ANALYSIS OF WEINSTEIN'S DIFFRACTION FUNCTION

Transcription

1 R898 Philips Res. Repts 30, 161*-170*,1975 Issue in honour of C. J. Bouwkamp ANALYSIS OF WEINSTEIN'S DIFFRACTION FUNCTION by J. BOERSMA Technological University Eindhoven Eindhoven, The Netherlands (Received December 13, 1974) Abstract The Wiener-Hopf solution of the reflection and radiation problems for an open-ended parallel-plane waveguide is nveniently described in terms of Weinstein's diffraction function. In this paper, the accuracy of a mmonly used approximation to Weinstein's function is examined. In addition, some new series expansions for Weinstein's function are presented. 1. Introduetion During the last decades the so-called Wiener-Hopf technique has proved its power in the solution of a broad class of diffraction problems. A survey of the early literature is presented in Bouwkamp's well-known review paper (ref. 1, 7). Later ntributions are reported in the books by Noble 2), and Mittra and Lee 3). The Wiener-Hopftechnique was originally developed by Schwinger, Levine, Copson, Heins and others, and independently in Russia by Fock and Weinstein (or Vatnstei'n). Weinstein's work of the years deals with the reflection and radiation problems for an incident mode travelling toward the open end of a semi-infinite parallel-plane waveguide or circular pipe. In a thorough treatment, detailed analytical and numerical results are presented for the modal reflection efficients of the reflected wave in the guide, and for the radiation pattern of the field radiated into the exterior free space. Weinstein's papers first became more readily available through a translation by Shmoys 4), a translation which was suggested by Bouwkamp. A full acunt of Weinstein's early work is to be found in part 1 of his recent book 5). Consider the reflection and radiation of an incident TM or TE mode from the open end of a semi-infinite parallel-plane waveguide with perfectly nducting walls. Referring to Weinstein 5), chapter 1, the solution to this problem can be expressed in terms of the auxiliary functions 1 J s 't" U±(s,p) = -- log [1 ± exp (ip s.)] ---- dr ~i ~'t"-s ro (1) where -1 ~ s ~ 1,p > 0, and the principal value of the logarithm is to be

2 162* J.BOERSMA taken. The path ro is the steepest-descent ntour Re (s r) = 1 that passes through the origin; ro intersects the real axis at an angle -n/4 and has asymptotes Re -r = ± tno The integrals U±(s,p) arise in the Wiener-Hopf procedure 'of factorization of certain analytic functions. Except for a slight change of notation, U+, U- are identical with the functions V, U, respectively, as defined in Weinstein 5), form. (10.07), (10.18). We shall call the functions U±(s, p) exact Weinstein functions. Under the transformation s -r = 1 + i t 2, the integral (1) reduces to in which the logarithm and square roots stand for principal values. The latter representation is not cited in Weinstein 5). Instead, Weinstein introduces what we shall call approximate Weinstein functions, defined by U±(s, - p) 1 f dt = -- log [1 ± exp (ip - P t 2 )] 2n i t - t (1 + i) s - (3) The underlying idea in this approximation is that for large p the main ntribution to the integral (2) arises from the vicinity of the saddle point t = O. In this vicinity all terms t 2 in the integrand are neglected, thus leading to (3). The present functions Ü+(s, p), a-cs, p) are identical with Weinstein's functions V(a, q), U(a, q), respectively, with a = s r'". q = p/2n; see Weinstein 5), form. (I0.08), (10.19). Weinstein did not examine the accuracy of the approximation a±(s, p), nor did he investigate whether the approximation is uniform in s. It is the aim of the present paper to fill this gap. Thus it is shown in sec. 2 that uniformly in s over the range -1 ~ s ~ 1. In addition we derive some further series expansions and approximations for the exact Weinstein functions U±(s,p) (sec. 3). Counterparts of these results, pertaining to the approximate functions a±(s, p), were provided in Weinstein 5), Appendix B. As an application, we present an improved approximation for the self-reflection efficient of an incident mode near cut-off in an open-ended parallel-plane waveguide. Due to the simultaneous analysis of the functions U+ and U-, a+ and Ü-, double signs ± or =F appear at various places. Throughout this paper it is understood that the upper sign rresponds to U+ or Ü+ and the lower to U- or Ü-. (4)

