OPTICA ACTA,
1980, VOL . 27, NO . 5, 587-610
Optimized amplitude filtering for superresolution over a restricted field I . Achievement of maximum central irradiance under an energy constraintt RICHARD BOIVINT and ALBERIC BOIVIN Groupe d'Optique Diffractionnelle, Antennes et Holographie/LROL, Departement de Physique, Universite Laval, Quebec, Canada G 1 K 7P4 (Received 23 May 1979 ; revision received 30 August 1979) Abstract. We consider an aberration-free optical system fitted with a passive amplitude filter, both rotationally symmetric, and prescribe M concentric zeros for the diffraction pattern in the image plane . We then determine the radial transparency law capable of generating these zeros while ensuring a maximum value of the ratio of central irradiance to total energy content in the pattern . This pupil function to appears as a finite series of M Bessel functions J° whose arguments involve the afore-mentioned zeros, while the series coefficients are obtained through the solution of a set of M simultaneous linear equations . Upon a simple reduction of the spread of the zeros, the solution of the above stated problem leads directly to the concept of superresolution over a restricted field . We display a few filters 0. giving rise to superresolution, along with the corresponding patterns . Our computations bring out a remarkable likeness between the optimized superresolving filter and the Zernike polynomial R2s ° . This kinship explains the low central intensity and the large sidelobe structure that are characteristic of limited-field superresolving patterns, when these are viewed over the whole image plane . We also show that beyond the annular dark zone created around the compressed core, pattern shape deviates little from the Airy structure . It is thus seen that filter ¢o generates the dark zone by absorbing most of the energy that would otherwise be spread within the circle bounded by the outermost prescribed zero . The accuracy required in the synthesis of filter cyo is evaluated and extensions of the method are discussed .
1 . Introduction It has been known for some time [1-6] that through proper amplitude filtering in the exit pupil of an optical system, the point response (the diffraction pattern) at focus may be shaped into a form giving rise to superresolution over a restricted portion of the field of the instrument . The pattern is forced to assume a series of zeros, the first one being such that the central core is narrower than that for the uniform pupil, while the remaining zeros are spaced around the core in such a way that a dark zone exhibiting low secondary lobes is produced . This concept of limited-field superresolution is due to Toraldo di Francia [1], who in 1952 showed how to generate the required pattern by adjusting the f Work supported by the Defence Research Board of Canada under Grant DRB 2801-24 and the National Research Council of Canada through a scholarship awarded to first author . Presented at the Boston Meeting of the Optical Society of America, 21-24 October 1975 . 1 Present address : Alcan Research and Development Center, P .O . Box 8400, Kingston, Ontario, Canada K7L 4Z4 . 0030-3909/80/2705 0587 $02. 00 ® 1980 Taylor & Francis Ltd
588
R . Boivin and A . Boivin
transparencies of a system of concentric narrow rings . Soon afterwards the same author [2] extended his approach to concentric annuli of finite width . The concept was revived much later and perfected by Frieden, using the finite Fourier and Hankel self-transforms . He devised rectangular [3] as well as circular [4] filters that could, in principle, generate a perfectly sharp impulsive response over the extent of the restricted field . One might speak of such filters as being impulse-generating, and of the corresponding diffraction patterns as being finite impulses, in the sense that the impulsive behaviour has a finite extent . For the rectangular case, Frieden [3] investigated numerically the truncated expansions in self-transforms that lead to the finite impulse . On the other hand Boyer [6] used the same approach in the circular case to explore the transition from incipient sharpening of the point-response core [7-11] to its further constriction, which leads to the limited-field hypothesis . Incidentally Boyer and Sechaud [5], by devising Taylor expansions around the pupil edge and centre, were the first to numerically obtain a filter function designed for limitedfield superresolution that was both mathematically continuous and rotationally symmetric . It turns out that, as a result of the constraints imposed on the superresolving pattern, most of the energy contained in it lies outside the dark annulus generated by the zeros . It would thus seem relevant to try and optimize in some way the energy content of the usable part of the pattern . In this respect, a result of great importance was obtained by Luneburg [7] in 1944 . He found what type of filter maximizes the ratio of the pattern's central intensity to its total energy content, under the constraint that the pattern assume one mandatory zero . Here we shall generalize Luneburg's approach to include the specification of an arbitrary number of zeros in the diffraction pattern . The maximizing filter thus obtained is thereafter applied to limited-field superresolution ; in this role it acquires remarkable properties, which we shall discuss . 2.
Statement of the problem Consider a rotationally symmetric, aberration-free objective whose exit pupil is fitted with an amplitude filter of radial variation law 0 (see figure 1) . In the
Figure 1 . Geometry of diffraction in the image plane of an aberration-free objective ; the exit pupil is fitted with an amplitude filter of arbitrary radial-variation law ¢ .
Optimized amplitude filtering for superresolution
589
low-aperture scalar approximation, the point amplitude response (or amplitude diffraction pattern) generated in the focal plane of this system is [10, p . 97] G(z) = 2
1 f 0
0(r)J0(zr)r dr .
(1)
(We disregard a constant geometrical and dimensional factor irrelevant to our discussion .) Here JN is the Bessel function of the first kind of order N ; r = p/a, where p is the pupil radial coordinate and a is the pupil radius ; z = 27r(a/ AR)a, where a is the radial coordinate in the image plane, A is the wavelength of the incident light issuing from point source Q and R is the conjugate image distance . A number of physically significant quantities derive from functions ! and G . The observable in the image plane is the irradiance pattern (or intensity diffraction pattern) I(z) = G(z)G*(z) = G(z) 12 . (2) Since J0(0) = 1, the central irradiance has the simple expression i 2 I(0)=4 0(r)r dr
(3)
0
The total energy E contained in pattern I(z) is defined as 00 E- I I(z)z dz = f I G(z) 12 z dz . 0
(4)
0
By Parseval's theorem [12], we have E=4F,
(5)
where 1
F= I I1(r)I 2 r dr 0
(6)
is the light flux coming through filter 4(r) ; the factor 4 in equation (5) is due to our introduction of a 2 in equation (1) . A very useful concept is the encircledenergy ratio Z
~(z)
_
Z
I I(z)z dz 0
4 f IG(z)I 2 z dz -
0
E
F
(7 )
which is the fraction of the light energy distributed over the image plane that is enclosed within the circle of radius z centred on the optical axis . In the special case of a uniformly transparent filter 4(r) = 1, the point amplitude response is Jl(z 2 f J,(zr)r dr=2 )=_A(z) . (8) z
0
By equation (3), the corresponding irradiance pattern A 2 (z) has unity for its central value ; moreover it contains a total energy E=2 for the flux F is then 2 . As for the encircled energy, its expression is [13] '9A(z) = 1 - J0 2 (z)
- J1 2(z) .
