# Diffraction Theory by Means of Singular Integral Equations VI Diffraction of Plane Waves by an Infinite Strip Grating.

код для вставкиСкачатьAnnalen der Phvsik. 7. Folae, Band 27, Heft 3, 1071, S. 257--288 .J. 21. Barth, 1,cipzig Diffraction Theory by Means of Singular Integral Equaiions VI’) Diffraction of Plane Waves by an Infinite Strip Grating2) By E. LCTERIJRQand K. WESTPBAHL With 20 Figures Abstract For an infinitc plane grating formed by strips antl gaps of equal width, a DIRICIII.ET boundary value problem for the HEuIloLTz equation is solved rigorously by function-throretic techniques. Plane wave excitation with an arbitrary angle of incidence is considered. The high frequency asymptotics of the solution is complctfily evaluated and compared with KIRCIIH~FF’S diffraction theory as well as with the asymptotics for a single strip. Extpnsivc numerical data arc laid down graphically. 1. Introduction This paper adds a further sample t o the small number of rigorously solvable boundary value problems in diffraction theory. I t may be considered as a heurist,ic method for the half plane to a systematic extension of SOMMERFBT.D’S non separable diffraction problem. We consider the diffraction of plane electromagnetic waves striking a strip grating in the plane z - = 0. The grating consists of an infinite set of identical perfectly conducting antl infinitely thin strips with edges parallel t o the y-axis. The periodically spaced strips have the width 2a in the direction of the x-axis and the spacing between any two adjacent &ips is also 2a (Fig. 1).The angle of incidence is 8,, and the electric vector is polarized parallel t o the edges of the strips. The case of normal incidence (ao = cos 8, = 0) has b e m treated rigorously b y AIDWI WIN and HICI\S111, whereas the independent treatment of WEISSTEIN [ 2 , 31 includes also thc discretr directions a. = zn/’.’ak ( k waw: n u m b e r , n integer). These authors consider the grating as an equivalent infinite waveguide of width 2a formed by tbc planes TC = 0, x = ’La. and a transverse &aphragm z :0 , 0 x n, a problem which is aintmable to the \VIE:SER-HOI>E’ technique (cf. also [ 1, 51). An approximate solution for a grating with the distances between the strips not equal t o their widths has been obtained by AGRAXOVJCH et al. [ 61 < < I ) This work is n seqiicl of the series publishetl in Ann. l’liys. 4 (1959) 283; 20 (1967) L 1 ; 21 (1968) 12; 25 ( l ! J i O ) 3 7 5 ; 26 (1971) 103. 2) Detlicatcd t o Prof. H. 110s~ on his retirement. 17 Ann. Yh,sih. 7 . E’olgr. l M . 27 258 Annalen der Pliysik * 7. Folge * Band 27, Heft 3 * 1971 and by MALIN [7]. The only other rigorous solution of a grating problem, known to us, is concerned with an infinite set of staggered equally spaced semi-infinite plates of zero thickness [9-171. The purpose of the present paper is twofold : (a) To produce a closed-form solution for a r b i t r a r y angles of incidence, by reducing the boundary value problem to a singular integral equation of the CAUCHY type, adapted by one of us to a class of two-dimensional diffraction problems [18]. 'T 'I Fig. 1. Cross section of the strip grating (b) To provide extensive numerical data for the rigorous solution as well as a thorough discussion of the high frequency properties of the solution including diffraction theory. its relation to KIRCHHOFF'S I n Sec. 2 the reduction of the boundary value problem to an appropriate transform of the induced surface singular integral equation for the FOURIER current density on the central strip is accomplished by means of a conformal mapping. The solution of the integral equation by means of a factorization theorem is provided in See. 3, and procedure in conjunction with LIOUVILLE'S supplemented by a complete evaluation of the high frequency asymptotics in See. 4. Most of our numerical results are laid down in See. 5 where the amplitudes of the first five propagating modes of the RAYLEIGH representation are graphically displayed for various angles of incidence. The comparison with KIRCHHOFF'S theory is accomplished via the asymptotic modes and via a properly defined form factor which is numerically evaluated. Finally, in Sec. 6 we use a modified optical theorem to evaluate analytically and numerically the transmissivity in comparison with its high- and low-frequency approximation. The appendices are concerned with infinite product decompositions and boundedness considerations, and give a table of phase functions. 