1994Stratton-Chu vectorial diffraction of electromagnetic fields

Vol. 11, No. 2/February 1994/J. Opt. Soc. Am. A 623 Stratton-Chu vectorial diffraction of electromagnetic fields by apertures with application to small-Fresnel-number systems Wenyue Hsu Department of Physics, Harvard University, Cambridge, Massachusetts 02138 Richard Barakat Aiken Computation Laboratory, Harvard University, Cambridge, Massachusetts 02138, and Electro-Optical Research Center, Tufts University, Medford, Massachusetts 02155 Recieved January 21, 1993; revision received August 16, 1993; accepted August 17, 1993 The Stratton-Chu theory of electromagnetic (EM) scattering is used to develop a Kirchhoff formalism of the diffraction of EM waves by an aperture. The theory is applied to the study of the diffraction of a polarized EM wave by small-Fresnel-number systems. It is demonstrated through sample numerical calculations that the vectorial aspects of the EM waves are important, especially for fields in the microwave regime. The focal shift is then calculated with the vectorial aspects taken into account. INTRODUCTION Scalar diffraction theory, usually associated with Fresnel and Kirchhoff,"' 5 has been used successfully in countless papers on optical imagery, laser resonator theory, optical signal processing, etc. The theory is based on several as- sumptions, of which the following are the most important: (A) The numerical aperture (NA) is reasonably small, or equivalently, the F-number is reasonably large. (B) The incident radiation is unpolarized. (C) The Fresnel number (N) is very large, where N a 2/Af, (1) A is the wavelength of the radiation, a is the radius of the aperture, and f is the focal length of the sys- tem. By very large we mean that N > 10. Relaxation of any of these three conditions invalidates the Fresnel-Kirchhoff anaylsis. Diffraction theory of high-NA systems operating in un- polarized light [i.e., condition (A) is violated, but condi- tions (B) and (C) are retained] was first investigated by Strehl'6 (see also Ref. 17-19 for additional research along these lines). Barakat2 0 is still a useful reference to many of these older papers. To the best of our knowledge no investigation has yet been carried to study the effect of a small Fresnel number (i.e., N < 10) in the large-NA regime but with light still unpolarized. The important situation in which only condition (C) is violated (i.e., N < 10) has been solved for all practical pur- poses by Li and Wolf 2126; see also Refs. 27-30. Here the effect of a small Fresnel number is to shift the maximum intensity along the paraxial axis from its values at very large Fresnel numbers. On relaxing both conditions (A) and (B), so that we have a polarized input and a large NA, we enter the realm in which the vectorial aspects of the incident wave make themselves felt. Note that we have to relax both condi- tions simultaneously, because relaxing only condition (B) while keeping condition (A) yields results very similar to those obtained by keeping both conditions (i.e., for polar- ized input waves to display vectorial properties in the image-receiving plane, large NAs are required). Such a full-vectorial approach was initiated by Ignatowsky3 1 32 over 70 years ago in two seminal papers, one dealing with aplanatic optical systems and the other dealing with a paraboloidal mirror. Unlike the scalar Kirchhoff dif- fraction integrals, the Ignatowsky-Kirchhoff diffraction integrals contain a factor (dependent on the angular coor- dinate) that varies with the type of optical system under consideration; the form of this factor must be derived from geometric-optics considerations. Ignatowsky did not attempt an ambitious numerical program (small won- der, because the research was carried out in 1919) but gave Bessel function expansions of the necessary inte- grals and discussed the behavior of the Poynting vector in the image field. The second scientist to attack this problem was Lune- berg3 (see also Klein and Kay4); he used a vector version of the second Rayleigh integral.3 3 Unfortunately, he con- fined his analysis to general considerations and did not really reduce his analysis to concrete terms. A substan- tial advance was carried out by Wolf,34 who rederived the Ignatowsky vector-diffraction integrals (for the aplanatic situation) by an integral representation of the image fields. Richards and Wolf"5 then made an extensive nu- merical study in this case by numerical evaluation of the various integrals, basically by the use of the Bessel func- tion expansion of Ignatowsky (see also Boivin and Wolf 36 and Carswell.37 ) Barakat3H has reexamined the paraboloi- dal mirror situation and a slight generalization of it in which the mirror has a central obscuration; that paper contains an extensive bibliography of research on the 0740-3232/94/020623-07$06.00 ©0 1994 Optical Society of America WZ Hsu and R. Barakat 624 J. Opt. Soc. Am. A/Vol. 11, No. 2/February 1994 vector problem (see also Ref. 20). Sheppard and his col- laborators have studied aspects of the vector case for use in confocal microscopy.3 9 This brings us to the last situation, in which all three conditions mentioned above are invalid. Such a situation arises in certain areas of microwave diffraction. The pur- pose of the present paper is twofold: 1. Development of another version of the vector Kirch- hoff diffraction theory by the use of the Stratton- Chu theory. 2. Application of the new diffraction theory to deter- mine the focal shifts induced by small Fresnel numbers. Representative numerical calculations have been carried out only in the microwave region because of the restric- tions that we choose small Fresnel numbers while keeping the angular aperture large. This requires the focal length to be comparable to the wavelength, which is the case in the microwave regime. On the other hand, the theory may also find its application in micro-optics. We wish to point out that the Kirchhoff theories, both scalar and vector, still require additional analysis, but not in the numerical sense because almost any situation that we can envisage can now be evaluated numerically.40 In- stead the problem is one of interpretation; one of the few papers that addresses this issue is by Shafer.4 Future papers in this series will be devoted to the imag- ing of extended objects, to the influence of wave-front aber- ration (both deterministic and turbulence induced), and to issues of confocal microscopy. DIFFRACTED VECTOR FIELDS IN THE FOCAL REGION Representation of vectorial-diffracted fields can be ob- tained by vectorial analogy of the Green's identity, which is used in the Helmholtz representation of the diffraction of scalar fields. In other words, one introduces a vector anal- ogy of the Green's function in free space, and the product of the fields and the vectorial version of the Green's function is constructed so that it leads to a term of the fields with delta support, such as E(r')8(3 )(r - r') or H(r')8(3)(r - r'), from which the fields in the image space can be expressed as surface integrations of fields. Two commonly used versions are the Stratton-Chu approach42 44 and the Franz approach.44 '45 The two versions can be shown to be equiva- lent to each other, but the former is more convenient for computational purposes because only first-order differen- tial operators are present. Because the electromagnetic fields satisfy the wave equation for a monochromatic field with wavelength A, we have the Green's theorem for the vector fields22 24 in the image space: E(r) = - [ik(no X H)G(r,r') + (no x E) x V'G(r,r') + (no E)V'G(r,r')]dS', (2a) H(r) = -ffs [ik(n X E)G(r,r') + (no X H) x V'G(r,r') Here exp(iklr - r) G~r~') = 4vIr - r'l (3) is the Green's function for the Helmholtz equation, k is the wave number, and unit vector no is the outer normal of the closed surface S, which encloses point r. The Kirchhoff approximation corresponds to setting