(9)
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R . Boivin and A . Boivin
Characteristics of the Airy pattern A 2(z) are frequently used as standards in the study of general patterns I(z) . Since equation (2) gives I(z) in units of 1(0) for 0(r)=1, the Strehl central-irradiance ratio [14], I( 0) I .., ~
S
(10)
I( 0 )10=1
comes out simply as S = I(0) .
(11)
T1" 0
(12)
A transmission ratio F1 0 = 1
may also be defined ; obviously T=2F.
(13)
Now consider the following problem . Suppose we require that the pattern G exhibit a priori a certain number M of specified zeros z i G(zi)=0, i=1, 2, . . ., M.
(14)
We restrict ourselves to all passive filters 0(r), meaning those subject to the requirement (15) 11(r) I < 1, and ask : which is the one, 0 o , that will maximize the ratio of the irradiance pattern's central value 1(0) = S to the total energy E enclosed in the pattern ? On account of equations (3), (5) and (6), the ratio to be maximized is i f
S E
2
e(r)r dr
0
f
10(r)1 2 r dr
0
Our problem appears as one of optimization of the quantity S/E under fixed constraints, given by equations (14) and (15) . However the second of these two requirements does not have to be taken account of during the process of finding the solution, as it may be satisfied afterwards . Indeed, any solution fo oc 0o we find after having dropped condition (15) may be properly scaled so as to meet that condition, without the value of ratio (16) being affected in the process . As a corollary to this, the problem may be further simplified if we normalize all potential solutions f so that they make the denominator of the right-hand side of equation (16) unity . We are thus led to reformulate our problem as follows. To find function fo such that if f = fo the value o f the functional i
U(f) _-
f
2
f(r)r dr
0
is maximized, given the following set o f (M + 1) constraints 1
2
f
0
f(r)J0(zir)r dr=0, i=1, 2, . . ., M
(18)
Optimized amplitude filtering for superresolution
591
and 1 I I f(r)I
2 r dr=1 .
(19)
0
3. Solution of the problem : the optimizing pupil filter The problem we have just stated can be solved through a straightforward application of the calculus of variations . We shall first assume that the solution is a real-valued function . Let us consider the set of real-valued squareintegrable functions f that meet requirements (18) and (19), without however maximizing functional (17), and that can be expressed as weak variations of fo . We may then write f(r) = fo(r) + Es(r),
(20)
where E is a small real number and s is an arbitrary real-valued perturbing function that is continuous, has continuous first and second derivatives and equals zero at r = 0 and r = 1 . Let us now form the new functional V(f)
=
1
0
f(r)r dr 2 + E pi 2 f f(r)J0(zi r)r dr+ v f [f(r)] 2 r dr, i=1 0 0 J
(21)
where the (M+ 1) constants µi and v are undetermined Lagrange multipliers . Replacing f by its expression (20) we obtain V=V(E)={
1
[fo(r)+Es(r)]rdr
M
f'
1
+2 L.i tii
1
f U 0 (r)
i=1
+Es(r)]Jo(zi r)r dr + v
0
+Es(r)] 2 r dr .
f U 0 (r)
(22)
0
Since every function f considered as a potential solution satisfies conditions (18) and (19), the last two terms in the right-hand side of equation (21) remain constant . Therefore V(f) will be maximum when the first term is maximum, that is when E = 0 . It follows that V(E) is maximum for e = 0, which means that d V(e) dE
e=0
Now from (22) we find d h(E)
= 2 1 [fa (r') + Es(r')]r' dr' M
+2
s(r)r dr
1
E pi 1
i=1
f
1
s(r)J0 (zir)r dr+2v
0
f
[f0(r)+es(r)]s(r)r dr ; '(24)
0
thus, according to (23), we have 2 f s(r) 0
C
f fo (r')r' dr'+ 0
p Jo (zir)+ vf0(r) i=1
r dr=0 .
(25)
I
Since s is arbitrary, the bracket in the above equation must be zero for any r, whence f,, is seen to satisfy the very simple integral equation M 1 - vf0 (r) = E p i Jo (zir) + 1 f 0 (r')r' dr' . i=1 0
(26)
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R . Boivin and A . Boivin
Once evaluated, the integral in this last equation appears as a constant number . We may thus immediately write the solution of equation (26) as fa(r)=a
biJo(z ir)1 .
(27)
C 1- t~ The factor a and the coefficients b i are combinations of the undetermined Lagrange multipliers v and /-1'i . We can find expressions for these new quantities by using the constraint equations (18) and (19), which fo satisfies by definition . Introduction of expression (27) for f o into condition (18) yields a set of M simultaneous linear equations in M unknowns (the b i ) E
b;
JJ(z;r)Jo (zir)r dr=
f
i=1
J0(zi r)r dr, i= 1, 2, . . ., M.
f
0
(28)
0
The integral Jo (xr)J0(yr)r dr-L(x, y),
I
(29)
0
appearing in the left-hand side of equation (28), is one of Lommel's integrals and has the simple evaluation [15] [xJ0(y)J1(x)-yJ0(x)J1(y)]/(x2-y2) if x0 y, L(x, y) =
(30) [Jo 2(x) + J12 (x)]/2
if x = y .
In this notation, and on account of equation (8), the system (28) becomes M
z i ) = J1 (zi )/ zi,
b;L(z;,
i =1, 2, . . ., M .
(31)
j=1 =1
As for factor a, condition (19) gives M
a -2 =1-2
1
bi i=1
f
M
J0(z i r)r dr+
0
M
b.i
E
i=1
1
b;
E
f
Jo (zir)J0(z ;r)r dr .
