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A new class of optimum amplitude filters
Volume
2 1, number
2
OPTICS
A NEW CLASS OF OPTIMUM AMPLITUDE
COMMUNICATIONS
May 1971
FILTERS
L.N. HAZRA Department of Applied Physics, University of Calcutta, 92 Aqharya Prafulla Chandra Road, Calcutta 700009, India Received
22 December
1976
This paper presents a new class of optimum amplitude filters for the maximization of the factor of encircled energy within a pre-specified central core area of the far-field diffraction pattern. The filters arc composed of finite uniformly transmitting zones with variations between them. The analysis is based on the known technique of indirect starch for the optimum filter by “the method of sets of screens” utilising the complete set of normal orthogonal Walsh functions as a system of base functions. Some numerical results are presented.
1. Introduction Amplitude filters on the pupil of an image forming system substantially modify its imaging characteristics. Since the pioneering works of Straubel [l] and Luneberg [2], the problem of determining optimum amplitude filters in accordance with some prespecified constraints on the image plane is receiving considerable attention [3-131 . As the far-field amplitude or intensity distribution is expressible as a finite Fourier transform of the pupil function or the anticorrelation function of the same, respectively, the problem is essentially one of optimization of a band-limited system. This way of looking at the problem emphasizes the universality and the general mathematical nature of the topic which arises in a variety of engineering fields, e.g., design of antennas, signal transmission systems, etc. [ 141 . An important problem of optimization often encountered in imaging systems is the suppression of side lobes of the far-field diffraction pattern. Variational methods have frequently been adopted for obtaining a solution to the problem. In order to bring forth clearly the motivation of our present investigation, we briefly point out a few salient features of the various approaches undertaken heretofore. Details of the methods are given in ref. [8]. The formulation developed by Luneberg [2] has led to a technique of “direct search” for the pupil function amongst the complete set of physically real232
izable pupil functions. For the determination of the optimum function, the method necessitates the solution of a linear homogeneous integral equation of the second kind. The kernel of this equation is, however, too complicated for the optimum function to be calculated otherwise than by numerical approximations which, in turn, introduce errors in the computed results. Al alternative attempt known as “the method of sets of screens” has been made wherein the optimum function is sought in an indirect manner. The method involves a finite series expansion of the pupil function in terms of a convenient set of base functions with known finite Fourier transforms. The solution is obtained via optimization of the coefficients of this expansion. Whereas in the earlier approach the optimum function is chosen out of the complete set of pupil functions, in the latter the scope of choice is limited as the solution obtained is best among the particular set of functions under consideration. In order to obtain sufficiently accurate and useful results, this inherent limitation can be circumvented by choosing progressively larger number of terms in the expansion. A continued search has also been undertaken to find out a suitable system of base functions which could provide the desired accuracy in the results with a smaller number of terms in the expansion so that the computations do not become unwieldy. Different sets of orthogonal polynomials and of some nonorthogonal functions like the lambda functions have
Volume
2 1, number
2
May 1977
OPTICS COMMUNICATIONS
been utilised [3-61 . In the sixties, Slepian’s introduction of prolate spheroidal wavefunctions in apodization studies opened up a new possibility in this theoretical search [9] . In the present communication, we propose to utilize Walsh functions in these studies. The complete set of normal orthogonal Walsh functions has provided a novel and very convenient analytic tool for the treatment of practical problems of information transmission systems [IS] . The application has been found to be so promising and encouraging that a host of suggestions for employing these functions in problems of allied fields are appearing at a high pace [ 161 . A pertinent example is a proposal for the utilization of Walsh functions in solving a variational problem in heat conduction [ 171. The rationale for our suggested application can be readily appreciated if one recognises the advantages to be gained in the subsequent process of practical fabrication of these filters. There exists a number of techniques for the fabrication of amplitude filters [ 18-201 ; nevertheless, one can very well surmise that all these processes will be considerably facilitated if one has to generate a filter composed of a finite number of uniformly transmitting zones with variations between them instead of a filter with continuously varying transmission. An obvious possibility is to approximate the known apodizing pupil functions by the Walsh functions which have the interesting property that a finite sum of a Walsh series expansion of a function yields a ladder-step approximation to the latter. The possibility has been explored and found to provide useful results as sought for [21]. However, the need for this further approximation can be obviated and the present paper reports a refined mode of analysis to obtain the desired results directly by utilizing the set of Walsh functions as a system of base functions in the “indirect search technique”.
