advertisement

2094 Salvatore Ganci J. Opt. Soc. Am. A/Vol. 3, No. 12/December 1986 Maggi-Rubinowicz transformation for phase apertures Salvatore Ganci Gruppo Nazionale Didattica della Fisica, ConsiglioNazionale delle Ricerche, Dipartimento di Fisica dell'Universitb, Via Dodecaneso, 33, 16146 Genova, Italy Received August 16, 1985; accepted July 28, 1986 The proof of the Maggi-Rubinowicztransformation is extended to a general phase aperture. General formulas are established for spherical and plane waves. JJ(an {IQexp[ik(rQ+ A)]I1iexp(ikr) INTRODUCTION +r | As is known,", 2 the Maggi-Rubinowicz transformation makes it possible to express the surface integral constituting Kirchhoff's formulation of the problem of diffraction in a line integral extended to the closed line constituting the boundary of the aperture. The derivation of this transformation (in the case of a spherical or plane wave) has been included in textbooks on optics for some time,3 -5 and the problem is considered here only in the special case when one has an aperture in an opaque screen. -1 rQ exp[ik(rQ + A)] exp(ikr)]ldxdy. LrJ (2) The integral extended to the aperture A can be considered in all aspects as the Kirchhoff solution of the problem of diffraction through an aperture A in an opaque screen, and thus Kirchhoff's boundary conditions are implicit: As the Kirchhoff integral can easily be applied to problems referring to phase apertures, it seems obvious that such problems On IA exp(ikrQ), - can also be dealt with by using the Maggi- UA(x, Y, 0) = Rubinowicz transformation. the plane x, y, let there be a constant phase jump. We can then describe the expression of the electric field U of the wave incident in the form U(x, y, 0) = Aexp(ikrQ), Qe A reQ -exp Q an Similarly, the integral extended to aperture B may be considered in all aspects as the solution of the problem of diffraction through an aperture B (complementary to A) in an opaque screen, and in this case also Kirchhoff's boundary conditions are implicit: - exp[ik(rQ + A)], (1) [ik (rQ + A)], B j 'P UB(X, Y, 0) = IJ {_,an Qe A O. 47r - U(xy,0) = | n Ir I{A n [f - I exp(ikrQ) rQ a an r 1 exp(ikr)I}dxdy exp(ikrQ)] - I exp(ikr) I- exp(ikr)I dxdy On r J 0740-3232/86/122094-07$02.00 QA OUB( o).o (x, y, 0) -exp(ikr) Qe B Q The diffracted wave field at the point P(x, y, z) is U(x, y, ) = QeB. UA(xy,0)=0, With reference to Fig. 1, a spherical wave (A/rQ)exp(ikrQ) is incident upon the plane x, y. Between surfaces A and B of A Qe B 0, MAGGI-RUBINOWICZ TRANSFORMATION FOR PHASE APERTURES Q rQ xlB 0) =0, QEA. To each of the previous cases one can apply the MaggiRubinowicz transformation, which, with reference to the geometry and notation of Fig. 1, gives 4irJA {[ OU (x, Y. 0)] O-an -UA(X, r exp(ikr) Y, 0)W © 1986 Optical Society of America 4 exp(ikr)]}dxdY Vol. 3, No. 12/December 1986/J. Opt. Soc. Am. A Salvatore Ganci 2095 S X J z ItI I / / II I I \S III II I I II II N Fig. 1. Diffraction geometry of the Maggi-Rubinowicz transformation applied to a phase aperture. Case of the spherical wave. Between surfaces A and B let there be a constant phase jump. r is the closed-line boundary of A and B; dl is the length of an infinitesimal element of r. The unit vectors nA and nB are orthogonal to rB and to dl. - IR exp(ikR) + UB1, A P J UB1, 4 IB 7rn {[an n and cos(nA, s) = -cos(nB, s). Let P J (3) f(n, s) = 4 I exp(ikr) (x, Y, 0) cos(nA,s) ( R Jr exp(ikR) + 1 - UB(X,Y "'" - IR ) a'n a [rr exp(ikr)I|dxdy P J exp[ik(R + A)]+ UB2, UB2, U(x,y,z) = 4ir [1- exp(ikA)] exp ik(rB + s)f(n, s)dl, Pe J r rBS A-expik(R + A) + A [1 - exp(ikA)] R r s exp 4r PEJ (4) A sin(rB, dl), 1 + cos(rB, s) JrI expik(rB+ s)f(n, s)dl, Pe J Pe J (7) where U A | 1 exp[ik(rB + s)] A sin(rB, dl)dl, 47r r rBs 1 + cos(rB, s) (5) UB2= A X sin(rB, dl)dl, The plane wave can be treated in the same way,6 and we obtain similar relations. More interesting (for extension to Fraunhofer diffraction) is the special case