何冠男毕业设计.docx

1068 J. Opt. Soc. Am. A / Vol. 12, No. 5 / May 1995 Moharam et al. Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings M. G. Moharam, Eric B. Grann, and Drew A. Pommet Center for Research and Education in Optics and Lasers, University of Central Florida, Orlando, Florida 32816 T. K. Gaylord School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332 Received August 24, 1994; accepted October 12, 1994; revised manuscript received November 7, 1994 The rigorous coupled-wave analysis technique for describing the diffraction of electromagnetic waves by peri- odic grating structures is reviewed. Formulations for a stable and efficient numerical implementation of the analysis technique are presented for one-dimensional binary gratings for both TE and TM polarization and for the general case of conical diffraction. It is shown that by exploitation of the symmetry of the diffraction prob- lem a very efficient formulation, with up to an order-of-magnitude improvement in the numerical efficiency, is produced. The rigorous coupled-wave analysis is shown to be inherently stable. The sources of potential numerical problems associated with underflow and overflow, inherent in digital calculations, are presented. A formulation that anticipates and preempts these instability problems is presented. The calculated diffrac- tion efficiencies for dielectric gratings are shown to converge to the correct value with an increasing number of space harmonics over a wide range of parameters, including very deep gratings. The effect of the number of harmonics on the convergence of the diffraction efficiencies is investigated. More field harmonics are shown to be required for the convergence of gratings with larger grating periods, deeper gratings, TM polarization, and conical diffraction. 1. INTRODUCTION Over the past 10 years the rigorous coupled-wave analysis (RCWA) has been the most widely used method for the accurate analysis of the diffraction of electromag- netic waves by periodic structures. It has been used successfully and accurately to analyze both holographic and surface-relief grating structures. It has been formu- lated to analyze transmission and reflection planar dielec- tric absorption holographic gratings, arbitrary profiled dielectric metallic surface-relief gratings, multiplexed holographic gratings, two-dimensional surface-relief grat- ings, and anisotropic gratings for both planar and conical diffraction.19 The RCWA is a relatively straightforward technique for obtaining the exact solution of Maxwells equations for the electromagnetic diffraction by grating structures. It is a noniterative, deterministic technique utilizing a state-variable method that converges to the proper so- lution without inherent numerical instabilities. The ac- curacy of the solution obtained depends solely on the number of terms in the field space-harmonic expansion, with conservation of energy always being satisfied. Our purpose in this paper is to present a detailed re- view of the RCWA and to provide a step-by-step guide for its efficient and stable implementation. A simple com- pact formulation for the efficient and stable numerical implementation of the RCWA for one-dimensional, rectangular-groove binary surface-relief dielectric grat- ings is presented. Formulations for TE and TM polar- ization and for the conical-diffraction configuration are included. It is shown that a very efficient formulation, with up to an order-of-magnitude improvement in the numerical efficiency, can be achieved by exploitation of the symmetry of the diffraction problem. The technique is shown to be fundamentally stable. The criteria for numerical stability are (1) energy conservation and (2) convergence to the proper solution with an increasing number of field harmonics for all the grating and the incident-wave parameters. Potential numerical difficul- ties can be preempted by proper formulation and nor- malization. Specifically, the nonpropagating evanescent space harmonics in the grating region must be properly handled in the numerical implementation. The effect of the number of terms in the field space-harmonic ex- pansion on the convergence of the diffraction efficiency is investigated. It is shown that for dielectric gratings, even very deep gratings, the calculated diffraction effi- ciencies always converge to the correct value as the num- ber of space harmonics increases. As expected, more field space harmonics are required for the convergence of gratings with larger grating periods, deeper gratings, TM polarization, and conical diffraction. 