(2.1) where f(r) is the circularly symmetric pupil function. The variable r is a fractional coordinate for a point on the pupil while p is a reduced diffraction variable given by p = (2n/A) (n’sin o’)t’,
(2.2)
where CY’is the semi-angular aperture of the optical system, 2n/h is the propagation constant and t’ is the geometrical distance of the point in the far-field plane under consideration from the centre of the diffraction pattern. The far-field intensity distribution is given by Z(P) = lFo7)12
(2.3)
and the expression for the encircled energy E(W) within a circle of radius W, concentric with the centre of the diffraction pattern in the far-field is 2ll
w
(2.4) where f3 is the variable of azimuth in the far-field plane. For the rotationally symmetric system under consideration the above reduces to W W E(W)=SI(p)pdp=SIF@)12pdp (2.5) 0 0 to within a multiplicative constant. By Parseval’s identity, the total transmitted flux is given by E(W)
=)If(r)12rdr.
(2.6)
0 The factor of encircled energy is defined by the ratio &( W) = E( W)IE(=J).
(2.7)
Using the set of radial Walsh functions p,(r) as the system of base functions, the pupil function f(r) may be expressed over the domain (0,l) as N-l
2. Mathematical analysis
(2.8) In what follows, we confine ourselves to circularly symmetric apertures. With the usual assumptions of scalar diffraction theory, the far-field amplitude distribution F(p) due to a point object positioned on the optical axis of the system is to a multiplicative constant given by
where
=2 j‘f(r)vnWrdr arz
(2.9) 0 and N represents the finite number of terms utilised 233
Volume
21. number
2
in the expansion. Fig. 1 depicts the first eight radial Walsh functions p,(r). Radial Walsh functions as a subset of the set of circular Walsh functions have beer derived earlier for the treatment of problems of optical imagery [22] . From (2.1) and (2.8), we obtain
C(W)=
N-l
N-l
N-I
N-l
c c amane,,(c n=o c amanemn’
m=O ,1=0
m=O
(2.12) where = i x,Wx,(PW~. 0 From the condition of orthogonality radial Walsh functions we have
emn(V
N-l
ml =
May 1911
OPTICS COMMUNICATIONS
ngo anx,(P)
(2.10)
where
(2.13) satisfied by the
Emn = k&m,, where a,,
Analytical expressions for xn@), the Hankel transforms of the radial Walsh functions are identical with those for the Fourier transforms of the unidimensional Walsh functions [23]. On algebraic manipulations of (2.5), (2.6), (2.7) and (2.10), we get
(2.14) is Kronecker delta defined as
6mn =l ’
m = n,
= 0,
m fn.
The condition
aw)
aam
for stationarity
_o ’
(2.15) of C(W)
m = 0, .. ..N-1.
(2.16)
leads to the following system of linear homogeneous equations
cn anemn(W)=P 1.0
Canernn, n
m =0 ,..., N-l, (2.17)
r-
0.2 -
Fig. 1. First eight radial Walsh functions,
234
g,(r);
n = 1, . . . . 8.
Fig. 2. Optimum pupil transmission functions (number of steps N = P = 8)which maximises the factor of encircled encr. gy within a circle of prespecified radius W concentric with the centre of the diffraction pattern; -: W = 0.0, ---: W = 2.0, -.-.-.: Iv = 3.0, . . . . . . . w = 4.0, xxxx: w = 5.0.
Volume
21, number
2
OPTICS COMMUNICATIONS
May 1977
lo-’
10+
I\
\
\ \ \ \ \
L
I
i=y \
‘,
i
.
1; :
16'
I
I
II :
I
! / :I
i
I
...
: ‘, : :
I ;
i\ \ \ , ‘. 1 \’ i h i
6
lo-
i ‘.,
i
.n : : 1 x
10-5 Fig. 3. Diffraction patterns corresponding to the optimum functions of fig. 2. -: . . . . . . W = 4.0, xxxx: W = 5.0. The scale of Z is linear as far asp = 6.0; a logarithmic of the side lobes.
W = 0.0, ----: scale is adopted
W = 2.0, -.-.--.: W = 3.0, for p > 4.0 for better display
235
Volume
21, number
2
where p is the stationary notation
OPTICS COMMUNICATIONS
value of C(W). In matrix
EWA=@A.
