results, the plots of |r|| | = f(T1 ) and M|| = g(T1 ) are shown in Figure 11.12.
S
1
n2 > n1
n2 > n1 M||
|r|||
0
TB
T1
90°
0
TB T1
90°
Figure 11.12. Modulus and phase angle for the reflection coefficient as a function of ș1 for a TM wave with n2 > n1 .
11.3.3.1.2. When n 2 n1 and T1 TA (kt is real), thus here with ș 2 ! ș1 For r&
a
normal
incident
n 2 - n1 0 and r& n 2 + n1
wave
(ș1
0D )
according
to
Eq.
(40)
r& is a real and negative magnitude. The dephasing (M||)
thus is equal to S. When ș 2 ! ș1 , (ș1 - ș 2 ) > 0, -S / 2@ and sgn > tan(ș1 - ș 2 ) @ is always negative; the result is that r|| goes from being negative to positive when > tan(ș1 + ș 2 ) @
Basic electromagnetism and materials
340
changes
sign.
This
also
happens
ș1
when
ș B such that (ș B + ș 2 ) =
S 2
.
Simultaneously, M|| goes from S to 0. n We still have tan TB = 2 though, and also can state that ș B șA . In effect, n1 n2
sin TB
> sinTB , so that șA > ș B . n1 cos TB A representation of this is given in Figure 11.13. sin TA =
= tan TB
It is worth noting that for the example of air and water, n1 = 1.5 and n 2 = 1 . § 1 · § 1 · We therefore have TA = Arc sin ¨ ¸ | 42 ° and TB = Arc tan ¨ ¸ | 34 ° . © 1.5 ¹ © 1.5 ¹
S 1
n2 < n1
n2 < n1 M||
|r|||
0
TB
Tl
T1
TB Tl
0
90°
T1
90°
Figure 11.13. Modulus and phase angle for the reflection coefficient as a function of T1 for a TM wave, with n2 < n1 .
11.3.3.2. When kt is complex ( n 2 < n1 and T1 > TA ) In this situation it is not an easy task to reuse the reasoning given in Section 11.3.2.2; the projections made demand a symmetry around the angle Tt { T2 , which has no real physical significance as the wave does not propagate only along the Oy axis. Mathematically, we can check that T2 is not a real angle, as in effect, n sin T1 sin T2 = 1 sin T1 = , so that with T1 >TA we have sin T2 > 1 . n2 sin TA In cos T2
Fresnel's
equations,
in
the
place
of
the
usual
equation,
1 sin ²T2 , now with sin²T2 > 1 we must bring in for the term cos T2 a
purely imaginary value given by cos T2
i sin ²T2 1 . The negative sign is
Chapter 11. Reflection and refraction in absorbent finite materials
341
required by the need to obtain at a later point an attenuation of the wave in medium (2). We also can state that cos T2
i
sin ² T1 sin ²TA
1 , and with
n2 n1
sin TA , Eq. (40)
gives: r&
sin²TA cosș1 + j sin²T1 sin ²TA sin²TA cosș1 - j sin²T1 sin ²TA
.
(43)
Once again the numerator is the conjugate complex of the denominator, for which z | r || |² = 1 . By writing r|| = exp(iM|| ) = , and by making z z = sin²TA cosș1 + j sin²T1 sin ² TA = ȡ exp(i ȕ) , we obtain: r|| = exp(iM|| ) =
z z
=
§ M& · = exp( i 2E) , so that with tan ¨ ¸ = tan E : ȡ exp(- iE) © 2 ¹ ȡ exp( iE)
§ M& tan ¨ © 2
· ¸= ¹
sin²T1 sin ²TA cosș1 sin²TA
.
(44)
§ M& · S TA , then tan ¨ ¸ 0 and M|| = 0 . If T1 , then 2 © 2 ¹ § M& · cos T1 = 0 , tan ¨ ¸ = f , and M|| = S (see also the plot on the shaded area of © 2 ¹ Figure 11.13).
If T1
11.3.3.3. Comment
TB
Figure 11.14. A Brewster angle incidence for a TM wave at an air/glass window.
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Basic electromagnetism and materials
To ensure that a ray passes through a glass/air interface, as shown in Figure 11.14, without loss by reflection, the wave should be TM polarized and undergo a Brewster type incidence as ª¬ r& º¼ 0 (such as is used in Brewster lasers with a Brewster T1 TB
window).
11.3.4. Reflection coefficients and energy transmission 11.3.4.1. Aide mémoire: energy flux through a surface The energy flux, associated with the propagation of an electromagnetic field in a G material, is equal to the flux traversing the surface given by the Poynting vector ( S ). Equation (33’) of Chapter 7, Section 7.3.3, shows that for a MPPEM wave in nonmagnetic materials the average value for the Poynting vector is given by G G G G G G 1 S R(E m u Bm ) , so that with ȦBm = k×E m [Eq. (13) of Chapter 7] we 2µ0 have
G S =
1
G G 2 k E m , and with
İ c = 0 , we can write that µ0Ȧ k0 1
2µ0 Ȧ G G İ0c k G 2 S E m . With our notations, and for incident, reflected, and transmitted 2 k0 wave, the associated average value of the Poynting vector is, respectively: G G G G G İ0c ki G 0 2 İ0c k r G 0 2 G İ0c k t G 0 2 Si Ei , Sr Er , S = Et . 2 k0 2 k0 2 k0
The energy flux across the surface 6 is the energy transmitted per unit time through the surface. In effect, it also represents the power transmitted by the wave under consideration.
T1 x
T1 n1
6
T2
y
h
G n12
n2
z
Figure 11.15. Reflection and transmission of an incident flux traversing 6 .
Chapter 11. Reflection and refraction in absorbent finite materials
343
The radiated power of the wave traversing an interface with surface (6) from a medium with an index denoted by n1 toward a medium with an index n2 (Figure 11.15) thus is given by:
G G G G 2 İ0c ki G 0 2 G İ c < Pi > = Si n12 ¦ = Ei n12 ¦ = 0 n1 Ei0 Ȉ cosș1 2 k0 2 G 2 G G G0 2 İ c k G0 G İ c < Pr > = Sr n12 ¦ 0 r E r n12 ¦ 0 n1 E r Ȉ cosș1 2 k0 2 G G G İ c k G0 2 G İ c G 0 2 k' < Pt > = St n12 ¦ 0 t E t n12 ¦ 0 E t 6 tz . 2 k0 2 k0
(45)
11.3.4.2. Equation for the reflection(R) and transmission (T) coefficients: total energy The titled coefficients are defined so as to give positive values, so that we have: Pt Pr R=and T = . (46) Pi Pi From this can be determined with the equations in Eq. (45) that: G0 2 G0 2 Er Et k 'tz k 'tz 2 2 R= = r and T = = t . G0 2 G 0 2 k n cosș k n cosș 0 1 1 0 1 1 Ei Ei
(47)
It is notable that in the case of the transmitted power, T z | t |² . This is due to the fact that the way in which the power is transported depends on the different cross sections, so that ¦ i = ¦ cos T1 for an incident wave and
¦ t = ¦ cos T2 for a transmitted wave (while for incident and transmitted powers, ¦ i = ¦ r = ¦ cos T1 ). With respect to the sum of the energy, we can calculate this by using a closed surface, such as a parallelepiped of a given height (h) such that h o 0, and then drawn around the level of the surface (Figure 11.15). In physical terms, for an isolated system, the incoming radiation power must be equal to that leaving, so: x the incoming radiation power is that of the incident wave, such that G G G n12 :
x the outgoing power radiated has two components:
Basic electromagnetism and materials
344
G x power radiated by the reflected wave with respect to - n12 G G Sr n12 ¦ = - < Pr >
G x the power radiated by the transmitted wave with respect to n12 G G
.
(48)
Comment: The transmission coefficient for a homogeneous transmitted wave (n1 n 2 or n1 ! n 2 and T1 TA ) . G G' In this case, k t is real and equal to k t . With k 'tz k 0 n 2 cos T2 , we have T
2
t
n 2 cosș 2 . n1 cosș1
(49)
11.3.4.3. Representation of the reflection coefficient as a function of the angle of incidence 1 1 n2 > n1 R R
§ n1 - n 2 ¨¨ © n1 + n 2
· ¸¸ ¹
2
RA
§ n1 - n 2 ¨¨ © n1 + n 2
R|| 0
TB
T1
90°
Figure 11. 16 a. Factors in the reflection of energy: R A = |rA |² and R || = | r|| |² as a function of
T1, for n2 ! n1 ( with rA and r|| from Figures 11.9 and 11.12)
T B = 57 ° for the air/glass interface.
0
· ¸¸ ¹
2
RA R||
TB Tl
T1
90°
Figure 11.16 b. Factors in the reflection of energy:
R A = |rA |² and R || = | r|| |² as a function of T1, for n2 n1 ( with rA and r|| from Figures 11.10 and 11.13). TB = 34 ° for an air/glass interface, and TA = 42 ° .
Taking the equation R = r
2
into account, we can use the representations given in
Figure 11.15 for when n 2 ! n1 , or n1 ! n 2 .
Chapter 11. Reflection and refraction in absorbent finite materials
For normal incidence, ș1 = ș 2 = 0, and R
At the same time, T
t
2
n2 n1
§ 2n1 ¨¨ © n1 +n 2
2
· n2 ¸¸ ¹ n1
§ n1 - n 2 ¨¨ © n1 + n 2
4n1n 2
n1 +n 2 2
345
2
· ¸¸ . ¹
1- R .
It can be seen in the plots of Figure 11.16 that when n 2 ! n1 the reflection is always partial and relatively weak, except in the neighborhood of the glancing angle ʌ (ș1 | ) . This is the reason why the air/water interface is always transparent 2 ( n 2 | 1.33) except when there is a glancing angle, or under conditions of reflection. Under a Brewster angle, only a wave polarized perpendicularly to the plane of incidence (TE waves) is partially reflected. At this incidence, the TM wave is entirely transmitted (R || = 0 when ș1 = ș B ).
polarized TM wave for 2nd mirror
1st mirror
Wave loss after 2nd reflection at Brewster angle TB
TB incident natural light
2nd mirror with incidence plane normal to that of the 1st
Figure 11.17. Malus's experiment.
Under natural light, the incident wave exhibits all polarization states, and for an incidence with the Brewster angle it is only waves polarized perpendicularly to the plane of incidence which is reflected (TE wave). It is this result that explains physically Malus's law, which demonstrates the polarization of light with the help of two mirrors set at a Brewster angle, so that the planes of incidence are orthogonal. The two successive reflections are at Brewster angles so that after the first reflection
346
Basic electromagnetism and materials
of natural light there remains only a component that is polarized perpendicularly to the plane of incidence of the first mirror (thus a TE wave for the first mirror). The incidence plane of the second mirror is perpendicular to that of the first, and the incoming wave is seen as a TM wave. Following a reflection at the Brewster angle on the second mirror, there is a complete extinction of the wave.
11.3.5. Total and frustrated total reflection
G G When n1 ! n 2 and ș1 > șA , we have k 'tz = 0 and k 't = k 'ty ey ; there no longer is a propagation along Oz and only propagation along Oy persists. As R = | r |² = 1 , we have T = 1 - R = 0 , while at the same time with t z 0 (see Eq. (36), for example). T1 > TA
G ki
TA
n1
k 't = k 0 n1 sinș1
T1
y
n2
G n 2 < n1
k ''tz
evanescent wave: - propagation along Oy - attenuation for Oz
z Figure 11.18. The evanescent wave and frustrated total reflection.
Therefore, even though there is a transmitted wave, as t z 0 , the transmission coefficient for energy is zero. This is because the propagation through G medium (2) can occur only at the interface (along e y ). The wave itself is evanescent (Figure 11.18 being the “completed” Figure 11.6) with wavevector G components ( k t ) given by Eq. (30’). By substituting the value for the components into Eq. (21), we find that the transmitted wave has the form: G G G0 E t (r,t) = E t exp( - k ''tz z) exp ªi kiy y - Ȧt º , ¬ ¼
with k ''tz = k 0 (n12sin²ș1 -n 22 )1/2
k0n2 (
n12 n 22
sin²ș1 - 1)1/2 .
G G 2 The intensity of the transmitted wave, proportional to E t (r,t) , is thus of the form:
Chapter 11. Reflection and refraction in absorbent finite materials
347
G 2 2z 1 1 n2 I t (z)= E0t exp(-2k ''tz )= I t (0) exp(- ) , with į = = ( 1 sin²ș1 - 1)- 1/2 į k 0 n 2 n 22 k ''tz where G represents the degree of attenuation (when z =
G 2
, the wave intensity is
divided by e). The wavelength in medium (2) is given by O 2 = į=
Ȝ0 n2
, so that also we have:
O 2 n12 ( sin²ș1 - 1)- 1/2 . 2 2S n 2
If T1 increases, sin T1 also increases, as (
n12 n 22
(50)
sin²ș1 - 1) , while (
n12 n 22
sin²ș1 - 1)-1/2
decreases with G and the evanescent wave penetrates less medium (2). Numerically, for the glass/air interface, and at what is practically a glancing ʌ angle, (ș1 | ), we find that G | 0.14 O 2 . 2
glass (1) (n1)
TA interface I
air (n2)
h|G
interface II glass (2) (n1)
Figure 11.19. Experimental demonstration of frustrated total reflection.
The presence of an electromagnetic wave in a thickness of the order of G of the second medium (such that T = 0) can be demonstrated by using the setup shown in Figure 11.19. At the level of the first interface (interface I) the wave penetrates to a depth of the order of G and is then transmitted to the level of the second interface II that is such that its index [glass (2) with index n1] is higher than that of the first medium (air with index n2). In other terms, following its passage from interface I, the wave is attenuated over a depth of h | G and what “remains”' at this depth is then transmitted to the level of the interface II, which in turn does not follow the conditions of total reflection. The result is that of frustrated total reflection.
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Basic electromagnetism and materials
In order to carry out such an experiment, a total reflection prism can be used with angles equal to 45°, which is greater than the 42° required for the glass/air interface ( ș1 = 45o > ș A ). By placing a second prism at a distance h | G from the first, as described in Figure 11.20, a frustrated total reflection occurs exhibited by the presence of part of the incident wave in the right-hand side. However, as mentioned above, the two prisms must be placed a distance apart given by h | G | 0.14 O 2 , which means that h | 0.1 µm for waves in the optical domain. This requires extremely delicate handling and can be made easier by the use of waves with wavelengths of the order of centimeters thus requiring h | several millimeters (which is easy with paraffin prisms). The results obtained for electromagnetic waves can be extended to waves associated with material particles. Frustrated total reflection can be considered that of a tunneling effect, in this case applied to photons. h
air
glass incident wave
T1 > TA
wave after frustrated total reflection
Figure 11.20. A total reflection prism used to demonstrate frustrated total reflection.
11.4. Reflection and Absorption by an Absorbing Medium 11.4.1. Reflection coefficient for a wave at a normal incidence to an interface between a nonabsorbent medium (1) (index of n1) and an absorbent medium (2) (index of n 2 ) The index of medium (2) is complex and given by n 2 = n '2 + i n '2' . For its part, the incident wave is assumed to be normal to the interface so that T1 = 0° and sin T1 = 0 , and the wavevector of the transmitted wave thus can be written, with k 'ty = k 0 n1 sinș1
0 , as:
Chapter 11. Reflection and refraction in absorbent finite materials 1/2 ' k tz = k 0 R n 22 - n12 sin²ș1 = k 0 n '2 ° G ° ' '' G k t = k t + ik t ez , where ® 1/2 °k '' = k Im n 2 - n 2sin²ș = k 0 n ''2 tz 0 1 1 2 °¯ G G G so that k t = k 0 n '2 + n ''2 ez k 0 n 2 ez . G G The EM field associated with the transmitted wave E t , Bt
349
G G0 E =E exp -k 0 n ''2 z exp i[k 0 n '2 z - Ȧt] °° t t G ® G kG ×E G °B = t t = n 2 eGz ×E . t t Ȧ c ¯°
thus is given by:
G G With the index n 2 being complex, Bt is no longer in phase with E t . Taking into account the form of the fields, we can equally state that this MPPEM wave Ȧ c propagates along Oz with a phase velocity given by vM = and is = ' k 0 n 2 n '2 attenuated exponentially along Oz in the absorbent medium (2). The condition of continuity at the interface of the tangential components for the G G fields E and B results in the same values as those cited previously for reflection and amplitude transmission coefficients, while nevertheless noting that n2 is in a complex form and that: r=
n1 - n 2
=
n1 - n '2 - in ''2 n1 +
n1 + n 2
n '2 +
in ''2
and t =
2n1 n1 + n 2
=
2n1 n1 + n '2 + in '2'
.
The arguments with respect to r and t, respectively, give the dephasing of the reflected and the transmitted wave with respect to the incident wave. For the reflection and transmission coefficients in terms of energy, we also obtain: R= r
2
2
n - n + n = n + n + n 1
' 2
1
' 2
2
''2 2 ''2 2
and
T=1-R=
4 n1 n '2
n + 1
n '2
2
+
. n ''2 2
11.4.2. Optical properties of a metal: reflection and absorption at low and high frequencies by a conductor See Problem 1 of the present chapter.
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Basic electromagnetism and materials
11.5. The Antiecho Condition: Reflection from a Magnetic Layer; a Study of an Antiradar Structure; and a Dallenbach Layer 11.5.1. The antiecho condition: reflection from a nonconducting magnetic layer 11.5.1.1. Reflection from a magnetic layer This follows on from the study in Section 11.3.2 on TE polarization by reflection at an interface between media (1) and (2) with the difference that here the media have a permeability different to that of a vacuum such that media (1) and (2), respectively, exhibit permeabilities µ1 and µ2. It is assumed that the media are sufficiently insulating so that the density of free charges is negligible so that VA = 0 and jA = 0. Under such conditions, the continuity conditions for the interface mean that G G G G G G B1t B2t E1t E 2t and H1t H 2t and in turn with in addition B1t = B2t as µ1 µ2
specifically to this case µ1 z µ2 (see also the end of Section 11.3.1). Equation (32) then becomes: G x along Ox: Ei0 + E 0r = E 0t in an unchanged relation due to the conservation of the continuity equation. By dividing the two terms by Ei0 , we also find 1 + rA = tA , (51) and x along Oy:
ki
Ei0 cosși -
kr
E 0r cosși =
kt
E 0t cosș t .
