This is to certify that the dissertation entitled Optimal Design of Diffractive Optics presented by Kai Huang has been accepted towards fulfillment of the requirements for Ph. D. degree in Mathematics. "' '\ Q (390 Major professor Date April 16, 2002 MS U is an Affirmative Action/Equal Opportunity Institution 042771 LIBRARY Michigan State University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE .A A '- i;‘;i* 'J mans ”'5 2 3 2007 6/01 cJCIRC/DatoDuepGS-DJS OPTIMAL DESIGN OF DIFFRACTIVE OPTICS By Kai Huang A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 2002 ABSTRACT OPTIMAL DESIGN OF DIFFRACTIVE OPTICS By Kai Huang In this dissertation, we consider two problems. Both problems deal with diffractive optics and design of grating structures. The first is known as the resonance design problem. In this case, we encounter one of the most interesting new developments in diffractive optics which is the integration of a zero—order grating and a planar waveguide to create a resonance. Such structures are able to yield ultra narrow bandwidth filters which potentially have many extraor- dinary applications. The main step in the design of such a grating structure is to find the resonant wavelength. For any fixed grating structure, calculation of the resonant wavelength is found by solving a nonlinear eigenvalue problem. The second of the two design problems focuses on nonlinear diffraction gratings. Here, we will consider a plane wave of frequency to] incident on a grating composed of some nonlinear optical material. The effects of the nonlinear material interacting with field give rise to diffracted waves with frequencies wl and (4)2 = 20.21. The creation of waves with the doubled frequency is a phenomenon unique to nonlinear optics known as second harmonic generation (SHG). The design problem in this context is to create a grating structure which enhances the very weak nonlinear Optical effects. ACKNOWLEDGMENTS I am most indebted to my dissertation advisor, Professor Gang Bao, for his support during my graduate study. I would like to thank him for suggesting the problem and giving invaluable guidance which makes this dissertation a reality. I would like to thank my dissertation committee members Professor Chichia Chiu, Professor Patricia Lamm, Professor Baisheng Yan and Professor Zhengfang Zhou for their valuable suggestions. My greatest thanks are reserved for my wife, Yinghui Li, and my family for their enduring patience, constant support and encouragement. iii TABLE OF CONTENTS LIST OF FIGURES INTRODUCTION 1 The diffraction problem 1.1 Preliminaries ................................. 1.2 Radiation condition and Grating formula ................. 1.3 Truncating of the domain .......................... 1.4 Variational formulation ........................... 1.5 Efficiency ................................... 1.6 Optimization of grating efficiency ..................... 2 Guided Mode Grating Resonance Filters (GMGRF) 2.1 Introduction ................................. 2.2 The reflectance in thin film ......................... 2.3 Guided Modes for the Slab Waveguide ................... 2.4 Design Methodolog ............................ 2.5 The dissipative diffraction problem .................... 2.6 Scattering frequency ............................. 2.7 Numerical solution for scattering frequency ................ 2.8 Design of GMGRF ............................. 2.9 Numerical example ............................. 3 Optimal design of nonlinear grating 3.1 Introduction .................................. 3.2 Modeling of the nonlinear scattering problem ............... 3.3 Optimal design ................................ 3.4 Numerical examples ............................. Appendix ...................................... BIBLIOGRAPHY iv 1.1 1.2 1.3 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 3.1 3.2 3.3 3.4 3.5 LIST OF FIGURES Geometry of Grating ............................. Binary Grating ................................ Geometry for the calculation of gradients ................. Waveguide (Single layer) ........................... Resonance example .............................. Propagation of an electromagnetic wave through a homogeneous film. . . Asymmetric reflectance curve ........................ Guided wave in the slab waveguide ...................... Design parameters .............................. Design methodology for GMGRF ...................... Resonance wavelength and singular wavelength .............. resonant wavelength ............................. eigenvalue but not resonant wavelength .................. reflectance curve for initial data ....................... reflectance curve ............................... reflectance curve for the design example. .................. Problem geometry. .............................. Cross section of a simple binary gratings .................. Numerical example 1 .............................. Numerical example 2 .............................. Numerical example 3 .............................. 4 13 15 18 19 20 23 24 27 30 Introduction The field of diffractive optics has developed over the past few decades in concur- rence with the semiconductor industry. The practical application of diflractive optics technology has driven the need for mathematical models and numerical codes which provide rigorous solutions of the electromagnetic vector-field equations for grating structures. Such models and codes allow scientists to predict performance of a given structure, and to carry out Optimal design of new structures. One of the most interesting new developments in diffractive optics involves the integration of a zero-order grating with a planar waveguide to create a resonance. Such structures take the form of a planar dielectric layer with the grating providing a periodic modulation of the dielectric constant in one or more of the layers. These grating structures can yield ultra narrow bandwidth filters for a selected wavelength. In practice, it is desirable to design the grating structures to focus the filtering prop- erties around a predetermined frequency. The design requirements are specified in terms of the resonant wavelength Am at which maximum reflectance occurs, the spectral bandwidth of the resonance AA , the polarization and angle of the incident wave, and possibly the “out-of-band” reflectance away from resonance. A great deal of computation is necessary to draw the reflectance curve response to many diflerent wavelengths. One of the problems is to find the resonant wavelength without draw- ing the reflectance curve. Instead of computing the entire reflectance curve, we will see that information about the location of the resonance frequency can be found by studying the scattering frequency which saves much in the way of computational cost. In another grating problem, we consider a plane wave of frequency wl incident on a grating or periodic structure consisting of some nonlinear optical material. The nonlinear material produces nonlinear optical interactions which give rise to diffracted waves at frequencies wl and (.02 = 2%. The creation of waves with the doubled frequency represents the simplest situation which is unique to nonlinear optics — second harmonic generation (SHG). An exciting application of SHG is to obtain coherent radiation at a wavelength shorter than that of currently available lasers. Unfortunately, it is well known that nonlinear optical effects, including SHG, are generally so weak that they can only be observed in the presence of very strong coherent electromagnetic fields such as lasers. Effective enhancement of nonlinear optical effects presents one of the most challenging