3 ANALYSIS OF WEINSTEIN'S DIFFRACTION FUNCTION 163* 2. Accuracy of the approximation U:!:(s, p) We first establish some simple properties of the Weinstein functions, It is obvious from (2) and (3) that U±(-s,p) = -U±(s,p), O±(-s,p) = -O±(s,p). (5) Both the exact and approximate functions are disntinuous at s = 0. Indeed, when applying PIemelj's formulas to the Cauchy-type integrals (2) and (3), it is found that U±(+0, p) = O±(+0, p) = t log [1 ± exp (i p)], U±(-O,p) = O±(-O,p) = -tlog [1 ± exp (ip)]. Hence, the approximation O±(s, p) bemes exact at s = 0. By means of Laplace's method in a properly modified form we derive asymptotic expansions for U±(s,p), O±(s,p), valid for large p provided that s is not close to zero. On expanding the integrands of (2) and (3) in Taylor series around t = 0, a term-by-term integration yields. (6) 1 + i <Xl U±(s,p) = -- J log [(1 ± exp (i p - P t 2 )] dt + ot«: 3 p- 3/2) 2ns -<Xl 1 + i S±(P) = + 0(S-3 p-3/2) 2n1/2 s pl/2., (7) where I (=F Iy' exp (i mp) S±(P) = -. m 3/2 Consequently, when s is away from zero the approximation O±(s, p) is rrect up to order p- 3/2. Next, we examine the overall accuracy of O±(s,p) over the range -1 ~ s ~ 1. In virtue of the symmetry of the integrands (2) and (3), we may set (8) (9) - l-lj sdt.<xl U±(s,p)=- log[l±exp(ip-pt 2 )] n U2-i~ o (11)

4 164* J.BOERSMA Consider now the difference where (12) 1- i f<xl S (t 2 - i) El = - log [1 ± exp (ip-p t2)] [( it2)-1/2-1] dt, :n; t 4-2 i t 2 - S2 o (13) We establish a number of auxiliary estimates valid for -1 ~ s ~ 1, P > 0, t ~O: [log [1± exp (ip-p t2)]1 ~ -log [1-exp (_pt 2 )], 1(1+ tit 2 )-1/2-11 ~-, 4 s t (t 2 - i) 1 Isli t t 1 t4 _ 2 i t? _ S2 ="2 t2 - i + i (I_S2)1/2 + t2 -i-i (I_S2)1/2 ~ 2-3/2Isll[1 _ (1 _ S2)1/2]-1/2 + [1 + (1 _ s2)1/2]-1/21 ~ 2-1/2, 1 st(t2-is2) 1&1 st(t 2 -i) 1&2-1/2 (t4 _ 1 2 i t2 - s2)(2t2 - is2) --=::: t 4-2 i t2 - S2 --=::: By means of these estimates it follows that eo 1 eo 1 1 :n; IE11~--flOg[1-exp(-Pt2)]tdt = -~- =-, 4:n; 8:n;p ~ m 2 48 p o na=! :n; IE21~- 12p This proves that indeed U±(s,p) = ij±(s,p) + D(P-1) (15) uniformly in s over the range -1 ~ s ~ 1. The error involved is less than (5:n;/48)p- \ however, this estimate is not claimed to be sharp. As a side result of the preceding analysis we have t 2 eo 1- ifs (t2 - i) U±(s,p) = - log [1± exp (ip-pt 2 )] dt + D(p-1), :n; t 4-2 i t 2 - S2 o (16)

5 ANALYSIS OF WEINSTEIN'S DIFFRACTION FUNCTION 165* again uniformly in s when -1 ~ s ::::;;; 1.Setting s = sin cp, -t:n; ~ cp ~ -i:n;, we substitute (t 2 - i) sin cp sin cp sin cp t 2-4i sin 2 tcp 2t 2-4i s 2 1cp then it is easily regnized that U±(sin cp,p) = O±(2sintcp,p)stcp + O±(2stcp,p) sin tcp + O(p-l), (17) uniformly valid for -i:n; ~ cp ~ t:n;.in a way the latter approximation is more natural than (15). Since 2 s tcp ~ 2 1/2, the send function ü= can be replaced by its asymptotic expansion (8), thus leading to _ 1+ i U±(sin cp,p) = U±(2sin!cp,p) stcp + _S±(p)p-1/2 tantcp + O(p-l) (18) 4:n;1I2 uniformly in cp over the range -i:n; ::::;;; cp ~ t:n;. 3. Expansions for the exact Weinstein functions U:J::.(s,p) the approximate functions O±(s,p) are mprehensively discussed in Weinstein 5), Appendix B. From this reference we quote the expansion _ ( sp1l2) Sp1l2 U±(s,p)=!log[l-exp(i(j)]+log (2(j)1/2 (2(j)1/2 1- i I:exp (i m (j) i + s p'!? S2 P + 2:n;1I2 m1/2 8 +I:[lOg ( 1 + [2 (2::: (j)]~/2 ) - [2 (2: =1: (j)jl/2 + is p1/ 2 ) is pl/2 ] + log ( 1- [2 (2 m st _ (j)f/2 + -[2-(-2-m-:n;--(j-)]-1/-2' (19) valid for s > 0, p > 0, p = n± :n;+ (j± where 0.~ (j±< 2:n; and n" (n-) is an odd (even) integer; for simplicity the superscript ± to (j has been suppressed in '(19). Starting from this expansion Weinstein deduces the approximation