of the fields in the aperture to be the same as the incident fields, with the fields on the surface of the screen set to be zero. The closed-surface integral reduces to the integral over the aperture. The approximation is mathematically inconsistent, but the inconsistency can be removed by adding a line integral.5 '4 2 4 3 This so-called saltus black screen problem6 is not an exact solution to our problem, but it is a good approximation in the case in which the wavelength is much smaller than the size of the aperture and in the forward direction of the incident field. The additional line integral is small compared with the other terms in the case in which the wavelength is much smaller than the other characteristic distances (i.e., the aperture radius, a, and the focal length of the lens, f ), so in our analy- sis we need to consider only the following integrals for the vector fields: E(r) = -ff [ik(no x Hi)G(r,r') + (no X Ei) X V'G(r,r') + (no Ei)V'G(r,r')]dS', (4a) H(r) = -ff [ik(no X Ei)G(r,r') + (no X Hi) V'G(r,r') + (no Hi)V'G(r,r')]dS', (4b) where the integral is over the exit pupil, A, of the optical system; the subscript i represents the incident fields; and no is the unit vector pointing away from the image space. As shown in Fig. 1, the coordinate of the point Q on the aperture is denoted by two angles, 0 and 0. The point P that we are considering is given by three cylindrical coor- dinates, p, ', and z. Q P z Fig. 1. Geometry of the problem. WZ Hsu and R. Barakat + (no - H)VG(rr')1dS'. (2b) Vol. 11, No. 2/February 1994/J. Opt. Soc. Am. A 625 The transport of the fields can be described by the us( ray-tracing procedures with the energy conservation la We construct the spherical coordinates (r' ' 4) for the ject space centered on the source rs, and (r, 6,4) for 1 image space centered on the image ri, the optical axis the -z axis, and measured from the -z axis. Consii a ray departing from rs along the direction ( 4) and riving at the exit pupil along the direction (6, 4): by glecting the losses in the optical system, we find that 1 amplitudes of the electric fields E0(6 4)) and Ei(6, 4) as ciated with the rays before and after the focusing are lated by IEi(0, 0)1 = Eo (0, o)J(dQf/df) 12 = IEo(O, )g(6), where d' = sin O'dO' do and dfl = sin 0 dO do are solid angles of the ray pencil in the object and in spaces, respectively, g(6) = (dff/dfl)"2 . Taking into count the vector nature of the fields and noting that 1 angle between the field vectors and the meridional ph of the ray remains constant, we can write the above eq tion in the following vector form: Ei(6, 4) = g(6)(¢oo + a0') Ei(ol ) where 6, 0 and are the unit vectors in the correspo ing directions. If the source is at infinity, the energy conservation I takes the form Ei (, 4) I = Eo(h cos 4) h sin P), h dh `2 = Eo(h cos 4, h sin ))jfgo(6) where h() is the height of the ray in the object space, fi constant that plays the role of a focal length, and go(6' geometric factor depending on the focusing system, is termined by fgo(6) = [h/sin 0(dh/d6)]"12. Thus for the cusing of a linearly polarized plane wave that is incident the z direction with electric field Eo, Eq. (6) has the fo Ei(0,0) = go(6)[)4 + 6(2 X )] Eo, Refer to Ref. 38 for further accounts of the factor go Without loss of generality, we assume that the plane w is polarized in the i direction, i.e., E0 = Eoi. Eq tion (8) then becomes Ei(0,4)) = go(6)Eo[(cos 6 cos 2 + sin2 )i - (1 - cos 6)sin 4 cos Oj + sin 6 cos ok], where we have used 4 = -sin Xi + cos4)j, 6 = -cos 6 cos4i - cos 6 sin oj - sin 6k. E(r) = -ikff [n o X (Ei X no) + (no x Ei) x' R + n0 exp(ikR)dS+ (no E)R] P R dS, H(r) = -ik ff [no X Ei + (no X (Ei X no) x R] exp(ikR) x 4rR dS. (12a) (12b) Here Ei is given by Eq. (9). Terms of the order of O(A/R) have been neglected, where R = r - r'I is the distance between point Q(r') and point P(r). The Cartesian coordinates of points Q on the screen and P in the image space can be