R904 Philips Res. Repts 30, 232*-239*, 1975 Issue in honour of C. J. Bouwkamp SOME REMARKS ON THE TREATMENT OF THE DIFFRACTION THROUGH A CIRCULAR APERTURE by Josef MEIXNER *) and Schiu SCHE Rheinisch-Westfälische Technische Hochschule Aachen Aachen, B.R.D. (Received January 14, 1975) Bouwkamp's thesis 1) is a cornerstone in the theory of spheroidal functions, in their numerical treatment and their application to a specific problem which was of great interest at that time. As a matter of fact, he treated the diffraction of scalar waves through a circular aperture from long wavelengths down to such wavelengths where the Kirchhoff-Rayleigh approximations are already quite good. At that time the numerical computations were quite cumbersome. With the advent of the present fast computers the situation has much changed. Not only can one easily carry through the calculations with an increased number of significant digits, one can also go to much smaller wavelengths as compared to the radius of the aperture. Our goal in this note is to extend Bouwkamp's results to wavelengths which are one half of his, but also to compare the exact solution, which is a series in spheroidal wave functions, with the two Kirchhoff-Rayleigh approximations, also expanded in spheroidal wave functions. We restrict ourselves, however, to the case of a hard screen with a circular aperture. The other case of an infinitely soft screen with a circular aperture can be treated in a similar fashion. The problem to be treated is the diffraction of a scalar plane wave by a rigid plane with a circular aperture A. The diffracting screen S is in the plane z = 0, the centre of the aperture is at x = y = 0, its radius is a. The plane wave exp (-ikz) is impinging from the half-space z ~ 0 normally to the plane z = O. The transmitted wave in z ;;:::: 0 can be written in terms of spheroidal functions as co "P(g, 'fj, (/)) = L C 2L S2L(4)(-i~; L=O iy) PS2L('fj; -y2) (1).) This paper is dedicated to C. J. Bouwkamp, from whose thesis the first author not only learned a little of the Dutch language but also got the confidence that it is worth while to study the spheroidal functions.
TREATMENT OF THE DIFFRACTION THROUGH A CIRCULAR APERTURE 233* with where Q2Lty) is given by (12). The spheroidal coordinates ç, 'fj, cp are defined by Furthermore x = a (ç2 1)1/ 2 (1 - 'fj2)1/2 cos cp, Y = a (ç2 + 1)1/2 (1 - 'fj2)1/2 sin cp, z = aç'fj. where k is the wavenumber, l the wavelength. The boundary conditions are 'b1p -=0 ons, 'bz (2) y = ka = 2najl (4) (3) 1p = 1 in A. (5) The problem is Bouwkamp's problem (lb) 1). The notation ofthe spheroidal functions is taken from ref. 2. We also introduce the Kirchhoff-Rayleigh approximate solution 1pK2, for Bouwkamp's case K2. It satisfies the conditions 'b1pk2 'b1pk2 -- = 0 on S, - = -i k in A. (6) öz' 'bz It can be expanded in spheroidal functions like in (1), but the coefficients C2L are to be replaced by where b 2L(y2) I (y2) C2LK2 = y2 (4L + 1) 0 2L 2 ' (7) ps2l(l; -y ) 1 12L(y2) = J'YJ PS2L('fJ; _y2) d'yj. o In the derivation of this expansion the Wronskian (ref. 2, p. 294) and (11) are used. Bouwkamp has compared the solutions 1p and 1pK2 by expanding them in spherical wave functions and by comparing the numerical values of the coefficients for y = 5 and y = 10. Quite interesting results are obtained if, instead, one compares the coefficients C2L and C2L K2 in the expansion (1) and in the corresponding expansion for 1pK2. This will be done for y = 10 and y = 20. The coefficients bn 2L (y2) in the expansion (8) PS2L('fJ; _y2) <Xl = L: bn 2L (y2) P2N('YJ), N=O (9)