(32)
0
i=1
After evaluation of the integrals involved, this becomes a 2 =2-2 i=1
bi Jl(zi) + zi
i=1
bi ~', b ;L(z; , z'i),
( 33 )
JAM
(34)
i=1
or, in view of equation (31), JAM a-2 = 2-2 M bi E
i=1
+
~M Lr
i=1
Zi
bi
zi
whence we obtain a=1/,,/P
(35)
with P=
2-
M
i=1
bi
J1(zi)
(36)
Zi
Our approach to the problem stated, via the calculus of variations, has yielded the integral equation (26), which the solution f o , assumed to be realvalued,, must satisfy as a necessary condition . However it might be argued that
593
Optimized amplitude filtering for superresolution
this condition is not a sufficient one for f o to be the solution . Furthermore nothing so far indicates that the solution should be real-valued at all . It thus remains to be firmly established that the f o expressed by equations (27), (31) and (35) is the actual solution to the problem, among all complex-valued, squareintegrable functions possessing a finite Hankel transform of order zero . Here we provide the necessary arguments, by generalizing a proof due to Wilkins [16] . Let us first determine the value of the functional U(f) (see equation (17)) when f = f o ; here of course fo , as specified above, meets requirements (18) and (19) . We find U(fo) = a2
f
0
[ 1-
b iJo(zir)
I
biJl(zi)l2
rdr} 2 =a2
C1-
i=1
M i=1
(37)
zi
and, in view of equations (35) and (36), this becomes (38)
U(fo )=P.
Now let us consider any one complex-valued function f amongst the set described above, and let us assume that it too satisfies conditions (18) and (19) . We may define functions v(r) = f(r)r112 (39) and (40)
w(r) =fa(r)r1j2 .
Applying Schwarz's inequality to v and w, we get 1
1
2
f
v(r)w*(r) dr
< f
0
1 Iv(r)I
2 dr
f Iw(r)
0
I 2 dr,
(41)
0
unless v and w are proportional, in which case the two sides of the inequality are equal. From equations (39), (40) and (27) we find 1 ' f v(r)w*(r) dr f(r)a b iJo(zi r) r dr 2 2= f 0 1- iE1 J
C
1
=a 2
f
M
f(r)r dr-
E
i=1
0
1
bi
f
2
f(r)J0(z i r)r dr
(42)
0
However by hypothesis f obeys constraint (18), whence every integral in the summation is worth zero . Thus we have 1 f
1
2
v(r)w*(r) dr
=a2
0
f
2
f(r)r dr
0
= U(f)1P_
(43)
As for the right-hand side of inequality (41), it is equal to 1 in view of the fact that both f and f,, satisfy condition (19) by hypothesis . Altogether we have U(f)/P< 1
(44)
U(f) < P.
(45)
U(f) < U(f o )
(46)
or, equivalently, Therefore,, by way of equation (38),
594
R . Boivin and A . Boivin
unless f = Hf o , where H is some complex proportionality constant . however, f is subject to constraint (19) we must have i i I If(r)I2 r dr= IH1 2 f Ifo(r)12 r dr= 1H12=1, 0
Since,
(47)
0
whence
H=exp (ih),
h real .
(48)
Equation (46) proves that function f o given by equations (27), (31) and (35) maximizes functional U under constraints (18) and (19) . That result must not come as a surprise, though, since it is a consequence of more general theorems, which pertain to matched filtering with constraints for band-limited signals [12, p. 186] . According to our reasoning of the previous section, we will obtain the passive filter 0 0 that maximizes the ratio S/E, under the set of constraints (14), by scaling f o so that the result satisfies condition (15) . To accomplish this, we simply define 0 . as 00(r)=G/C)
E b i Jo(zi r)
C1-
(49) I,
==1
where constant C is the maximum of the absolute value of the bracket,
fo
/a .
4. The optimized diffraction pattern To obtain an expression for the amplitude diffraction pattern G0(z) generated ./Ca of equation (49), we substitute 0. in the diffraction by the passive filter 00 =f integral (1) . Subsequent use of results (8) and (29) yields G0(z)=2(1/C) [J1~z)-
1 b i L(zi , z)] .
(50)
In view of the fact that L(x, 0)=J1(x)/x (see equation (30)), the central value So of the irradiance pattern I0 (z) = I Go(z)1 2 is S o =- IGo(0)1 2 =4(1/C)2 [ -
J1zzi)l2. 1m1
bi
(51)
Making use of equation (36), we get the simple expression S o = 4(P/C) 2.
(52)
We could have obtained that result also by realizing that S o = 4 U(fo lCa) and by using equation (38), together with the fact that U(f /t) = I 1/t I2 U(f ), where t is any non-zero constant . As for the total energy E o contained in the irradiance pattern, we have 1
1
Eo-4 1 1c o (r)1 2 r dr=4(1/Ca)2 f I fo(r)1 2 r dr. 0 0' On account of equation (19), which
fo
(53)
satisfies, we find
E0 = 4(1 /Ca)2 = 4P/C2
(54)
./E S .=P.
(55)
and therefore
595
Optimized amplitude filtering for superresolution
The quantity P is seen to occur frequently in our analysis ; table 1 summarizes some of the results where it is involved . M
Table 1 . The value of some quantities of interest, in terms of P=}Constant C is the maximum of Ifo/a
=I -
~',
M
bjJo (z;r)
1=1
I.
Value
Quantity Normalization factor Maximum value (18) and (19)
a
U(fo)
bsJl(zt)/zt . i=1
for function fo of functional
1 / -,/P
U(f)
under constraints P
Central value S o of the optimized pattern Io(z)
4(P/C)2
Total energy Eo contained in the optimized pattern Io(z) Ratio So/Eo
4P/C2
P
5. Special cases 5 .1 . Only one zero z i - P is specified In this case the problem amounts to that already solved by Luneburg [7], and later reconsidered by Barakat [17], Wilkins [16] and others . The optimizing function fo takes the simple form f o (r) = a[1-bJo (Sr)],
(56)
while system (31), for the sole coefficient b, degenerates to bL(p, P)=Jl(fl)lfl ;
(57)
using (30) we obtain b-
2J1(p)lfl
(58)
J02(fl) + J12(fl)
As for factor a, it is still computed as in equation (35), with P derived from equations (36) and (58), that is _ 2J1 2(8)/fl2 (59) PJO2(p)+J12(p)' The optimizing passive filter
4 . is, from (56),
0o(r) = (1/C)[1- bJo(flr)]
(60)
and the optimized pattern G o is G o (z)=2(11C) [j1(z)-bL(P, z)] .
(61)
Numerical evaluation shows that if fi is smaller than 3 . 8317, the first zero of the Airy pattern, then 11- bJ0 (f r) I is maximum at r = 1 . Accordingly in that
596
R . Boivin and A . Boivin
particular instance we have C =
11- bJo(#) I =1- bJ0(fl)
(62)
since fo/a then happens to be positive at r=1 . 5 .2 . Each zero is a root ei of J1 If zi=Si such that J1(ei )=0, equation (30) yields L(zj, zi) = L(e2, ei)=
0
if
j # i,
Jo 2(e i )/2
if
j =i,
(63)
1
whence system (31) degenerates to a sequence of N equations in one unknown biJo2(ei)=2J1(ei)/e =0, i=1, 2, . . ., M.