tasks in nonlinear optics. The dissertation is organized as follows. In chapter 1, we describe the direct diffraction problem. Some basic results that will be used in the next two chapters are provided . The formulas for the partial derivatives of the reflection and transmission coeflicients with respect to the parameters of a binary grating profile are introduced. The formulas will be used in determining the thickness of grating in resonant filter design. Chapter 2, presents the resonant filter design problem. First we consider how to find a good initial guess of grating thickness and period. Then, we try to find the resonant wavelength for a given grating structure. Finally, we consider the design problem. Also, numerical examples are presented . Chapter 3 discusses the nonlinear grating design problem. In section 3.2, we present the nonlinear scattering problem. Subsequently, in section 3.3, the perturbed diffraction problem with respect to smooth variations of the interfaces is studied and a gradient formula is derived. Numerical examples are given in section 3.4. CHAPTER 1 The diffraction problem 1 . 1 Preliminaries Suppose that the whole space is filled with non-magnetic material with a permittivity function 5, which in Cartesian coordinates (r1,xg,x3) does not depend on $3, is periodic in $1, and homogeneous above and below certain interfaces. In practice, the period A of optical gratings under consideration is comparable with the wavelength A = 27rc/w of incoming plane optical waves, where c denotes the speed of light. Suppose also that the media are nonmagnetic; i.e., the magnetic permeability constant p is a fixed constant everywhere. The upper interface is denoted by S +, the lower interface is denoted by S ‘. The medium between 5+ and S ' is inhomogeneous with s = Eo($1,$2), and we assume that the function so is periodic in $1, 80(x1,:r2) = 50(x1 + A, 3:2) and piecewise constant. Above the surface 8*“ and below the surface S ‘, the media are assumed to be homogeneous with e = 5*. Assume the grating is illuminated by a monochromatic plane wave El : Aeiaxl—iflxge—iwt HI :: Beiarl—ifirge-iwt with [3 ¢ 0. Here, the complex amplitude vector A is perpendicular to the wave vector k = it $2 I‘+ G+ Figure 1.1. Geometry of Grating (a, —fi, 0) and B = (wp)k X A. The incident. wave (Ei, Hi) will be diffracted by the grating. In region 0“ above the grating surface 5'“, the total fields will be given by Eup = Ei + Erefl Hup 2 Hi + Hrefl. In region 0‘ below the grating surface S‘, the total fields will be given by Edown = Erefr Hdown = Hrefr. ’ Dr0pping the factor exp(—iwt), total fields satisfy the time-harmonic Maxwell equations V x E 2: iwuH (1.1) V-sE = 0 (1.2) VXH = —inE (1.3) V-H II o (1.4) Additionally, the tangential components of the total fields are continuous when crossing an interface between two continuous media V x (E1 — E2) = 0, u x (H1 — H2) = 0 on 3+,S‘ (1.5) where u is the unit normal to the interface 3*. E and H can be represented as the superposition of solutions corresponding to the TE case (field transverse electric), where E1 = (0, 0, A3) exp(ia:1:1 - ifixg), Hi 2 —(w,u)—l(—fiA3,aA3,0) exp(ia:r1 — iflzg), and to the TM case (field transverse magnetic ) with Ei = (A1, A2,0) exp(z'a:r1 — iflrg), Hi = (0, 0, ,8/11 + (1A2)exp(z'ax1 — iflzg). Denote by ui the normed transverse component Ei -x3 for TE or Hi ox3 for TM, i.e. ui = exp(iaa:1 — ifirg) with a = wJEsin 0,fl = uq/E cos 6. The functions ui (2:1, 3:2) and uo(:rl, 9:2) equal to either the transverse component E - x3 for TE or H - x3 for TM in Git and Go, resp. The boundary conditions (1.5) are translated into transmission conditions for the unknowns u2t and no in the following way: 0 TE mode: . . + 1 81/ 61/ u_ — 9—2.1; — 211—0. on S— _ uo, 6V _ 01/ ' 5 0 TM mode: we +29) _ Laue 5+ 61/ _ so 611 LEE — £9312. on