(2.18)
May 1977
level ladder-step filters by binary filters has been developed. Preliminary experimental results are quite promising and we intend to publish a fuller report soon.
With the help of expression (2.13) EwA=$4
=vA.
(2.19)
Acknowledgements
The maximum value of the factor of encircled energy corresponds to the dominant eigenvalue of the above characteristic-value equation and elements of the corresponding eigenvector correspond to the requisite optimal values of the coefficients of expansion (2.8), of course, to within a multiplicative constant.
The author is indebted to Professor M. De for his sustained guidance and encouragement. Thanks are due to the computer centre, Calcutta University, for assistance in the computational work. Acknowledgement is also made to the University Grants Commission, India for sponsoring a scheme during the tenure of which this work is done.
3. Numerical results and discussion
References
For any particular apodization interval, the expressions of the previous section can be utilized to design optimum amplitude filters with some prespecified number of admissible step variations in transmission. Incidentally it may be recalled that, by virtue of the dyadic algebra followed by Walsh functions, the resulting number of steps P on utilising a finite number of terms N in expansion (2.8) is determined by the relationship p= 2P’
for
2p’-1
2p’,
(3.1)
where p’ is any positive integer. We present some typical results with N = P = 23. All the computations were performed on an IBM 1130 computer. The dominant eigenvalue and the corresponding eigenvector of the characteristic-value equation (2.18) were determined by the Jacobi technique [24]. In general all iterates were taken such that the final answers were computed to an accuracy of lo@. Fig. 2 gives the optimum transmission functions for four different prespecified radii on the farfield plane which are given by the lower bounds of the apodization intervals: 2 G p < + CQ,3 < p < + m, 4 < p < + 00and 5 < p < + 00.The transmission values were normalised by the boundary condition imposed by the passivity of the optical system If(r)1 G 1.
(3.2)
The corresponding diffraction patterns are git-en in fig. 3. A convenient technique for simulating these grey236
[1/ R. Strauble, Pieter Zeeman Verhandelingen
(Nijhoff, The Hague, 1935). [2] R.K. Luneberg, Mathematical Theory of Optics (Brown Univ. Lecture notes, 1944; Univ. Calif. Press, Berkeley, 1964). [3] P.M. Duffieux, C.R. Acad. Sci. Paris 222 (1946) 1482. [4] II. Slevogt, Optik 4 (1949) 349. [5] G. Lansraux, Rev. d’optique 26 (1947) 24. [61 E. Wolf, Repts. Progr. in Physics XIV (1951) 95. [71 R. Barakat, J. Opt. Sot. Am. 52 (1962) 264. 181 P. Jacquinot and B. Roizen-Dossier, Progress in Optics, Vol. III, ed. E. Wolf (North-Holland, Amsterdam, 1964). 191 D. Slepian, J. Qt. Sot. Am. 55 (1965) 1110. 1101 B. Roy Frieden, J. Opt. Sot. Am. 58 (1969) 402. [Ill A.M. Clements and J.E. Wilkins Jr., J. Opt. Sot. Am. 64 (1974) 23. [I21 T. Asakura and T. Ueno, Nouv. Rev. Optique 5 (1974) 349. [131 M. De, L.N. Hazra and P. Sengupta, Optica Acta 22 (1975) 125. [I41 G.C. Temes, V. Barcilon and F.C. Marshall, Proc. IEEE 61 (1973) 196. by OrthogI151 H.F. Harmuth, Transmission of Information onal function (Springer Verlag, Berlin, 1972). Bibliography on Walsh [I61 J.N. Bramhall, An Annotated and Walsh Related Functions (Silver Spring, Md.; Appl. Phys. Lab., The John Hopkins Univ., 1974). [I71 C.F. Chen and C.H. Hsiao, J. Franklin Inst. 300 (1975) 265. [I81 P. Jacquinot, J. Phys. Rad. 11 (1950) 361. [I91 P.L. Ransom, Appl. Opt. 11 (1972) 2554. PI C.I. Abitbol and J.J. Clair, Opt. Acta 22 (1975) 145. WI L.N. Hazra and A. Banerjee, J. Opt. 5 (1976) 19. WI M. De and L.N. Hazra, Optica Acta (in press). Compatibili~231 G.S. Robinson, IEEE Trans. Electromagn. ty, E MC-16 (1974) 183. of numerical methods 1241 S.S. Kuo, Computer applications (Reading, Mass.; Addison-Wesley, 1971).