(52) Ȧµ1 Ȧµ1 Ȧµ2 With the media being nonconducting, i.e., V | 0, we can state that in each medium, according to Eq. (5) of Section 8.5.1, k i = k r = Z İ1 µ1 and Oy becomes:
H1 µ1
k t = Z İ 2 µ 2 . The preceding equation with respect to
Ei0 cosși -
H1 µ1
E 0r cosși =
With Eq. (6) from Section 8.5.2, i.e.,
H2 µ2
1
H1
Z1
µ1
E 0t cosș t . , and
1
H2
Z2
µ2
, we have:
0 Z1-1Ei0 cosși - Z1-1 E 0r cosși = Z-1 2 E t cosș t .
By dividing the two terms by Ei0 , we obtain: Z1-1cosși - Z1-1 r A cosși = Z-1 2 t A cosș t , so that with the same notations as those used in Section 11.3.1, (1 - r A ) Z1-1cosș1 = t A Z-1 2 cosș 2 ,
(52’)
Chapter 11. Reflection and refraction in absorbent finite materials
351
From Eqs (51) and (52’), we can determine using a method analogous to that in Section 11.3.2 (where Z1-1 was substituted for n1 and Z-1 2 for n2): cosș1 cosș 2 Z1 Z2 rA = , so that following multiplication of top and bottom with cosș1 cosș 2 + Z1 Z2 Z1Z2 we have: rA
Z2 cosș1 - Z1 cosș 2 Z2 cosș1 + Z1 cosș 2
, from which according to Eq. (47) we have:
RA =
Z2 cosș1 - Z1 cosș 2 Z2 cosș1 + Z1 cosș 2
2
.
(53)
G G As the incidence is normal (with directions k i and ez being merged) the notion of incidence plane loses its significance (otherwise defined by the vector G G G directions k i and n12 { ez ), and we can state for the energy reflection factor for whatever polarization direction of the incident wave, that: R =
Z1 - Z2 Z1 + Z2
2
.
(54)
1.5.1.2. The antiecho condition Here the reflected wave is annulled so that R = 0. For this to occur, taking Eq. (54) into account being applicable for a great distance where the wave exhibits a normal İ İ2 , and hence: incidence on the target, it suffices that Z1 = Z2 , that is, 1 µ1 µ2
İ1r
İ 2r
. (55) µ1r µ2r So that this condition is true, a specific sort of coating must be applied to medium (2). If medium (1) is air, then Eq. (55) means that between the dielectric permittivity and magnetic permeability of the second medium, there is a simple equation given by H r = µr . (55’) Comment: the antiecho condition for EM waves is just one part of the conditions under research in optics in order to find a material that is completely anti-reflective.
352
Basic electromagnetism and materials
In such a case, the condition R = 0 means that R =
n1 - n 2
2
n1 + n 2
0 , so that
n1 = n 2 . This condition has no sense in itself, as it would mean that the media (1) and (2) would be identical and that therefore there would be no interface. One method though is to insert between the two media a layer of a given thickness and index n such that n1 < n < n 2 and then find the conditions for which a destructive interference is formed between the two reflected waves, one being at the interface of the materials with indices n1 and n and the other being at the interface of the materials with interfaces n and n 2 .
11.5.2. The Dallenbach layer: an anti-radar structure 11.5.2.1. The stealth concept
G Ei G ki O
G Et G kt
d
G Er
air (H0, µ0)
G kr
y dielectric (H) and/or magnetic (µ) medium which is slightly conducting, although not equal to zero
metal
z Figure 11.21. Antiradar structure.
Assuming that a wave with a frequency denoted by Z has a normal incidence to a metallic surface, the reflection is total. In order to make the metallic layer stealthlike, it is covered with a dielectric and/or magnetic material adapted so as to satisfy the antiecho condition for an air/coating interface. In addition, the wave transmitted in the layer also must be totally attenuated in order to suppress the total reflection that would be produced by the metallic layer. The attenuation is obtained with a slightly conducting layer of sufficient thickness (d) as indicated in Figure 11.21.
Chapter 11. Reflection and refraction in absorbent finite materials
353
11.5.2.2. Condition (1): antiecho at the air/dielectric-magnetic interface The classically used absorbent materials are generally composites based on carbon, iron carbonyls, or ferrites (see also Chapter 4, Section 4.2.5.2). These materials do, however, suffer from certain inconveniences such as their weight or their mechanical rigidity due to the high number of charges necessary to reach the required absorption. These composite materials are presently being replaced by conducting polymers, which can be acquired through doping a conductivity appropriate to the system.
11.5.2.2.1. Equation for the reflection coefficient In the system under study, the conductivity (ı) is such that V z 0 and in the layer k = k t so that from Eq. (5) of Chapter 8, Section 8.5.1, where “electrokinetic” notation was used to establish the equation we have: 1/ 2
ı · § k t = Ȧ µİ ¨ 1 - i (56) ¸ . Ȧİ © ¹ Under normal incidences, the preceding Eq. (52) can be written for the air/layer Ȧ interface (where for air k i = k r = k 0 = ): c k0 µ0 k0 µ0
Ei0 -
k0
E 0r =
µ0
kt
1 - r =
µ
k0 r
µ0 k0 µ0
+
kt µ
E 0t . By dividing the two terms by Ei0 it is determined that:
t , from which with the help of Eq. (51) we determine that: kt µ , so that also r kt
µk 0 - µ0 k t µk 0 + µ0 k t
µ
µr µr
Ȧ c Ȧ c
- kt .
(57)
+ kt
11.5.2.2.2. Approximate calculation using Ȟ = 5 GHz, H r = 15, V = 5 x 10-1 :-1m-1 = 5 x 10-3 cm-1 ) Figures 11.22a and b give a qualitative indication of the evolution of the conductivity and dielectric permittivity of polyaniline films with varying levels of doping as a function of frequency. On going from A to D the plots are for increasing levels of doping. It can be seen that for sufficiently high levels of doping, quite high values can be attained.
Basic electromagnetism and materials
354
V(:-1cm-1)
Hr
D
Vdc
105
10-3
D C
Vdc
(b)
103
10-6 (a)
B
A
10-9 1 kHz
1 MHz
C
B 10
frequency 1 GHz
A
frequency
1 kHz
1 MHz
1 GHz
Figure 11.22. Evolution of V and Hr’ as a function of frequency for doped polyanilines.
Using ı Ȧİ
=
the
values
given
5 x 10-1 2ʌ x 5 x 109 x 15 x 8.85 x 10-12
in
Eq.
(56)
for
kt ,
the
| 1.2 x 101 is very much less than 1. With a
ı · § limited development, from Eq. (56) for k t we obtain k t Ȧ µİ ¨ 1 - i ¸. 2Ȧİ ¹ © ı By making E , we thus can write that: 2Ȧİ Ȧ (56’) k t Ȧ µİ 1 - iȕ µ r İ r 1 - iȕ . c The reflection coefficient given by Eq. (57) then takes on the form:
r
µr -
µr Hr (1 iȕ)
µr +
µr H r (1 iȕ)
term
1, so that finally r
1+
İr µr İr µr
(1 iȕ) . (1 iȕ)
Numerically, as E | 10-1 << 1, we have (1 - iE) | 1 , and r can be reduced to:
Chapter 11. Reflection and refraction in absorbent finite materials
355
2
ª İr º 1«1» µr µr » İ « (58). The condition R | « r| 0 thus gives r = 1 , » µr İr «1+ İ r » 1+ « µr µr »¼ ¬ so that H r = µr = 15 . We once again find the condition of Eq. (55'), which is quite normal as the numerical calculation lead us to neglect the component due to conductivity, just as was assumed hypothetically in order to arrive at Eq. (55'). İr
11.5.2.3. Condition (2) to predict total reflection at a metal: attenuation and depth of penetration of the wave into the coating G G The transmitted wave is given by E t = t Ei exp(i[Ȧt - k t z]) using electrokinetic notation. According to Eq. (56’) kt is of the form k t = D -iJ, with Į = and Ȗ = Ȗ=
Ȧ c
Ȧ c
Ȧ c
µr İr
ȕ µ r İ r , so that:
µr İr
ı 2Ȧİ
we finally have J
=
ı
µr
2cİ 0
İr
ı 2cİ 0
. With the preceding condition (1), that is, H r = µr ,
. The wave is in the form:
G G 1 E t = t Ei exp - Ȗz exp(i[Ȧt - Dz]) , so that by making į = , we have Ȗ G G § z· E t = t Ei exp ¨ - ¸ exp(i[Ȧt - Dz]) . © į¹ The wave is attenuated little by little along its propagation through the coating. When z = G, the wave amplitude is divided by e = 2.7 . This attenuation is obtained at a penetration depth (G) such that: 1 2cİ 0 į= , Ȗ ı which here in numerical terms means that G | 1.3 cm .
11.5.2.4. Conclusion It is notable that the thickness calculated is for ı measured at a given frequency, while the law for the evolution of current as a function is frequency is typically of
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Basic electromagnetism and materials
the form V(Z) = V(0) + AZs (see also Figure 11.22 for a sufficiently doped material such as curve C or D). The absorption thus is limited to a certain band width. In order to absorb over a large enough band, a structure based on one single layer (called the Dallenbach layer) has to be improved upon by using multi-layer absorbing structures, in which each layer is optimized to obtain a minimum reflection coefficient for a given band width. A final restriction is imposed by the necessity of having thick enough layers to be practicable, to the extent that absorbing paints might be used.
11.6 Problems 11.6.1. Reflection and absorption at low and high frequencies by a conductor The optical properties of a metal are treated through a series of questions (and answers!) as given below and follow on from the problem studied in Section 8.6.2, concluding with a look at the optical properties of a metal as derived from the relationships for dielectric permittivities as established in Section 8.6.2 from the equation for electronic polarization. Recalling Section 8.6.2, the equations for the complex dielectric permittivity of a metal were: i V(0) x H r = 1 [question 3(c)] Eq. (1); ZH0 [1 iZW] x
Hr = 1 - i
Z2p W Z[1 iZW]
[question 3(d)] Eq. (2).
The metal under consideration is copper, and in the equations for Hr:
x W is the relaxation time, which is such that W = 10-14s ; x V(0) is the conductivity for a steady (DC) state, which will be taken equal to 6 x 107 :-1m-1; and Nq² x Zp is the plasma angular frequency, that is defined by the equation Ȧp 2 = , mİ 0 and is typically Ȧp 1016 rad s-1 . It is worth noting that H0 =8.85 x 10-12 MKS . The incident wave at medium (1) is a MPPEM wave polarized in parallel with G G Ox in the form E 0 = E 0 exp(-ikz) in electrokinetic notation as the applied field was G G written in the form E E 0 exp(iZt) from the beginning of the problem in Chapter 8, Section 8.6.2. The propagation of the wave thus is much as usual, along positive values of z. The wave is considered to arrive at a normal incidence, so T1 = 0 and the reflection is detailed at the air/metal interface where each, respectively, have indices n1 such that n1= 1, or more simply n = n' - i n'' .
Chapter 11. Reflection and refraction in absorbent finite materials
357
1. This part considers the optical properties of a metal at low frequencies. 1 (a) This domain is defined by Z << . From Eq. (1) for H r , give the simplified W form which H r takes. (b) By using orders of magnitude of different parameters, show that the expression for H r can be reduced to a single and explainable term. (c) Calculate the complex index for n n' - i n'' (d) Determine the reflection coefficient for the amplitude r (which is thus in the n -n following the preceding Section 11.4.1). Equally, evaluate the form r = 1 n1 +n reflection factor in terms of energy. (e) Give k as a function of n inside the metal. Study the form of penetration of the electric wave in the metal by writing the form of the wave in there as G G G0 G E t = E 0t exp(iȦt) = E t exp(i[Ȧt-kz]) , where the complex amplitude E 0t is G G0 thus of the form E 0t E t exp(-ikz) . In order to do this, determine the depth (G) of the penetration of the wave, which we will define here by using the term z for attenuation in the form exp( ) . G (f) Establish the form of the energy transmission factor (T). Give this as a function of n' or n", and then as a function of G (the so-called Hagen-Rubens equation). Give a numerical value for G and then also for T given that Q = 1 GHz. From this result, make a conclusion.
2. These questions now concern a zone of slightly higher frequencies which are 1 defined by Z << . W (a) Using Eq. (1) for H r , give the simplified form of H r in this domain. (b) When Z << Zp , determine the form of H r as well as that of the MPPEM wave in the metal. Calculate the reflection coefficient for a normal incidence at an air/metal interface. (c) When Z ! Zp (high frequencies), give the range of variation possible for H r . Give a value for H r when Z !! Zp , as well as that of the reflection coefficient for an incidence normal to the air/metal interface.
3. Give a recapitulative scheme of the reflection and transmission properties of a metal in the EM spectrum.
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Basic electromagnetism and materials
Answers 1. Low frequencies Z < (a) We use H r = 1 -
i
1
W iV(0)
ZH0 [1 iZW]
ZW<<1
,
1
and with
Z W<< 1 , iZW | 0 (negligeable with respect to 1), we have H r = 1 - i
V(0) ZH0
.
(b)
V(0) = 6 x 107 :-1 m-1 , 1 Z << = 1014 s-1 , Z H0 << 103 W ª V(0) º « » «¬ ZH0 »¼ i min With
V(0) ZH0
V(0)
V(0)
¬ªZH0 ¼ºi max
103
½ ° ¾ ° ¿
= 6 x 104 >> 1 .
always being greater than 1, in the low-frequency range we have: Hr | - i
(c) Given H r
n = n' - i n'' =
2
n | i V(0) ZH0
i
e
V(0)
V(0)
ZH0
ZH0
i
e
V(0) ZH0 S 2
S 4
.
, from which we deduce that i
, so that with e
S 4
S S 1 = cos - i sin = (1 - i) , we 4 4 2
obtain: n = n' - i n'' =
V(0) 2ZH0
(1 - i) , from which n' = n'' =
V(0) 2ZH0
.
ª V(0) º V(0) >> 1 >> 1 and n' >> 1 just as n'' >>1 . (d) We have « » ZH0 «¬ ZH0 »¼ i min
Chapter 11. Reflection and refraction in absorbent finite materials
The result is that r =
n1 - n n1 + n
=
1 - n' + in''
|-
1 + n'- in''
n' - in''
eiS and on reflection
1
n' - in''
359
there is a dephasing by S while the reflecting power is in the neighborhood of 1, i.e., R = |r²| | 1 . The ability to reflect energy is given by R= r
2
1- n + n = 1+ n + n '
2
'
''2
2
n '2 +n ''2
n '2 +n ''2
''2
=1
where n' >>1 . (e) With k =
Z c
Z
n = k' - i k'' , we have k''
c
n''
Z
V(0)
1
Ȧ ı(0)
c
2ZH0
c
2İ 0
E 0t eiZt
A wave that propagates in the form E t
.
E 0t eikz eiZt can be written for
this frequency range as E = E 0t ei(Zt k ' z) e k '' z . The equation carries a term for the propagation as in ei(Zt k ' z) , and a term for attenuation as in e k ''z that also can be
rewritten as e
z G
where G
1 k ''
c
2İ 0 Ȧ ı(0)
. With G representing the depth at
which the electric wave is attenuated by the ratio
1 e
, for distances greater than
several G , the wave is practically zero. As G
c
2İ 0 Ȧ ı(0)
, we can see that as Z increases, G decreases. For the
highest frequencies in this region ( Z d
1
| 1014 Hz ) the EM waves are localized at
W the surface of the conductors, in an effect called the skin effect.
(f) The equation for T is deduced from that of R, as in T = 1 - R = With n'' = n' = n >> n1 = 1 , the result is that T
2 n
2Ȧį
2ʌ
2ʌ
c Ȧ which is known as the Hagen-Rubens equation.
k0
2 k 0 į , so that with Ȝ 0
c
(n1 + n')2 + n ''2
.
, so in addition T 0 (as n >>
1). In more precise terms, when n' = n' = n , and with G T
4n1n'
1
c 1
k ''
Ȧ n''
, we also have
, we have T
4ʌ į Ȝ0
,
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Basic electromagnetism and materials
The energy received by the metal is transmitted by carriers to the lattice that dissipates the energy through the Joule effect. So, the factor T is often called the absorption power of the metal. In numerical terms, for copper, V(0) = 6 x 107 :-1m -1 , Z = 2 S 109 rad/s, H0 = 8.85 10-12 MKS so that G As O 0
c Ȟ
c
2İ 0
2.06 µm .
Ȧ ı(0)
0.3 m (region of centimeter wavelengths), the Hagen-Rubens
formula gives T | 8 x 10- 5 o 0 , to which R | 0.99990 o 1 . Hence the use of microwave frequency radars for police use in determining car speeds!
2. (a) This part uses for H r the general formula (2) obtained from question 3(d) of Section 8.6.2, for metals, as in: Z2p W Hr 1 i . Z[1 iZW] When ZW>> 1 , we have H r | 1 i (b) When Z
n
2
Z2p Z²
i
2
that n' = 0 and n'' =
Zp2 Z² Zp
Z2p Z²
Z2p W iZ² W²
1
Z2p Z²
.
1 , we have H r =H r"
Z2p Z²
< 0 . For the index then
must be true. The result is that n = n' - i n'' = ± i
Zp Z
so
(only the positive solution is physically acceptable and it Z represented the only available absorption of the wave in the medium). With k' = 0 , the wave defined by form E t
Et
E 0t eiZt
E 0t eikz eiZt takes on he
E 0t e k ''z eiZt , and as there is no longer a term for the propagation, in
effect the signal no longer propagates through the metal. The wave oscillates in a standing position in the neighborhood of the interface, and in practical terms the wave is reflected, as shown in the following calculation. Zp 1 + in'' ( | 1 with n'' 1 ), which is such In effect, r 1 - in'' Z that |r| =1, so R = |r²| =1.