(64)
Since JO j ) A 0, we must conclude that in this case every coefficient bi is worth zero . Furthermore from equation (36) we find that P= ; thus (6
=
fo(r)=a=-\/2
(65)
c o(r) = 1 .
(66)
and accordingly That result had to be expected, since the prescribed zeros were those of the Airy pattern, generated by the uniform filter 0 = 1 . It is interesting to note that whatever the number of zeros ei specified, and no matter in what sequence they are given, the result of the optimization process comes out as the unit pupil filter . 5 .3 . Each zero is a root Ai of JO If z i = A, such that J0(X1 )=0, equation (30) yields 0
if
j :A i,
L(z; , zi) = L(a;) A )=
( 67 ) J12(Ai)/2 if j =i, whence system (31) degenerates once again to a sequence of N equations in one unknown 2JlA ili) , i=1, 2, . . ., M. (68) biJ12 (Ai)= 2
Since J1 (Ai ) 0 0, we have b i = J1( 2 x)
(69)
x
As for factor a, equations (35) and (36) give a=V/2 C Altogether we find that in this case f (r)=,/2
1-4 C
i=1
1 J-i12 1-4 1 T2 i=1
1 J-112 C1-2 =Jo(Air)1 Ai2 t1 A .Ji(A .) J~
(70)
(71)
597
Optimized amplitude filtering for superresolution
whence 0 o(r)=(1/C) C1-2 =1
(72) AA( ti),
and M
G0 (z)
2(1/C) Jj(z)-2 i
( , z)1
By the very conditions imposed on G0(z), we have Go(A) z :A a2 , the L function becomes L(A2, z)= ~'Jl (')J(z) , z # a2, A,2 - Z2
=-
(73)
0. At other abscissae ( 74)
whence M
1
G0(z)=2(1/C) Jl ( z)-2Jo(z) :Y,
A, 2
z
z5 a2 .
Z2]'
(75)
Let us now examine what happens when M= oo . The series
um(r)
E J0 (fir)
=2
it i AA
(76)
R)
has the property that [15, p . 18] u.(r) =1,
0,
(77)
while obviously UM(l) =
for any M. C=1 and
0
(78)
Therefore from equation (72) we find that in the limit considered, 0,
0,
1,
r=1 .
0o(r)=
(79 )
It thus seems that when the complete collection of zeros A 1 , A2, A3, . . . is prescribed, the optimization process yields an infinitely thin annulus as the required filter . This is consistent with the fact that the pattern Jo (z), whose zeros are the A2 s, is generated by the delta function S(r - 1) . But delta functions, since they have unit energy while being infinitely sharp, must be considered as active filters . On the other hand 0 0 , as given by (79), is a passive filter. Being an infinitely thin annulus, it lets out no energy and accordingly we should have G 0 (z) - 0 when M= oo . Indeed, consider the result [18]
_ J11+1(z) __
J„(z)
a0 i=1
1 +- + Y z - cop, 2 W1) v, 2 i=1
1
- 1 ,
z+Wv,2
(80)
Wv,i
where the w„ 2, i= 1, 2, 3, . . ., are the consecutive zeros of the Bessel function of any order v . By grouping the terms on the right in one single summation, we obtain (for z, v real and v > - 1, which implies that w,,,2 is real [18, p . 482])
v+i
z
Jy ( z)
= 2z L.. 2 i=1 2'2 -WV,2
598
R . Boivin and A . Boivin
or, equivalently t, 00
1 Jv+1(z) Z =2J„(z) Y,
2- z2
Making v = 0
we
(82)
find that °°
1
2Jo(z)
JAZ)
A,2-Z2='
t-i
z ,
( 83 )
whence equation (75) yields G0(z)=_0 as expected . However for M large but still finite, the shape of pattern G0 (z) (that is, the function G0(z)/G .(0)) should approximate J0(z) fairly well . The question arises as to whether series (71) for fjr), which satisfies condition (19), could possibly represent the required delta function adequately when M= oo . This is a delicate question, because an indeterminacy of the type 0/0 is then involved if r < 1 ; it arises from equation (77) and the following property of the A i s [19] 00
A,1= 2
(84)
==1
[which also proceeds from equation (83) when z is made equal to zero] . However on account of (78) the second bracket in equation (71) equals unity at r = 1, and as M increases the first bracket gets ever closer to zero . Therefore f .(l) grows without limit, which at first sight appears to be consistent with deltafunction behaviour . In view of all this, a more refined investigation is needed before we may reach a conclusion about the nature of expansion (71) in the limit when M= oo . 6. Application to superresolution and numerical examples To show one application, among others, of the optimizing filter 0', we use it to obtain superresolution over a central portion of the image plane . For this purpose, we follow the well established procedure of specifying a first zero z l that decreases the radius of the diffraction pattern's central core below its value e, - 3 . 8317 for the Airy pattern, and subsequent zeros z 2 < z3 < . . . < z M that constrain the amplitude oscillations between them to sufficiently low levels . In socalled optical units, the width w of the annular dark zone thus generated between z=z1 and z=z M is w=z M -zl, (85) and accordingly the diameter d of the limited field where superresolving imagery could be performed is ( 86) d=2z1 +w=z M +zl . The number n of point response cores that can be aligned in the limited field without mutually overlapping, will thus be the integer part of
k = 2 =1+Zz . 1
( 87)
1
f We wish to thank Professor Jean-Louis Lavoie, Department of Mathematics, Universite Laval, for having brought to our attention result (80) and its consequence (82) . Equation (83) was also derived by Boivin [10, p . 197] in a different context .
Optimized amplitude filtering for superresolution
599
Clearly, perfect resolution of two point-sources requires that w >, 2z1 .
( 88)
An obvious measure of the degree of superresolution attained is the value P ' of the radius of the superresolving pattern's central core in units of 61
P' =z1/e1,
(89)
. 0
(That quantity corresponds to Frieden's `resolution enhancement' [3] .) gain g in resolution may be defined as the inverse of P ' g=_fl'-1=z1,
The
(90)
g> , .