2 1, number
2
OPTICS
A NEW CLASS OF OPTIMUM AMPLITUDE
COMMUNICATIONS
May 1971
FILTERS
L.N. HAZRA Department of Applied Physics, University of Calcutta, 92 Aqharya Prafulla Chandra Road, Calcutta 700009, India Received
22 December
1976
This paper presents a new class of optimum amplitude filters for the maximization of the factor of encircled energy within a pre-specified central core area of the far-field diffraction pattern. The filters arc composed of finite uniformly transmitting zones with variations between them. The analysis is based on the known technique of indirect starch for the optimum filter by “the method of sets of screens” utilising the complete set of normal orthogonal Walsh functions as a system of base functions. Some numerical results are presented.
1. Introduction Amplitude filters on the pupil of an image forming system substantially modify its imaging characteristics. Since the pioneering works of Straubel [l] and Luneberg [2], the problem of determining optimum amplitude filters in accordance with some prespecified constraints on the image plane is receiving considerable attention [3-131 . As the far-field amplitude or intensity distribution is expressible as a finite Fourier transform of the pupil function or the anticorrelation function of the same, respectively, the problem is essentially one of optimization of a band-limited system. This way of looking at the problem emphasizes the universality and the general mathematical nature of the topic which arises in a variety of engineering fields, e.g., design of antennas, signal transmission systems, etc. [ 141 . An important problem of optimization often encountered in imaging systems is the suppression of side lobes of the far-field diffraction pattern. Variational methods have frequently been adopted for obtaining a solution to the problem. In order to bring forth clearly the motivation of our present investigation, we briefly point out a few salient features of the various approaches undertaken heretofore. Details of the methods are given in ref. [8]. The formulation developed by Luneberg [2] has led to a technique of “direct search” for the pupil function amongst the complete set of physically real232
izable pupil functions. For the determination of the optimum function, the method necessitates the solution of a linear homogeneous integral equation of the second kind. The kernel of this equation is, however, too complicated for the optimum function to be calculated otherwise than by numerical approximations which, in turn, introduce errors in the computed results. Al alternative attempt known as “the method of sets of screens” has been made wherein the optimum function is sought in an indirect manner. The method involves a finite series expansion of the pupil function in terms of a convenient set of base functions with known finite Fourier transforms. The solution is obtained via optimization of the coefficients of this expansion. Whereas in the earlier approach the optimum function is chosen out of the complete set of pupil functions, in the latter the scope of choice is limited as the solution obtained is best among the particular set of functions under consideration. In order to obtain sufficiently accurate and useful results, this inherent limitation can be circumvented by choosing progressively larger number of terms in the expansion. A continued search has also been undertaken to find out a suitable system of base functions which could provide the desired accuracy in the results with a smaller number of terms in the expansion so that the computations do not become unwieldy. Different sets of orthogonal polynomials and of some nonorthogonal functions like the lambda functions have
Volume
2 1, number
2
May 1977
OPTICS COMMUNICATIONS
been utilised [3-61 . In the sixties, Slepian’s introduction of prolate spheroidal wavefunctions in apodization studies opened up a new possibility in this theoretical search [9] . In the present communication, we propose to utilize Walsh functions in these studies. The complete set of normal orthogonal Walsh functions has provided a novel and very convenient analytic tool for the treatment of practical problems of information transmission systems [IS] . The application has been found to be so promising and encouraging that a host of suggestions for employing these functions in problems of allied fields are appearing at a high pace [ 161 . A pertinent example is a proposal for the utilization of Walsh functions in solving a variational problem in heat conduction [ 171. The rationale for our suggested application can be readily appreciated if one recognises the advantages to be gained in the subsequent process of practical fabrication of these filters. There exists a number of techniques for the fabrication of amplitude filters [ 18-201 ; nevertheless, one can very well surmise that all these processes will be considerably facilitated if one has to generate a filter composed of a finite number of uniformly transmitting zones with variations between them instead of a filter with continuously varying transmission. An obvious possibility is to approximate the known apodizing pupil functions by the Walsh functions which have the interesting property that a finite sum of a Walsh series expansion of a function yields a ladder-step approximation to the latter. The possibility has been explored and found to provide useful results as sought for [21]. However, the need for this further approximation can be obviated and the present paper reports a refined mode of analysis to obtain the desired results directly by utilizing the set of Walsh functions as a system of base functions in the “indirect search technique”.