234* JOSEF MEIXNER AND SCHIU seas where P2N('fJ) = Legendre polynomial of degree 2N, with the normalization condition (10) N=O have been calculated for l' = 10 and l' = 20. The eigenvalues of the differential equation for PS2L('fJ; -y2) have been taken from the tables 3). Thus the calculation, which otherwise used Bouwkamp's method of continued fractions, could be considerably simplified. These tables which also contain S2L W (-io; i1'), j = 1, 2, provided a useful check of our numerical results through the two relations (11) (12) Table I contains values of PS2L(0; _1'2), ps2l(i; _1'2), h02l(1'2) and 12L(1'2) for l' = 10 and l' = 20. In table II we give the values of the coefficients C2L> C2L K2 and their contributions to the transmission coefficient, 2 1 a K2 = _ Ic K212 2L r2 4L + 1 2L (13) For the same values of l' the transmission coefficients D 2 and D K 2 are obtained by summing these contributions over L. For the rigorous solution we obtain Dz(10) = 1-0 002 073 135, (14) D 2 (20) = 1-0 000 618 554, while the asymptotic formula 4) yields 1 cos (2r - n/4) 7 sin (21' - n/4) D2(r) = 1-- 4 2 + 4 1/2 5/2 + 64 1/2 7/2 + l' n v n l' 1 sin (41' - n/2) + + O( -9/ 2 ) 161'4-64 n 1'4 l' Dz(10),.., 1-0 002 070 475, Dz(20),.., 1-0 000 618531. (15) (16)
TREATMENT OF THE DIFFRACTION THROUGH A CIRCULAR APERTURE 235* TABLE I Numerical values of the spheroidal functions PS2L(0; _1'2), ps2l(i; -y2), the first term of its expansion in Legendre polynomials and the integral 12L(1'2) 2L ps2l(o; -100) ps2l(i; -100) bo 2L (100) 12L(100) 0 0 000 813 921 4 352285688 0 460356 895 0 411482831 2-0 013 195170 1 816161 791-0,223335757 0 148035309 4 0 142454 360 1 168436570 0 194170 124 0 074325 127 6-0 395987554 0 803293 693-0 122258400 0 011760 887 8 O 342 507 590 0 958 505058 0 017189123-0 010927848 10-0 279 938 449 0 986319157-0 001055765-0,005341160 12 0 245901 337 0 993493 761 0 000045 337-0,002499 185 14-0 222983981 0 996477924-0,000 001 436-0 001 439 872 16 0 205954660 0 997 922 363 0 000000 035-0,000 944 328 18-0'192564821 0 998694 079 ~ -7.10-10 -0 000 667 045 20 0 181634169 0 999 137644 ~ 1.10-11 -0 000 494 893 2L ps2l(o; -400) ps2l(i; -400) bo 2L (4oo) 12L(400) 0 0 000000 052 6 242859069 0 320483936 0 304019055 2-0 000001 774 2'712803726-0 147765526 0 124509 834 4 0 000046161 1 954587247 0'Ü4468122 0 083438 551 6-0 000 809 642 1 559665055-0 100331086 0 060619398 8 0 009967 985. 1 289243 860 0 095016531 0 043866912 10-0 086201 577 1 040623 680-0 098 943 970 0 028665 777 12 0 296814884 0 764317794 0 082089 832 0 008735551 14-0 302758317 0 903 357 521-0 020041 794-0 004415302 16 0 249059 344 0 959284260 0 002084752-0 003 211 008 18-0 220365401 0 976307 870-0 000 165798-0 001 614650 20 0 201421 516 0 984 894 928 0 000010 568-0,000929 794 22-0 187345805 0 989 826 232-0,000 000 553-0 000 614 259 24 0 176226 112 0 992 865 877 0 000000 024-0 000 440 693 26-0 167095555 0 994836701 ~ -9.10-10 -0,000 332 791 28 0 159392860 0 996165426 ~ 3.10-11 -0 000 260410 30-0,152762497 0 997089 973 ~ -8.10-13 -0,000209 269 \