2. Reduction of the boundary value problem to a singular integral equation Let (z, x ) and (r, 8) be Cartesian and polar coordinates, respectively, in a plane perpendicular to the infinite set of strips forming the grating and let m S = u n= -m S,, S, = (-a + 4an 5 z 5 a + 4an, z = 0) (1) be its cross section (Fig. 1).Let v be the total electric field polarized parallel to the edges of the st,rips (omitting the time factor exp (-id)). Then, according to [l], v is uniquely determined by the following data,: LUNEBURG and WESTPFAHL: Diffraction of Plane Waves by an lnfinite Strip Grating 259 (a) Field equation (V2 + k2)v(z, z ) = 0 (k wave number). (b) Boundary conditions w(z,S O ) = 0 ( x E 8). (c) Radiation condition (d) Edge condition O w = O(p-l'z) (e = distance from the edges --f 0). wo is the incident field for which we take the plane wave wo(x,z ) + (2d) ~ (ikjaox 1/1 - at z ) ] (ao = cos 6,), (3) propagating perpendicular t o the edges with an angle of incidence Go (cf. Fig. 1). By means of HUYCEN'S principle (HELMHOLTZ'S formula) we obtain an integral representation for v(x,z ) which satisfies (2a, c), viz. w(z, z ) = exp = vo(x,z ) where 3, Y(+,) 1 +- sJ;y(x'la0) H:l'(k = i, rv,(x, -0) - w,(x, +ON V(T - x')2 +z2)ax', E s) (4) (5) is the induced surface current density on the strips. Due t o the periodic structure of t h e grating the current densities on different strips are related by the FLOQUET condition y(x 4anla0) = y(z/ao)e42en0n(1x1 < a ; n = 0, &l,& 2 , ...) E = ka and subject t o the edge condition (2d),viz. (6a) (Gb) N x o n : fi= Ef/&. 1/1- a2> 0 for a real and la/< 1. The branch cuts are taken along (1,icu) and (-1, -iw). cf. Fig.%. 17* 3, 4, 260 Annalen der Physik * 7. Folge * Band 27, Heft 3 * 1971 I n the last equation use has been made of the FOURIER representation of the HANKEL-function H t ’ , and we have introduced the FOURIER transform of the current density on the single strip So: U ji(cyIcyo) = / y(xlao)e--ikmx dx. --a For reasons of convergence the infinite sum has to be ident,ified with m 2 e--4i€(n-aa,)n = 1 cot 2 4 3 - ,I). 2 n= (9) -m where the hooks indicate that the path of integration in (4“) has to pess above ( ), respectively below ( ), the poles of the cot-function. Inserting (9) in (4”) we thus obtain v n (10) The path of integration is sketched in Fig. 2. (In order to simplify our analysis n integer, corresponding we temporarily exclude the values a,, = & 1 nn/2&, + I - I I -7 ” 111 111 L I I I I I t 1 I I I Fig. 2. Integration path for Eq. (10) t o the coincidence of a pole of the cot-function with the branch points m = & 1. I n the final results, however, these values may be included by a limiting process.) Applying the boundary condition ( 2 b) for IC t So to (10) yields the dual integral equations inversion of (8). where the second equation results from FOURIER I n order t o indicate the method of getting singular integral equations of the CAUCHYtype we begin with the (rather trivial) Eq. ( l l b ) , where we replace cy b y a’, multiply by exp ( - ikmz) and integrate over the 2-intervals (-00, -a) and LuNEsuRa and WESTPFAHL:Diffraction of Plane Waves by an Infinite Strip Grating 261 ( a , co), respectively. The results are the singular integral equations5) -m f y(&jag) -m m da' a' - a = 0, $ ij(a+x0)e-iaa' --m da' ___ -0 a' - a (12a, b) or, equivalently, The equations (12 a) and (12 b) are the necessary and sufficient conditions for their integrands being holomorphic functions in the upper and lower &-half plane, respectively, and vanishing there a t infinit;y (cf. [19] p. 27). Thus @(also) is an entire function (13) or, more precisely, 1 holomorphic i (13a) (13b) whereby the holomorphy, of course, is already implied by the representation (8) and the exact behaviour a t infinity is implied by the edge condition (7). Proceeding in a similar manner with ( l l a ) we find the integral equation y(alao)e " i E 1 = 0(a-'l2) (la1+oo) in the upper/lower a-half plane, Our goal is now t o convert (14) into such a singular integral equation which admits only an entire function as solution. To this end we remark that (14) is equivalent t o sin &(a- a ) +-,--sin &(a- a , ) } ~~da' - STC--------'-, sln &(a - a,) a' - a a - a, (15) where t'he point a lies outside the integration oontour. This may be seen by subtracting from (14) the term vanishing b y virtue of (13). Bearing in mind (13b) we can deform the path of integration into the loops C and 6 encircling the branch cuts (1, ioo) and (-1, -ioo), respectively, ~. 