1
It is the number of compressed cores (including fraction thereof) that can be aligned in one Airy disk . Table 2 shows the values of a first set of five zeros z i , a second set of three zeros and a third set of four zeros, together with the corresponding coefficients bi, which are the solution of the set (31) of simultaneous equations . These synthesis parameters zi and bi completely define filter 0 0 , disregarding the scaling factor C. Table 3 lists the value of C for the three cases, along with that of each one of the performance parameters S o , Eo , P(= S o /Eo ), g, w, d and k . The set (31) of simultaneous equations was solved by means of the Gauss elimination algorithm with partial pivoting [20] . All computations were carried out on a Wang 720-C series 12-digit programmable calculator . The values of the zeros in table 2 are the exact ones that were fed into the calculator, while those of the coefficients b i and of C, So, Eo and P have been rounded at the eighth digit ; a two-decimal accuracy has been given for the other quantities . For graphical purposes it is convenient to define a normalized amplitude diffraction pattern G(z) __ G(z)/G(0), (91) which obviously gives the diffracted amplitude in units of its value on the optical
Table 2 .
Sets of zeros zi selected for computation of the optimizing filter responding coefficients bi of expansion (49) . First set
Second set
7o,
with cor-
Third set
zl
1 .915850
1 .915850
3 .193080
Z2
3 .50780
4 .10
5 .550
Z3
5 .086750
5 .90
Z4
6.661850
-
Zr,
8 .235320
-
8.420 11 .10316
b1
2.0847002
1 .6396060
2 .1857049
b2
-1-8295022
-1-0370643
-2-3702677
b3 b4 b5
1 .0782272 -4.1452369 x 10 -1 _2 8.2798476 x 10
4.7236280 x10 -1
1 .9391646
-
-1 .6214321
-
-
600 Table 3 .
R . Boivin and A . Boivin Value of the scaling constant C for filter ¢o and of some parameters describing 96's performance . First set
C So Eo
P g w d k
1 . 9493034 2 .4826084 1 .6166078 1 .5356900 2.00 4.75 8 .58 2 .24
x x x x
Second set 10-1 10 -8 10 -1 10- '
9.2306703 1 .2530898 2 .4254250 5 .1664750 2 .00 3 .98 7.82 2 .04
x x x x
10 -2 10 -4 10 -1 10-4
Third set 1 .3495944 1 .5574213 x 10 -2 1 .8493954 x 10 -1 8 .4212455 x 10 -2 1 .20 7 .91 14 .30 2 .24
axis z = 0 . (Strictly speaking, the above definition of G is useful for real-valued functions G only . Should G be complex-valued, separate normalizations could be made for Re G and Im G .) Similarly we define a normalized irradiance pattern
I(z) _- I(z)/I(0) .
(92)
The dashed curve in figure 2 is the Airy pattern A2(z), while the solid curve is the normalized irradiance pattern 10 (z) that is generated by the filter 0 0 corresponding to the first set of zeros of table 2 . These are the first five roots of the equation J1 (gz) = 0, where g = 2 . 00 is the gain (see table 3) . That method of specifying the zis was originally suggested by Boyer and Sechaud [5], though only for the first few ones . Indeed, it is clear that the procedure does not control adequately the intensity level of the outermost sidelobes when M exceeds a certain threshold (in this case, 4) . Here the fourth sidelobe is in fact so high that it cannot be considered a part of the dark zone required for limited-field superresolution ; therefore we must take w = z M_1 - z1 and this is reflected in the particular values of w, d and k given in table 3 for this case . To a certain extent the problem can be managed if some zeros of the equation J0(gz) = 0 are interlaced between those of the preceding one . This generalizes Toraldo di Francia's original method [1] of placing the zeros . But as more and more of them are added to widen the dark zone and/or control sidelobe intensity within it, the central irradiance decreases very rapidly . This well-known phenomenon [1-6] also occurs if the dark zone is widened towards the optical axis rather than in the opposite sense . Actually, on account of its very nature, limited-field superresolving imagery does not require the dark annulus in the diffraction pattern to be arbitrarily wide, especially if the object to be superresolved is a two-point one (e .g . close binary stars) . Moreover the intensity of the sidelobes within the annulus need not be as constrained as is the case for, say, the first two sidelobes in figure 2 . Suppose that we keep the resolution gain at 2 . 00, as in figure 2 . Then we could trade a reduction of the width of the dark annulus along with an enhancement of the sidelobes in the annulus for an increase of central irradiance . We would achieve this by prescribing less zeros around the core, say two, and separating these
Optimized amplitude filtering for superresolution
601
1.0
0.8
1
0.6
1 1 1 1
I 0.4 1 0.2
11 2 1 \ \\
0 .0 Figure 2 .
1
2 3
8
Curve 1 : normalized irradiance pattern Io(z) generated by filter 9o of figure 3 . Curve 2 : Airy pattern A 2(z) .
slightly more, up to the point where a sufficient yet not excessive rise of the sidelobes had occurred . Of course by spreading out the zeros that make up the annulus, we would simultaneously avoid too drastic a reduction of its width . The second set of zeros of table 2 has been chosen in that spirit ; the normalized pattern 10 (z) that corresponds to it is shown in figure 4 (solid curve) . From the numbers given in table 3, it appears, that the restricted field is still wide enough to accept two compressed cores without overlap . True, sidelobe intensity in the dark zone, relative to that of the central core, has risen to about 15 per cent, but 1 .0 0 .8 0 .6 0.4 0.2 0 -0.2 -0.4 -0 .6 -0.8 i
-1 .0 0
L
0.2
0.4
0.6
0.8
1 .0
r Figure 3 . Curve 1 : filter #0(r) computed for the first set of zeros . polynomial R10 (r) .
Curve 2 : Zernike
602
R . Boivin and A . Boivin
I
Figure 4 . Curve 1 : normalized irradiance pattern 10(z) generated by filter Oo of figure 5 . Curve 2 : the same function evaluated with slightly inaccurate parameters zi and bi for ¢ o .
Figure 5 .
Curve 1 : filter Oo(r) computed for the second set of zeros. polynomial R6 0(r) .