TABLE 11 I~ Numerical values of the expansion coefficients C 2L and C 2L K2 and of the partial amplitudes 0'2L and 0'2L K2 2L I Re C 2L (lo) Im C 2L (IO) C 2L K2 (10) 0'2L(10) 0'2L K2 (10) 0 4-352285688 -0-000002137 4-352401 752 0-378847814 0-378868020 2-9-080 806 226 0-004 979 179-9-102046386 0-329 844 266 0-331388994 4 10-441289884 -0-882796710 11-116176621 0-243 999 699 0-274598 628 6-2-810603098 4-631 536125-2-326953993 0-045154795 0-008330331 8 0-004188 186-0-261 203 400-0-333152343 0-000 080 288 0-000 130 577 10-0-000 000 170 0-001 875538 0-012006 169 0-000000 003 0-000 000 137 12 1_10-12 R:j -0-000005375 - 0-000285116 2 _10-14 R:j 7 _10-11 ~ 14 R:j -2 _10-18 R:j 0-000 000 008 0-000006017 4 _10-20 R:j 2 _10-14 2L Re C 2L (20) Im C2L(20) C K2 2L (20) 0'2L(20) 0'2L K2 (20) 0 6-242859069 R:j -1 _10-14 6-242859074 0-194866447 0-194866447 2-13-564018632 R:j 2 _10-10 -13-564019325 0-183 982601 0-183982620 ê 4 17-591285225 -0-000 000 225 17-591333171 0-171918509 0-171 919446 '"o 6-20-275645720 0-000103700 -20-277 720 483 0-158 116081 0-158148442 ~ 8 21-917 129240-0-018944 799 21-984170125 0-141282621 0-142 148 158 10-21-775712475 1-298117512 -22-894893 665 0-113301 610 0-124803 847 12 9-301429747 -9-550634073 9-382221 747 0-035546241 0-017 605 217 14-0-081 299914 1-457132152 1-136 306 065 0-000367214 0-000 222 619 16 0-000025 193-0-028 240 605-0-092113 310 0-000 000 121 0-000 001 286 18-0-000 000 002 0-000 260 356 0-004058 181 R:j 9 _10-12 0-000 000 002 20 R:j 5 _10-14 -0-000001415 - 0-000163624 R:j 2 _10-16 R:j 3 _10:-12 22 R:j -5 _10-19 0-000 000 005 0-000006 174 R:j 3 _10-21 R:j 4 _10-15 '-< 0 i:tj '" "1 is: i:tj ~ := ~ '"o
TREATMENT OF THE DIFFRACTION THROUGH A CIRCULAR APERTURE 237* The transmission coefficient of the Kirchhoff approximation is D K 2(10) = 1-0 006683313, D K2 (20) = 1-0 006301917. It is obtained by summing the individual contributions from the rigorous expression DK2(y) = 1_ J 1 (2y), Y (17) in table II, but also (18) where J 1 (2y) is the Bessel function of the first order. The calculations have been carried through on the CD 6400 computer of the "Rechenzentrum der Rheinisch-Westfälischen Technischen Hochschule". All digits given in the tables except for the last digit are considered to be safe. The surprising result is that DK2, although pretty close to D2 in numerical value, has an asymptotic expression which departs from 1 by a term of order y-3/2 in contrast to (15) while, on the other hand, there is such close agreement between the contributions (Jo and O'OK2. One expects that both can be expanded asymptotically in descending powers of y. The close numerical agreement at least to 10-4 for y = 10 and to 10-9 for y = 20 would, however, indicate, that the two asymptotic expansions agree in quite a number of leading terms. Therefore we have studied the situation by comparing the expression with e2l K2 for large positive values of y. The quantity e 2L ' is obtained from e2l in (2) by using (19) and noting that for ~ = 0, y = 20 and L = 0, 1 the