Chapter 11. Reflection and refraction in absorbent finite materials
(c) If Z > Zp , so that that 0 <
Z2p Z²
Z2p Z²
< 1 , and with
Z2p Z²
361
> 0 , we can state more succinctly
< 1 and 0 < Hr < 1 , and that H r is positive, real, and between 0 and 1. Z2p
1 , we have H r = H r | 1 . Thus the metal behaves Z² as if it is a vacuum without charges simply because the latter cannot follow such a high frequency. Z As H r = n² , we have n n c 1 ; k' = n c and n cc 0 just as k''= 0 (no c n1 - n 1 1 0, and R = 0 and T = 1 , and the transmission absorption). So r = n1 + n 1 1 is perfect. When Z Zp , so that
3. To sum up the characteristics of a metal: Radio frequency
µ-wave
IR
1 Question 1.
Z <<
1
so is near low
IJ
Z >> 1/W in the region of higher frequencies visible UV Ȧp=1016 rad/s 1014 s-1
1/ W <
2(c) Z Zp THF region
2 (b) Hr < 0 :
(high energy)
frequency region
n'= 0 and k'= 0
H r | 1 and the metal acts as
k' = k'' z 0
a standing wave oscillates without propagation. Total reflection and
For a good metal
(V (0) >> ZH 0 ) dephases by S at reflection and R | 1 . Transmitted wave attenuated, and by z | G (skin thick) is practically zero (more so when Z is high, as is µwave). Frictional forces nonnegligible. Absorption and Joule effect transmits energy through network.
if a vacuum, with
R = 0, T = 1 and total transmission
R=1 Z > Zp , 0
W
same as plasma where friction is negligible
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Basic electromagnetism and materials
11.6.2. Limited penetration of Hertzian waves in sea water Here the system is based on medium (1) which is air (which in electromagnetic terms is assumed to be the same as a vacuum) separated by a horizontal plane (Oxy) from medium (2) which is sea water. For Hertzian frequencies (the maximum of which is considered to be 100 MHz) we can take for the real parts of the dielectric permittivity (Hr) and the conductivity (ı) values measured for a continuous stationary regime, as in H r = 81 and V= 4 :-1m-1 . In addition, medium (2) has a permeability equal to µ0 and is assumed to be non-magnetic.
1. The Maxwell equations (a) Give the equation for the total current in medium (2). (b) Write the four “Maxwell equations” for medium (2). (c) Using a rapid calculation (resembling that used for a vacuum), establish from the Maxwell equations the equation for the propagation of an electric field (equation of partial derivatives followed by the electric field). (d) Toward which forms does this equation tend on going from a nonconducting medium to a medium where conductivity predominates? 2. The question concerns the search for a solution to the electric field in the form of G GG G G a MPPEM wave where E = E 0 exp(i[Ȧt - k r ]) and k is the vector of the complex wave. (a) With the help of the equation for propagation, establish the equation that brings together k 2 , İ r , and ı . (b) From the general equation that exists for wavenumber and index, give the equation that ties n 2 , İ r , and ı .
3. Numerically compare the value of two terms that intervene in n 2 . From this determine the approximate and literal form for n (the imaginary number which is such that n = n' - i n'' ) that can be expressed as a function of (
V
2
1
ZH0 ) 2 .
4. This question concerns, within the approximation of the preceding question, the propagation of a wave with an incidence normal to the air/sea interface. (a) Give the theoretical form of the electric field that can be expressed as a function of the parameter: G= (
c Zn c
)=(
2 µ0 ZV
1
)2 .
(b) At which order of depth (when Q = 1 MHz and Q = 100 MHz ) does the wave penetrate the water. Give a conclusion from the result.
Chapter 11. Reflection and refraction in absorbent finite materials
363
(c) What is its phase velocity (literal expression given as a function of the frequency Ȟ and then a numerical value for when Ȟ = 100 MHz Q = 100 MHz ). Answers 1. (a) Taking into account the data given for the problem, for the medium with a real conductivity (ı) and a real dielectric permittivity (Hr), we can state that the total current density in the medium is a sum of the conduction current and the G G G G G G wD G wE ıE + H0 H r . displacement current, so JT = Jc + J D = ıE + wt wt (b) Here, with UA 0 , G G divE = 0 (1), divB = 0 (3) G G JJGG wB JJG G G G İ r wE rotE=rot B=P0 JT = µ0 ıE + (2), . (4) c² wt wt (c) The calculation of the rotational of Eq. (2) gives: G G JJJGG JJG JJG G wE İ r w ²E w JJG G + rot(rotE)=- rot B , so that ǻE= µ0 ı . (5) c² wt² wt wt (d) If the medium is a nonconductor, we immediately find (with V = 0) the classic G JJJGG İ w ²E r 0 , where H r = n² (see the relation d'Alembert equation, as in ǻE c² wt² obtained Section 7.2.1.5 for nonabsorbing materials) If the term due to conductivity is dominant, i.e., the medium is well conducting, G JJJGG wE 0. the propagation equation takes on the form ǻE - µ0 ı wt
2. (a) By looking for a solution to the general Eq. (5) of the form GG G G E = E 0 exp(i[k r - Ȧt]) , we end up with the following equation to verify G G JJJGG G2 G G G wE w ²E 2G (where ǻE = i²k E = - k E , iZE , and Z2 E ): wt wt² İ k 2 = r Z2 - i Z µ0 V . c² 2
Accordingly, afterward it can be verified that k, just as k 2 ( z k = k k ) are complex.
364
Basic electromagnetism and materials
(b) With the general equation introduced into that for absorbent media [Eq. (25) of Ȧ Chapter 7] as in k = n , we obtain c c² 2 ı n2 = k = Hr - i . Ȧ² İ0Ȧ
3. Numerically, and for each of the two terms: x for the first term, H r = 81 ; x for the second term, the minimum value is obtained at a maximum value of Z given in the Hertzian domain, which is Zmax = 100 MHz = 108 Hz . Thus, ª ı º | 719 . « » ¬« İ 0 Ȧ ¼» min
ı
H r and the conducting component is largely dominant, so İ0Ȧ that in practical terms we can state that: ı ı § ʌ· § ʌ· n² | - i exp ¨ - i ¸ from which , so that with i exp ¨ - i ¸ , then n² | İ0Ȧ İ0Ȧ © 2¹ © 2¹ can be deduced that ı ı § ı § ʌ· ª ʌº ª ʌº· n| exp ¨ - i ¸ (1 - i). ¨ cos « - » + i sin «- » ¸ İ0Ȧ İ0Ȧ © 2 İ0Ȧ © 4¹ ¬ 4¼ ¬ 4¼¹ With n = n' - i n'' (using electrokinetic notations), we finally have: So numerically,
ı
n' = n'' |
2 İ0Ȧ
.
ı
H r , is the same as neglecting the İ0Ȧ displacement current with respect to the conduction current. The complex index (n) which is tied to the complex relative permittivity by the equation n² = H r is Comment: The numerical condition,
n² = H r = H r - i
ı
| -i
ı
. The complex relative İ0Ȧ İ0Ȧ permittivity is practically purely imaginary as is the case for a conductor subject to therefore
such
that
low frequencies for which exactly n' = n'' =
ı(0) 2 İ0Ȧ
is found [see also the preceding
exercise, questions 1(b) and (c) with concerning the notation: V { V(0) ].
Chapter 11. Reflection and refraction in absorbent finite materials
365
4. (a) For a normal incidence, (Ti = 0) , with n1 denoting the index of medium (1), we have for the transmitted wave: 1
k 'ty = k 0 n1 sinși = 0 , k 'tz = k 0 R(n²-n12sin²și ) 2 k ''tz =
1 2 k 0 I(n²-n1 sin²și ) 2
k 0 n'
k 0 n''
By using electrokinetic notation, the wavevector for the wave transmitted in medium G G G G G G (2) is thus k k t = k 't - ik ''t ez = k 0 (n' - i n'')ez = k 0 n ez = k t ez .
By
G
making 2
G
İ0
c
µ0 Ȧı
ʌıȞ
1
1
k 0 n'
k 0 n''
, and k t =
1 į
,
-
so i į
that
with
n' = n'' |
ı 2 İ0Ȧ
; the form of the electric field in medium
(2) is therefore: GG G G G G z z E = E 0 exp(i[Ȧt - k r ])= E 0 exp(i[Ȧt - k t z ])= E 0 exp (- ) exp(i[Ȧt - ]) . G G The depth of the penetration is of the order of G . The z noted above is equal several G and the transmitted wave is practically zero. (b)
In c
terms
of
actual
numbers,
8
3.10
,
when
Q = 100 MHz = 108 Hz
to
and
İ0
= 2.52 10-2 m = 2.52 cm . Ȟ ʌıȞ 10 When Q = 1 MHz , (O = 300 m) , we find G = 25.2 cm . To conclude, we can see that for a depth z of the order of several G, being here at most a meter or so, the wave signal is practically all absorbed, and it therefore is not possible to communicate using Hertzian waves with a submarine. In effect, underwater communications are established using sonar with acoustic waves. c 1 , we have , and with n' = (c) The speed of the phase is given by vM n' k0į Ȝ=
=
vM = ZG=
8
= 3 m , we find that G
4ʌ Ȟ µ0 ı
c
. Thus, if Q decreases, vM decreases also, and we arrive at a
dispersion of waves. In numerical terms, when Q = 100 MHz , we have vM = 1.6 x 107 m/s .
Chapter 12
Total Reflection and Guided Propagation of Electromagnetic Waves in Materials of Finite Dimensions
12.1. Introduction As described in Chapter 11, there are two forms of total reflection: 1 x a reflection at a vacuum/perfect conductor interface where < Ȧ <Ȧp and R = 1, IJ with Ȧ < Ȧp and ı(0) >> Ȧİ 0 , so that R | 1; and x a reflection between two dielectrics such that n1 > n 2 and ș1 > șA . The superposition of the incident wave with its reflected wave can lead only to propagation, on average, when parallel to the surface. The resulting wave is in effect guided. By having a second interface, as shown in Figure12.1a, the propagation can be channeled between the two surfaces and the system constitutes a wave guide.
(b)
(a)
Figure 12.1. Wave guides.
Wave-guides can be made with different geometrical configurations: x with a rectangular cross section as shown in Figure 12.1b formed of two parallel metallic planes and generally used for guiding waves with wavelengths of the order of centimeters;
368
Basic electromagnetism and materials
x with a circular cross section so that the signal propagates in a central dielectric core which is bound within a metal surface to give a coaxial cable; x for the optical domain, the phenomenon of total internal reflection provided by two dielectrics is used, so that the internal dielectric at the heart of the cable has an index higher than that of the external dielectric which makes up the gain. When the cross section is circular the result is an optical fiber, and when the cross section is rectangular, the result is an optical guide. In the latter case, in order to assure wave guiding, i.e., confinement of the optical wave, different geometries are used such as the buried guide shown in Figure 12.2 a or the strip guide in Figure 12.2 b.
(a)
(b)
Figure 12.2. (a) a buried guide; and (b) a strip guide.
If the extremities of the guides are closed, then we end up with resonance cavities that are used in oscillators and lasers. This chapter will look first at the form of the electromagnetic (EM) wave between two conductors in a coaxial cable. In a second part it will then describe the metallic total reflection for a perfect conductor along with the generation of stationary waves. Following this there will be a study of the propagation of a wave between two plane conductors which will then yield a more general description of the properties of a wave guide, most notably those termed buried optical guides. Generally, metallic wave guides are hollow and have constant cross-sectional widths and are used for the propagation of EM energy of relatively high frequency, such as microwaves with attenuations being less than those for wires. It is worth noting that attenuation in wave guides is due to small imperfections in the conducting walls or imperfect characteristics of the conductor and dielectric losses in the insulator in the case of coaxial cables. These losses will not be covered in any detail, although losses due to material characteristics in an optical wave guide will be discussed.
Chapter 12. Total reflection and guided propagation
369
12.2. A Coaxial Line 12.2.1. Form of transverse EM waves in a coaxial cable Here the coaxial cable is assumed to have infinite length and has a structure, shown in Figure 12.3 a, made up of two conducting surfaces (C1) and (C2) which are cylindrical and around the same axis Oz. The C1 has a radius denoted by a, and C2 is assumed to have walls thick enough so that its internal radius (b) and its external radius (e) are a < b < e . It also is assumed that the two conductors are separated by a dielectric that exhibits no losses and has an absolute permittivity denoted by H .
dielectric
y
a
x
b
e
y
G eș
z
G er
M T
x
Figure 12.3(a) Cross section of a coaxial cable.
Following the application of a sinusoidal current—which circles in one sense with respect to Oz in the internal conductor (radius = a) and on passing to the external conductor circles in the opposite sense—there is between the two conductors a radial and symmetrical electric field that has a form given by G G E (r,z,t) = E0 (r) exp i[Zt - kz] er . It is supposed that the amplitude of the electric field at the surface of the internal core is given by E a . Initially assuming that the two conductors are perfect G and for a point (M) with cylindrical coordinates r, T, z, we will look for forms of E G and B , along with the intensity.
12.2.1.1. Form of E 0 (r) For a point M in a dielectric, where there is no real charge, the Maxwell-Gauss G equation is written as div E 0 . Changing this to cylindrical coordinates, in that G 1 w rE r G w Ez 1 w ET with E having components E r , E T 0 div E wz r wr r wT
370
Basic electromagnetism and materials
and E z
0,
this
equation
can
be
reduced
1 w rE r
to
r
wr
0,
so
that
r E 0 (r) = constant = a E a (value of constant of r is r a ). From this can be deduced that G a Ea a Ea G E 0 (r) = , and E (r,z,t) = exp(i[Zt - kz])er . (1) r r 1 The amplitude of the electric field varies by in the dielectric and passes, r a between the core and the gain, from the value E a to E a . From this can be b deduced the graphical representation given in Figure 12.3 b for perfect conductors G which exhibit internally E= 0 . E0
Ea a b
Ea
a
b
e
r
Figure 12.3(b). Field within a coaxial cable.
G 12.2.1.2. Form of B
G JJG G wB , is used but with the Here the Maxwell-Faraday equation, as in rot E = wt cylindrical coordinates: JJG G § 1 wE z wE T · G § wE r wE z · G § 1 wrE T wE r · G rot E = ¨ ¸ ez . ¸ er ¨ ¸ eș ¨ wz ¹ wr ¹ wș ¹ © r wș © wz © r wr As E T = E z = 0 , and E = E r is independent of T, we have more simply: wE r G wB G wB G wB G eș =- r er - ș eș - z ez . By identification according to the components wz wt wt wt and following a simple integration, we obtain:
Br = constant , Bz = constant .
Chapter 12. Total reflection and guided propagation
371
Solutions that do not represent propagation are rejected (see Section 6.3.1 of this volume) and constants are assumed to be equal to zero, so that B = BT = - ³
wE r
dt = jk ³ E0 (r) expi(Ȧt-kz) dt=
k
wz Ȧ In terms of vectors, we thus obtain: G a k G B= E a exp >i(Ȧt - kz) @ eș . r Ȧ
E 0 (r) exp(i[Zt - kz]) .
(2)
G G G The B field thus is orthoradial. In addition the E and B fields are in phase E Ȧ and orthogonal. The ratio of their amplitude is such that = , but as a difference B k 1 of plane waves, their amplitude varies by . r
12.2.2. Form of the potential, the intensity, and the characteristic impedance of the cable 12.2.2.1. Form of the vector potential G The vector potential, denoted by A(r,z,t) and directed along Oz, is such that G JJJG G wA G B = rot A , so that with respect to eș it is given by Bș = . By integrating with wr respect to r, we have k dr A = - ³ Bș dr = - a E a exp >i(Ȧt-kz) @ ³ , from which Ȧ r G G ª k º A(r,z,t) = «- a E a Ln r + Cte » exp >i(Ȧt-kz) @ ez . ¬ Ȧ ¼ By taking the origin of the vector potentials as r = a (on the core of the central k conductor), the constant can be fixed as: constant = a E a Ln a , so that finally, Ȧ G a º G ª k A(r,z,t) = « a Ea ln » exp > i(Ȧt-kz) @ ez . r ¼ ¬ Ȧ
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Basic electromagnetism and materials
12.2.2.2. Form of the scalar potential
G JJJJJG G wA wV wA Along Oz the equation E = - gradV gives 0 = , so that with the wt wz wt wV wA a º ª k == - jZ « a E a ln equation for A, we have » exp >i(Ȧt-kz) @ . By wz wt r ¼ ¬ Ȧ integrating with respect to z and also by taking the origin of (scalar) potentials as r = a, then it is possible to state that: a º ª V(r,z,t) = « a E a Ln » exp >i(Ȧt-kz) @ . r ¼ ¬
12.2.2.3. Form of the intensity Ampère's theorem applied to a circular cross section of the cable and passing through a point M is written as G G G G G G G § G wE · G v³ B.d l = ³³6 ¨¨ µ0 j + µ0 İ ¸¸ d6 ³³6 µ0 j + iZµ0 İE d6 . With E being radial wt ¹ © (C) G G and E A 6 (section perpendicular to the plane of the circle), Ampère's theorem can be written as G G G G v³ B.d l = ³³6 µ0 j d6 = µ0 I .
(C)
The result is that vM =
Ȧ k
=
c n
=
c İr
I=
2ʌ r B µ0
. Given the expression for B, and with
, we obtain
I = 2 ʌ a c İ 0 İ r E a exp >i(Ȧt - kz) @ .
(3)
Equation (3) also can be rewritten as I = I0 exp(i[Zt - kz]) , with I0 = 2 ʌ a c İ 0 İ r E a .