Curve 2 : Zernike
Optimized amplitude filtering for superresolution
603
this is certainly made good by the fact that the central irradiance has increased by four orders of magnitude ! Figures 3 and 5 show the optimizing filters 00 (r) corresponding to the first two sets of zeros, together with the Zernike radial polynomial [21] R 2M °(r) . In both cases the two curves shown are obviously very much alike, but we notice that the resemblance is somewhat closer in the case of figure 3 . Other computations have shown that, as a general rule, the higher the constraints imposed on the superresolving pattern (that is, the greater g and w and the lower the sidelobe amplitude within the annular dark zone), the greater the similarity between filter q o(r) and the Zernike polynomial R 2M°(r) . As we shall see further on, this has profound consequences . In order to evaluate the,precision required in the synthesis of filter 00 for the second set of zeros, we rounded the coefficients bi and the zeros zi defining 0. at the fourth digit, then inserted the resulting slightly inaccurate filter into the diffraction integral . (Should a rounded z i have been equal to the exact one, we introduced an error of plus or minus one unit at the fourth-digit position . The rounded parameters were thus z1 =1 . 9166, Z2 =4-1010, Z3 =5-8990, b 1 =1 . 640, b 2 =-1-0370,b,=047240 .) The pattern obtained is the dashed curve shown in figure 4. Obviously the imparted inaccuracy has brought about large and detrimental modifications of the sidelobes, even though at the scale of figure 5 one could hardly make out the inaccurate filter from the exact one . Another negative aspect is that in the process the value of S o has been almost halved from about 1 .25 x10-4 to 0 . 659 x 10 -4 . It may be asked, then, what degree of accuracy is required in the z is and b i s so that no appreciable departure from the desired result will occur in the point response function . In this specific case, we found that rounding the synthesis parameters at the sixth digit has no visible effect at the scale of figure 4. Further tests have shown that, more generally, the required accuracy increases when the constraints imposed on the superresolving pattern are raised . On the whole our observations corroborate the conclusions of Frieden [22] and of Boyer and Sechaud [5] in that one very high price to be paid for limited-field superresolution is the extreme precision required in the synthesis o f the ad hoc filters. Figure 6 displays the irradiance pattern I0(z) produced by the filter 00 of figure 5, together with that, Iz (M>(z), arising from use of the Zernike polynomial R2 .0(r) as the pupil filter IZ(M)(z)= IGz (M) (z)1 2, (93) with Gz(M)(z)==2
1 f R2M°(r)Jo(zr)r
0
J2M+1(Z) dr=2( -1 )M (94) z
In this case M=3 so that the pattern is actually 4[J7(z)/z]2 . Note that the z variable ranges from 0 to 20 here, and that the ordinate is in units of the centre of the Airy pattern ; for these reasons it might be said that the, point of view adopted here is macroscopic, whereas it is microscopic in figure 4 . The two curves are strikingly similar, with main differences consisting of a bodily shift of the sidelobes and the presence, in the superresolving pattern, of a miniscule central core . Of course the resemblance observed in figure 5 between filter 00 and polynomial Rg explains that similarity. In fact it reveals, in the case of the
604
R . Boivin and A . Boivin 0.007 0.006
Figure 6 . Curve 1 : overall view of irradiance pattern Io (z) produced by filter ¢o of figure 5 . Curve 2 : pattern [-2J,(z)/z]2 arising from the filter ¢(r)=R8 (r) . Curve 3 second ring of the Airy pattern A 2(z) .
optimized pattern I0(z) at least, the origin of two well-known characteristics of limited-field superresolution, namely the very low central intensity of the pattern and the huge sidelobes lying outside the annular dark zone . Indeed, since the filter is more or less a perturbation of the Zernike polynomial R 2 M (r), the pattern as a whole is a perturbation o f that pattern, Iz (M>(z), generated by the aforementioned polynomial and having a perfectly dark centre (if M > 0) . By reversing that principle, we see that ~ o (r) will look like R 2 M (r) provided I0(z) bears enough resemblance to Iz (M>(z) . The criterion determining the degree of that resemblance would appear to be the size of the central core relative to that of the first sidelobe outside the annular dark zone : the smaller the core, the closer the resemblance . The lone peak in figure 6 between z = 7 and z =10, approximately, is the second bright ring of the Airy pattern. It gives an idea of the physical scale of 0.15 0.10 0.05
G o
I -0.05 -0.10 -0 .150 Figure 7 .
1
I I I I I
1I v 2 4 6 8
10 12 14 r 16 Z
18 20
Curve 1 : amplitude response Go(z) generated by filter Oo of figure 5 . Airy response A(z) .
Curve 2
Optimized amplitude filtering for superresolution
605
the superresolving pattern's central core . Figure 7 shows a superposition of the point amplitude response O0(z) generated'b y the filter ~'o of figure 5 and the Airy response A(z), in the interval 0<,z<,20 . Beyond abscissa z = z 11 = 5 . 9, the two curves undergo similar amplitude variations ; moreover the bodily shift between oscillations decreases steadily . Further plotting shows that farther out still along the z axis, the two curves preserve their tendency to merge . Therefore, whether or not there is a superresolving filter in the pupil, to a good approximation regions of the image plane that are sufficiently far from the optical axis always display the Airy pattern fringes . Another way of putting it is that the energy content of the superresolving pattern beyond the annular dark zone is comparable to that o f the Airy pattern in the same region . In the present instance this can readily be verified . Of the energy E=2 enclosed in the Airy pattern, the fraction that lies beyond abscissa z M =S •9 (which defines the region of interest) is 1- fA (5 . 9) 10 per cent (see equation (9)) . As for the superresolving pattern, figure 6 allows us to state that practically all of its energy E 0 is located beyond z = zM = 5.9 . From table 3, we see that this is 0-243/2 12 per cent of the Airy pattern's total energy, which is very close to 10 per cent indeed . What this really means is that through the interposition of superresolving filter 00 in the pupil, actually very little energy is pushed outside the dark zone z < z M generated in the image plane (in this case 2 per cent) . In other words the filter creates its superresolving effect by absorbing a lot of energy (in this case 88 per cent of the total energy reaching it!) . Thus the energy trade-off that governs limited-field superresolution occurs mainly in the filter itself . An immediate corollary is that if laser illumination is resorted to as a means of compensating for the low central intensity of the superresolving pattern, efficient cooling techniques will be required for the filter . One alternative would be to devise equivalent phase-modulating transparent filters, perhaps by generalizing the Wilkins screen [9] to limited-field superresolution, or by using recent results concerning the exchange between phase and modulus in the pupil [23, 24] . On the other hand in eventual astronomical applications the problem of low central intensity in the point response would be alleviated through use of cascaded image intensifiers, under the limitations set by noise propagation . As we have seen, if the constraints imposed on the diffraction pattern are the least high, serious problems plague limited-field- superresolution, even though optimized : tolerances in the synthesis of the filter are extremely tight, and the usable part of the pattern is almost devoid of energy . One way out of these problems would be, very simply, to relax the constraints, even to the point where the limited-field concept is left out . This, in the main, is the approach suggested by Boyer [6] . Figures 8 and 9, which pertain to the third set of zeros of table 2, will illustrate the idea . The four zeros have been chosen so that the three sidelobes between them have about the same amplitude, roughly that of the first sidelobe of the Airy pattern (see figure 8) . The gain g has been lowered to 1 .20 ; because of this the pattern is surprisingly well-behaved beyond z = z 1 , so much so indeed that it would not seem necessary to restrict imaging to a limited field at allt . Note how closely the pattern follows the curve A(gz) for z < 5 . The central intensity is worth about 1 . 6 per cent of the Airy pattern's t Provided that one is ready to accept a maximum sidelobe of 16 per cent intensity relative to core, which is typical of the J02 (z) pattern .