2. FORMULATION The general three-dimensional binary grating diffraction problem is depicted in Fig. 1. A linearly polarized elec- tromagnetic wave is obliquely incident at an arbitrary an- gle of incidence u and at an azimuthal angle f upon a binary dielectric or lossy grating. The grating period L is, in general, composed of several regions with differing refractive indices. The grating is bound by two differ- ent media with refractive indices nI and nII. In the for- 0740-3232/95/041068-09$06.00 1995 Optical Society of America Moharam et al. Vol. 12, No. 5 / May 1995 / J. Opt. Soc. Am. A 2 gx j U Fig. 1. Geometry for the binary rectangular-groove grating 3. PLANAR DIFFRACTION: TE POLARIZATION The incident normalized electric field that is normal to the plane of incidence is given by Einc, y ? expf2jk0nI ssin u x 1 cos u zdg , (3) where k0 ? 2pyl0 and l0 is the wavelength of the light in free space. The normalized solutions in region I s0 , zd and in region II sz . dd are given by EI, y ? Einc, y 1 P Ri expf2j skxi x 2 kI,zi zdg , (4) i EII, y ? P Ti exph2j fkxi x 2 kII, zi sz 2 ddgj , (5) i where kxi is determined from the Floquet condition and is given by kxi ? k0fnI sin u 2 isl0yLdg (6) and where ( 1k0fn,2 2 skxiyk0 d2g1/2 k0 n, . kxi , diffraction problem analyzed herein. kL ,zi ? 2jk fsk yk d 2 n 2 g1/2 k . k n 0 xi 0 , xi 0 , mulation presented here, without any loss of generality, the normal to the boundary is in the z direction, and the grating vector is in the x direction. In the grating region s0 , z , dd the periodic relative permittivity is expand- able in a Fourier series of the form , ? I, II . (7) Ri is the normalized electric-field amplitude of the ith backward-diffracted (reflected) wave in region I. Ti is the normalized electric-field amplitude of the forward- diffracted (transmitted) wave in region II. The magnetic ´sxd ? X ´h exp j 2ph ! , (1) fields in regions I and II may be obtained from Maxwells equation h L ! where ´h is the hth Fourier component of the relative permittivity in the grating region, which is complex for lossy or nonsymmetric dielectric gratings. For simple grating structures with alternating regions of refractive indices nrd sridged and ngr sgrooved the Fourier harmonics are given by sinsphf d H ? vm = 3 E , (8) where m is the permeability of the region and v is the angular optical frequency. In the grating region s0 , z , dd the tangential electric ( y-component) and magnetic (xcomponent) fields may be expressed with a Fourier expansion in terms of the space- ´0 ? nrd2f 1 ngr 2s1 2 f d, ´h ? snrd2 2 ngr 2 d , ph (2) harmonic fields as Egy ? X Syi szdexps2jkxi xd , (9) i where f is the fraction of the grating period occupied by the region of index nrd and ´0 is the average value of the relative permittivity, not the permittivity of free space. The general approach for solving the exact electro- Hgx ? 2j e0 m0 !1/2 X xi szdexps2jkxi i xd , (10) magnetic-boundary-value problem associated with the diffraction grating is to find solutions that satisfy Maxwells equations in each of the three (input, grat- ing, and output) regions and then match the tangen- tial electric- and magnetic-field components at the two boundaries. For the case of planar diffraction sf ? 0d where e0 is the permittivity of free space. Syi szd and Uxi szd are the normalized amplitudes of the ith space- harmonic fields such that Egy and Hgx satisfy Maxwells equation in the grating region, i.e., Egy z ? j vm0 Hgx , (11) the incident polarization may be decomposed into a TE- and a TM-polarization problem, which are handled H ? j ve ´sxdE 1 Hgz . (12) independently. Here all the forward- and the backward- diffracted orders lie in the same plane (the plane of inci- dence, the x z plane). For the general three-dimensional problem sf f i 0d, or conical diffraction, the wave vectors of the diffracted orders lie on the surface of a cone, and the perpendicular and the parallel components of the 0 gy z x Substituting Eqs. (9) and (10) into Eqs. (11) and (12) and eliminating Hgz , we obtain the coupled-wave equations Syi z ? k0Uxi , electric and the magnetic fields are coupled and must be obtained simultaneously. The three cases are considered separately in Sections 3 5. Uxi z kxi 2 ! ? k0 Syi 2 k0 X ´si2pd Syp , (13) p 1070 J. Opt. Soc. Am. A / Vol. 12, No. 5 / May 1995 Moharam et al. , or, in matrix form, or, in matrix form, " Syy sz0d #" 0 I #"Sy # ? (14) " di0 # " I 1 # fRg ? "W WX #"c1 # , Uxy sz0d A 0 Ux which may be reduced to jnI cos u di0 2j YI V 2VX c2 (21) f 2Syy sz0d2g ? fAgfSy g , (15) where z0 ? k0z and A ? Kx 2 2 E , (16) where E is the matrix formed by the permittivity har- monic components, with the i, p element being equal to ´si2pd; Kx is a diagonal matrix, with the i, i element being equal to kxiyk0; and I is the identity matrix. Note that A, Kx , and E are sn 3 nd matrices, where n is the num- and at z ? d n X wi, m fcm1 exps2k0qmdd 1 cm2g ? Ti , (22) m?1 n X vi, m fcm1 exps2k0qmdd 2 cm2g ? j skII, ziyk0dTi , m?1 (23) or, in matrix form, ber of space harmonics retained in the field expansion, "WX