(3’)
12.2.2.4. Characteristic impedance (Zc) We have Zc =
Va -Vb
Va (a,z,t) -Vb (b,z,t)
I
I
. By using the expressions for V and
§b· §b· a E a Ln ¨ ¸ Ln ¨ ¸ a © ¹ , so that Z = © a ¹ . (4) for I we can directly obtain Zc = c 2ʌc a İ 0 İ r E a 2ʌc İ 0 İ r
Chapter 12. Total reflection and guided propagation
With İ 0 =
1 9
36 ʌ 10
1
S.I. , and c = 3 x 108 m/s , we have 2ʌc İ 0
60
Zc =
İr
§b· Ln ¨ ¸ . ©a¹
60
373
, from which
(4’)
12.2.3. Electrical power transported by an EM wave µ0
From Ampère's theorem we obtained B =
2ʌr
I , so that from Eqs. (2) and (3'), it is
G µ G possible to state that B= 0 I0 ^exp[i(Ȧt - kz)]` eș . 2ʌr
Using an equation developed in Section 12.2.1.2, i.e., we have E = B
c İr
=
µ0
c
2ʌr
İr
I=
1 2ʌr İ 0 c
G E=
B
=
Ȧ k
= vM =
c İr
,
I , so that in vectors:
İr
I0 2ʌr İ 0 c
E
İr
.
G G G E×B , so that by taking the real The Poynting vector therefore is in the form S = µ0 G G G 1 G solutions for E and B gives us S I02 cos²(Ȧt-kz) ez . 2 2ʌr İ 0 c İ r
The average value of cos²(Zt-kz) with respect to time is G S
1
G I02 . ez .
1 2
, so that:
8ʌ²r² İ 0 c İ r The average power transported by the EM wave is given by the flux of the average Poynting vector through a ring composed of the circles with radii a and b. G G I02 I02 b b 2ʌ rdr P = ³³ S .dȈ= = Ln . ³a r² a 8ʌ² İ 0 c İ r 4ʌ İ 0 c İ r
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Basic electromagnetism and materials
12.2.4.
Conductor with an imperfect core that exhibits a resistance and the attenuation length Power dissipated along the length of an element (dz) of a cable with resistance given 1 dz by dR = is dp = I²(z,t)dR , so that with Eq. (3’), we have ı ʌa² 1 dz dp = I0 ² cos²[Zt - kz] . The average power loss with time for a length dz is ı ʌa² thus dP =
1 1 I02
dz . ı ʌa² 2 The loss relative to the power is finally given by:
1 1 I02 2İ c İ r ı ʌa² 2 dz= 0 dz . b I02 b ı a² Ln Ln a a 4ʌ İ 0 c İ r
dP P
By introducing Eq. (4) for the characteristic impedance we also have: dP 1 dz . Integration with respect to z gives for the z abscissa P ʌa²ı Zc § z P(z) = P(0) exp ¨ ¨ ʌıa²Z c ©
· ¸¸ . ¹
§ z· This equation is in the form P(z) = P(0) exp ¨ - ¸ , with L = ʌıa²Zc where L is © L¹ the attenuation length of the cable, i.e., the distance over which the power is divided by e | 2.7 .
In numerical terms, with a = 10 mm and b = 40 mm , V = 5 x 107 :-1m-1 , and H r = 4 , we have Zc =
60 2
ln 4 | 42 : , and L | 660 km .
12.3. Preliminary Study of the Normal Reflection of a Rectilinearly Polarized MPPEM Wave on a Perfect Conductor, Stationary Waves, and Antennae 12.3.1. Properties of a perfect conductor and equations of continuity at the surface As shown in Chapter 11, problem 1, for relatively low frequencies, such that 1 Z << , and with the condition that the continuous conductivity is sufficiently W
Chapter 12. Total reflection and guided propagation
375
large so that V(0) >> ZH0 , and that within a range of frequencies that does not go too high ( Z << Zp ), the reflection at a conductor can be assumed to be total, as in r | - 1 , with as a consequence a dephasing equal to S. A medium is assumed to be a perfect conductor if its conductivity is quasiG G infinite. The result is that Ohm's law for a volume of the conductor, written j = ıE , G shows how j stays infinite as V o f (perfect conductor), and the field in the G conductor, which is here denoted by E C , can be equal to zero. With respect to the magnetic field, the Maxwell-Faraday relation, which is G JJG G G wB , indicates that B = constant . A static magnetic field cannot written rot E= wt exist without a distribution of permanent currents, and in the absence of their G application (as is assumed here) the field in the conductor denoted BC can be equal only to zero. In other words, there is no magnetic field in a perfect conductor under a G G varying regime. Finally, as E C = 0 and BC = 0 , the EM wave cannot penetrate into a perfect conductor under a varying regime. In addition, the volume of a perfect conductor can be assumed to be electrically neutral. The charges are sufficiently mobile so that an excess in one given charge, for example, positive, is immediately compensated for by the movement of opposing charges. Further details can be found in Section 1.3.5. In very simple, and perhaps too simple terms, the charges can be assumed to be infinitely mobile (as V = qnµ , so if V o f , then the mobility µ o f ), so that there can be an instantaneous return to neutrality. Finally, any charges can only appear at a discontinuity such as a surface of the metal where from the point of view of a given charge, there cannot materially exist opposite charges in a vacuum (an excess of charges of a given type thus is possible at the surface). The contents of the envelope of the conductor are electrically equivalent to a vacuum and the laws of continuity at the vacuum/conductor interface only bring in the permittivity and permeability of a G vacuum, along with the possible surface charge (Vs) and surface current ( js ) densities. G G By denoting as n 21 ( { -ez as indicated in Figure 12.4) the normal that goes from medium (2) (the metal) to medium (1) (the vacuum or air), we can apply to the vacuum/metal interface the following classic conditions of continuity (generally cited in the first years of university courses):
G G °E1t = E 2t G ®G °¯B1n = B2n
G ıs G G G G n 21 , so with E C = E 2 = 0 °E1n - E 2n = İ0 G G ® G G , so with BC = B2 = 0 G °G ¯B1t - B2t = µ0 js ×n 21
ıs G G n 21 °E1n = İ0 ® G G °G ¯B1t = µ0 js ×n 21
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Basic electromagnetism and materials
12.3.2. Equation for the stationary wave following reflection The system under study is that of a vacuum/metal interface in the presence of a monochromatic (with angular frequency denoted by Z) EM wave undergoing a plane incidence (with respect to the normal at the interface so that ș1 = 0 ° ) and progressing along Oz as shown in Figure 12.5 a with the incident wavevector G G k i = k 0 ez . The wave also is polarized rectilinearly along Ox and is assumed to be G G of the form Ei = Ei0 exp[i(k 0 z - Ȧt)] ex . The wave reflected into the vacuum will have the same angular frequency as the incident wave (due to the properties of the media) and the wavevector of the G reflected wave ( E r ) will have the same modulus ( k 0 ) as the incident wave. Following a reflection under a normal incidence, the wavevector of the reflected G G G wave is according to the first Snell-Descartes law given by k r = - k i = - k 0 ez . In G G G addition, div E r = 0 , from which k r E r = 0 , showing that the reflected wave also is transversal (parallel to Oxy). In the medium (1) (vacuum) the resultant wave thus is G G G E1 = Ei + E r and is parallel to Oxy (in a tangential plane).
G Ei medium (1) vacuum
G kr
x
G ki
G Bi G Br
medium (2) metal
G n21
z
G Er y
Figure 12.4. Vacuum/metal interface.
As mentioned above, with the conductor being perfect, the fields that have been G G denoted E C and BC in the conductor are zero. At z = 0 (where G G G G E 2t (z = 0) = E Ct 0 ), we can state that E1t = E 2t = 0 , so that G G G 0 = E1t (z = 0) = Ei (z = 0) + E r (z = 0) , from which : G G E r (z = 0)= -Ei (z = 0) .
Chapter 12. Total reflection and guided propagation
377
The reflected wave is polarized parallel to the incident wave, that is to say that along G ex with an amplitude such that E 0r = - Ei0 . Thus, the reflected wave has the form:
G G E r = - Ei0 exp[i(- k 0 z - Ȧt)] ex .
G E1
In medium (1) (vacuum), the resulting wave thus is given by: G G Ei E r G = Ei0 ^ exp[i(k 0 z - Zt)] - exp[i(-k 0 z - Zt)] ` ex . G 2i Ei0 sin k 0 z exp[-iZt] ex
signal t1
t2 > t1 z
(a) progressive wave
signal t1 t2 > t1
z
(b) stationary wave
Figure 12.5. (a) Progressive; and (b) stationary waves.
The physical solution for the resultant wave in a vacuum thus is finally given by G G G E1 = R(E1 )= 2 Ei0 sink 0 z sinȦt ex where the spatial and temporal dependences are now separated. The phase velocity thus is zero, and the wave is stationary, as shown in Figure 12.5 b (see also Section 6.2.5).
12.3.3. Study of the form of the surface charge densities and the current at the metal 12.3.3.1. The surface charge density of zero
G G The normal components of the electric field are zero in medium (1) ( E1 // ex ) as in G medium (2) (where E C = 0 ), and the normal component of the electric field does G G not give rise to any discontinuity, so E 2n = E1n (= 0) , where Vs = 0 .
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Basic electromagnetism and materials
12.3.3.2. Form of the surface current density G 12.3.3.2.1. Equation for js As can be seen in Figure 12.4, which schematizes the results obtained for a reflected G G wave, following reflection Bi and Br remain in phase. In medium (1) the resultant magnetic field thus is given by: G G G G G 1 G G ª k i ×Ei º + ª k r ×E r º , B1 = Bi + Br = ¼ ¬ ¼ Ȧ ¬ G G G G G so that with - k r = k i = k 0 ez and -E 0r ex = Ei0 e x we have G G G e ×e B1 = z x k 0 Ei0 exp ª¬ik 0 z º¼ +exp ª¬-ik 0 z º¼ exp > -iȦt @ . Ȧ Z With k 0 = , we obtain for the physical solution for a vacuum c G G 2Ei0 G B1 = R(B1 ) = e y cos k 0 z cos Ȧt . c
The magnetic field thus is directed parallel to the interface. G G As in a metal BC = 0 (and B2t = 0 ), the tangential component of the magnetic field undergoes a discontinuity at the interface. Therefore, there must appear a G surface current density ( js ) at the interface so that the equation for continuity (end of preceding Section 11.3.1) G B1t G G G G G G G z=0 ez × js . B1t = µ0 js ×n 21 µ0 js × ez , so that z=0 P0 G G G By multiplying by vectors the right-hand side by ez and on noting that ez A js , we obtain: G B1t G 2Ei0 G z 0 G js = ×ez = ex cos Ȧt . µ0 P0 c
G G G Of note then is that js // Ei ( //ex ) and the current density is collinear to the incident electric field. Physically, the current density is the result of the polarization of the metal by the electric field.
Chapter 12. Total reflection and guided propagation
379
12.3.3.2.2. Comment As Figure 12.4 illustrates, the magnetic field vectors of the incident and reflected waves are parallel to the interface. Their resultant normal component in medium (1) G thus is zero, thus assuring the continuity of the normal component of B , as in G G medium (2) the magnetic field is zero ( BC = 0 implies that B2n = 0 ). 12.3.3.3. Applications
This property is the basis of the use of metallic antennae for detecting EM waves. With the antenna lines parallel to the direction of polarization of the incident wave, the current density detected is at a maximum and directly proportional to the intensity of the incident electric field. If the lines of the antenna are not parallel to the incident wave, then the detected current density is no longer proportional to the intensity of the component of the electric field at a parallel incidence to the antenna lines, as shown in Figure 12.6.
0 Eix
Ei0
detected current density 0 proportional to Eix
0 Eiy
Figure 12.6. Relation between the disposition of the antenna lines and the detected current.
With a metal never being a perfect conductor, in effect the current circulates over a thickness several times greater than the “skin” thickness ( G ). If the thickness (d) of the metal is such that d < G , a part of the wave in fact can be transmitted. However, if d >> G , the metal becomes a real obstacle to the propagation of the EM wave and will create an electromagnetic shield.
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Basic electromagnetism and materials
12.4. Study of Propagation Guided between Two-Plane Conductors: Extension to Propagation through a Guide with a Rectangular Cross Section 12.4.1. Wave form and equation for propagation between two conducting planes
z a y O
y
x
Figure 12.7. Propagation between two-plane conductors.
This section considers a wave propagated along Oy in a vacuum without attenuation between two-plane and perfect conductors separated by a given distance (a) and assumed to be of great dimensions (with respect to a) in directions Oy and Ox. G With the system being invariant with respect to Ox, the field ( E ) does not G depend on x and it can be assumed that E is a monochromatic wave in the form
G G E = E m (z) exp[i(k g y - Zt)] ,
(1)
where kg is a positive constant. Equation (1) is valid uniquely between the two-plane conductors. Above and G below these planes, the electric field is zero (metallic planes inside which E= 0 ). Additionally, in a plane normal to the direction of propagation, the wave is not uniform and cannot be assumed to be planar. Between the two plane conductors, dominated by a vacuum, the equation for G JJJGG 1 w ²E propagation is ǻE = 0 [from Eq. (5) of Chapter 6]. Taking Eq. (1) into c² wt² account and by making a calculation resembling that in Section 11.2.6, we have
G d²E m (z) § Ȧ² ·G +¨ - k g ² ¸ E m (z) = 0 . dz² © c² ¹
(2)
Chapter 12. Total reflection and guided propagation
381
The tangential component of the electric field, situated in a plane parallel to Oxy, exhibits two components denoted Ex and Ey which are such that the limiting G G conditions are [with E m (0) = 0 and E m (a) = 0 ] E mx (0) = E my (0) = 0 , and E mx (a) = E my (a) = 0 . G For its part, the continuity of the normal component of B is Bmz (0) = Bmz (a) = 0 . (4)
(3)
12.4.2. Study of transverse EM waves Here the possibility of the transverse wave propagating at a velocity (c) along Oy with, on the one side vM = c (nondispersion solution as vM = constant), and on the other vM =
Ȧ kg
[equation for vM in agreement with Eq. (1)], we have k g =
Ȧ c
.
We shall now determine the corresponding exact form of the electric and magnetic fields.
G 12.4.2.1. Form of the electric field (given by E m (z) ) Equation (2) for propagation thus is reduced to
G d²E m (z) dz²
G G G integrations with respect to z give E m (z)= C1 z + C2 , (5) G G where C1 and C2 are two constant vectors.
= 0 . Two successive
The
conditions
E mx (0) = 0 and E my (0) = 0 , respectively, give: C2x = 0 and C2y = 0 and similarly, from E mx (a) = E my (a) = 0 we can deduce that C1x = 0 and C1y = 0 . G G Finally the vectors C1 and C2 only have components in the z direction, and are of G G G G the form C1 = C1z ez and C2 = C2z ez . Eq. (5) can therefore be rewritten as G G G E m (z)= C1z z + C 2z ez = E m (z) ez . (6)
G dE m (z) Gauss's equation gives then for its part div E= 0 , where =0, dz dE m (z) = C1z . From this can be deduced that C1z = 0 , such that while Eq. (6) gives dz G G G E m (z) = C2z ez = E m ez
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Basic electromagnetism and materials
G where E m is a constant. For this type of wave, E m (z) is a constant vector parallel G G to ez . The electric field, also parallel to ez , thus is in the form G G E = E m {exp[i(k g y - Zt)]} ez .
(7)
12.4.2.2. Form of the magnetic field With the wave assumed to be monochromatic, the Maxwell-Faraday equation makes G G wB G it possible to state that rotE= =- (- iȦB) , from which can be deduced directly wt JJG G G i that B = - rotE . Ȧ G From Eq. (7) for E , we can deduce that: G w G G rotE = E m exp[i(k g y-Ȧt)] ex ik g E m exp[i(k g y-Ȧt)] ex , with the result that wy
G i²k g G B=E m exp[i(k g y-Ȧt)] ex , so with k g = Ȧ G E B = m exp[i(k g y-Ȧt)] c
^
`
Ȧ c G ex .
we have (8)
We can note that this result is identical to that which would be obtained had we G G G k g ×E used the equation for plane waves as in B = , which a priori cannot be used Ȧ here directly as the wave cannot be assumed to be completely planar. Nevertheless, G G Eqs. (7) and (8) demonstrate that here again E and B are transverse, and hence the term “transverse electromagnetic wave” or TEM wave.
12.4.2.3. Comment: The rectangular cross-sectional wave guide If we envisage that in a rectangular guide, as shown in Figure 12.1 b with an axis as in Figure 12.7, there is a wave that propagates at a given speed (c), then the limiting conditions imposed on that wave are: x in one part by the parallel metallic planes with sides z = 0 and z = a , which result in the electric field taking on a form given by the equation (see also results from Section 12.4.2.1): G G E = E m {exp[i(k g y - Zt)]} ez G G where E m is a constant and E also is parallel to ez (at whatever value of x);
Chapter 12. Total reflection and guided propagation
383
x and in another part, fixed by the second system of metallic planes parallel at the sides to x = 0 and x = b with respect to Ox (0 being on the side of the plane Oyz and b on the side of the plane to which it is parallel), impose that for x = 0 and G x = b , E= 0 so that E m = 0 . This trivial solution is not a physical one, as the possibility of propagation at the speed c along Oy is not available to a guide with a rectangular cross section.
12.4.3. Study of transverse electric waves (TE waves) Here we study the possibility of a wave propagating in the guide plane Oxy. The wave is thus polarized in accord with the breakdown given by: G G G E = E x ex + E y e y ,
(9)
G G G and has a form given by Eq. (1), as in E = E m (z) exp[i(k g y - Zt)] = E(y,z) .