imaginary part is about 10-11 times the real part or less. For y = 40 this would be true even for L = 0, 1,...,6 (see ref. 3). We therefore examine the asymptotic expansion of e 2L '(y) 1 [ps2l(1; _y2)y K(y) := e 2L K2(y) = y2 b02l(y2) 12L(y2) (21) for large positive y and moderate values of L. An asymptotic expansion of the functions PS2L(17; _y2) has been given in ref. 5. Although the normalization (10) drops out in (21), we give the expansions of the individual terms in (21) with regard to the normalized functions PS2L(17; -y2). They are, with p = 2L + 1,
238* JOSEF MEIXNER AND SCHIU SCHE (22) (23) (_I)L I2L(Y) '" 2 )1/2 [ 3p r«. ( Y (4L + 1) 1-4y 8p2 + 5 P ] ------ (26 p2 + 57) +.... 32 y2 128 y3 (24) Introducing these expansions into (21) yields K(y) = 1 + O(y-4), as y_oo. (25) Our conjecture is that the non-occurrence of terms of the orders y-l, y-2, y-3 is not accidental, and that all higher powers of y-1 also drop out. But no general proof is available at the present time. This does not mean that C2L' and C2L K2 are equal, rather is it expected that the O(y-4) can be replaced by O(yn exp (-y» with some value of n, which may depend on L. Bouwkamp has also considered the other Kirchhoff-Rayleigh approximation Kl which has the boundary conditions 'ljjkl = 0 on S, 'ljjkl = 1 in A. (26) It can also be easily expanded in spheroidal functions 00 V)K1= :L C2L+1K1 S2L+1(4)(-i~; iy) PS2L+1(1]; _y2) (27) L=O with 0 ~ ~ < 00, 0 ~ 1] ~ 1, where b12l+ 1(y2) 1 C2L+IK1=-ty2(4L+3) Jps2L+l(1];-y2)d1]. (28) _ PS2L+ 1(1 ; _y2) 0 At first sight it does not seem possible to relate simply the individual terms in the expansions (7) and (28). However, at large distance from the aperture, ~» 1, we have Moreover, for large positive values of y we have, apart from terms of order yn exp (-y) with some number nel) (29)
TREATMENT OF THE DIFFRACTION THROUGH A CIRCULAR APERTURE 239* The square root of (4L + 1)/(4L + 3) enters these expressions due to the different normalization factor of ps2l and PS2L+1. Thus in the far field corresponding terms in both Kirchhoff expansions Kl and K2 become equal to any order y-m (m = 1,2,3,... ) as y -)0 00. But how fast this approach to equality is, depends on the value of L. We have chosen y = 10 and the numerical results of Bouwkamp 1) in order to calculate the contributions of the partial spheroidal waves to the total transmission coefficient for the case Kl. Their consecutive values are 0 37869; 0 32323; 0 18358; 0 01460; 0 00009 of which four decimals are considered to be correct. These values should be compared with the 0"2L(10) and the 0"2L K2 (10) in table 11. REFERENCES 1) C. J. Bouwkamp, Thesis Groningen, J. B. Wolters Uitgevers-maatschappij N.V., Groningen-Batavia, 1941 (English translation in IEEE Trans. Antennas and Propagation AP-18, 152-176, 1970). 2) J. Meixner and F. W. Schaefke, Mathieu'sche Funktionen und Sphäroidfunktionen, Springer, Berlin, 1954. 3) S. Hanish, R. V. Baier, V. L. Van Buren and B. J. King, Tables of radial spheroidal wave functions, Vol. 4, N.R.L. Report 7091, Naval Research Laboratory, Washington O.C., 1970. 4) D. S. Jones, Proc. Camb. Phil. Soc. 61,223-245, 1965. 5) J. Meixner, Z. angew. Math. Mech. 28, 304-310, 1948.