6 06
R . Boivin and A . Boivin
Figure 8 . Curve 1 : normalized amplitude response 00(z) produced by filter qo of figure 9. Curve 2 : Airy response A(z) . Curve 3 : function A(gz) with g=1 .20 .
1 .0 0.8 0.6 0.4
950 .2 0 -0.2 -0.4 -0.6 0
0.2
0.4
0.6
0.8
1 .0
r Figure 9 . Curve 1 : filter #o(r) computed for the third set of zeros . polynomial R e °(r) .
Curve 2 : Zernike
central intensity, and this is certainly a far cry from the previous situation of approximately 0 . 013 per cent . But the most interesting feature here is the fact that a three-digit accuracy on the b tis and z1s is sufficient to guarantee the expected results . In fact rounding these parameters at the third digit entails, as the sole noticeable effect on the pattern at the scale of figure 8, an enhancement of roughly 10 per cent of the first sidelobe's amplitude . (A decrease in the central intensity S o also occurs, from about 1 . 56 x 10-2 (see table 3) to 1 . 53 x 10 -2 , but this is utterly negligible .) Hence, in the context of a holographic method for the
Optimized amplitude filtering for superresolution
607
synthesis of filter 0. that we shall propose shortly, it would seem possible to construct the filter of interest here . That filter is shown in figure 9, together with the Zernike polynomial RS(r). The large discrepancies observed between the two curves are consistent with the fact that in this case the pattern G0(z) is certainly very different from that due to the Zernike polynomial ; indeed, the former pattern has its central core higher than any sidelobe (see figure 8), whereas the latter, 2J9 (z)/z, has a perfectly dark centre . Other examples of filter 0 0 and pattern G 0 will be found elsewhere [25] . 7.
Conclusions In this paper we have determined the passive pupil filter io which maximizes the ratio of the diffraction pattern's central intensity to its total energy content, under the constraint of specified zero crossings . Through application of this optimizing filter 00 to superresolution over a restricted field, we have gained new insight into that physical process . The low central intensity and large sidelobes characteristic of highly constrained limited-field superresolving patterns were shown, in this case, to be the consequence of the great resemblance that exists between filter /' and the Zernike radial polynomial having the same number of zeros . There is good reason to believe that more generally, any type of passive continuous amplitude-filter designed for limited-field superresolution will exhibit such a likeness with a Zernike polynomial . We also elucidated the mechanism through which filter qo creates a dark zone for superresolution : it absorbs most of the energy that would otherwise lie in the z < z M part of the Airy pattern. Oddly enough though, 0 0 does not modify the far zone z > z M of the pattern very much from the Airy structure . As in previous work, filter synthesis accuracy was identified as one of the more serious problems afflicting the concept of limitedfield superresolution . We therefore investigated a superresolving configuration of relatively modest resolution gain, that might lend itself to future experimentation (see third set of zeros in table 1 and figures 8 and 9) . Our filter 0. has one disadvantage with respect to Frieden's impulsegenerating filter : the zeros in the annular dark zone do not come about naturally . Given the first zero z l (that is, given the gain g), suitable positions for the next zeros must be found by trial and error . On the other hand, precisely because the possibility of moving the zeros around has been preserved, our method offers greater flexibility . Indeed, it allows independent control of gain, sidelobe amplitude and limited-field extent (if one chooses to retain the concept), so that an arrangement of the zeros can always be found that will satisfy the design requirements for pattern shape . Moreover use of filter 00 guarantees that the pattern is optimized with respect to one energy criterion ; this feature does not exist in the Frieden scheme . The impulse-generating filter could however be viewed as a very efficient means of finding initial positions for the zeros . These could be modified soas to improve pattern shape in whatever respect, and then the final positions would be used in our own scheme . The second paper in this series is devoted to that approach . The mathematical problem considered here can be generalized to finite Hankel transforms of arbitrary non-negative integer order N. The diffraction integral is then replaced by i 9r(z)=2 f p(r)JN(zr)r dr, N=0, 1, 2, . . . . (95) 0
608
R. Boivin and A . Boivin
In terms of the functions LN (z)=2NN!
JN()
z
(96)
,
equation (95) may be written as i G(z)=2
f
o(r)L N(zr r2N+1 dr,
(97)
0
where (98)
G(z) -2NN! T ' (N )
z
and ~(r) _
(N ) r
( 99 )
Because JN(0)=0 for N= 1, 2, 3, . . ., Hankel transforms V(z) for such values of N are null at z=0 and thus it would be meaningless to try and maximize their central intensity . However functions G(z) given by (98) and obtained through the LN (zr) transformation have no such limitation on their central value, since LN(O) = 1 for all N. Hence for the generalized diffraction patterns (97) an optimization problem similar to the one stated at the end of § 2 may also be formulated . The functional whose value must be maximized is then i U(f)=
f(r)r
2N+1 dr
2 (100)
0
and the constraints imposed on f become i I
f(r)LN (zir)r2 N+ 1 dr=0, i=1, 2, . . ., M,
(101)
0
with i I I
f(r )I2 r2N+1 dr=1 .
(102)
0
Finally, going over to a simpler geometry, we may also carry out for systems involving rectangular apertures the optimization achieved in this paper . The relevant diffraction integral is simply the unidimensional Fourier transform i G(y)=
f -1
¢(x) exp (ixy) dx .
(103)
Barakat [17] has solved the problem for one prescribed zero, and a generalization to M zeros is straightforward . We have completed the mathematical analysis for both problems outlined above and shall publish our results in the near future . These problems have more than academic interest, because they are intimately connected with the possibility of expressing the Fourier self-transforms over the unit interval and the unit circle in terms of their zeros [26] .