W #"c1 # " I # with the ith row of the matrix corresponding to the ith space harmonic. The s2n 3 2nd matrix in Eq. (14) thus VX 2V c2 ? j YII fT g , (24) becomes an sn 3 nd matrix in Eq. (15). We solve the set of the coupled-wave equations by cal- culating the eigenvalues and the eigenvectors associated with the matrix A. The simplification step taken from Eq. (14) to Eq. (15) effectively reduces the overall com- putational time of the eigenvalue problem by a factor of 8. Moreover, for symmetric gratings, the matrix A is symmetric for dielectric or Hermitian for lossy binary gratings. Hence a significant enhancement in the com- putational efficiency and a reduction in the computer memory requirement can be achieved by use of an ap- propriate eigenvalue software package. The space har- monics of the tangential electric and magnetic fields in the grating region are then given by n where di0 ? 1 for i ? 0 and di0 ? 0 for i fi 0 and X, YI, and YII are diagonal matrices with the diagonal ele- ments exps2k0qmdd, skI,ziyk0d, and skII,ziyk0d, respectively. Equations (21) and (24) are solved simultaneously for the forward- and backward-diffracted amplitudes Ti and Ri. Numerical overflow is successfully preempted by the normalization process; i.e., at both boundaries Eqs. (19) (23) the arguments of the exponential are al- ways negative. One may significantly improve numeri- cal efficiency by eliminating Ri from Eqs. (19) and (20) and Ti from Eqs. (22) and (23), solving the resulting set of equations for the cm1 coefficients, and then substituting these coefficients back into Eqs. (21) and (24) to calculate Ri and Ti. However, attempts to solve Eq. (24) for cm1 and cm2 in terms of Ti and then substitute for cm1 and Syi szd ? P wi, m hcm1 exps2k0qmzd c 2 in Eq. (21) to determine T and R will probably cause m?1 m i i 1 cm2 expfk0qmsz 2 ddgj , (17) n numerical errors. This is due to possible zero columns on the left-hand sides of Eqs. (21) and (24), which result from very small terms in the diagonal matrix X when some of Uxi szd ? P vi, m h2cm1 exps2k0qmzd m?1 1 cm2 expfk0 qmsz 2 ddgj , (18) the generally complex eigenvalues have a large positive real part. The diffraction efficiencies are defined as kI, zi ! , where wi, m and qm are the elements of the eigenvector matrix W and the positive square root of the eigenvalues DEri ? Ri Rip Re k0nI cos u of the matrix A, respectively. The quantity vi, m ? qmwi, m is the i, m element of the matrix V 5 WQ, where Q is a diagonal matrix with the elements qm. The quantities DEti ? Ti Ti p Re kII, zi ! . k0nI cos u (25) cm 1 and cm2 are unknown constants to be determined from the boundary conditions. Note that the exponential terms involving the positive square root of the eigenvalues are normalized to prevent possible numerical overflow, as is shown below. We calculate the amplitudes of the diffracted fields Ri and Ti (together with cm1 and cm2) by matching the tangential electric- and magnetic-field components at the two boundaries. At the input boundary sz ? 0d n The sum of the reflected and the transmitted diffrac- tion efficiencies given by Eq. (25) must be unity for loss- less gratings. This sum is independent of the number of space harmonics retained in the field expansion, which de- termines the accuracy of the individual diffracted orders. 4. PLANAR DIFFRACTION: TM POLARIZATION The incident normalized magnetic field is normal to the di0 1 Ri ? P wi, m fcm1 1 cm2 exps2k0qmddg , (19) m?1 plane of incidence and may be written as j fnI cos u di0 2 skI, ziyk0dRi g n ? P vi, m fcm1 2 cm2 exps2k0qmddg , (20) m?1 Hinc, y ? expf2jk0nIssin u x 1 cos u zdg . (26) The normalized solutions in region I s0 , zd and region II sz . dd are given, respectively, by Moharam et al. Vol. 12, No. 5 / May 1995 / J. Opt. Soc. Am. A 4 , HI, y ? Hinc, y 1 P Ri expf2j skxi x 2 kI, zi zdg , (27) i Sxi szd ? n P vi, mh2cm1 exps2k0qmzd m?1 HII, y ? P Ti exph2jfkxi x 1 kII, zi sz 2 ddgj , (28) i 1 cm 2 expfk0qm sz 2 ddgj , (38) where kxi , kI, zi , and kII, zi are defined as in Eqs. (6) and (7). Ri is the normalized magnetic-field amplitude of the ith backward-diffracted (reflected) wave in region I. Ti is the normalized magnetic-field amplitude of the forward- diffracted (transmitted) wave in region II. The magnetic- field vectors in the two regions can be obtained from Maxwells equation 2j !where wi, m and qm are the elements of the eigenvector matrix W and the positive square root of the eigenvalues of the matrix EB, respectively. The quantities vi, m are the elements of the product matrix V ? E21WQ, with Q being a diagonal matrix with the diagonal elements qm. The quantities cm1 and cm2 are unknown constants to be determined from the boundary conditions. Again, note that the exponential terms involving the positive square root of the eigenvalues are n