of a plane monochromatic 1 exp[ik('r + s + )]cos(nB, s) 4T r rBs where R is the length of the geometrical ray from S to P. wave normally incident upon a phase aperture. With reference to the geometry and notation of Fig. 2, the expressions of the bound- 1 + cos(rB, S) (6) ary waves UB1 and UB2 are 2096 J. Opt. Soc. Am. A/Vol. 3, No. 12/December 1986 Salvatore Ganci X J I'., I I I/ P I '% I I j II 1 % % Fig. 2. Diffraction geometry of the Maggi-Rubinowicz transformation applied to a phase aperture. Case of a plane monochromatic wave of amplitude A traveling in the direction of the z axis (normal incidence). The unit vectors nA and nB are orthogonal to dl and to the z axis. UB1 = A - I 4w ir EXAMPLES OF APPLICATIONS IF cos(nA,s)1 exp(iks) _ dl, - sI1+ (8) cos(s,z)J The Single Phase Slit UB2 =A [-exp ik( + ) cos(nB,s) sA 1 + cos(s z) 47r dl. (9) First, we consider the classical problem of a half-plane in the special case of a normally incident plane monochromatic wave. With reference to Fig. 3, let a, Let f(n; s) = cos(nA, s)/1 + cos(s, z), which again is 3,and y be the direction cosines of s and 1, 0, and 0 be the direction cosines of n. The Maggi-Rubinowicz transformation gives UB = 1[ ) UB=II exp (iks) A exp(ikz) + A [1 - exp(ikA)] 4w j[ U(X' ,yZ) exp(iks)f(n, s)] dl, P J fF1exp(iks) eS A7r (10) =A A exp ik(z + A) + - 4 r Jr, [1 - exp(ikA)] 4w Jr[-exp(iks)f(n, s)]dl, PC J A simple inspection of Eqs. (7) and (10) shows that one does not have a diffracted field whenever the phase jump is 2mwr (m = 0, 1, 2, .. .). On the other hand, we have the maximum A 1 fs (1 + Cos UB- 4r l -exp(iks) 2 22 JrLXs fX +f, A 4w from the same transformation applied to the corresponding aperture in an opaque screen because of the factor [1 - exp(ikA)]. (11) )1 dl. 32]- frequencies), Eq. (11) gives phase jump is (2m + 1)7r (m = 0, 1, 2, ... ). The Maggi- Rubinowicz transformation applied to a phase aperture thus 2(1- a2 + If we consider the diffracted wave field at P(x, 0, z) in the approximation so- 7r/2, and let f = IX and f = /X (spatial contrast of the fringes in the observation plane whenever the differs 1dl a a +'Y 4wr .rS dl fx f+7/2[ exp (ik So (fx2 + fy 2 ) i- 7 /2l cos / 1 + os l do. cos J (12) The method of stationary phase7 is applied to Eq. (11), and we find for the boundary wave at P the remarkable result Vol. 3, No. 12/December 1986/J. Opt. Soc. Am. A Salvatore Ganci Y So z P Fig. 3. Diffraction geometry of the Maggi-Rubinowicztransformation applied to the half-plane problem. A X B I~'I2 . - A I _ _ _ So _ _ _ 1 -- _ - S -o. - so2 --- -I/2 B Fig. 4. Diffraction geometry of the Maggi-Rubinowicz transformation applied to a single phase slit. -I 2097 2098 J. Opt. Soc. Am. A/Vol. 3, No. 12/December 1986 Salvatore Ganci A y %s S 'A I x t I Fig. 5. A 1 UB 2- S A polygonal phase aperture with a constant phase jump between A and B. exp[i(kso + r/4)]. (13) Obviously, if the direction cosines of n are -1, 0, and 0, Eq. (13) changes sign. With reference to Fig. 4, the single phase slit has a transmission function Polygonal Phase Aperture In the special case of the Fraunhofer approximation, the Maggi-Rubinowicz transformation is a powerful method for calculation of the diffracted wave field through a polygonal aperture in an opaque screen.8 The boundary wave from the ith side is simply the unidimensional Fourier transform of the x projection of this side [Eq. (4) of Ref. 8] or the unidimensional Fourier transform of the side if this side is t(x, ) = {exp(ikA), orthogonal to the x axis [Eq. (6) of Ref. 8]. In the special case of a polygonal phase aperture (Fig. 5), [x: > 1/2 Equation (10) and relation (13) directly applied to this case give UB = UBI + UB2 = A [1 2w the factor [1 - exp(ikA)] is taken into account in the use of Eqs. (4) and (6) of Ref. 8. An interesting application is given in the following subsec- tion. exp(ikA)exp(ir/4) Young's Double-Phase-Aperture Interference x [exp(iksol)- exp(ikO2)]. In