6 166* J.BOERSMA _. n i I-i U±(s,p) = log (<51/ /2 Sp1/2) fj Sp1/2 + O(e) (20) 4 2 in which fj = -,(t)/7(, 1 / 2 = and s = max (S2p, <5). The approximation (20) applies when s p1/2 is small and p is close to an odd (even) multiple of 7(,. In this section we shall derive unterparts of (19), (20), pertaining to the exact Weinstein functions U±(s,p). It is assumed that 0 < s :::;;1, p > 0 and, just provisionally, p =1= n± 7(,where n+ (n-) is an odd (even) integer. Consider the derivative ö U±(s, p)/"ös as obtained by differentiation of (1) with respect to s. The resulting integral is transformed into "öu±(s,p) P! exp(ipcosl') --- = =f - dr + os 27(, 1 ± exp (ip s r) To SP! exp(ipsl') dr =f- 2n 1 ± exp (i p s r) sin. - s To (21) through an integration by parts. Now, the first integral in (21) can be reduced to :7(,I (=F ly' f exp (i mp s r) dr = tpi (=f 1)111 Ho(I)(mp) TO m=1 m=1 by calling upon a well-known integral representation for the Hankel function Ho(l). The send integral in (21) is evaluated by ntour integration and residue calculus. To that purpose, nsider the integral along the closed ntour r that nsists of r o, the line Re = --tn and a segment of the line Im l' = -R where R is chosen such that no poles of the integrand are passed through. Then the ntribution of the line segment tends to zero when R --+, whereas f -1< < exp (i p s r) dr i = =f (t7(,- arcsin s). 1 ± exp (i p s r) sin. - s (1 - S2)l/2 Notice that the latter integral is to be understood as a Cauchy principal value when the lower sign applies. The ntour r encloses infinitely many poles given by s. = m n/p where m is an odd (even) integer dependent on the prevailing upper (lower) sign in the integrand. A finite number of poles lies on the real axis between -1-7(, and 0, the remainder is located on the negative imaginary axis. The residues at these poles may be evaluated in a standard

7 ANALYSIS OF WEINSTEIN'S DIFFRACTION FUNCTION 167* manner. Notice that the pole 7: = -tn lies on I', hence, its residue should be unted half. In so doing, the ultimate result is found to be 00 in which Bo = 1, Bm = 2 for m =1= 0 and N+ (N-) denotes the set of odd (even) integers ~ O. The square root (p2 - m2 n2)1/2 is positive imaginary if m rc b-p, The derivative (22) is to be integrated with respect to s. Then, in virtue of the initial value (6), one gets the expansion U±(s,p) = tlog [1 ± exp (ip)] _.!!... [tn-s- (1_s2)1/2 (tn-arcsins)] + 2n 00 + t sp L (=F l)m Ho(1)(mp) + m=1 valid for 0 < s ~ 1, p > 0, p =1= n= n, The latter restrietion can be dropped, since the expansion (23) tends to a finite limit when p -+- n± n, as will be shown below. We now set p = n± n + <5± where ~ <5± < 2n and n+ (n-) is odd (even). For simplicity the superscripts ± to nand <5 are suppressed. Then for n ~ 1 the expansion (23) can be reduced to U±(s, p) = t log [1 - exp (i <5)] - ip [tn - s - (1 - S2)1/2 (tn - arcsin s)] + 2n I ( 00 SP ) sp + tsp (_l)mn Ho(1)(mp) + log 1 + (2 2 2)1/2 (2 2 2)1/2+ P -n n p -n n +t '" Bm [log (1 + SP) - SP]..f ; (p2_m2n2)1/2 (p2_m2:n;2)1!2 men± m*n (24)

8 168* J.BOERSMA Guided by the asymptotic behaviour of the Hankel function, we write pl/2 L (_I)mn H o (1)(mp) = I-i exp(imp)] L [ n1/2 m1/2 m=1 = (_I)mn pi/2 H o (I>(mp) (_I)mn exp (i mp) +~ 1/2' l-il n m m=1 then the first series is absolutely nvergent for p > 0, since its general term is O(m-3/2). The send series is reduced to. L 1 - i exp (i m <5) 1 - i = _- exp (i <5) q)(exp (i <5),!, 1) nin mln n1n m=1 where q) stands for Lerch's transcendent, see e.g. Erdélyi et al.6), sec From the latter reference we quote the expansion 1- i 1- i L,(!- r) -- exp (i <5) q)(exp (i <5), t, 1) = 2 1/2 {J-1/2 + _- (i {JY, n1/2 n1/2 r! r=o valid for 0 < {J < 2n where, denotes Riemann's zeta function. Combining the preceding results, it follows that p1/2 L (_I)mnHo(1)(mp)=21/2{J-1/2_(I-i){ln+O({J1/2) (25) where the efficient {ln is given by m=1 For numerical purposes the Hankel function may be replaced by the first two terms of its asymptotic expansion, thus leading to