S— 6— all _ 60 all , u+ + u1 = no, on 5*, u_=u0, We shall assume throughout that the optical index of the grating materials is such that the 5 satisfies x/EI > 0, (1.6) Re \/€—‘ > 0, Im F 2 0, (1.7) Re ‘/€0(:r1,:r2) > 0, Im ‘/€0(x1,:1:2) 2 0. (1.8) It is easily seen that the Maxwell’s equations become Helmholtz equation. TE case: Au + when = 0 (1.9) TM case : 1 V- —Vu +w2u=0 1.10 (#5 ) ( ) 1.2 Radiation condition and Grating formula It is obvious that the incident field ui satisfies the following condition: ui(:l:1 + pA, 3:2) 2 ui(:rl, 3:2)6'7’“ (2.1) Suppose now that u is a solution of (1.9) (resp. (1.10)). Then by the periodicity of the index 5, every field 110’) of the form ”(”00 = ”($1 + 1>/\,5172)e"‘1"°‘A for p E Z is a solution of the same problem. Of course, the associated diffracted field no”) has the same behavior for 2:2 —> 00 as u, and there is no physical criterion to eliminate some of these solutions. In order for the problem not to have-infinitely many solutions, we will look only for a-periodic solutions, i.e. solutions it such that u(:1:1 +pA,a:2) = u(a:1,.r2)eipA°' V1) 6 Z. (2.2) As usual, diffraction problems (1.9) and (1.10) are not well-posed and must be completed by a radiation condition. Since the grating is unbounded in the 351- direction, the classical Sommerfeld condition is not appropriate. Because the domain is unbounded in the zg-direction, a radiation condition on the scattering problem must be imposed at infinity, namely the diffracted fields ui remain bounded and that they should be representable as superpositions of outgoing waves. Define the coefficients 2 0,, = cr+p—A1 (2.3) e:- = fi§(a)=e%£|w2u£*—aili (2.4) where 11,, z arg(w2p.€ — 0,2,) E (—7r, 7r] is an argument of the complex number. Define the finite sets of indices Pi = {p e z : 3;: e R}. (2.5) The a-periodicity condition will allow us now to write a radiation condition in the rug—direction. Indeed, if the diffracted field u is a-periodic, it means that the function ”($1,132) 2 “($1,332)€"iax1 (2-6) is periodic. Moreover, since ui are analytic above 5'” (resp. below S ‘) by the classical regu- larity results for the Helmholtz equation, the Fourier decomposition of 1) leads us to the so-called a-periodic Fourier decomposition of u: + _. + 1'0 $1+il3+32 + to: 11-il3+1'2 u ($1,32)—ZA1,€ P P +Bpe P P pez for 272 > max 3+ u-(xl,$2) : z A;etap$1-1fip $2 + Bp-etapxl-Hfip :52 p62 for 132 < min 5" The physics of the problem imposes the obvious condition that the diffracted field remains bounded as l$2l -—> 00. Thus, we will insist that u2t satisfy the outgoing wave condition (OWC) B: = 0, i.e. they are composed of bounded outgoing plane waves h1(?i: . . + u+(x1, 2:2) 2 Z A;e‘apxl+'5p “’2 for 2:2 > max 3+ pez u—(r1,$2) = Z A;e'apx‘_'fip “:2 for $2 < min S" pEZ Since D: is real for at most finitely many p, there are only a finite number of prop- agating plane waves in the sums of (2.7). Note that physically the case [if = 0 corresponds to a plane wave propagating parallel to the grating. We assume that B: 75 0. i.e. (wuss) e (a + 37(1)? (27) . . + u+ = z sameness: pEZ . . + . = Z Age'apxfi‘flp ””2 “ outgomg waves ” p€P+ - . + + Z Age‘ai’xfl‘fip 1'2 “ evanescent waves ” p¢P+ n-(elnz) = ZA;ei“PI“w;Iz pEZ = Z A;em”‘"'5p “’2 “ outgoing waves ” pEP— + Z A176”? “"5? $2 “ evanescent waves ” P93?