5) We use the notation 1 + 1 - ni S(n' - a), 1 1 97 = 9 _ _ - ni a(&' - a ) a-a a' - QL a' - a or, in integrated form, a'-F - CAUCHYprincipal values being taken for the respective first terms on the r.h.s. (cf. [MI). 262 Annalen der Physik * 7. Folge * Band 27, Heft 3 * 1971 (cf. Fig. 3) : += ( or for values of a on the bra.nch cuts6) c-L‘ +(a’la,) 1- I - cos &(a - no) + sin &(a - cl,) -~ dn‘ sin ~ ( n x,) a’ - a cos &(a’- a,) ~~ sin &(a- a,) a - 01, - . (l6b) The solution of this singular integral equation is indeed an entire function as may be seen by regarding (16a) as an integral representation for which satisfies (12) by means of (16b). Now, substituting’) we conformally map the cut a-plane onto the region from above by the contour D defined as R- of the !-plane, bounded D passes from ca exp (37ci/4) through the points - 1, 0, 1t o 00 exp (- ni/4)and is symmetrical t o the origin (cf. Fig. 4). Equation (16 b) now reads 1 where a and 00 are rational functions of 48 sin &(a - a,) /?and / I r ,respectively, cf. (17b). (19) f we mean t h a t principal values are t o be taken on either side of the branch cuts. ‘j) By 7) 1/1.11> 0 for a real and > 1. LijNEBURG and WESTPFAHL: Diffraction of Plane Waves by a n Infinite Strip Grating 263 I Fig. 4. Integration path in the B-plane The more general case of distances 2b between the edges of two adjacent strips (not equal t o their width 2a) leads t o ihe singular integral equation (7 == kb) which, for the special cases 7 = 1 E (I - 4p p -- (19a) 1, 2, . . .) can be written in the form sin c(z - __ no) - 1 = z - xo . Our method, however, works only for 1 = 1, i. e. a = b. 3. Solution of the singular integral equation Let us define t,wo sectionally holomorphic functions (20b) and denote by C“(Blno) and S*(Pla,) their boundary values, if /Iapproaches D from the semi-infinite regionss) R’ (as well as their holomorphic branchesin R*). From (IS) we obtain the two equivalent functional equations 1 x - a, - 1 (Y -- xo (B E D ) ( 2 1 % b) (upper and lower signs, respectively) with L(ocJa,)= (a- a,) cot &(a- no). 8) R+ is the complement of R- in the B-plane. (22) 264 Annalen der Physik * 7. Folge * Band 27, Heft 3 * 1971 Using the fact that L(a[ao)as well as D is symmetrical with respect to /3 we conclude for the increment of arg ( L )along the path D arg -wajao)lD = 0, (23) which is the well known condition for the unicrueness of the solution of the RIEMANN-HILBERT problems (21) (cf. [I91 p. 85 [20] p. 1 2 7 ) . The essential step towards their solution is the factorization -&la,) L-(Pla,) = L+(Pi.o) with L*(P1a0)being holomorphic and without zeros in R* or, more explicitly, L+(Bla,) = L.-(-Pla,). Infinite product expansions of L*(PIao)are given in App. A. Using we rewrite equation ( 2 l a ) in the form and decompose the term on the r.h.s. of (27) into: Defining the holomorphic functions - 11- a! 2i ~ 1 + "0 L-(Bol"o) 1 ~ B- a in R', we arrive a t the homogeneous RIEMANN-HILBERT problem With (cf. Appendix B) and WESTPBAHL: Diffraction of Plane iVaves by an Infinite Strip Grating 1,LXE:BI*RG 'k - (where the r.h.s. applies t o both L+ and L-) and L G bounded for whereas C' (B[ao)and X-(plao) are bounded everywhere, we obtain 265 -+ 00, (32) The sectionally holomorphie function m p a01 = Q=(B/no) (p E R ' 1 (31%) vanishes for )3 --f 00 and is holomorphic in the p-plane punctured a t /3 = f1. A t these points s2(plao)cannot have a pole nor a branch point due to (32) and the holomorphic character of f2-'(/3iao)in conjunction with the continuity condition (30). There remains only the possibility of a reinoveablc singularity. By LrouVILLE'S theorem we conclude SZ-L(p:KO) = a-(p'a,)= 0 , (34) and find from (29) and llerein C-(Plao) and SJ (B/ao)e m be derived either in a similar manner from (21b), or simply by using the oddness of C(p/ao)and X(p'ao)with respect t o p and property (2513) of L i ( P a , ) . Sow, the difference of the boundary values of C*(p do)or S ' ( ~ ' K according ~) t o (20) yields (30) which by inserting (35) can be written as: (37) From (26a) we obtain, replacing p' by -/3' and adding : (38) Returning to the a-plane by means of (17) and writing L+(P,a,)= LIJI(aln,) (39) we find LIJI(Cxja,) (xE sheet I, 11). (39') = esp C'fC Lr(ala0)and