the Fraunhofer approximation this is SO Sol S02 With reference to Fig. 6, a double-square phase aperture of width is represented. Between A and B and A' and B let there be a constant phase jumps kAl and kA2. Equation exp(iks0 2 ) = exp i(so, + (10), in which the line-integral calculation is performed, yields a set of Fourier-transform relations as observed in the previous example. Let f and f, be the spatial frequencies, defined by Xlxfx). Then f = cos O/X xO/Xz, UB = A ~Xs 0 [1 - exp(ikA)] exp [ik (0 + 2 Sin(wxf 1JfX fx = Cos 0- tx,/Xz, X ) f + 4 y = coso/X (14) where xOis the abscissa of the observation point P. The diffracted wave field at point P is the superposition of Yo/Xz. The boundary wave from each side of the phase aperture in the direction r identified by giving it the spatial frequencies fA,and fy following Eqs. (4) and (6) of Ref. 8. Let K = - 2X exp(ikz)exp [ik xo2 +Y02 UB and the geometrical wave A exp(ikz), or A exp[ik(z + A)]. This conclusion is fully consistent with the results obtained by using the Kirchhoff or the Rayleigh-Sommerfeld formulation of the problem. UB1 a = - K _______ /x2 + exp(-iwf,1)exp(iwf.,d)1 sin(wf1), 7f., wy Vol. 3, No. 12/December 1986/J. Opt. Soc. Am. A Salvatore Ganci UB2a = K 2 2099 U(P) = UG + [1 - exp(ikA)] UBA + [1 - exp(ikA2 )] UBA', exp(iwf 1)exp(iwfd)lsin(fyl) + / 2fX2 7r~~fy 1 + fy 2 (15) where UB3.= K 2 f exp(iwfxl)exp(irfxd)l f , wfXl f/ + f2 a Inwl) +f2 exp(-iwf.l)exp(iwfd)l UB4.= -K f2 Pc L 'A exp(ikz), UG = A exp[ik(z + Al)], P E LA A exp[ik(z + A 2 )], P E LA' and srf UB1 = -K 2 +/2 exp(-irfxl)exp(-irfxd)l 7rfxl~w/~ fX2 + f,2 UB 'K2 UB2a = _____ K2+ f 2 UBA=K exp(-i7rfd)2 Sin(lf) sin(f) U~~exn(-Lwfd'fIlr~f wf l exp(iwx/1)exp(-iwfd)l sin(wfyl) UBA'= K w(i fd)12 sin(rf) sin(1/fy) Simple inspection of Eq. (15) shows that one does not have a diffracted field whenever exp(ikA,) = exp(ikA 2 ) = 1. If exp (ikA,) = exp(ikA2 ) = exp(ikA), the boundary-wave con- UB3.'= K 2 /2+ f 22exp(iwfx1)exp(-iwrfd)1 wrfl ) tribution to the diffracted wave field is 2 UB4 a.=-K /X2+ exp(-irfx1)exp(-irfxd)l nfw/fy 4Y2 UB = K[1 - exp(ikA)]l 2 lfx wl/X s fY cos(7rfxd). (16) wl/y For each aperture, the line integral in Eq. (10) is the sum Aside from the amplitude and phase factors, this expression of the boundary wave is the classical result of the Fourier- UBA = UB1a + UB2a + UB3a + UB4a, optics treatment of Young's double-slit experiment. 9 A simple inspection of Eq. (16) shows a maximum of fringing effects in the diffracted wave field when exp(ikA) = -1. UBA' = UBla. + UB2a + UB3a'+ UB4a'- The diffracted wave field at point P is y I I X i L la A1 2a 3a I I I I - L I II ", ZI- LA Fig. 6. Diffraction geometry for Young's double-phase-aperture constant phase jump beween A' and B. interference. kAl is the constant phase jump between A and B; kA2 is the 2100 J. Opt. Soc. Am. A/Vol. 3, No. 12/December 1986 Salvatore Ganci REFERENCES AND NOTES 6. The corresponding explicit expression of the boundary wave can 1. G. A. Maggi, "Sulla propagazione libera e perturbata delle onde luminose in un mezzo isotropo," Ann. Matematica 16, 21-48 (1888). 2. A. Rubinowicz, "Zur Kirchhoffschen Beugungstheorie," Ann. Phys. 73, 339-364 (1924). 7. M. Born and E. Wolf,Principles of Optics, 6th ed. (Pergamon, 3. A.Rubinowicz,Die Beugunswelle in der Kirchhoffschen Theorie der Beugung, 2nd ed. (Springer-Verlag, Berlin, 1966). 4. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), pp. 449-453. 5. A. Sommerfeld, Optics (Academic, New York, 1954), pp. 311318. be found in Ref. 3, Chap. III. Oxford, 1980), pp. 752-753. 8. S. Ganci, "Simple derivation of formulas for Fraunhofer diffrac- tion at polygonal apertures from Maggi-Rubinowicztransformation," J. Opt. Soc. Am. A 1, 559-561 (1984). 9. One of the referees suggested this example and kindly drew my attention to the fact that this treatment may have application in problems involvingsegmented optics or phased telescope arrays.