(2.1) where f(r) is the circularly symmetric pupil function. The variable r is a fractional coordinate for a point on the pupil while p is a reduced diffraction variable given by p = (2n/A) (n’sin o’)t’,
(2.2)
where CY’is the semi-angular aperture of the optical system, 2n/h is the propagation constant and t’ is the geometrical distance of the point in the far-field plane under consideration from the centre of the diffraction pattern. The far-field intensity distribution is given by Z(P) = lFo7)12
(2.3)
and the expression for the encircled energy E(W) within a circle of radius W, concentric with the centre of the diffraction pattern in the far-field is 2ll
w
(2.4) where f3 is the variable of azimuth in the far-field plane. For the rotationally symmetric system under consideration the above reduces to W W E(W)=SI(p)pdp=SIF@)12pdp (2.5) 0 0 to within a multiplicative constant. By Parseval’s identity, the total transmitted flux is given by E(W)
=)If(r)12rdr.
(2.6)
0 The factor of encircled energy is defined by the ratio &( W) = E( W)IE(=J).
(2.7)
Using the set of radial Walsh functions p,(r) as the system of base functions, the pupil function f(r) may be expressed over the domain (0,l) as N-l
2. Mathematical analysis
(2.8) In what follows, we confine ourselves to circularly symmetric apertures. With the usual assumptions of scalar diffraction theory, the far-field amplitude distribution F(p) due to a point object positioned on the optical axis of the system is to a multiplicative constant given by
where
=2 j‘f(r)vnWrdr arz
(2.9) 0 and N represents the finite number of terms utilised 233
Volume
21. number
2
in the expansion. Fig. 1 depicts the first eight radial Walsh functions p,(r). Radial Walsh functions as a subset of the set of circular Walsh functions have beer derived earlier for the treatment of problems of optical imagery [22] . From (2.1) and (2.8), we obtain
C(W)=
N-l
N-l
N-I
N-l
c c amane,,(c n=o c amanemn’
m=O ,1=0
m=O
(2.12) where = i x,Wx,(PW~. 0 From the condition of orthogonality radial Walsh functions we have
emn(V
N-l
ml =
May 1911
OPTICS COMMUNICATIONS
ngo anx,(P)
(2.10)
where
(2.13) satisfied by the
Emn = k&m,, where a,,
Analytical expressions for xn@), the Hankel transforms of the radial Walsh functions are identical with those for the Fourier transforms of the unidimensional Walsh functions [23]. On algebraic manipulations of (2.5), (2.6), (2.7) and (2.10), we get
(2.14) is Kronecker delta defined as
6mn =l ’
m = n,
= 0,
m fn.
The condition
aw)
aam
for stationarity
_o ’
(2.15) of C(W)
m = 0, .. ..N-1.
(2.16)
leads to the following system of linear homogeneous equations
cn anemn(W)=P 1.0
Canernn, n
m =0 ,..., N-l, (2.17)
r-
0.2 -
Fig. 1. First eight radial Walsh functions,
234
g,(r);
n = 1, . . . . 8.
Fig. 2. Optimum pupil transmission functions (number of steps N = P = 8)which maximises the factor of encircled encr. gy within a circle of prespecified radius W concentric with the centre of the diffraction pattern; -: W = 0.0, ---: W = 2.0, -.-.-.: Iv = 3.0, . . . . . . . w = 4.0, xxxx: w = 5.0.
Volume
21, number
2
OPTICS COMMUNICATIONS
May 1977
lo-’
10+
I\
\
\ \ \ \ \
L
I
i=y \
‘,
i
.
1; :
16'
I
I
II :
I
! / :I
i
I
...
: ‘, : :
I ;
i\ \ \ , ‘. 1 \’ i h i
6
lo-
i ‘.,
i
.n : : 1 x
10-5 Fig. 3. Diffraction patterns corresponding to the optimum functions of fig. 2. -: . . . . . . W = 4.0, xxxx: W = 5.0. The scale of Z is linear as far asp = 6.0; a logarithmic of the side lobes.
W = 0.0, ----: scale is adopted
W = 2.0, -.-.--.: W = 3.0, for p > 4.0 for better display
235
Volume
21, number
2
where p is the stationary notation
OPTICS COMMUNICATIONS
value of C(W). In matrix
EWA=@A.