LI1(a'a0) are analytic functions of ix on the two-sheeted RIEMANX surface of 1/m2 - 1. L' is holomorphic and without zeros on sheet I (which we 266 Annalen der Physik * 7. Folge * Band 27, Heft 3 * 1971 identify with the originally introduced cut a-plane) and is meromorphic on sheet I1 while LI1 has analogous properties with sheet I and sheet I1 interchanged. These functions are related by LI(nja,)LII(ajol,)= ( a - a,) cot &(a- a,). (Infinite product expansions of are given in App. A.) It's not hard to conclude from (39') that the quantities + 1 L1(ala,) LI1(a/ao) are holomorphic (a2- 1)-1/2 [L'(ala,) - Lrl(a/ao)] within C and Thus, the solution (37) in the a-pIane reads and (40) (41a) (41b) or, by virtue of (40), This solution, indeed, is an entire function: Eq. (42b) tells us that y(aIao)is holomorphic in the cut &-plane, but according to (42a) and (41a) there are no outs a t all. The behaviour (13b) a t infinity is demonstrated by means of the ~ / ~la1 ) -+ 00, proved in App. B. relatiorl L1(ala,)= O ( O L for For completeness we finally indicate the verification of the dual integral equations ( l l a , b). Denoting the 1.h.s. of ( l l a ) by J [ y ] ,viz. -m --m we insert y(alao)from (42b) and obtain by shifting the path of integration from the real axis to C and C, respectively, by virtue of (40) : CtC The contour integral vanishes since by means of ( 4 1b) the integrand is holomorphic within the loops C and C. I n order to verify ( l l b ) for 2 > a and x < - a , we insert y(ocIa,) from (42a) and shift the path of integration to C and 6, respectively: The contour integrals vanish by virtue of (41a). LUNEBURG and WESTPFAHL: Diffraction of Plane Wavee, by an Infinite Strip Grating 267 4. High frequency asymptotics The purpose of this section is to obtain a n asymptotic representation of our solution (42) useful for the high frequency case E 9 1. Writing Eq. (40) in the form (43) where the upper signs refer to n E C, the lower ones to (39‘) In L1(clan)= In [-;(a’ - %,,)I 01 E 6 , we obtain from dx‘ - C (44) with By means of the indentities we obtain (Y3 (7 This result has been achieved by deforming the contours C, C of the integrals containing ~n [(1/v2i)(k1/1Tir p‘1- & F 1/ K 7 vi+pLyo)] 11_- into the contours -C, -6 on the other sheet of tlhe RIEMANN surface of - a2 and redefining the radicals, i. e. replacing V1-y n by - 1’1 a. With the aid of (47) we find C-t c 268 Annalen der Physik * 7. Folge * Band 27, Heft 3 * 1971 The last form has been arrived a t by residue calculation since there is no contribution from the arcs closing the path C C a t infinity. With (48) equation (44) reads + ___ _ _ (1/1+ a 1 1- a, + 1/1_-_ a 1/1+ a,) -~ LI(a]a,)= (I@) (49) ( a € sheet I). We now expand the logarithm in the integrand of (45) into a n infinite series converging on C: x exp [- with 7 = 2&(2n nl(alao)- nl(-a/-ao)] + 1) and We decompose the last integrand according to ewa __ -..--. v>; eirp' _ _ _ _ _am' ___ 1 _ - 2ni a' - or C 1 2ni $1- _ eZqa' _ _ _ _dor'_ _ a' - a (53) C 9 where and f mean that the loop C in the &'-plane is distorted by the pole a' = a approaching C from the interior and exterior, respectively. The first term on the r.h.s. of (53) is holomorphic within C and the second term can be expressed as a FRESNEL integral Inserting (53) and (54) in (52) we obtain (55) 1 B 1) The asymptotic expansion') ( q la - 1 then yields (this term by term integrated series being again asymptotic) 9) Representations by means,of asymptotic expansions are denoted by the symbol &. LUNEBURG and WESTPFAHL: Diffraction of Plane Wevee by a n Infinite Strip Grating 269 or, by residue calculation, finally where the phase functions a t ( % are ) given by a (x) ezn(ex+l/l-z/!) 1 - cc In elin%% c -;7pi -___-- ((%a) ?L=O(,n with ((iOb) x = (&/7C)(1 - ao) or (cf. [21], p. 27) ((iOc) From (GO) it follows a l ( x )= a l ( x mod 1) (tila) at(. -k 112) = - a&) I m a J x ) = (-1)1Re a,(l - x ) . (61b) (61c) Therefore, in order to calculate a,(%), i t is sufficient t o know R e a l ( % )for 0 5 x 5 112. Tables for the cases 1 = 0 and 1 = 1 are given in App. C. It should be noted that the derivative dao/dx does not exist for the discrete values x = n/2 ( n integer, 20). These points characterize the change of the n-th order right hand modes from the evanescent to the propagating form (cf. See. 5). More explicitly we obtain for e l l - 011 ~- nI(aIa,)L+&- I + %Ix [a,(.) &-I/% $ 1 the following approximation +- i(l:+ I - % L)al(z)s-3/2+ -- 0(&-"2)] (62) and finally, taking into account (49) lo), ~-~_ _ _ ____ LI(a~a,)-=(tiV2i)(1/1+a1/1-Mo+~~1-011/l+OIo) >< exp 1") to { - 1 2 -- [bo(oljao) &-1/2 + b,(alao)&-3/q + I (63) 0(&-j/2) Expanding the exponential we obtain the correct asymptotic series for L1(r]a,)u p O(E-2). 