9 ANALYSIS OF WEINSTEIN S DIFFRACTION FUNCTION 169* eh) cm i i fjn = = n 1 / 2 8 n 3 / 2 n n (27) It has been checked that the relative error involved is equal to n=": On substitution of (25) into (24), it is easily seen that all singularities cancel and the expansion (24) remains finite at (j = 0, i.e. at p = n n. In addition we deduce the approximation SP ) ni I-i, ( (2p - (j)1/2 4 2 n U±(s p) = log (j1/ fj S pl/2 + O(e) (28) in which e = max (S2 p, (j). The present approximation applies when S pl/2 is small and p is close to n± st, n± ~ l. The case n: = 0 rresponding to 0 < p < 2n, needs a separate investigation. Employing Poisson's summation formula, it can be shown that m=1 m=1 (29) valid for 0 < p < 2n where l' is Euler's nstant. The latter result is inserted in the expansion (23) for U-Cs, p). Then by re-expansion for small p, we obtain the approximation U-(s,p) = t log [p (1 + s)] _ ni + ip[s(iog 4n+ 1-1' + ni) + 4 2n p 2 + (l_s 2 )l/2 (tn-arcsins) ] + 0(p2). (30) As an application of (28), (30), we nsider the reflection of an incident mode at the open end of a semi-infinite parallel-plane waveguide of width a. Referring to Weinstein 5) and Boersma 7), the self-reflection efficient ofthe on mode (n = 0, 1, 2,... ) is given by (31) where ~n = (k 2 - n 2 n 2 fa 2 )1/2 and k is the wavenumber. In (31) the upper sign applies for n odd and the lower for n even. Let now the incident mode

10 170* J.BOERSMA be near cut-off, i.e. k a - approximation n n = (j is small. Then, applying (28) we deduce the e; = -exp [-(1 - i) fjn (2(j)1/2 + (::) 1/2 + O«j)J valid for n ;;:::1, with fjn given by (26) or (27). By extending (28) up to order e3/2, it is found that the error term in (32) is actually O«j3/ 2 ). The case n = 0 needs a separate treatment based on (30). Thus for small k a we get ROD = -exp [i : a (log :: y + :i) + O(k 2 a 2 ) J in agreement with the results of Weinstein 5), secs 6 and 9. A similar approximation can be derived for the self-reflection efficient of the TEon mode (n = 1, 2, 3,... ) near cut-off. Starting from the exact representation for the reflection efficient 5,8), it is found that (32) (33) Rnn = -exp -(1 - i) fjn (2(j)1/2 - en [ 2(j )1/2 ] + O«j3/ 2 ) (34) The present approximations for Rnn are to be mpared with the previous approximate result 5,3) Rnn ~ -exp [- (1 - i) fj (2(j)1/2], fj = -C(t)/nl/ 2 = 0,824, (35) which is based on the use of approximate Weinstein functions O±(s, p). Approximations for the self-reflection efficient of an incident mode near cut-off play an important role in the ray-optical analysis of open resonators 9,10). It is suggested that such an analysis can be improved by utilizing the new approximate formulas (32), (34) instead of (35). REFERENCES 1) C. J. Bouwkamp, Rep. Progr. Phys. 17, , ) B. Noble, Methods based on the Wiener-Hopf technique, Pergamon Press, London, ) R. M ittr a and S. W. Lee, Analytical techniques in the theory of guided waves, Macmillan, New York, ) L. A. Vajnshtejn, Propagation in semi-infinite waveguides, Six papers translated by J. Shmoys, New York Univ., Inst. Math. ScL, Div. Electromagnetic Res., Res. Rep. No. EM-63, ) L. A. Weinstein, The theory of diffraction and the factorization method, The Golem Press, Boulder, ) A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Trimi, Higher transcendental functions, McGraw-HiIl, New York, 1953, Vol. I. ') J. Boersma, Ray-optical analysis of reflection in an open-ended parallel-plane waveguide. I. TM case, SIAM J. appl. Math., to appear. S) J. Boersma, Proc. IEEE 62, , ) L. A. Weinstein, Open resonators and open waveguides, The Golem Press, Boulder, ) L. W. Chen and L. B. Felsen, IEEE J. Quantum Electronics QE-9, , 1973.