represented by the angles 6 and for Q, and cylindrical coordinates p, a, and z for P can be represented as XQ = f sin 6 cos , yQ = f sin 6 sin , ZQ = -f cos . X = cos', yp = p sin y, ZQ = Z, (13) (14) where f is the focal length. Using the approximation Jr - r = [f 2 + Z2 + p2 + 2fz - 2fz(1 - cos ) - 2fp sin cos(y - )]1/2 1 [p 2 -2fp sin cos(y-)) 2 (f + z) -2zf(1-coseA)] + ... , (15) (7) we can simplify Eqs. (12) further. As a consequence of these manipulations we obtain the fields, represented in is a terms of the integral over the solid angle extended from ), a point P to the aperture, as de- in E=- Eof 2 expWiF) ff sin i'g(O~x kfin iA(z + f) JJ Zexp{i+ f arm (8) X p sin e~ cos(y - 4 + z(1 - cos a)]} cat X [(cos if cos2 4) + sin2 4)i (1Oa) (lOb) + (1 - cos i)sin 4 cos 4j + sin e~ cos 4k]ddo, (16a) H = Ef( expf) sin l'go(O)exp{-i kf X [p sin e cos(y - ) + z(1 - cos,&A X {(1 - cos 1)sin 4) cos Oi + [cos e~ - (1 - cos t)sin2 4]j + sin e sin 4)k}did4, (16b) where The corresponding magnetic field, Hi, can be obtained from E by kp2(D =k~z P 2(z + f) (17) Hi = Ei X no. (11) On substituting Eqs. (9) and (11) into Eq. (4), we can write the fields in the image space as Equations (16) and (17) represent the approximate solu- tion to the vector diffraction problem in the image space in the context of the Stratton-Chu approach. W Hsu and R. Barakat 626 J. Opt. Soc. Am. A/Vol. 11, No. 2/February 1994 It is convenient to introduce the dimensionless vari in the image space22 : z UN= 27rN , VN = 2 N (f )P, explicitly showing the Fresnel number dependence. the new variables UN and vN, the electric and mag fields at point P(r) become E= i f 1 - exp(i) sin g0(e)jA 2'irN/ JJ X exp(-i f [VN sin e cos(y - )- UN(1 - cos) x {[cos e + (1 - cos ')sin 2 O]i + (1 - cos a)sin k cos /j + sin e cos Okditdo Lo-= 00 go(i9)sin (1 + cos O)exp[if2(1 - cos )UN] (24a)(18a) X J vN sin ,)di, (18b) L 1 f go()sin2 & exp[if2(1 - cos )UN] With X aN sin (24b) L2-f go()sin 0(l - cos i)exp[iL (1 - cos )uN] X J2( vN sin W.)dit. (19a) H = of1 - UTN) exp(iD) sin&go(ie) X exp( - i f N sin, cos(y - k) - -uN(l - cos'&)iaL aJ X {(1 - cos i)sin 4 cos i + [cos i' + (1 - cos t9)sin2 0]j + sin te cos Ok}dWd), (19b) where 1D now is given by 27rN f2 N2 2 TN-2N =-) + 2(2N- VN - (20) Following Ignatowsky,3 13 2 the double integrals in Eq. (18) can be reduced to single integrals by the use of the identi- ties J27, Jo cos no exp[ip cos(4 - y)]d = 2irinnj(p)cos ny, r2 J sin n exp[ip cos(A - y)]do = 2sinJn(p)sin n'y, (21) where Jn is the first-kind Bessel function of order n. Thus the Cartesian components of the diffracted electric- and magnetic-field vectors at point P are E = -iA(Lo + L2 cos 2y), H = -iA(L 2 sin 2 y), E = -iA(L 2 sin 2y), E = -A(2L, cos y), H =- iA(Lo - L2 cos 2y), H2 = -A(2L, sin y). (22) The factor A is A = 'TEof( f)ex(i4I), and the L functions are defined as (23) (24c) Thus the fields in the image space are represented by the L functions, which depend on the Fresnel number N explicitly in the integrals through UN and vN. According to Eqs. (18) and (24), when z changes sign, the L functions are not simply related to their complex conjugates. Addi- tionally, we note that A(-z) # A(z) and 1D(-z) =# cF(z), hence in general the diffracted electromagnetic fields do not have symmetry about the focal plane as they do in the usual optical imaging situation, for which the Fresnel number is essentially infinite. In the latter case, the fo- cal length is so big that we can neglect terms of the order of O(z/f), and consequently there is a simple relation be- tween the L functions and their complex conjugates evalu- ated at symmetric positions on the opposite sides of the focal plane, so that EX(-z,p,y) = -EX*(z,p,), H(-z,p,y) = -H*(ZP.y), EY(-z,p,y) = -E*(z,p,y), H(-z,p,y) = -HY*(ZP.y), EZ(- z, p, y) = EY*(z, p, y), HZ(-z, p, y) = HZ*(, p, y). (25) ENERGY DENSITIES OF DIFFRACTED FIELDS We now calculate the time-averaged energy densities for the diffracted fields. The