270 Annalen der Physik * 7. Folge * Band 27, Heft 3 * 1971 with1I) (64b) If a. can be written as a,"; = nn/2&( n = 0, 1, 2, ...) (cf. [3]) the asymptotic expression simplifies considerably. Indeed, due to (45) and (60) we have12) nI(a1at) = ( - 1 ) T k a,(%,) = (-1)n nI(n/O) (65a) a,. (65 b) For the corresponding discrete angles of incidence the current densities on all strips become identical (amplitude a n d phase, cf. (6)). We are now in a position t o write down the (asymptotic) high frequency development of our solution (42) in terms of elementary functions and the phase factors a l ( x )for ~ lf l a1 9 1 and 811 f a0l 9 1: + +(-a/-.,) Y(ala0)= i ( + O ) with +(also) + 2 ?jn(ala0) &-n/2 n = O , I , ... where the first five Gn are explicitly given by11) .~ ?j0(a/ao) = - v1 ___ e-i&(%-%o) + a1/1 - aoP- 01 - LYo e-i&(a-a,) ?j1(aIao)= 11) 12) - IT+; IF-;,a(7 Notation: a: = a L ( x * ) ,xi = (&in) (1'f: -ao). Notation: x, = (&in) (1 - a:), az = -aL(&/n). 271 LUNEBURG and ~VESTPFAHL:Diffraction of Plane Waves by an Infinite Strip Grating @,a)l,.( 1 ,-ze(.-r.) ~~ ~ p - - 1 16 V T ; - - - ~ a l / l - a. (67 e) 1 Note that only yo(aiao)is peaked a t a = ao. It is instructive t o compare this result with the high frequency asymptotics of the corresponding solution for a single strip (width Za, cf. [ZL']) : - pe-n/r) ~- -~ dl s\/'; G 11 + a0 ,-i&+n,)E-3/2 + O(&-6/0-). (68) (1- a ) (1- ao) There is agreement of the dominant term of both solutions, this term referring t o the two (noninteracting) half planes complementary to the strip. The higher order terms for the single strip describe the interaction of these two half planes, whereas for the grating there is a predominant interaction with the other strips. 5. Node representation and comparison with Kirchhoff's theory 5.1. Exact modes The only singularities of the integrand of the field representatiorl (10) are the poles of the cot-function. We thus obtain by residue calculation the well expansion known RAYLEIGH 272 dnnalen der Physik * 7. Folge * Band 27, Heft 3 * 1971 Fig. 5 Fig.5--12. Amplitudes ITn\of propagating RAYLEICH modes versus &in= 2 a / l for 8 different angles of incidence t?o (2a is the common width of strips and gaps, I the wavelength) Fig. 6 - LUNEBURQ and ~VESTPFAHL: Diffraction of Plane ITaves bp an Infinite Strip GratJing 273 25 2 2,s 3 3,5 E,&, n h Fig. 7 f lT,i Fig. 8 with (an,,= 1 for 72 = 0 and is zero otherwise) 4 274 Annalen der Physik * 7. Folge * Band 27, Heft 3 * 1971 Fig. 9 t IT,I 0.6 0.5 - 0.4 - 0,3- These representations tell us that through any point ( T , 6) travels a finite number of propagating modes (undamped plane waves) corresponding to Ia,&l5 1, i.e. t o n- 5 n 5 n+ with l 3 ) f [ ( 2 & / 4 (1 TaCx0)l. the modes (la,J > 1) is damped and 721 = (72) The rest of can be neglected for kz 1. I n the high frequency domain E & 1 (where, by virtue of (72), these sums become practically useless) the approximation of Sec. 5.3 applies. 13) [a] means the integer part of a L~NEBUR and C WESTPFAHL: Diffraction of Plane Waves by an Infinite Strip Grating 275 0 ' 0.5 15 1 2 2.5 3 3;5 €_la, 5-A 4 Fig. 11 / 4', =ZOO -7---'-' 2 2.5 3 4 3.j L=&r h Fig. 1 2 The solution for the complementary excitation (viz. the case in which the magnetic field G is parallel t o the edges of the strips) is obtained in a similar boundary value problem) or by applying the manner (solution of a NEUMANN rigorous version of BABINET'S principle to the above solution. The result is m G(r, 8)= 11= 2 +n(Lyo) e ? k r c o s ( e - 8 , ) -_ - 1 - eakreos(rP-rP,) with (z > 0) (73 a) m +2 n= -m n (a O ezlircos(iY+rP,,) ( z < 0) = ,a , - %(a,) = - ( E l l n R n ( 4 . (73b) 0 this agrees with the results of BALDWIN and HEINS[l]and WEINSTEIN [2, 31 (of. App. A). The amplitudes IY',l, lT.tll and IT*.zl of the first five propagating modes are shown in Fig. 5-12 for 8 different angles of incidence as functions of E ( E 5 4n). TnbO) For a. 18* = 276 Annalen der Physik * 7. Folge * Band 27, Heft 3 * 1971 By virtue of (6Yb) the propagating forms of themodesT,,,begin a t E = nlnl/ 2 ( 1 T a0). With increasing E the amplitudes lTol, ITzn]and IT2n+l/approach values l / 2 , 0 and l/nl2n+ll, respecbively asymptotically the KIRCHHOFP (cf. Sec. 5.2.). It should be noted that for ( n / s ,a,J E T , where T is the triangle { ( l , O ) , (4/3, l/3), (2,O)) in the ( n / s ,&,)-plane, the amplitudes of TI and Y'-l are exactly equal, cf. Fig. 6. This statement, which may be verified from (71b) and App. A, corresponds physically to the situation where for ( n / s ,ao) E T the modes T , and T-,are (apart from T o )the only propagating mode forms. 