(2.18)
May 1977
level ladder-step filters by binary filters has been developed. Preliminary experimental results are quite promising and we intend to publish a fuller report soon.
With the help of expression (2.13) EwA=$4
=vA.
(2.19)
Acknowledgements
The maximum value of the factor of encircled energy corresponds to the dominant eigenvalue of the above characteristic-value equation and elements of the corresponding eigenvector correspond to the requisite optimal values of the coefficients of expansion (2.8), of course, to within a multiplicative constant.
The author is indebted to Professor M. De for his sustained guidance and encouragement. Thanks are due to the computer centre, Calcutta University, for assistance in the computational work. Acknowledgement is also made to the University Grants Commission, India for sponsoring a scheme during the tenure of which this work is done.
3. Numerical results and discussion
References
For any particular apodization interval, the expressions of the previous section can be utilized to design optimum amplitude filters with some prespecified number of admissible step variations in transmission. Incidentally it may be recalled that, by virtue of the dyadic algebra followed by Walsh functions, the resulting number of steps P on utilising a finite number of terms N in expansion (2.8) is determined by the relationship p= 2P’
for
2p’-1
2p’,
(3.1)
where p’ is any positive integer. We present some typical results with N = P = 23. All the computations were performed on an IBM 1130 computer. The dominant eigenvalue and the corresponding eigenvector of the characteristic-value equation (2.18) were determined by the Jacobi technique [24]. In general all iterates were taken such that the final answers were computed to an accuracy of lo@. Fig. 2 gives the optimum transmission functions for four different prespecified radii on the farfield plane which are given by the lower bounds of the apodization intervals: 2 G p < + CQ,3 < p < + m, 4 < p < + 00and 5 < p < + 00.The transmission values were normalised by the boundary condition imposed by the passivity of the optical system If(r)1 G 1.
(3.2)
The corresponding diffraction patterns are git-en in fig. 3. A convenient technique for simulating these grey236
[1/ R. Strauble, Pieter Zeeman Verhandelingen
(Nijhoff, The Hague, 1935). [2] R.K. Luneberg, Mathematical Theory of Optics (Brown Univ. Lecture notes, 1944; Univ. Calif. Press, Berkeley, 1964). [3] P.M. Duffieux, C.R. Acad. Sci. Paris 222 (1946) 1482. [4] II. Slevogt, Optik 4 (1949) 349. [5] G. Lansraux, Rev. d’optique 26 (1947) 24. [61 E. Wolf, Repts. Progr. in Physics XIV (1951) 95. [71 R. Barakat, J. Opt. Sot. Am. 52 (1962) 264. 181 P. Jacquinot and B. Roizen-Dossier, Progress in Optics, Vol. III, ed. E. Wolf (North-Holland, Amsterdam, 1964). 191 D. Slepian, J. Qt. Sot. Am. 55 (1965) 1110. 1101 B. Roy Frieden, J. Opt. Sot. Am. 58 (1969) 402. [Ill A.M. Clements and J.E. Wilkins Jr., J. Opt. Sot. Am. 64 (1974) 23. [I21 T. Asakura and T. Ueno, Nouv. Rev. Optique 5 (1974) 349. [131 M. De, L.N. Hazra and P. Sengupta, Optica Acta 22 (1975) 125. [I41 G.C. Temes, V. Barcilon and F.C. Marshall, Proc. IEEE 61 (1973) 196. by OrthogI151 H.F. Harmuth, Transmission of Information onal function (Springer Verlag, Berlin, 1972). Bibliography on Walsh [I61 J.N. Bramhall, An Annotated and Walsh Related Functions (Silver Spring, Md.; Appl. Phys. Lab., The John Hopkins Univ., 1974). [I71 C.F. Chen and C.H. Hsiao, J. Franklin Inst. 300 (1975) 265. [I81 P. Jacquinot, J. Phys. Rad. 11 (1950) 361. [I91 P.L. Ransom, Appl. Opt. 11 (1972) 2554. PI C.I. Abitbol and J.J. Clair, Opt. Acta 22 (1975) 145. WI L.N. Hazra and A. Banerjee, J. Opt. 5 (1976) 19. WI M. De and L.N. Hazra, Optica Acta (in press). Compatibili~231 G.S. Robinson, IEEE Trans. Electromagn. ty, E MC-16 (1974) 183. of numerical methods 1241 S.S. Kuo, Computer applications (Reading, Mass.; Addison-Wesley, 1971).