12.4.3.1. Polarization direction of the electric field Under these conditions, Gauss's equation gives: G div E= 0 dE x dE y dE z = + + = ik g E y Ey = 0 dx dy dz =0 (as E x = E x (y,z) )
=0 (as E z = 0 )
G G Equation (9) thus is reduced to E = E x ex and the electric field is polarized along Ox perpendicularly to the direction of propagation Oy. The wave is termed “transverse electric” and denoted TE. The electric field can be written: G G G E = E mx (z) ex exp[i(k g y - Zt)] = E mx (z)exp[i(k g y - Zt)] . 12.4.3.2. Solutions for the propagation equation Equation (2) for propagation becomes: d²E mx (z) § Ȧ² · +¨ - k g ² ¸ E mx (z) = 0 . dz² © c² ¹
(11)
(10)
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Basic electromagnetism and materials
12.4.3.2.1. First case Ȧ § Ȧ² · If ¨ - k g ² ¸ = - D ² < 0 (where D is real), so that k g > , the solution to c © c² ¹ Eq. (11) is: E mx (z)= C1 exp(Dz) + C2 exp(-Dz) . With E mx (0) = 0 and E mx (a) = 0 , we can deduce that, respectively,
½° ¾ 2C1 shDa = 0 C1 exp(Da) + C2 exp(-Da) = 0 °¿ C1 = - C2
With shDa z 0 , we deduce that C1 = - C2 = 0 , so that E mx (z)= 0 , and there is no possible physical solution in this case. 12.4.3.2.2. Second case Ȧ § Ȧ² · If ¨ - k g ² ¸ = K² > 0 (with K being real), then k g < and the solution to Eq. c © c² ¹ (11) is given by E mx (z)= C1 exp(iKz) + C2 exp(-iKz) .
(12)
The limiting conditions, E mx (0)= 0 and E mx (a)= 0 , respectively, give:
½° ¾ 2iC1 sin ka = 0 , C1 exp(ika) + C2 exp(-ika) = 0 °¿ mʌ so that K = , with m being a whole nonzero number. a Substituting these results into Eq. (12), we gain: C1 = - C2
E mx (z)= 2 i C1 sin
mʌ a
z.
(13)
G G The electric field, according to Eq. (10), is of the form E = E mx ex exp[i(k g y - Zt)] , G mʌ G so that with Eq. (13), we find that E= 2 i C1 sin z e y exp[i(k g y - Zt)] . a By choosing the origin of the phases as being at y = 0 at a time t = 0 , then it is
Chapter 12. Total reflection and guided propagation
385
imposed that 2 i C1 is real, as in 2 i C1 = E° , and we have
G mʌ · G § E= E° ¨ sin z ¸ ex exp[i(k g y - Zt)] . a ¹ ©
(14)
12.4.3.3. Proper vibrational modes With K =
mʌ a
, we thus have: Ȧ² c²
from which Z = c k g2 + m²
= k g2 +
m²ʌ² a²
,
(15)
ʌ²
. The equation Z = f(k g ) is not a linear relation and a² the system undergoes dispersion. ʌc When m = 1 and k g o 0 , Z o Zc = (Figure 12.8). a Sc For any value of m and and k g o 0 , Z o Zm = mZc = m . a
Z m=4
4Zc
Z
Z = ck
m=3 3Zc m=2 2Zc
Ȧc =
ʌc
m=1
a k3
k2
k1
kg
Figure 12.8. Dispersion plots for Z = f(k).
For a given value of Z, there are many possible values of k g : k1 , k 2 , k 3 . They are associated with whole values of m which are of a limited number as in the
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Basic electromagnetism and materials
second case,
Ȧ² c²
-
m²ʌ² a²
= k g ² > 0 , so that m<
Ȧa
m²ʌ² a² Ȧ
<
Ȧ² c²
, which is:
= . ʌc Ȧc For each value of m, or of kg, there is associated a solution of the electric field that assures a propagation of the field without modification of its intensity along the mʌ guide (for given values of m and z, the intensity is given by E° sin z = Cte , a y ). These are the guide modes, and the different values of k g (k1 , k 2 , k 3 ...) are called the proper modes of spatial vibration of the guide. Only signals with angular frequencies higher than Zc =
ʌc a
(termed the cutoff
frequency) can propagate in the guide. This angular frequency corresponds to a wavelength in the vacuum given by 2ʌc Ȝ 0c = = 2a . The condition Z t Zc also can be written as O 0 d O 0c = 2a . Ȧc
G 12.4.3.4. Form of B The magnetic field can be taken from the electric field given in Eq. (14) with the G JJG G G wB help of the Maxwell-Faraday relation, rotE = = i ZB , so that: wt ° °0 G i JJG G °° -i wE mS 0 mSz B = - rotE ®( ) ( x )=- i( ) E [cos ( )] exp[i(k g y - Zt)] . (16) Ȧ aZ a wz ° Z ° -i -wE x mSz )= - (k g /Z) E 0 [sin ( )] exp[i(k g y - Zt)] °( ) ( a wy °¯ Z
The E = E x is in phase with Bz and in the squared phase, or quadrature, of the G longitudinal component By z 0 ; B therefore is not transverse and the wave is simply a transverse electric wave and not a TEM.
Chapter 12. Total reflection and guided propagation
387
12.4.3.5. Propagation of energy G G G G With E = E m exp(i[k g y - Ȧt]) and B = Bm exp(i[k g y-Ȧt]) , the average over time for the Poynting vector is given by (see also Section 7.3.3) :
G S =
G G* R(E m ×Bm ) .
1 2µ0
Taking the form of the electric and magnetic fields into account, the direct G G* calculation of E m ×Bm shows that the vector has a purely imaginary component with respect to Oz. The associated average value, which corresponds to the real part, G G* is therefore zero. With respect to Oy, the component of E m ×Bm , however is real so that the propagation of the energy is through Oy, the direction of wave propagation.
12.4.3.6. Phase and group velocities
The phase of the wave is given by (k g y - Zt) and its phase velocity is given by vM =
Ȧ kg
. With k g2 =
Ȧ² m²ʌ² Ȧ ʌc , we have vM = , , and as Zc = 1/ 2 c² a² a § Ȧ² m²ʌ² · ¨ ¸ © c² a² ¹
therefore vM =
c
>c
§ m 2 Ȧc2 ¨1 ¨ Ȧ2 ©
For its part, the group velocity is vg = obtain
2Ȧ c2
.
· ¸ ¸ ¹ dȦ dk
. By differentiating Eq. (15), we
dȦ = 2k g dk g , from which: vg =
dȦ dk g
= c²
kg Ȧ
=
c² vM
.
With vM > c , then vg < c while noting that vg vM = c² also is verified.
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Basic electromagnetism and materials
v vM
TE wave
c
TEM wave
vg
Z
Zc
Figure 12.9. Plots of vg = f(Z) and vM = g(Z).
Variations in vg and in vM as a function of Z are indicated in Figure 12.9 for m = 1 . When Z >> Zc , so that also O 0 << 2a , we have Z o ck , and vg just as vM tends toward c. The wave guide is no longer dispersing. This is found in metallic guides with “a” being of the order of a millimeter, the optical waves which are such that O 0 << 2a is true propagate with the same speed as if in a vacuum (TEM wave).
12.4.4. Generalization of the study of TE wave propagation to a guide with a rectangular cross section, and the physical origin of the form of solutions for the electric wave 12.4.4.1. Generalization of the study of TE wave propagation to a guide with a rectangular cross section
z a O G E
kg
y
b x
Figure 12.10. Propagation in a guide with rectangular cross section.
Chapter 12. Total reflection and guided propagation
389
In place of studying the propagation of a TE wave, polarized with respect to Ox, between only the metallic planes parallel to Oxy as noted above, here the propagation is through a wave guide with a rectangular cross section so that it has two supplementary metallic sides at 0 and b with respect to Ox and parallel to Oyz (Figure 12.10). The form of the preceding solutions (Section 12.4.3) is still, a priori, valid, as the conditions required for the continuity of the tangential component of the electric field in the planes parallel to Oxy and on the sides z = 0 and z = a still should be true, so the solution given in Eq. (14), i.e., G mʌ G E= E° sin( z) ex exp[i(k g y - Zt)] a
must still be a solution, and the procedure therefore is to simply find out the consequences of the supplementary limiting conditions due to the presence of a second system of metallic planes, as in: E t (0) = E t (b) = 0 , where Et represents the tangential composition of the electric field in the planes Oyz of the sides x = 0 and x = b . G G In effect, as the solution for the TE wave for E is parallel to ex , the tangential component of the electric field in the planes parallel to Oyz is zero, which accords to the new limiting conditions for whatever value of b. This latter dimension of the guide therefore plays no role in the solutions for the electric field, which thus remain unchanged.
Note: However, to state that the solutions for a system based on two parallel planes also are solutions for a rectangular guide does not mean that in the latter system there are not other solutions for transverse waves. To the contrary, we can show that the general solutions for the transverse electric waves (TE waves and as a consequence with > E @y = 0 and By z 0 ) give rise to components along Ox and G Oz. For this, it would be correct to look for solutions for E in the form:
E x =E mx (x,z) exp[i(k g y - Zt)] G °° E ® Ey = 0 ° °¯ E z = E mz (x,z) exp[i(k g y - Zt)] G (and thus with E // Oxz , the solutions would again correspond to those for a TE wave).
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Basic electromagnetism and materials
For the equation of propagation, we therefore look for solutions to a variable separated such that, for example E mx (x,z) = f(x).g(z) . This can be done by proceeding as was the case for the parallelepiped potential box. The rather long G calculations show that in general terms E has the components: z· § § x· E x = A x sin ¨ mʌ ¸ cos ¨ nʌ ¸ exp[i(k g y-Ȧt)] , and a¹ © © b¹ z· § § x· E z = A z cos ¨ mʌ ¸ sin ¨ nʌ ¸ exp[i(k g y-Ȧt)] , a¹ © © b¹ where Ax and Az are complex constants. The corresponding wave is denoted TEmn and Eq. (15) for the dispersion is replaced by : Ȧ² c²
= k g2 +
m²ʌ² a²
n²ʌ² b²
.
(15’)
In even more general terms, if the vacuum in the guide is replaced by a dielectric without loss in permittivity (H), then Eq. (15') would m²ʌ² n²ʌ² . become Ȧ²İµ0 = k g2 + a² b² Thus Eq. (15’) shows that when b > a , the mode associated with the lowest angular frequency is the wave TE01 (such that E z z 0 while E x = E y = 0 ). When a > b , the low frequency mode, also called the dominant mode, corresponds to the TE10 wave ( m = 1, n = 0 ), which is thus such that E z = E y = 0 , and: §ʌ · E x = A x sin ¨ z ¸ exp[i(k g y-Ȧt)] . ©a ¹ The latter equation is the form of the TE wave studied thus far, which we will continue to use in its general form (TEm0 wave polarized with respect to Ox) as an example solution in order to interpret the physical origin. 12.4.4.2. Physical origin of the solution form of a TE wave: TEm0 wave given to support argument for a wave polarized with respect to Ox G mʌ G G E= E mx (z) ex exp[i(k g y - Zt)] = E° sin z ex exp[i(k g y - Zt)] . a
Chapter 12. Total reflection and guided propagation
391
Here we will study a monochromatic plane EM wave rectilinearly polarized with respect to Ox in a guide that is alternately reflected against the planes sides z = 0 and z = a parallel to the plane 0xy and as shown in Figure 12.11.
z M
a P
GT ki
Q
G kr
O x x
y
Figure 12.11. Propagation of a monochromatic plane EM wave with successive reflections on plane sides z = a and z = 0.
At any point (Q), the wave is the result of incident and reflected waves that have fields in the form: GG G G G G G G Ei = Ei0 ª¬sin(k i r - Ȧt) º¼ ex and E r = E 0r ª¬sin(k r r - Ȧt) º¼ ex . In a vacuum, we have: G G Ȧ k i = k r = k 0 = , and Descartes first law gives Ti = Tr = T , for which: c
0 G ° k i ®k 0 sinT °k cosT ¯ 0
0 G ° k r ® k 0 sinT °- k cosT ¯ 0
G The wavevector k i of the incident wave depends on the direction of the incidence and belongs to a plane parallel to Oyz, and is by consequence normal to G G the direction Ox of wave polarization. It is worth noting that in fact k i = k 0 , and that G by denoting the wavevector of the oblique wave in the guide as k 0 , we can state that: G G G k 0 = k 0 sinș e y + k 0 cosT ez ). The dephasing by S of the reflection at the perfect conductor (due to the limiting condition at the metallic plane, at z = 0 or z = a, where the resultant tangential field must be equal to zero) gives: E 0r = Ei0 e jS = -Ei0 .
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Basic electromagnetism and materials
At the point Q, the resulting wave is thus: G G G E = Ei + E r
G G = Ei0 [sin(y k 0 sinT + z k 0 cosT - Zt)]ex - Ei0 [sin(y k 0 sinT - z k 0 cosT - Zt)]ex
With sin p - sin q = 2 cos
p+q 2
sin
p-q 2
, we obtain:
G G E = 2 Ei0 {sin([k 0 cosT] z)} {cos([k 0 sinT]y - Zt)}ex .
(17)
This form of the solution of the wave contains two parts: x one part due to propagation and associated with the term cos([k 0 sinT]y - Zt) , which shows that the propagation is carried out with a wavevector given by G G G G G k g = k 0 sinT e y (such that k g .r = y k 0 sinT ) where k g thus represents the G definitive wavevector which “pilots” the wave guided along e y ; and x a part due to a stationary term given by sin([k 0 cosT]z ). This term corresponds to G G G G a wavevector k s = k 0 cosș ez and as such must verify k s .r = [k 0 cosT] z . The wavevector G G G k 0 = k g +k s .
G G G k 0 = k 0 sinș e y + k 0 cosT ez
therefore
is
such
that
With respect to Oz, the presence of a node (E = 0) at z = a demands that sin([k 0 cosT]a) = 0 so that ( [k 0 cosT]a) = mS , from which with k 0 cosT = k s we have k s =
mʌ a
. From this can be deduced that k s = k 0 cosT =
that in addition cosT = m
ʌc Ȧa
=m
Ȧc Ȧ
Ȧ c
cosT =
mʌ a
, so
. This occurs for values of Tm of T such that:
Tm = Arc cos m
Ȧc Ȧ
.
(18)
We also can state that following Pythagoras's law as shown in Figure 12.12, Ȧ² m²ʌ² that k 02 = k g2 + k s2 , which means that , which is an equation = k g2 + c² a² identical to that of Eq. (15).
Chapter 12. Total reflection and guided propagation
393
z a G ks
G k0 T G kg
O x x
y G
Figure 12.12. Breakdown of the vector k 0 such that k 02 = k g2 + k s2 .
With k g = k 0 sinT , we can deduce that for varying values of m there are corresponding different values of T which can be denoted by Tm, and are such that: Ȧ²
= k 02 sin²ș m +
m²ʌ²
, so that in addition, 1 = sin²ș m + m²
c² a² equation equivalent to Eq. (18), as in: Tm = Arc sin
1 m²
Ȧc2
.
Ȧ2
Ȧc2 Ȧ2
, hence the
(19)
In numerical terms, for a value of a of 3 cm, we can determine that Ȧ c Qc = c = = 5 GHz . When Q = 12 GHz (which verifies Z > Zc ), we obtain 2ʌ 2a for when m = 1 that T1 | 66 ° and when m = 2 , we have T2 | 37 ° . Finally, it is the angle of incidence (i') at the entrance of the guide that determines the mode of propagation (detailed further on Section 12.5.2 and Figure 12.17). 12.4.4.3. Physical representations
G ks
z a P O
x x
T G - ks
G G k 0 = ki G kG g kg
G kr
y G
Figure 12.13. Geometrical representation of the k vectors.
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394
12.4.4.3.1. Wave breakdown The superposition zones of the incident wave with the reflected wave on the metallic planes give (see Figure 12.13): x with respect to the Oy axis, a progressive wave with a doubled amplitude due to the superposition of two progressive waves with the same amplitudes and both G with the same wavevectors given by k g , such that for the incident wave G G G G k g = |k i | sinT e y and for the reflected wave k g = |k r | sinT e y ; and x with respect to the Oz axis, a stationary wave that by superposition of the two progressive waves with the same amplitude but propagating in opposing senses G such that the wavevector of the incident wave is k s and for the reflected wave G is - k s . 12.4.4.3.2. Different speeds z N G a ki P
M T H
O xx
Q G kr y
N’
Figure 12.14. Definition of the points P, M, N, and H used to illustrate the various speeds.
Figure 12.14 shows an incident ray along PM which is a progressive wave such that PM = c W . We thus have: kg PM §S · = cos ¨ - ș ¸ = sinș = x , so that with (see Section 12.4.3.6) k0 NM ©2 ¹ kg = PM NM
Ȧ vM
, and k 0 =
= sinș =
c vM
Ȧ c
then:
. From this can be deduced that NM =
which in turn means that NM = vM W . x
PH PM
= sinș , from which PH = PM sinT = NM sin²T , so that
§ c PH = (vM W) ¨ ¨ vM ©
2
· c² ¸ = IJ = vg IJ . ¸ v M ¹
PM sinș
=
cIJ c/vM
,
Chapter 12. Total reflection and guided propagation
395
The three speeds, c, vM, and vg, respectively, correspond to the following: x c is the speed of point P on the zigzag trajectory of incident and reflected waves Ȧ (k i = k r = k 0 = ) ; c x vg is the displacement speed from P to H, in other words the displacement of P along a trajectory globally parallel to Oy, such that vg < c (the resultant speed given by vg of the point P on the resultant direction thus is inferior to the speed along the instantaneous zigzag trajectory); and x vM is the speed of a point N with respect to the direction Oy. The N has no material reality and as it physically represents none of the characteristics of a wave it is not subject to relativistic physical laws. To give an idea of what it represents, it can be thought of as a “running shadow” on the line z = a of the point P following its illumination from N'. This means that vM > c simply states that the immaterial shadow is faster than the material point that moves at the speed c. 12.4.4.3.3. Representation of the resultant amplitude for the resultant progressive wave According to Eq. (14), the resultant wave is given by: G mʌ G G E= E° sin( z ) ex exp[i(k g y - Zt)] = E0x ex exp[i(kg y - Zt)] , a mʌ where E 0x = E° sin( z) . a
a
(2nd case)
z
(1st case)
z
+a/2
y
O E 0x -a/2
O m=1
m=2
m=3
m=4
Figure 12.15. Representation of the amplitudes E 0x
f (z) for two axial systems where the
origin O can be taken on the lower metallic plane or as between the two metallic planes.