5.3. KIRCHHOFF modes and high frequency modes For the sake of simplicity we confine ourselves t o normal incidence (ao = 0) and consider KIRCHHOFF'S formula ( Lis the union of the grating slits)14) wK(z,Z ) = i O ( Z )v,,(x, Z ) - 4 J (u0 8, L + uo2)HI:) (k l ( z - x')~+ z 2 )dx' or We have to compare these coefficients with the asymptotics of Sec. 4 applied t o the rigorous coefficients (71) : As expected, KIRCHHOFF'S formula yields only the dominant term of the correct high frequency asymptotics. 5.3. Form factor I n this section we point out that our result may also be interpreted in terms of a finite N-strip grating in the E N % 1approximation. To begin with we formula for a finite gratJing (consisting of N strips write down KIRCHHOFF'S 11) O(2) = (1/2) (1- 2/12]). LUNEBURG and WEESTPFAHL: Diffraction of Plane JVaves by a n Infinite Strip Grating 277 placed symmetrically to the origin, N odd) vF(x, = eik(wA$izqz) (78%) which by the method of steepest descent reduces to (x t i z = r 0 < 6 < n): eiA, r $ as, I< iki cos(4--it0) vlv+ e 8 + i(sin 6 + sin 8,) sin [c(cos ;c 1ioy79, - Y - - On the other hcnd our rigorous representation (4") for an infinite grating may by means of (in the sence of distributional convergence) - be written in the form -__ v(z, z ) eik(w-Lv1-4?:z) The comparison with (78a) suggests t o consider the expression Vlv(z, z ) = eih(lo7t dl-niz) as a n approximation for a finite grating (at least for N $ 1)taking into account the interaction of its individual elements. Thus, we obtain by the method of steepest descent ( r 9 a N , 0 < 8 < n): vAT(r,8)--f ezkrcos(-it--4,) Hence, the form factors multiplying the common grating pattern are to be defined as follows: KIRCHHOPF'S theory : F"(BJ6,)= i(sin 6 + sin 8,) sin cos -- 8 6 )] 8, [ ~ ( C O S - cos --~79 -- cos 278 Annalen der Physik * 7. Folge * Band 27, Heft 3 * 1971 our theory: P(616,)= ,(cos 6lcos 79,). (83b) I n KIRCHHOFF’S theory the form factor is given by the FRAUNHOFER diffraction pattern of a single element (strip). However, in view of the high frequency Fig. 13-16. Grating form factor IF1 for 4 different angles of incidence 8, and 4 different values of E versus the diffraction angle 8 Fig. 13 Fig. 1 4 L~NEBIJRG and WESTPFAHL: Diffraction of Plane Waves by an Infinite Strip Grating 279 Fig. 15 relations (66), (67) and (tjS), in our theory besides the dominant term 8 po($j]$jo) = 2 i sin ( + 8, sin [e(cos 8 - cos So)] 1 - ~~ cos 6 - cos8, 2 6 - 6, cos [E(COS 6 -- cos .a,)] sin ( 7 cos ) 6 - cos 8, (8312) + of the form factor, wich is also given by the diffraction pattern of a single strip (and agrees with KIRCHHOFF’s expression in the vicinity of the direction of incide nee), there are’ higher order terms etc. which take into account the mutual interaction of the other strips. (These remarks indicate that our approximation yields the dominant term of the EN $ 1 approximation even for N = 1.) The interaction is described by the phase factors a , which involve a summation over the grating elements, cf. (60 a). 280 Annalen der Physik * 7. Folge * Band 27. Heft 3 * 1971 Fig. 17. Comparison of form factors IF1 and jFKl versus 6 for F = 47c and;? different angles of incidence 8, = 90" (--) and 8, = 30" ( - - -). The arrows indicate the diffraction orders n The absolute value of the form factor P(B16,) is shown in Fig.13-16 for 4 different angles of incidence and for various values of c as function of the dif0 ) ~KIRCHHOFF'S value lFK(6[6,)1 fraction angle 0.Fig. 1 7 compares ~ F ( 8 ~ 8with for Oo = 90" and 30" in the frequency domain E = 4n. The agreement decreases with increasing @-distance from the direction of incidence and mith increasing 6,-deviation from normal incidence. The peaks of the common grating pattern G (viz. the orders of diffraction) are marked by arrows. The most striking fact: in contrast t o KIRCHHOFF'S theory the spectra of order 2% ( n $. 