Optimized amplitude filtering for superresolution
609
Nous considerons un systeme optique parfait muni d'un filtre d'amplitude pupillaire passif, tous deux de revolution . Imposant a l'avance M zeros concentriques a la figure de diffraction focale, nous determinons la loi de transparence radiale capable de produire ces zeros, tout en assurant la valeur la plus elevee possible au rapport de l'intensite centrale a 1'energie totale disponible dans la figure . Cette fonction pupillaire ¢ o se trouve exprimee par une serie finie de M fonctions de Bessel J° , dont les arguments font intervenir les zeros precites, alors que les coefficients de cette meme serie sont donnes par la solution d'un systeme de M equations lineaires simultanees . Moyennant une simple reduction de 1'etalement des zeros la solution du probleme ci-dessus mene directement au concept d'hyperresolution en champ restreint . Nous presentons quelques exemples de filtres ¢o et de figures operant en regime hyperresolvant . Nos calculs revelent une remarquable ressemblance entre le filtre hyperresolvant optimise et le polynome radial de Zernike R2M° . Cette parente explique la faible intensite centrale et l'imposante structure de lobes lateraux caracteristiques des figures hyperresolvantes en champ restreint, lorsque celles-ci sont envisagees dans 1'ensemble du plan image . Nous demontrons aussi qu'au deli de la zone sombre annulaire creee autour du pic central retreci, la forme de la figure s'ecarte assez peu de la structure d'Airy . Il s'avere ainsi que le filtre Oo produit cette zone sombre en absorbant la majeure partie de 1'energie qui autrement serait repartie dans le cercle delimite par le zero assigne le plus eloigne . La precision requise dans la synthese du filtre ¢ o est evaluee et certains prolongements de la methode sont discutes . Wir betrachten ein abbildungsfehlerfreies optisches System, ausgestattet mit einem passiven Amplitudenfilter-beides rotationssymmetrisch-und betrachten M konzentrische Nullstellen des Beugungsbilds in der Bildebene . Dann bestimmen wir den these Nullstellen erzeugenden radialen Verlauf der Filtertransparenz bei gleichzeitiger Sicherstellung eines Maximalwerts des Quotienten aus zentraler Bestrahlungsstarke gebrochen durch den gesamten Energiestrom im Beugungsbild . Diese Pupillenfunktion ¢o erscheint als endliche Reihe von M Besselfunktionen Jo, deren Argumente die oben erwahnten Nullstellen beinhalten, wahrend die Koeffizienten der Reihe sich als Losung eines Systems von M gleichzeitig linearen Gleichungen ergeben . Vber eine einfache Reduktion der Nullstellenverteilung fiihrt das oben formulierte Problem sofort zu dem Konzept der Vberauflosung innerhalb eines beschrankten Bildfelds . Wir zeigen einige Oberauflosung liefernde Filter ¢ o and die zugehorigen Beugungsbilder . Unsere Rechnungen offenbaren eine bemerkenswerte Ahnlichkeit zwischen dem optimierten Uberauflosungsfilter and dem Zernike-Polynom R2M° . Diese Verwandtschaft erklart die fur die Uberauflosung im beschrankten Bildfeld charakteristische niedrige zentrale Intensitat and die ausgepragte Struktur der Nebenmaxima, wenn man das gesamte Bildfeld betrachtet . Wir zeigen ferner, daB auBerhalb der um den komprimierten Kern liegenden ringformigen dunklen Zone die Form des Beugungsbilds nur wenig von der Airy-Form abweicht . Daraus ersieht man, daB der Filter qo die dunkle Zone hauptsachlich durch Absorption der sonst in diesen Bereich flieBenden Energie erzeugt . Die zur Synthese der Filter ¢o notwendige Genauigkeit wird bestimmt and Verallgemeinerungen der Methode werden diskutiert .
References TOItALDO DI FRANCIA, G., 1952, Supplto nuovo Cim ., 9, 426 . TORALDO DI FRANCIA, G ., 1952, Atti Fond . Giorgio Ronchi, 7, 366 . FRIEDEN, B . R ., 1969, Optica Acta, 16, 795 . FRIEDEN, B . R ., 1971, Progress in Optics, Vol . IX, edited by E . Wolf (Amsterdam North-Holland), p . 311 . BOYER, G ., and SECRAUD, M., 1973, Appl. Optics, 12, 893 . BOYER, G . R., 1976, Appl . Optics, 15, 3089. LUNEBURG, R. K., 1944, Mathematical Theory of Optics (Providence : Brown University Press), p . 391 . OSTERBERG, H ., and WILKINS, J . E ., JR ., 1949, J. opt . Soc . Am ., 39, 553 . WILKINS, J . E ., JR ., 1950, J . opt. Soc. Am., 40, 222. BoIVIN, A ., 1964, Theorie et Calcul des Figures de Diffraction de Revolution (Quebec Les Presses de l'Universite Laval ; Paris : Gauthier-Villars), p . 383 .
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[11] CLEMENTS, A . M ., and WILKINS, J . E ., JR., 1974, Y. opt . Soc. Am ., 64, 23 . [12] PAPOULIS, A ., 1968, Systems and Transforms with Applications in Optics (New York McGraw-Hill), p. 143 . [13] GRAY, A ., MATHEWS, G . B ., and MACROBERT, T. M ., 1962, A Treatise on Bessel Functions and their Applications to Physics (New York : Dover), p . 193 . [14] BORN, M ., and WOLF, E ., 1959, Principles of Optics (London : Pergamon Press), p . 461 . [15] BOWMAN, F ., 1958, Introduction to Bessel Functions (New York : Dover), p . 9 . [16] WILKINS, J . E ., JR ., 1963,,x. opt . Soc . Am ., 53, 420. [17] BARAKAT, R., 1962, J. opt. Soc . Am., 52, 264. [18] WATSON, G . N ., 1966, A Treatise on the Theory of Bessel Functions (Cambridge University Press), p . 498 . [19] HOCHSTADT, H ., 1961, Special Functions of Mathematical Physics (New York : Holt, Rinehart & Winston), p . 58 . [20] MCCRACKEN, D . D ., and DORN, W . S ., 1964, Numerical Methods and Fortran Program[21] [22] [23] [24]
ming (New York : John Wiley & Sons), p . 231 . ZERNIKE, F ., 1934, Physica, 's Grav ., 1, 689 ; see also [10, p . 169] . FRIEDEN, B . R ., 1970, Appl . Optics, 9, 2489 . VILLENEUVE, J . E ., 1980, Ph .D . Thesis, Universite Laval . BOIvIN, A ., and VILLENEUVE, J . E ., 1977, Y. opt. Soc . Am., 67, 1437A.
[25] BoIVIN, R ., 1976, M .Sc . Thesis, Universite Laval. [26] BoIViN, A ., and BoIVIN, R., 1978, Y. opt. Soc . Am ., 68, 1453A .