time-averaged electric and magnetic energy densities are W 1% e167(E * E*),167r 1 Wm (H H*). (26a) (26b) From Eqs. (22), we obtain We - 1A [11 2 + 1L212 + 4LiI2 cos2 y16r1 + 2 Re(LoL2 *)cos(2y)], Wm -'A [L0 12 + IL2 12 + 4L 12 sin2 y162r - 2 Re(LOL2*) cos(2,y)] . (27a) (27b) Note that both the electric and magnetic energy densities are dependent on the polar angle y because of the vectorial W Hsu and R. Barakat Vol. 11, No. 2/February 1994/J. Opt. Soc. Am. A 627 nature of the fields. On the other hand, the total energy density of the diffracted fields is independent of the polar angle y and is given by W = We + Wm = A (Lo 2 + L212 + 21L 2). (28) As we have shown above, the absolute value squared of the L functions and the amplitude, A, are not even functions of z, and thus the energy densities are not symmetric about the focal plane, z = 0. On axis p = 0, so that N = 0. Because J1(O) = 0 and J2 (0) = 0, we have L1 = L2= 0. Then, the electric field is polarized in the i direction and the magnetic field is polarized in the j direction on the axis. The correspond- ing electric and magnetic energy densities are the same, We = Wm = ILo2 (29)167r (9 and the total energy density is W = 1 r Lol,2 (30) 87T where Lo on the axis is given by, because JO(0) = 1, Lo = I go(O)sin 1a(1 + cos 1a)exp[if2(1 - cos a)uN]d (31) The z dependence of A and UN gives rise to the so-called focal-shift effect; that is, the position of the maximum en- ergy density is not at the Gaussian focus z = 0. Instead, it is shifted toward the aperture by an amount depen- ding on the focal length and Fresnel number for fixed wavelength. SPECIAL CASES So far our only approximation to the saltus problem of the diffracted vector fields in the image space is relation (15), the paraxial approximation for the distance between point P in the image space and point Q on the screen. In prac- tical situations, however, there are many restrictions on the parameters, especially the Fresnel number, N. In this section, we consider two typical situations, that is, dif- fraction for visible lights and diffraction for microwaves. For visible light, the normal focusing systems have very large Fresnel numbers because of the fact that the wave- length is too small compared with the aperture size. To have small Fresnel numbers, say of the order of one, we must use a very long focal length such that the ratio a/f is of the same order of A/a. Because the upper limit 0 of the L integrals is determined by the angular aperture, i.e., sin 00 = a/f, it is very small for all practical purposes. For an estimate of the order of magnitude, take A = 500 nm and f = 200 cm. For N = 1, we have a = 1 mm, and thus 0 = 0.0005 rad. Even for N as high as 100, 0 is only 0.005 rad. The upper limit 0 of the integrals in Eqs. (24) can then be taken as much smaller then 1 for visible light diffraction with small Fresnel numbers, and we can use the approximation for small angles, sin , 1 - cosi_ e 02/2, (32) to write the L functions in Eqs. (24) as Lo = jI 2go(t9) Df exp (i f UN 2)JO(f vN)dt* L = J go(i)9}2 exp i2 2 ja vN dis G 3 f2 2J( fv)d 2= go (a)- exiUN vN dJ2 (Lfexp a af N~Yd (33a) (33b) (33c) The factor go() depends on the imaging system. For an aplanatic system,3 '3 5 (34a)go(le) = _OS1; whereas for a parabolic mirror,32 38 2 =1 + COS 1 When the angle , as measured in radians, is much smaller than 1, the difference between the two go factors is small. In fact, for an angle & that is smaller than 0.01 rad, the difference between the go factors is less than 0.005%. Ad- ditional numerical calculations of the L integrals with go() given by Eqs. (34a) and (34b) have been conducted separately, and they show negligible difference. By using the asymptotic expansion of the Bessel functions for small arguments, we can estimate the integrals and show that L1 and L2 are of higher orders of small quantity 00 than is Lo. Only Lo is significant in this case, and consequently there is only one signif