0) do not vanish in our approximation, judged both from the (asymptotic)mode and from the form factor aspect. 6. Transmissivity and reflectivity The transmissivity t and reflectivity Q of the grating are defined by L ~ V E B U Rand G WESTPFAHL: Diffraction of Plane Waves by a n Infinite Strip Grating 281 where the summation extends over all propagating modes. The individual terms of the sunis represent the relative power of the trainsmitted and reflected modes, of order n (i. e. the power leaving a unit area of the grating in thedirection 6,, dividcd by the power arriving a t a unit area due t o the incident wave with the angle of incidence 8,). We now proceed to derive a modified optical theorem appropriate t o grating problems. From (G9a) we obtain for z = 0, 1x1 5 a the identity (85) which me substitute in (8) :15) Since y n is real for n- 5 TL 5 n and purely imaginary otherwise we obtain I m V(+O) = 1 ?)+ 1 2 r /Y("%laO)l2. ~ = n -Jn (87) Taking into account the definition (701 ) of Tn(c~o), we rewrite ( M a ) with the aid of (87) in two equivalent forms (mod fied optical theorem): 1 n+ C 1 - iv(~n/.o)12. 16E2Y, n==%-Y n =I- ~ The reflectivity @(ao) follows from the law of conservation of energy Tho) 3- e b o ) = 1 (89) which is easily verified by means of (84), (87) and (70b). Substituting from (42) we obtainI6) This yields the high frequency asymptotics (with the aid of the develop1 a,) % 1: ments of Sec. 4) for ~ ( & t(aO) with II . 1 2 ~ + 4&(11 -T I m exp (bo~-lI2+ bl&--3'2) + O ( E - ~ ' ~ ) ~ - 010) (91) = a. in (G4). a* denotes the complex conjugate of a. From (Tla) follow simple relations betwcen the total transmitted and reflected power and the amplitude of the dominant mode : 16) 4 4 = Re To(oc,), e b o ) = - Re Ro(oco). (90') Annalen der Physik 09 ~ t T 08 07 - * 7. Folge * Band 27, Heft 3 * 1971 4-900 j11._ 80° !/ 60O Fig. 18. Transmissivity t versus c / n = 2a/A for 8 different angles of incidence 6,.(2aIA = strip widthlwavelength) The low frequency approximation follows from (90) and (A9) by an &-power expansion for E < 1: 2(ao)F y -cos (C1&+ C,&3 + C3&5)+ 0(&8) 2 ( 92) and m ((n)= 2 k= 1 (94) LUNEBURG and WESTPFAHL: Diffraction of Plane Waves by an Infinite Strip Grating 283 I n Fig. 18 we have plotted t versus e l n _< 4 for 8 different angles of incidence. The comparison with the high frequency asymptotics is shown in Fig. 19: There is excellent agreement for (e/n)(1 - ao) 0,8. The also shown low frequency approximation improves a n old result due t o LAMB[23]. 2 t &=goo 5 / 0.6 0.6 0.4 0,2 /// I Fig. 19. Test of the quality of the high and low frequency approximations (91) and (92), respectively, for the transmissivity t (dashed curves) for 3 different angles of incidence @,, Appendix A . Infinite product factorization I n order t o decompose L(alao)= (a - ao)cot &(a- ao)according t o (24a), we use the infinite product expansions 284 Annalrn der Physilr * 7. Folge * Band 27, Heft 3 * 1971 and with In ( A l ) and (A2) the last product is to be extended over the four possible comand - signs. Hence binations of the + 2 + (4.).( x%o X%(% - l/Wx % b o - ( 4 4 ( n - l i 2 ) ) + nni4 x 3 . 0 - nn/4 and or, by making use of (A3), and passing t o the a-plane ~ -t ~ _ _ 1/1 - 2 4 1 q= a O ) / n n ] (where sum and product extend over the two terms with upper and lower sign,, ”) +(a),X ~ ( C U> ) 0 for 01 respectively. > 1;the branch cuts are taken along (1,ioo) and (-1, -ice)> L ~ K E B U and R G WESTPFAHL: Diffraction of Plane IVaues by an Infinite Strip Grating 285 respectively) and ~ ~ _ ~ ~ ~ ~~ _ ~ _ _ I m !,'I - %(I a,)/nn: 5 0. + a,)/;d we find simply For n 2 2 4 1 &,(&,) E _ (AGb) 11- [a&(& 5 n/2(l + ao)there is a furt,her simplificatjion: I&,(al&o)i = For a = a. and _ - ao)/n;d]2. (L47) (A8a) = e-i'%(ao) with and with For = 0 t>hequantity LI acquires the explicit form (AlOa) (AlOb) Finally, using a somewhat more explicit notation suggested by (AG) the equivalence (2.5 b) in the &-planereads as follows = LI (01, - v1- azla,). (All) Appendix B. Houndednoss considerations The investigation of the behaviour of L-(/ljoc,'l in the vicinity of /l = 00,0 and 0 == 1,which are boundary points of the coniour D ,requires by (17) the examination of LI(alix,) in the vicinity of a = & 1and m , respectively. To this end -~ we represent (50) as (0 5 arg l/afz - 15n/a) Obviously we have %I( - 1I a,) = 0. 