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Basic electromagnetism and materials
Whatever the y point of the trajectory, the amplitude ( E 0x ) of the wave only depends on z and the excited m mode (which is fixed by the choice of the incidence angle). So, Figure 12.15 represents the form of the electric field amplitudes associated with the various possible modes, and E 0x = f(z) is represented in the plane Oxz on the y = 0 side. It is notable that the wave is polarized with respect to Ox, so for the representation that has to be in the 0xz plane, the Oy axis is directed toward the back of the figure. Comment: Role of the choice of the origins for the guide planes with respect to stationary solutions In general terms, the propagation is given by Eq. (11) so that with d²E m (z) Ȧ² - k g ²= K² and E mx = E m (as E // Ox ), then +K² E m (z) = 0 . The c² dz² general solutions, which also can be written as E m (z) = A cos(Kz) + B sin(Kz) , can be placed into one of the two following forms, depending on the how the guide planes parallel to Oxy are assumed to be placed:
z
z a
+ a/2
0 x x
O x x
scenario 1
O
y
scenario 2
y
- a/2
(a)
(b)
Figure 12.15. bis. The origin (O) placed (a) on a plane; or (b) midway between planes.
1. At z = 0 and z = a as shown in Figure 12.15bis, scenario 1, which is the same as the representation used up till now, so that the limiting conditions are thus: x E m (0) = 0 , for which A = 0 , so that E m (z) = B sin(Kz) = E 0 sin(Kz) , with B = E 0 (constant); and x E m (a) = 0 ,
for
which
so
that mʌ
, and finally E m = E 0 sin z a a = 0 , which is a nonphysical solution).
m = 1, 2, 3...., so K = that E m
mʌ
E 0 sin(Ka) = 0 ,
Ka = m S with ( m = 0 means
Chapter 12. Total reflection and guided propagation
397
The form of the wave is thus: § mʌ · E = E m exp[i(k g y - Zt)]= E0 sin ¨ z ¸ exp[i(k g y - Zt)] ), © a ¹ which is what has been used until now.
2. At z =
-a
and z =
+a
(scenario 2 in Figure 12.15bis). The limiting conditions 2 2 thus are now: a) a a a a a E m ( ) = A cos(K ) + B sin(K ) = 0 = E m (- ) = A cos(K ) - B sin(K ; 2 2 2 2 2 2 Two solutions are possible for A and B, either: a x B = 0 and A z 0 ( A = constant denoted E 0 ) with cos(K ) = 0 , so that 2 a ʌ ʌ Mʌ K = + nS , for which K = [1 + 2n] = with odd M, and 2 2 a a Mʌ E m = E 0 cos z with odd whole M; or a a x A = 0 and B z 0 ( B = constant denoted E0 ) with cos(K ) = 0 , so that 2 a ʌ Mʌ with an even value of M. K = nS , for which K = 2n = a a 2
The form of the wave E = E m exp[i(k g y - Zt]) thus is given by either: § Mʌ · z ¸ exp[i(k g y - Zt]) with a whole odd value for M; or x E = E 0 cos ¨ © a ¹ § Mʌ · x E = E 0 sin ¨ z ¸ exp[i(k g y - Zt]) with M being whole and even. © a ¹ We can verify that the plots of Em(y) in scenario (1) where m = 1, 2, 3, 4... or for scenario (2) with the two forms of solutions for even or odd values of M, where M = 1, 2, 3, 4... ( { m ) , are identical, which is reassuring as the problem is the same! (This remark also can be made in terms of the quantum mechanics of wells with symmetrical potentials, where the origin can be taken either as at an extremity of the well where there is a potential wall, or at the center of the well.)
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398
12.4.4.4. Multimodal fields
It should be noted that it is not only the field corresponding to a given mode that can be guided, meaning in other words those fields that have a transversal intensity (with respect to Oz) independent of the axial position (with respect to Oy) as shown in Figures 12.16 a and b corresponding, respectively, to the given modes (using notation of scenario 1): x complex amplitude, with m = 1 E10 = 2 Ei0 sin([k 0 cos T1 ]z) exp(i[k 0 sinT1 ]y) , so that its intensity is independent of y; and complex amplitude, with m = 2
x
E 02 = 2 Ei0 sin([k 0 cos T2 ]z) exp(i[k 0 sinT2 ]y) , so that its intensity is independent of y. The fields simply have to verify the limiting conditions (E = 0 on metallic planes) and thus can correspond to the superposition of several modes. Thus in the case given in Figure 12.16 c, taking into account the analytical forms of the fields, it is possible to see that the transversal intensity (with respect to Oz) of the wave is dependent on the axial position (with respect to Oy) in which it is placed. G In effect, the complex amplitude of the wave E1,2 , as a superposition of the modes m = 1 and m = 2 , is in the form 2 Ei0 ^ sin([k 0 cos T1 ]z) exp(i[k 0 sinT1 ]y) + sin([k 0 cos T2 ]z) exp(i[k 0 sinT2 ]y)` , and its intensity gives rise to a variation in its distribution with respect to Oy as k 0 sinT1 z k 0 sinT2 . z
z y
y (a) m = 1
z
(b) m = 2
y (c) mode m = 1 and mode m = 2
Figure 12.16. (a and b) Guided waves corresponding to a mode; and (c) without a mode.
12.4.4.5. Comment: TM modes
TM modes occur when it is the magnetic field that is applied with respect to Ox. They can be studied in the same way as TE modes and as such can be guided. The electric thus has components with respect to Oy and Ox.
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399
12.5. Optical Guiding: General Principles and How Fibers Work 12.5.1. Principle Optical fibers are used, by way of a support, to guide an incident luminous wave from its point of injection at an entrance face to its exit point at the other. Typically, there are two main types of fiber: fibers with a jump in indices and fibers with a gradient in indices. This text will limit itself to detailing the first type, which are constituted of a core made from a material with a circular cross section and a given index (n) surrounded by a gain with an index denoted n1 such that n > n1 . Generally, the whole structure sits in air, which has an index given by n' | 1 . The luminous wave injected at the entrance face of the fiber is under an angle i', which is such that n' sin i' = n sin i . This wave meets the core/gain interface at an ʌ angle of incidence given by Ti = T , such that i +T = . When T >TA where TA is 2 a limiting angle above which the phenomenon of total reflection occurs when n > n1 , the wave makes no penetration of the gain material and is guided along the fiber in a zigzag trajectory associated with successive total reflections at the core/gain interface, and as illustrated in Figure 12.17. The calculations carried out below in the plane of the longitudinal section of the fiber are equally valid for the longitudinal plane section of a symmetrical rectangular guide where the Or axis is simply replaced by a Oz axis.
r
lost ray
n1 n’
O
T’ i
guided ray
T
i’
n (> n1) n1 L Figure 12.17. Trajectories when T > TA and T’ < TA .
y
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Basic electromagnetism and materials
12.5.2. Guiding conditions 12.5.2.1. Condition on i
The limiting angle TA is defined by n sinTA = n1 , wherein at the entry face an angle i has a value given by iA =
ʌ 2
- TA . A condition for the total reflection T t TA is
therefore i d iA . In addition, by moving TA = obtain cos iA =
ʌ 2
- iA into n sinTA = n1 , we directly
n1
. n This condition for wave guiding thus brings us to a condition for the incidence at the n entry face, which must be such that i d iA , so that cos i t cos iA = 1 , which n n1 . means that i d iA = Arc cos n All rays with i ! iA have at the core/gain interface an incidence such that T' < TA . A fraction of the wave is transmitted toward the gain, and following several successive reflections, the remaining fraction of the wave in the core becomes smaller and smaller so finally it is extinguished along its journey. In numerical terms, with n = 1.50 and n1 = 1.49 , we obtain iA = Arc cos
n1 n
= 6.62 ° .
When n = 1.60 and n1 = 1.50 , we have iA = Arc cos
n1 n
= 20.36 ° .
12.5.2.2. Condition on i’ (acceptance angle): numerical aperture
The condition T t TA makes it possible to state that: cosT d cosTA = ʌ
1 sin ²șA =
1
n12 n2
.
-T , from which with n' sin i' = n sin i 2 ʌ n n' sin i' = n sin( - T) = n cosT , so that sin i' = cosT and the preceding equation 2 n' therefore leads to
We also have i =
Chapter 12. Total reflection and guided propagation
sin i' =
n n'
cosT d
n n'
cosTA
n n'
1
n12
n § n² - n12 ¨ n' ¨© n 2
n2
1/2
n - n 2
2 1
n'
, so that:
1/2
n - n sin i' d 2
1/ 2
· ¸ ¸ ¹
401
2 1
.
n'
To conclude, all rays introduced under an angle of incidence i' verify the preceding inequality and therefore can propagate. The quantity given by
NA = n 2 - n12
1/2
is the numerical aperture of the system, which although here is an optical fiber, also can be applied to a guide structure. The acceptance angle is by definition: 2
i'A
1/2
n - n = Arc sin 2 1
n'
.
In numerical terms n = 1.50 - n1 = 1.49 - n' = 1, so the NA = 0.173, so that i'A = 9.96 ° . When n = 1.6 - n1 = 1.5 - n' = 1, and the NA = 0.557, so that i'A = 33.83 ° .
12.5.3. Increasing the signals Once a luminous impulsion is injected into a fiber, various pathways can be followed, as in: x the shortest trajectory which is along the pathway Oy and corresponds exactly to c the length (L) of the fiber. With the speed of propagation being given by vM = , n nL ; the pathway time (ty) is t y = c x zigzag trajectories which each have a given value for the angle i and have a L > L . With the pathway speed still being pathway length (Li) given by Li = cos i c L nL vM = , the pathway times (ti) are now given by t i = i = > t y (ti is n vM c cos i great when cos i is small, so that i is great).
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Basic electromagnetism and materials
The difference in propagation times thus is given by: i² · § 11 ¸ nL§ 1 · nL¨ 2 ¸ | n L i² . ¨ 't i = t i - t y = 1¸ | ¨ i² c © cos i c ¨ 1 ¸ 2c ¹ ¨ ¸ 2 ¹ © In numerical terms, when L = 1 km, n = 1.5 and i = 6 ° < iA , so that 't i = 28 ns. Therefore, an impulsion emitted at y = 0 at a time t 0 = 0 over an interval of time assumed to be Gt | 0 will be emitted at y = L starting from the moment t 0 + t y = t y (for the wave that takes the shortest pathway) up to the moment t 0 + (t i )max = (t i )max where (t i ) max is the pathway time for the longest possible trajectory. This latter value comes from the highest possible value of i, i.e., i = iA . The corresponding interval in time therefore is given by 't iA =
nL 2 iA where 't iA 2c
is the duration time of the emission at the exit. The interval in time (T) between two impulsions at the entrance which is required so that they do not mix at the exit therefore must be such that T t 't iA , which is the same as stating that the frequency of the impulsions must be lower than 1 1 Q iA = (we can state that we should have T = t 't iA so that 't iA Q
Q
1
't i A
=QiA . and in terms of actual numbers, this means that for the
aforementioned values, QiA | 30 MHz ).
12.6. Electromagnetic Characteristics of a Symmetrical Monomodal Guide Here we look at a buried guide, which for technological reasons is simplified to the symmetrical structure shown in Figure 12.18. - a/2
y
+ a/2
z
n
G E x
Figure 12.18. Buried guide.
n1
Chapter 12. Total reflection and guided propagation
403
The propagation of an electromagnetic wave with an angular frequency given by Z is along Oy and is polarized with respect to Ox. The field is reflected at the interface, between a medium with an index given by n and another medium with an index denoted by n1, along planes parallel to Oxy with the EM wave alternating its -a +a and z = . This configuration has the reflection at the planes defined by z = 2 2 same geometry as that given in Section 12.4, and as that presented in Figure 12.19a, making direct comparisons possible between systems that only differ by virtue of their planes of interface. The metallic planes, which are imposed as a limiting G condition where E= 0 , are now replaced by dielectric diopters with the aforementioned index n1 where n1 < n (and as a consequence, with different limiting conditions). In addition, the two dielectrics are assumed to be nonabsorbent. In the zones (1), (2), and (3) the electric field can be written by notation as G 1 G 2 G 3 E , E , and E , respectively, and in a manner analogous to that of Section 12.4, taking into account the symmetry of the problem, the solutions for the TE wave polarized with respect to Ox are looked for in each of the three zones in the form: G p p G E =E mx (z) ex exp[j(k yp y - Ȧt)] where p = 1, 2, 3 characterizes the zone under consideration. G 3 zone (3), index n1 < n, and field E
z +a/2 O x x
T i
G 2 zone (2), index > n1, and field E
y
- a/2 G 1 zone (1), index n1 < n, and field E Figure 12.19(a). Propagation in a symmetrical dielectric guide.
12.6.1. General form of the solutions The equations for continuity of the tangential component for the different field G 1 G 2 G 3 E , E , and E directed exactly along Ox, true for all moments t and whatever values of y, lead to k y1 = k y2 = k y3 = k y [see also Section 11.2.3, concerning Eq. (7) or (7’) with respect to Oy].
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With the media being nonabsorbent, H rp = n p ² , we then have: G p G JJG G p n p2 w E p wE rot B = µ0 İ 0 İ rp = . By taking the rotational of the two terms c² wt wt G p JJG G p wB , and by using the preceding equation, we obtain of the equation rot E = wt an equation for propagation in a medium with an index denoted np [see Chapter 7, Eq. (5) this volume, and with µr = 1 ] G p JJJGG p n 2p w ²E ǻE =0. c² wt² With the wave being monochromatic, we also can state, using Eq. (7") again from Chapter 7, G p JJJGG p Ȧ2 2 w ²E ǻE + np = 0. wt² c² Taking into account the form of the required solution, this equation gives [using a calculation analogous to that yielding Eq. (11) of Section 12.4.3.2]:
p
d²E mx (z) § Ȧ² 2 · p + ¨ n p - k y ² ¸ E mx (z) = 0 . dz² © c² ¹
§ Ȧ² · n 2p - k y ² , and as ¨ n 2p - k y ² ¸ can be, a priori, positive or c² c² © ¹ negative, and Į p can be real or purely imaginary, under these conditions the Making Į 2p =
Ȧ²
solutions are in the general form:
p
E mx (z)= A exp(jD p z) + B exp(-jD p z) .
12.6.2. Solutions for zone (2) with an index denoted by n kz { D G k i
T
ky { E G
Figure 12.19 (b) Geometrical component of k .
Chapter 12. Total reflection and guided propagation
In this zone, the wavenumber is given by k =
nȦ c
405
and its projection along the axis
of propagation Oy is k y = k sinT = k cos i { E (by notation) which is a constant bound to the propagation phase. Ȧ² 2 n - k y ² = k² - k² cos²i = k²(1 - cos²i) = k²sin²i , so The result is that Į 2 ² = c² that D 2 = ± k sin i . The sign is not important with respect to the general solution given by E (2) mx (z) , which contains both signs as exponentials, so by notation we use
D 2 = k sin i = k cosT = k z { D . This magnitude is finally tied to the amplitude of a stationary wave due to reflections at the interfaces (see the metallic guide in Sections 12.4.2 and 12.4.3). Therefore, in the zone (2) we have, with D = k sin i ,
2
E mx (z)= D1 exp(jD z) + D2 exp(-jD z) where D1 and D2 are the constants to be determined for the limiting conditions. In fact, given the symmetry of the problem, we should have D1 = ± D2 , so that D1 = D and D2 = HD with H = ±1 . In effect, given this symmetry of the guide, with respect to the z axis and for two points symmetrical placed about O located by z = z 2 and z = - z 2 , the intensity of the wave should be the same (in quantum mechanics, the probabilities of presence would be stated to be equal). Thus, | D1 exp(jD z 2 ) + D 2 exp(-jD z 2 )|² = | D1 exp(-jD z 2 ) + D2 exp(jD z2 )|² , so that | D1 + D2 exp(-2jD z 2 )|² = | D1 exp(-2jD z 2 ) + D2 |², for which |D1|² = |D2 |² , and D1 = ± D2 . Finally, G 2 G E = > D exp(j Į z) + İ D exp(-j Į z) @ ex exp[j(k y y-Ȧt)] .
12.6.3. Solutions for the zones (1) and (3) In the titled zones [zones (1) and (3)], we have D1 ² = D3 ² = n12 Ȧ²
n12 n 2 Ȧ²
n12
c²
n²c²
n²
n12 Ȧ² c²
- k y ² , where
k² , so that k y = k cos i is always true. The result is that
Basic electromagnetism and materials
406
§ n2 · D1 ² = D3 ² =k² ¨ 1 cos ²i ¸ , ¨ n² ¸ © ¹
and as: x x
n12
< 1, n² and on the other, in order to guide the incident ray (i) to the entry face shown in Figure 12.19a, we should have i d iA as a result of Section 12.5.2.1 which uses the same geometry, so that n cos i t cos iA = 1 n 2 §n · We have ¨ 1 cos ²i ¸ < 0 , so that we can make: ¨ n² ¸ © ¹ on the one hand, n > n1 ,
§ n2 · D1 ² = D3 ² =k² ¨ 1 cos ²i ¸ = j² J ² < 0 , for which D1 = D3 = jJ, where: ¨ n² ¸ © ¹
n2 J = k cos²i - 1 > 0 . n²
1
In zone (1) we thus find that E mx (z)=C1 exp(-j D1 z) + C3 exp(j D1 z) , and in this zone where z < 0 , so that the wave is evanescent (corresponding to the guiding condition as in T > TA or i < iA ), we should find C3 = 0 [If not, the component G 1 C3 exp(j D1 z) yields for E a contribution of the form C3 exp(j² J z) = C3 exp(-J z) , which will diverge when z o -f to give the amplification term rather than the term for attenuation as required]. In zone (1), the solution thus is in the form E1mx (z)= C1 exp(-j D1 z) , so that:
1
E mx (z)= C1 exp(J z) . In the zone (1), the final expression for the wave is: G 1 G E = C1 exp(Ȗ z) ex exp[j(k y y-Ȧt)] . This wave is evanescent with respect to Oz and progressive with respect to Oy. It is
Chapter 12. Total reflection and guided propagation
407
in effect an inhomogeneous MPP wave that has an amplitude that is dissimilar at different points along the plane of the wave defined by y = constant. Similarly, in zone (3), we have:
3
E mx (z)= C1 exp(-j D3 z) + C3 exp(j D3 z) and in this zone where z > 0, the physically acceptable solution (yielding an attenuation and not an amplification) is E3mx (z)= C3 exp(j D3 z) , so that:
3
E mx (z)= C3 exp(- J z) . In zone (3) then, the final equation for the wave is given as G 3 G E = C3 exp(-Ȗ z) ex exp[j(k y y-Ȧt)] which also is an evanescent wave with respect to Oz and a progressive wave with respect to Oy. It is worth now looking at the relationship between C1 and C3 due to the symmetry of the guide. With the problem being one of symmetry around O with respect to the z axis, we should have the same wave intensity at two symmetrical points, one located with z = z1 and in zone (1), and the other located with z = z3 = - z1 and in zone (3). So, |C1 exp(J z1 )|²
½° C1 = ± C3 = {|C3 exp(-J z3 )|²}z3 = -z1 = |C3 exp(J z1 )|² ¾°¿
By making C1 = C and C3 = HC with H = ± 1 , then we will have as solutions for zone (1): G 1 G 1 E mx (z)= C exp(J z) , and E = C exp(Ȗ z) ex exp[j(k y y-Ȧt)] .