286 Annalen der Physik * 7. Folge * Band 27, Heft 3 * 1971 h Now, if p(t) E H ( l ) l s )on the closed arc ab, the CAUCHYtype integral h behaves like where Qo(z) is bounded a t z = a for L > 1 / 2 (cf. [19] p. 53ff. or [20] p. 73ff). Since (Bl)is of the form of (B3) with p(z) E H(1)we conclude nI(a/ao) = O(1) (a --f 1). I n order t o examine the behaviour of nl(alao)in the vicinity of stitute a’= l/y‘, LY = l / y and consider the behaviour of c*3, (B5 ) we sub- in the vicinity of y = 0. This integral is of the type h a h where p(z)is HOLDERcontinuous on the closed arc ab. For z + a it behaves like @ ( z ) = - v(a) ---;In ( z - a ) 2na + Q0(z) (B8) with Qo(z) bounded a t z = a (cf. [19, 201 1. c . ) . Since the logarithm in (B6) vanishes for y -+ 0 it follows nI(alao)= O(1) (la/-+ 0 0 ) . H ( 1 ) means the class of HOLDERcontinuous function with the index 1. (B9) LUNEBURQ and WESTPFAHL: Diffraction of Plane Waves by en Infinite Strip Grating 287 Collecting results we obtain taking into account (49) which, in the b-plane, correspond t o I n view of (25b) this behaviour is also valid for L-f-(/?lao). Appendix C. Table of phase functions I n Table 1 and Fig. 20 the real part of the first two phase functions a o ( x ) and a,(%),cf. (GO), is given for 0 5 ~t l / 2 . continuation Table 1 x 0,00 0,01 0,02 0,03 0,04 0,05 0,OG 0,07 0,08 0,09 0,10 0,11 0,12 0,13 0J.4 0.185 Oil6 0,17 0,18 0,19 0.20 -0,6736 -0,4422 -0,3390 -0,2708 -0,2144 -0,1651 -0,1218 -0,0828 -0,0471 -0,0137 0,0177 0,0471 0,0745 0,1003 0,1247 0,1481 0,1707 0,1927 0,2142 0,2350 0,2552 0,2746 0,2932 0,3110 0,3281 0,3446 -0,441 -0,474 -0,496 -0,516 -0,531 -0,543 -0,551 -0,557 -0,562 -0,564 -0,564 -0,562 - 0,558 -0,552 -0,544 -0,534 -0,524 -0,512 -0,500 -0,487 -0,472 -0,457 -0,439 -0,421 - 0,401 -0,379 0,26 0,27 0,28 0,29 0,30 0,3i 0,32 0,33 01,34 0,35 0,36 0,37 0,38 0,39 0,40 0,41 0,42 0,43 0,44 0,45 0,46 0,47 0,48 0,49 0,3608 0,3765 0,3921 0,4075 0,4228 0,4378 0,4526 0,4671 0,4812 0,4949 0,5081 0,5210 0,5336 0,5460 0,5583 0,5707 0,5830 0,5950 0,6068 0,6183 0,6297 0,6411 0,6524 0,6624 0,50 0,6736 -0,356 -0,332 -0,306 -0,280 -0,254 -0,227 -0,199 -0,171 -0,142 -0,112 -0,081 -0,049 -0,016 0,018 0,053 0,088 0,124 0,161 0,199 0,238 0.277 0:317 0.357 0;400 0,441 References [l] BALDWIN, G. L., and A. E. HEINS,Math. Scand. 2 (1954) 103. [2] WEINSTEIN, L. A., Zh. Tekn. Fiz. 25 (1965) 841, 847. [3] WEINSTEIN. L. A., The Theory of Diffraction and the Factorization Method. Boulder: The Golem Press 1969. [4] GNATO OW SKI, fir. S.. Dokl. Akad. Nauk SSSR 22 (1934) 21. [5] KARP,S. N., and W. E. WILLIAMS, Proc. Camb. phil. Soc. 63 (1957) 683. [6] AORANOVICH, 2. S., V. A. MARCHENKO and V. P. SHESTOPALOV, Zh. Tekn. Fiz. 32 (1962) 381. 288 Annalen der Physik * 7. Folge * Band 27, Heft 3 * 1971 [7] MALIN,V. V., Radiotekhn. Electr. 8 (1963) 211. [8] CARLSON, J. F., and A. E. HEINS,Quart. Appl. Math. 4 (1947) 313. [9] HEINS,A. W., and J. F. CARLSON, Quart. Appl. Math. 5 (1947) 82. [lo] HEINS, A. E., Quart. Appl. Math. 8 (1950) 281. [ll] BERZ,I?., Proc. ZEE 98 (1951) 47. [12] WHITEHEAD,E. A., Proc. I E E 98 (1951) 133. [13] HEINS, A. E., J. Math. Mech. 6 (1957) 401, 629. [14] MEISTER,E., Arch. rat. Mech. Anal. 10 (1962) 67, 127. [15] BAKLANOV, E. V., Dokl. Akad. Nauk SSSR 163 (1963) 570. [16] IGARASHI,.A.,J. phys. Soc. Japan 19 (1964) 1213. [17] IGARASHI, A., J. phys. SOC.Japan 26 (1968) 260, 607. [IS] WESTPFAHL,K., Ann. Physik, Leipzig 4 (1959) 283. [19] GAKHOV, F. D., Boundary Value Problems. Oxford: Pergamon Press 1966. [20] MUSCHELISCHWILI, N. I., Singuliire Integralgleichungen. Berlin : Akademie-Verlag 1965. [21] ERDELYI,A., W. MAGNUS,F. OBERHETTINGER and F. G. TRICOMI,Higher Transcendental Functions Vol. I. New York: McGraw-Hilll953. [22] LUNEBURG, E., u. K. WESTPFAEL,Ann. Physik, Leipzig 21 (1968) 12. 1231 LAMB,H., Hydrodynamics. Cambridge: University Press 1932. Die numerischen Rechnungen wurden auf der IBM 7040-Anlage des Rechenzentrums der Wniversitat Freiburg durchgefiihrt. F r e i b u r g / B r ., Institut fur Theoretische Physik der Albert-Ludwigs-Universit at. Bei der Redaktion eingegangen am 5. Februar 1971. Anschr. d. Verf.: Dr. E. LUNEBURG, Deutsche Forschungs- und Versuchsanstalt fur Luftund Raumfahrt (DFVLR), BRD-8031 Oberpfaffenhofen, Post WesslinglObb. Prof. Dr. K. WESTPBAHL, Inst. f. theoret. Physik d. Univ. Freiburg BRD-78 FreiburgIBr., Hermann-Herder-Str. 3

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