On zone (3), the solutions will be the same: G 3 G 3 E mx (z)= H C exp(- J z), and E = İ C exp(- Ȗ z) ex exp[j(k y y-Ȧt)] .
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408
12.6.4. Equations for the magnetic field Zone (1) G JJG G (1) wB(1) , we have : From Maxwell-Faraday law, which states rot E = wt JJG G (1) rot E =
0 G (1) °° G (1) wB (1) (1) and = jZB Ex / z = E w w J ® wt ° (1) (1) °¯- wEx / wy=- jk y E
0 G (1) 1 °° B = ®-j J E (1) Z° (1) °¯- k y E
Zone (3) 0 JJG G (3) °° rot E = ®wEx (3) / wz =- J E (1) ° (1) (3) °¯- wEx / wy=- jk y E
and -
G (3) wB wt
G (3) = jZB
0 G (3) 1 °° B = ® j J E (3) Z° (3) °¯- k y E
Zone (2) 0 ° ° (2) ®wEx / wz =- jD > D exp(j Į z) - İ D exp(-j Į z) @ exp[j(k y y-Ȧt)] and ° (2) (2) °¯- wEx / wy=- jk y E 0 G (2) G (2) 1 °° = jZB B = ®D > D exp(j Į z) - İ D exp(-j Į z) @ exp[j(k y y-Ȧt)] Z° (2) °¯- k y E
JJG G (2) rot E =
-
G (2) wB wt
12.6.5. Use of the limiting conditions: determination of constants §Ȗa· § Įa· Here we make G = exp ¨ ¸. ¸ and µ = exp ¨ j © 2 ¹ © 2 ¹ The continuity of the tangential components of E at z =
a 2
are written:
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409
a a E (3) ( ) = E (2) ( ) , 2 2 from which a a a H C exp(-J ) = D exp(jD ) + H D exp(-jD ) , 2 2 2 so that HC H = D( µ + ). (1) G µ With the media being nonmagnetic, we can equally write that the continuity of a are such that we have the tangential components for B at z = 2 a a By (3) ( ) = B y(2) ( ) , from which 2 2 jJHC exp(-J a/2) = D > D exp(j Į a/2) - İ D exp(-j Į a/2)@ , so that: j J H C/G = D [µ - H/µ] D .
(2)
We thus obtain two linearly independent equations that allow a determination of the unknowns C and D. The other equations of continuity give for their part equations leading to the same relations as Eqs. (1) or (2). If the reasoning permitted by the effect of symmetry had not been possible, then there would have been four constants, namely C1,C2, D1, and D2, which these additional equations would have permitted to determine. a For example, at z =- , we have: 2 a a E (1) (- ) = E (2) (- ) , 2 2 for which a a a C exp(-J ) = D exp(-jD ) + H D exp(jD ) , 2 2 2 so that C D = + HµD . G µ By multiplying by H we find Eq. (1).
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Basic electromagnetism and materials
12.6.6. Modal equation 12.6.6.1. Placing into equations First, we write the Eqs. (1) and (2) for H = +1 , as in: C/G = ( µ + 1/µ) D
(1’)
j J C/G = D [µ - 1/µ] D
(2’)
the ratio (2’)/(1’) gives
jĮa
jJ=D
µ² - 1 µ² + 1
, so that
Ȗ
2
1e
Į
j
jĮa
e 2
jĮa
tan
jĮa
Įa 2
, and therefore
e 2 e 2 tan
Įa
=
2
Ȗ Į
.
Similarly, when H = -1 , we obtain: - C/G = ( µ - 1/µ) D (1’’) - j J C/G = D [µ + 1/µ] D
(3)
the ratio (2’’)/(1’’) gives
(2’’) jĮa
jJ=D
µ² + 1 µ² - 1
, for which
Ȗ
j
Į
e
2
jĮa
e 2
jĮa
jĮa
e 2 e 2 Įa
tan
Equation (3) gives Į a 2
1 x
=
ʌ 2
Į Ȗ
.
(4)
Į a
§Ȗ· = Arctan ¨ ¸ + rS , where r is a whole number t 0 as 2 ©Į¹
> 0 . So, we have
Arctan x + Arctan
2
=-
-1 , so that Įa t an 2
,
Da 2
=
§Į· - Arctan ¨¨ ¸¸ + r S , by using the equation 2 ©Ȗ¹
ʌ
§ 1 · ʌ 1 ʌ ( tan( - y) = gives - y = Arctan ¨¨ ¸¸ by 2 tan y 2 © tan y ¹
making x = tan y ). We deduce that: §Į· ʌ + Arctan ¨¨ ¸¸ = (2r + 1) 2 2 ©Ȗ¹
Į a
[ r is an integer t 0 ] .
(5)
Chapter 12. Total reflection and guided propagation
411
From Eq. (4) we pull out: §Į· Į a Į a > 0 , we thus have = - Arctan ¨¨ ¸¸ + s S , and with s integer > 0 as 2 2 ©Ȗ¹ § Į · 2 s ʌ . + Arctan ¨¨ ¸¸ = 2 2 ©Ȗ¹
Į a
(6)
The solutions from Eqs. (5) and (6) can be regrouped into a single expression, as in: §Į· qʌ + Arctan ¨¨ ¸¸ = , 2 2 ©Ȗ¹
Į a
(7)
where q is an integer and q t 1 as r t 0 and s > 0.
12.6.6.2. General equation and solutions n2 n In addition, we made D = k sin i , and J = k cos²i - 1 , with 1 = cos iA . The n² n left-hand side of Eq. (7) is by consequence a function of the angle i, and Eq. (7) can be rewritten: a f(i) = k sin i + Arctan 2
sin i cos²i - cos²iA
q
ʌ 2
.
(8)
Equation (8) is a modal equation that permits a determination of q when i [0, iA ] . For this interval, sin i and
1
are strictly increasing functions of i, just
cos²i - cos²i A like the arc tangent function. Indeed, f(i) is also a strictly incremental function, so that for given values of a and q, there is a single value of i as a solution to f(i). The maximum value of the function f(i) is obtained for the maximum acceptable value of i, that is i = iA . For this value, we therefore have: a a ʌ ʌ [f(i)]max = k sin iA + arc tanf = k sin iA + = q . 2 2 2 2 The solution q = 1 therefore is always obtained, even when a o 0 , as it suffices to take i o iA as the arc tangent function tends toward S/2.
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Basic electromagnetism and materials
12.6.6.3. Monomodal solution To remain in a monomodal regime where q = 1, the solution q = 2 must remain unattainable. For this to be true [f(i)]max < S must be imposed, so that a ʌ ʌ nȦ 2ʌ . With k = k sin iA < , which means that a < a1 = = n k0 = n 2 2 k sin iA c Ȝ0 and sin iA =
n2 1 cos ²iA = 1 1 , we obtain: n2 a1 =
Ȝ0 2 n 2 - n12
.
In numerical terms, with O 0 = 1.3 µm (a typical telecommunications wavelength), n = 1.50 and n1 = 1.49 . For the monomodal condition, where q = 1 uniquely, the guide must have a width less than a1 = 3.76 µm. When O 0 = 1.55 µm, n = 1.50 and n1 = 1, we obtain a1 = 0.7 µm .
12.6.6.4. Multimodal solutions When a > a1 , the modal Eq. (8) in effect permits several different values of q as solutions. In order to find the general solution to Eq. (8), we can write that with 2ʌ sin²iA - sin²i , we have k= and cos²i - cos²iA Ȝ a sin i ʌ ʌ sin i + Arctan q , Ȝ 2 sin²iA - sin²i so that on applying Arctan x + Arctan
1
=
ʌ
, we find that x 2 sin²iA - sin²i a ʌ ʌ sin i - q-1 = Arctan . Ȝ 2 sin i By setting m = (q - 1) and as q t 1 , m therefore is an integer such that m = 0, 1, 2, 3.... , and so on using the fact that x = Arctan y , we have y = tan x , and 1/2
ʌ · § sin²iA § ʌa · tan ¨ sin i - m ¸ = ¨ - 1¸ 2 ¹ © sin²i © Ȝ ¹
.
(9)
The two terms in this equation are functions of the variable sin i, and the solutions can be found at the intersection of the plots representing:
Chapter 12. Total reflection and guided propagation
413
ʌ· § ʌa x the function g(sin i) = tan ¨ sin i - m ¸ ; and 2¹ © Ȝ 1/2
§ sin²iA · - 1¸ x the function h(sin i) = ¨ © sin²i ¹
.
In reality, the function g(sin i) can be more simply seen as the function given by: § ʌa · 1. g1 (sin i) = - cotan ¨ sin i ¸ when m is odd (as © Ȝ ¹ ʌº ª ªʌ º tan « x - » = - tan « - x » = - cotan x ); and 2¼ ¬ ¬2 ¼ ʌa § · 2. g 2 (sin i) = tan ¨ sin i ¸ when m is even as tan > x ± ʌ @ = tanx . Ȝ © ¹ The resolution of Eq. (9) thus leads to plotting, as a function of sin i, the two systems given by: ʌa ʌ x the plots for 2 defined for when 0 < sin i< when m = 0 , Ȝ 2 ʌa 3ʌ S < sin i< when m = 2 , etc..., so that Ȝ 2 Ȝ Ȝ 3Ȝ 0 < sin i < ( m = 0 ), < sin i < ( m = 2 ), and so on; and 2a a 2a ʌ ʌa x the plots for 1 defined for when < sin i< S ( m = 1 ), 2 Ȝ 3ʌ ʌa < sin i< 2S ( m = 3 ), etc..., so that 2 Ȝ Ȝ Ȝ 3Ȝ 2Ȝ < sin i < ( m = 1 ), < sin i < ( m = 3 ), and so on. 2a a 2a a The solutions for sin i in each interval precisely defined by the value of m Ȝ for the solution), thus can be found as shown in (hence the notation sin i m = m 2a 1/2
§ sin²iA · - 1¸ , which is a Figure 12.20 at the intersection with the plot of h(sin i) = ¨ © sin²i ¹ function that decreases monotonically with sin i. When i = iA , this function cancels out so that h(sin iA ) = 0 .
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Basic electromagnetism and materials
Figure 12.20. Graphical solutions for the modal equations obtained by the intersection of the plots h(sin i) and g(sin i).
For given values of a, O, n, and n1, for each value of m there is a corresponding value of I denoted Im and between 0 and iA . Associated with Im is the wavevector with a component nk0 cos Im with respect to the axis of propagation along Oy. Em = k y = nk 0 cos i m is the propagation constant. With respect to the axis Oz, the component of the wavevector associated with Im is for its part given by k z = nk 0 sin i m { D m . kz nk0 n 1k 0 n k0 sin il
Q m = 3 (q = 4)
n k0 sin i3
m = 2 (q = 3) m = 1 (q = 2) i3
m = 0 (q = 1)
il nk0 n k0 cos il = n1k0 n k0 cos i3
E = ky = n k0 cos i
Figure 12.21. Determination of Im angles corresponding to m modes.
Chapter 12. Total reflection and guided propagation
415
n1
, Em varies between nk 0 and n1k 0 . n The im angles corresponding to the m modes, as well as the corresponding components given by k y { Em and k z { D m , are shown in Figure 12.21. As cos i m varies between 1 and cos iA =
12.6.6.5. Number of nodes From Figure 12.20, each interval associated with a mode has a width given by O/2a, which means that for a width defined by L as a function of sin i such that L = sin iA , the number of possible modes (M) is equal to the first integer immediately above the sin iA sin iA . By convention, we can write that M . number given by the ratio Ȝ/2a Ȝ/2a 12.6.6.6. Cutoff frequency Returning to Eq. (8) and the reasoning used in Sections 12.6.6.2 and 12.6.6.3, for a ʌ ʌ mode given by q (= m + 1) to not be affected, then [f(i)]max < q = (m + 1) 2 2 a ʌ 2ʌ must be true, so that k sin iA < (m + 1) , and from which with k = n and 2 2 Ȝ0 n1 n
= cos iA =
Ȝ 0m > Ȝ 0mc =
1 - sin²iA included we can deduce that sin iA = c (m + 1)
2a n 2 - n12
n 2 n12 n
and
.
12.6.7. Comments: alternative methodologies 12.6.7.1. Comment 1: by analogy to solutions for potential wells The type of mode, whether even or odd, can be apprehended directly through Eqs. (3) and (4) which are identical to those found in a resolution of potential wells with the given width as a. Į a Ȗ a and Y = ; Considering Eq. (3), and by making X = 2 2 Eq. (3) can be written as Y = X tan X Similarly, Eq. (4) gives Y = - X cotan X . This then yields:
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Basic electromagnetism and materials
X²+Y² =
a² 4
a² ª n2 º « k² sin²i + k² cos²i - k² 1 » 4 «¬ n² »¼ § n 2 · a² a² = k² ¨ 1 - 1 ¸ = k 02 (n² - n12 ) . ¨ ¸ n² ¹ 4 4 ©
>Į² + Ȗ² @ =
For a given guide, with indices n and n1 and of dimension a, it is possible to state that X²+Y²= R² is a constant characteristic of the guide. The solutions for X and Y can be found at the intersection of the plots of the preceding Y = f(X) with a circle of radius R. From these solutions D can be determined, thus i, and then, as will be detailed in Section 12.6.8, the distribution of the field that varies with cos Dz (for even modes corresponding to the solutions to the equation Y = X.tan X ) or with sin Dz (for odd modes corresponding to solutions to the equation Y = - X cotan X ). 12.6.7.2. Comment 2: methodology from principles of optics
We have assumed up till now that for a wave to be guided that (1) it has to undergo a reflection at an incidence such that T > TA at an interface between two dielectrics that make up the guide; and (2) the equation for the propagation should be verified using the limiting conditions of the problem. It is by this route that we have selected solutions corresponding to the modes. For a given mode, the solutions are such that the transversal intensity of the wave is independent of the position in Oy of the resultant direction of propagation. The second condition can be directly replaced due to the fact that for the wave to propagate it must be part of the same system of plane waves, meaning that after several reflections the planes of the waves orthogonal to the direction of propagation are conserved. This is the same as saying that the dephasing between the wave that propagates along AB (subject to dephasing of MA and MB on reflection at A and B) and the wave that propagates directly from A to C is a multiple of 2S, as shown in Figure 12.22. Thus: 2ʌ k.AB - k.AC + MA + MB = [AB - AC]+ 2 MA = 2S m . (10) Ȝ In effect, the two interfaces are identical (the same dielectrics with indices n and n1) and the wave is polarized perpendicularly to the plane of the figure so that MA = MB = MA , and therefore we either have MA + MB =2MA , or MA as determined in Section 11.3.2.2 with Eq. (37). ʌ This with (i + T) = , gives: 2
§M tan ¨ A © 2
1/2
sin²ș - sin²șA · ¸= cosș ¹
1/ 2
§ sin ²iA · 1¸ = -¨ sin ²i © ¹
.
(11)
Chapter 12. Total reflection and guided propagation
C
dielectric with index = n1
A
417
A’
i i
T
a
O
dielectric
dielectric with index = n1
with
index = n
B
Figure 12.22. Conservation of wave planes in guided propagation.
With [AB - AC] = 2 a sin i
(classic dephasing calculated, for example, in
establishing Braggs law, so that here AB = AB - AC = 2ʌ Ȝ
a sin i
a sin i
and AC = AB cos 2i , to give
[1 -cos 2i] = 2a sin i ), then Eq. (10) gives:
2 a sin i + 2 MA = 2S m , which in turn yields: ʌ · MA § ʌa . ¨ sin i - m ¸ = 2¹ 2 © Ȝ
By taking the tangent of this equation into which is also plugged in Eq. (11), we obtain the preceding Eq. (9), as in: 1/2
ʌ · § sin²iA · § ʌa tan ¨ sin i - m ¸ = ¨ - 1¸ 2 ¹ © sin²i ¹ © Ȝ
.
12.6.8. Field distribution and solution parity It is worth looking at the distribution of the electric field in each of the zones (1), (2), and (3) for the possible values of q.
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Basic electromagnetism and materials
q=1 m=0
z + a/2
q=2 m=1
q=3 m=2
q=4 m=3
y
- a/2
Figure 12.23. Distribution of the electric field for various m modes.
12.6.8.1. In zone (2)
The form of the field established from Section 12.6.2 is such that E 2mx (z)= ª¬ D exp(jĮ q z) + İ D exp(-jĮq z) º¼ . 12.6.8.1.1. Even solutions (as cosines) When H = +1 , we have:
2 E mx (z)= ª¬ D exp(jĮq z) + D exp(-jĮq z) º¼ v cos Įq z = cos [(k sin iq )z] .
In addition, H = + 1 corresponds to Eq. (3) which leads to q = 2r+1 [Eq. (5)], with r an integer t 0. The r = 0 gives q = 1 so that m = 0; r = 1 gives q = 3 so that m = 2, while r = 2 gives q = 5 so that m = 4, and so on (see also Figure 12.21 and 12.23). 12.6.8.1.2. Odd solutions (as sins) When H = -1, we have:
2
E mx (z)
ª D exp(jĮ q z) - D exp(-jĮq z) º v sin Į q z = sin [(k sin iq )z] . ¬ ¼
Here, H = - 1 corresponds to Eq. (4) which leads to q = 2s (Eq. (6)), with s being an integer > 0, so that for s = 1, we have q = 2 and m = 1, and when s = 2 we have q = 4 and m = 3, and so on (see also Figures 12.21 and 12.23).
Chapter 12. Total reflection and guided propagation
419
12.6.8.2. In zone (1)
The form of the field established in Section 12.6.3 is such that when z < - a/2 ,
1
E mx (z)= C exp(J z) , and with J as the extinction coefficient, we can state that: n2 J = k cos²i - 1 = k 0 n cos²i - cos²iA n² = k 0 n cos iA
cos²i cos²iA
- 1= k 0 n1
cos²i cos²iA
-1
12.6.8.3. In zone (3)
The form of the field established in Section 12.6.3 is such that when z > a/2 ,
3
E mx (z)= H C exp(-J z) and J takes on the same form as above. 12.6.8.4. The Goos–Hänchen effect
We can note that in the presence of a dielectric (with an index = n1) in place of the plane conductors, the electric field no longer gives rise to nodes (E = 0) at z = ± a/2 (see Figure 12.23) and the penetration to a depth of the order of G | 1/J of the ray undergoing a total reflection is called the Goos–Hänchen effect.
12.6.9. Guide characteristics 12.6.9.1. Form of the signal leaving the guide If a >> O , numerous values are possible for q, which results in discreet but close values for the injection angle (iq). The group of modes allows the guide to function in multimode. To each value of iq there is a different path length in the guide along with an associated time (tq), and under multimodal regimes, there is a temporal increase of an impulsion leaving a wave with respect to that at the entrance, in what is an effect termed “intermodal dispersion” (Figure 12.24).
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Basic electromagnetism and materials
I
I all modes
all modes
t mode 1 mode 2
t
mode 1
mode 2
mode 3
t
at entrance to guide
mode 3
t at a given time (t) the exiting signal is weaker than the starting signal
Figure 12.24. Comparing guide entrance and exit signals.
In effect, for a given value of q, i is dependent on Z [as k =
nZ c
intervenes in
the modal Eq. (8)], and E = k cos i thus is dependent on Z just as is vg =
dZ
. dE The pathway time also is dependent on Z (and hence also the wavelength), and hence the slight increase in the impulsions with each mode. This is termed the “intramodal effect”, which is superimposed on that of dispersion due to a variation of the index with wavelength. The result is a chromatic dispersion. 12.6.9.2. Nature of losses in a guide
Losses in a guide have various origins and can be organized into two main classes. They are, first, characterized by a attenuation coefficient, or second, an attenuation factor. 12.6.9.2.1. Attenuation coefficient and the linear attenuation factor The transmitted power (P) for a penetrated length (L) varies in accordance with the equation P(L) = P(0) exp(- 2J L) . P(0) is the power introduced at the entrance point, and J here represents the linear attenuation coefficient for the amplitude.
Chapter 12. Total reflection and guided propagation
421
For its part, the linear attenuation factor (D) is defined with the help of the expression: P(L) = P(0)10-ĮL/10 , so that D =
10 L
log10
P0 P
where D is expressed in decibels per meter (dB m-1). As an example, a fiber with 0.22 dB km-1 transmits around 95 % of the energy over 1 km. 12.6.9.2.2. Losses due to the physical configuration of the guide These losses can include: x losses due to a defective guiding (in principle, these losses do not count for guided modes); x losses due to curves (unnatural losses due to deformations of the fiber) for example, when the radius of curvature is less than a centimeter, the higher modes can be refracted within the gain; x losses at joints, which in turn can be divided into two groups: 1. losses due to poor alignment of adjacent components; 2. Fresnel losses, associated with the reflection of the injection signal at the entrance face (as in Figure 12.25). Under normal incidence, using classic 2
experimental
conditions,
we
have
ª n - n1 º R= « » , ¬« n + n1 ¼»
so
that
with
n = 1.5 and n1 = 1, we have R | 0.04 = 4 % . Thus the losses by Fresnel reflections are given by KFresnel = 10 log10 (1-R) = - 0.18 dB .
injection guide with index = n medium with index = n1 reflection Figure 12.25. Fresnel losses.
12.6.9.2.3. Losses due to the material making up the guide Losses due to the properties of the guide material can be divided into two physical phenomena, absorption and diffusion. First, the losses due to absorption (which also are further detailed in Chapter 10 of this volume, or Chapter 3 of the second volume entitled Applied Electromagnetism and Materials) which can come about:
422
Basic electromagnetism and materials
x in the visible domain (zone 3) due to electronic absorption; x in the infrared domain (zone 4) due to network vibrations (vibrations of atoms or ions depending on the nature of the guide and/or fiber). In polymer based materials, the wavelength for the fundamental absorption by the C-H group is approximately 3.3 µm and over distances of any consequence, the window of transparency is in fact limited to 0.8 µm due to absorptions associated with harmonics. This phenomenon resembles absorptions due to hydroxyl bonds (-O-H) that exhibit a transparency window up to 2 µm, which includes in particular two transmission windows at 1.3 and 1.55 µm used in optical telecommunications. The use of fluorinated polymers permits an extension of this region of transparency; x because of a wide number of impurities that present absorption characteristics, and hence the necessity of using highly purified materials. For silicon-based systems, absorptions due to impurities including hydroxyl groups are situated around the above-mentioned windows at 1.3 and 1.55 µm. The improvement of manufacturing techniques has meant that the presence of water has been reduced to less than 1 part per 107 and accordingly the performances of silicon-based fibers have increased considerably to less than 0.2 dB km-1 at 1.55 µm. The use of fluorinated glasses can extend the transparency window to 5 µm, and although the absorption bands are always present, they are well shifted to longer wavelengths. D (dB km-1) peaks due to OH- ions
10
infrared absorption (network vibrations)
1 Rayleigh diffusion increases at shorter O
0.1
0.5
1
1.5
2
O(µm)
Figure 12.26. Plot of D = f(O) showing the origin of losses in a silicon based fiber.
Chapter 12. Total reflection and guided propagation
423
Second, losses due to diffusion that can be further separated into two groups: x extrinsic diffusion which itself can be due to two possible causes, the first being O tied to the possible inclusion of dust (of a size greater than ) and the second 20 being associated with the presence of microcrystallites. It is evident therefore that there is an interest in using extremely clean and amorphous materials. x intrinsic diffusion due to Rayleigh diffusion (see also Chapter 10) which is caused 1 by interactions of light with materials. It varies with respect to so that the O4 effect becomes more important at shorter wavelengths. Thus, at very short wavelengths, there is a deviation in the plot of D = f(O ) due to an attenuation of the system (see Figure 12.26 for silicon fiber optics).
12.7. Problems Monomodal conditions By way of recall, the total reflection at the interface between two dielectric media brings into consideration two different and real indices denoted here as n and n1 and such that n > n1 . We thus have n1 = n - 'n . x
+a
i0 n0
n1 = n - 'n
n guided wave O i T -a n1 evanescent wave z
y
Gc
The figure shows the configuration of injection and propagation of light in a medium with an index denoted as n. When T > TA , where TA represents the limiting angle (and iA is the corresponding angle on the inside of the injection surface), G electromagnetic theory tells us that the transmitted wave ( E t ) in the medium with index n1 and for the zone defined by z > a/2 , is in the form:
Basic electromagnetism and materials
424
G JJG E t =E 0t exp(- Jz) exp(i[k y y - Zt]), avec J = k 0 n1 (cos²i/cos²iA ) - 1]1/2 .
1. Show that the intensity of the wave in the medium with index = n1 can be written § 2z · in the form I t (z) = I t (0) exp ¨ ¸ . Give an equation for G, which then can be used © G ¹ to give an approximate value, assuming that T is sufficiently close to S/2 to state that sin²T | 1 . 2. Determine the condition for 'n that permits a weak penetration of a wave (evanescent) into the medium with an index = n1. The condition for weak penetration is fixed using the condition G d Gc | 3 O1 (wavelength in a medium with an index = n1). In numerical terms, determine the condition on 'n when n | n1 | 1.5 . 3. Monomodal condition. In this chapter it is shown that: a < a1 =
Ȝ0 2 n 2 -n12
. With
'n being very small, which can be verified later on, and with the same numerical characteristics as given before in that O 0 = 1.3 µm and a = 5 µm , give the numerical condition that 'n must verify so that the guide is monomodal. Conclude.
Answers JG 1. We have I t (z) = E t
2
JJJJG 2 § 2z · = E 0t exp(- 2 Jz) = I t (0) exp ¨ ¸ with © G ¹ 1/ 2
§ cos²i · G= = 1¸ . ¨¨ ¸ Ȗ k 0 n1 © cos²iA ¹ cos²i as a function of T, we have in one Expressing cos²iA 1
1
part, cos i = cos (S/2 - T) = sin T , and in the other cos iA = cos (S/2 - TA ) = sin TA =
§ n2 · ¨ G= sin²ș 1¸ ¸ k 0 n1 ¨© n12 ¹ 1
1/ 2
n1 n
, with the result that
§ n2 · ¨ | 1¸ ¸ k 0 n1 ¨© n12 ¹ 1
1/ 2
.
Chapter 12. Total reflection and guided propagation
2. With k 0 n1 =
Ȧ c
· Ȝ § n2 G | 1¨ 1¸ 2 ¸ 2ʌ ¨© n1 ¹ 1/ 2
§ n2 · ¨ 1¸ 2 ¸ 6S ¨© n1 ¹ 1
425
n1 , G < 3 O1 gives rise to the condition:
1/ 2
< 3 O1 , and the condition for weak penetration therefore is 2
2 n2 2 ǻn n 'n § 1 · , so that ¨ ¸ < 1= 1 1 | . 2 2 n1 n1 n1 © 6S ¹
We therefore have as a condition, finally, 'n >
n1 72 ʌ 2
.
In numerical terms with n1 = 1.5, we have 'n > 0.0021 .
3. The monomodal condition can be written as Ȝ0 Ȝ0 Ȝ0 , so that: a < a1 = = | 2 2 2 2 ' 2 n -n1 2 2n n 2 n1 +'n - n1 1
'n <
Ȝ 02 1 8n1 a²
.
In numerical terms, a = 5 µm, n1 = 1.5, and O 0 = 1.3 µm, we have 'n < 0.0056 . To conclude, the two conditions brought together at 'n have a common domain, namely 0.0021 < 'n < 0.0056 . In this region, the conditions of weak penetration and monomodal guiding are simultaneously true.
Index
a absorbent media 212, 317 absorption (atom, molecules) 298 absorption (conductor) 356 absorption (zone) 234 acceptance angle 400 Ampere (theorem) 28 Amperian currents 101 antennas 280, 379 antiecho condition 351 antiferromagnetism 150 antiradar structure 352 attenuation 215, 326, 355 attenuation length 374, 420 b B (vector B) 106, 109 Barkhausen effect 148 bound charge 40 breakdown potential 119 bremmsstrahlung 297 Brewster (angle) 341 buried guide 368, 402 c cable (coaxial) 369, 371 Cauchy (formula) 244 cavity (cubic) 72 cavity (molecular) 82 cavity (spheric) 62, 112 charges equivalent ( to a polarization) 45
charges (nature) 40 circular polarization 188 Clausius-Mossotti equation 67, 71 coaxial line 369 compensation charge 40 complex notation 192, 268 conductivity 20, 119, 254 conservation of charge 17 continuity (relations) 53, 107, 168, 374 Coulomb (gauge) 272 Coulomb (law) 8, 58 Coulomb (theorem) 56 cubic cavity 72 Curie (ferroelectric temperature) 128 Curie’s law 134 Curie-Weiss 135 curl 3 current density 16 current density at an interface 19 current sheet 24, 378 current (vitual) 116 cylindrical coordonates 3 cylinder (with surface currents) 114 d Dallenbach layer 350 Debye (field) 82 Debye (formula) 71 decibel (dB attenuation) 421 depolarizing field 42 diamagnetism 131
428
Basic electromagnetism and materials
dielectric (materials) 42, 119, 228 dielectric characteristics 123 dielectric function 240, 248 dielectric withstand strength 119 diffusion (by bound electrons) 291, 309 diffusion (by free electrons) 292 diffusion mechanisms 289 dipolar charges 40 dipolar radiation 267 dipole transition moment 305 dispersion 210, 242, 244, 251, 255, 256, 385 dispersing media (abnormally, normal) 211 displacement (electric, vector D) 51 displacement current 162, 163, 170 divergence 2, 49 dipole field 275 dipoles 40 divergence 2, 49 domain (ferroelectric) 126 drift velocity 21 Drude model 20 Drude-Lorentz 231 dynamic systems 50 e effective field 43 electret 123, 151 electric displacement 52 electric field (vector E) 9, 51 electrical discharge 119 electronic polarization 65, 228, 260, 263 electronic transition 309 electrostatics and vacuums 8 electrostatics of dielectrics 39 elliptical polarization 188 emission 267, 298 energy (EMPW) 215 equation (of wave propagation) 173, 206, 380 equivalent charges 44 evanescent wave 346
excitation vector 106, 109 external field 43 extinction index 215 extrinsic diffusion 423 f ferrimagnetism 150 ferroelectrics 126 ferromagnet (soft and hard) 145 ferromagnetism 137, 145 fibers 399 field distribution 395, 418 flux of current density 18 flux of the vector 6 Fresnel (equations) 331 Fresnel losses 421 frustrated total reflection 347 g Gauss theorem 10, 52 gradient 1 guide 368, 380, 420 guide modes 386 guided propagation 417 guiding conditions 400 h H (vector) 106 Hagen – Rubens (equation) 359 half-wave antenna 283, 285 heterocharge 40 Hertzian dipole 281 homocharge 40 hysteresis (loop) 126, 142 i images (electric) 77 impedance characteristics 257 indices 212, 214 induction 158 insulator 121 integrated transformations 6 interaction (EMW-materials) 227 interfacial polarization 65 internal field 82
Index
intrinsic diffusion 423 ionic polarization 65, 228 j Jones (representation) 195 k k (wave number) 178, 213 k (wave vector) 179, 191 l Langevin (function, theory) 67, 131 Laplace (equation) 14 laplacian 2 laplacian vector 2 Larmor equation 279 Larmor (precession) 98 leak current 206 Lenz (law) 159 linear materials 130, 319 l.h.i. dielectrics 58, 229 Lorentz gauge 268 losses in a guide 420 m magnet 151 magnetic doublet 93 magnetic field 25, 82, 92 magnetic (materials) 89, 119, 129, 256 magnetic mass 110 magnetic moment 89, 94, 97 magnetization (primary) 139 magnetization intensity 100 magnetostatics (vacuum) 24 Malus’s experiment 345 mass (magnetic) 110 Maxwell’s equation 167 Maxwell-Ampere 28, 161 Maxwell-Faraday 159 metals 229, 252, 263 modal equation 410, 414 mode of propagation 393, 414 modes (standing) 179, 386 molar polarizability 67, 71
429
molecular field 135 monomodal 402, 412 MPPEMW 186 multimodal 398, 412 n numerical aperture 400 o Ohm’s law 20 Ohm’s law (limits) 22 Onsager (field) 82 operators 1 optical guiding 399 oscillating charge 280 Ostrogradsky theorem 7 p paramagnetism 132 permittivity (dielectric) 52, 207, 213, 232, 258 piezoelectret 124 plasma (medium) 228, 245 Poisson’s equation 26, 160 polarization (induced) 63, 66 polarization (orientation) 65, 67, 228 polarization (TE) 332 polarization (TM) 332 polarization (vector) 43, 57 polarization charge distribution 46 polarization current 50 polarizability 64 polarized dielectrics 44, 258 polymer 121 potential (scalar) 7, 40, 274 potential vector 8, 89, 101, 274 potential well 415 power 216, 373 Poynting (vector) 216, 220 PPEMW 180 propagation 173, 206, 273 propagation (guided) 367 pseudo scalar potential 93
430
Basic electromagnetism and materials
r radiation power 219, 278, 282 radiation zone 277 RAQSS 157 Rayleigh diffusion 289, 423 reaction field 83 reflection 317, 356 reflection (coefficient) 331, 343 reflection (total) 329, 367 refraction 56, 317 refraction index 215 resonance 229 retarded potentials 268, 273 rotational 2, 30 rotational transition 307 Rutherford diffusion 294 s scalar potential 7, 40 slowly varying rate 158 Snell-Descartes law 322, 325 solid angle 10 solution parity 417 space charge 40 speed (group) 210, 387 speed (wave phase) 194, 210, 387, 394 sphere (charged) 33, 111 sphere (dielectric) 61 spherical cavity (empty) 62 spherical coordinates 3 stationary regimes 17 Stokes’ theorem 7 strip (dielectric) 75 strip guide 368 surface charge 42, 48 surface current density 24, 378 susceptibility (dielectric) 58, 222 synchrotron radiation 297 t Thomson model 81 transition rules 302 transmission (coefficient) 331, 343 treeing 120
tube (of electric field) 15 v vacuum-metal interface 376 variable slow rate 157 vector circulation 3 vector potential 8, 26, 27 vectorial analysis 1 vectorial integration 3 vibrational transition 308 voltammeter (electrostatic) 123 w wave (direct) 175, 192 wave guide 367, 403, 423 wave (homogeneous) 327 wave (inhomogeneous) 327 wave (longitudinal) 176, 240 wave (monochromatic) 178, 191 wave number 178, 213 wave penetration 355, 359, 362 wave (planar) 176 wave (planar progressive) 178 wave polarized 186, 193 wave (progressive) 178, 395 wave (rectilinear polarized) 183, 199 wave (retrograde) 175, 192 wave (spherical) 177 wave (stationary) 179, 377 wave (transverse) 176, 181, 241, 381, 383, 388 wave TEM 193 wave vector 179, 191, 320 Weiss (domains) 148