“ Each term of the outgoing waves in the above represents a prOpagating plane wave, which is called the diffracted wave in the p-th order. Thus we have derived the following grating formula: 7r ap=a+p— A or . . /\ 51nd,, = smfl + pX Let H362) denote the restriction to {2 of all functions in the Sobolev space HfOC(R2) which are A-periodic in :51. Since we look now for a-periodic fields, the problems can be written on a cell of the grating and we introduce O = {(231, 3:2) : $1 6 (O, A)}. Define no, 2 tie—““1. It is easily seen that if u satisfies (1.9) then ua satisfies Aau + when = 0 in R2 where AG, is defined by A0, = (V0,)2 = (Bx, + in)2 + (81.2)? Let us consider the following problems: TE problem Pm: find 11. e 11,;(52) ~ Aau + w2p€u = 0 in $2 (2.8) udif = u — ui has the form (2.7) for |z2| big enough (2.9) TM problem Pm: find u e H;(§'2) Va - (inn) + wzu = 0 in O (2.10) udif = u — ui has the form (2.7) for |r2| big enough (2.11) 9 1.3 'I‘runcating of the domain We introduce the following notations: Fix number b > maxS+ and —b < minS'" and let $2 = (0, A) x (—b,b),Qi = (mafia, = nnGo,I‘+ = {x2 =b}m’2,1‘- = {2:2 = —b}n(2. We introduce the following space: H’(I‘i)= {12: Z 121,,e'(2p"/A)‘”1 Z,(l + 4p21r2/A2)’ lvpl2 < 00} (3.1) pEZ We define the operators Ti: Ti : H%(I‘i) —> H-%(r*) Ti: v- — Z '12,,e:‘(2""/M“1 —+ Ti( (=12) Z ifiivp e'm’mr/Mxl (3.2) p62 1262 which are periodic pseudo-differential operators of order 1. It is easy to check that Bu 6—71 2 T_u on I“ (3.3) g; = T+u—2B63"‘.’6boe'la“l on I“+ (3.4) Thus, the problem can be formulated as following: TE problem Pélg: find u 6 H1362) Aau+w2peu = 0 in n (3.5) anu = Wet—22131624131" on P“ (3.6) Bn'u = T'u on I" (3.7) Similarly, the TM problem can be formulated as following: TM problem Pg)”: find u 6 H1162) Va - (iVQU) + wzu = 0 in 0 (3.8) Bnu = T+u — 2iflle'ifi‘b on D“ (3.9) Bnu = T‘u on I" . (3.10) 10 Problems PTE (resp. PTM) and P125. (resp. Pg," are clearly equivalent in the sense of the following proposition. Proposition 1.3.1 If u is a solution of PTE (resp. PTM ) such that ii = uln E H;(Q), then it is a solution of PTE (resp. PTM). Conversely, if it is a solution of Pf~E(resp.P%E), it can be extended to a solution it of PTE ( resp. PTM). 1.4 Variational formulation From the TE problem P7313, integration by parts results in the variational relation [Va-Vv—wz/teuv— 9217- 221sz an an n N I‘- By (3.3) and (3.4), we have the variational formulation for the TE diffraction problem BTEW, 7)) = LTEU (4-1) BTE(u, v) :2 [(Vau - Vat—j — (Knew?) — /T+uv — [Two (4.2) 9 W" I“ LTEv :2 - f2ifie‘ifibt7 (4.3) I‘+ Similarly, the TM diffraction problem can be formulated as follows. Find it E HI}(Q) that satisfies BTM (u, v) z LTMv (4.4) BTM(u, v) :2 [(ivau - V017 — w2uv) — 1: /T+uv — —-1:/T—uv (4.5) n “8 #6 r+ ”5 I‘- 1 . LTMv :2 ___+ / 2i,Be_'3bv (4.6) #5 r+ We have the following theorem: Theorem 1.4.1 [2] The problem P793 (PfiM ) is well-posed for every value of k except maybe for a discrete set of values It. 11 1.5 Efficiency Let u be the solution of the TE or TM variational problem (4.1) or (4.4). The reflection and transmission coefficients are determined by traces of u on the artificial boundaries I‘i: A+ = (2a)“le‘w’ib/ue4m‘, I‘+ A; = —e_2wb + (27r)—le'ifib/u, I‘+ A; = (2a)“le-ifi5b/ue—inx‘. I‘— Then the reflected an transmitted efficiencies in the TE case are defined by :i: e; = knew, p 6 P, (5.1) and in the TM case by fid- e; = FPIA;|2, p E P+, (5.2) _ 5+ fl" _ _ ep : Ffilflp [21 p E P a (53) For lossless gratings, i.e. all optical indices are real, the principle of conservation of energy then, in either case, yields the relation 2 e;+ 2 e; = 1. (5.4) pEP'i‘ pEP‘ 1.6 Optimization of grating efficiency Consider a binary grating profile I‘ which is composed of a finite number of horizontal and vertical segments and is determined by the height d and by , say m +1 transition 12 $2 (1 b d 2m Em 2m 21 2,- 0 t1 tj A :51 —b Figure 1.2. Binary Grating points 0 = to < t1 < < tm_1 < tm = A. Since to and tm are assumed to be fixed, we write I‘ = I‘(t1, - - - ,tm_1,d) Assume that the number of transition points is fixed and, for given numbers cf 6 {—1, 0, 1}, define the functional J(F) 2 J(t1,'°°,tm_1,d) I: ZCEB: (6.1) We consider the following minimization problem. Find a binary grating profile P0 such that min J(I‘) = J(r°) (6.2) (t1,---,tm_1,d)eK where K is some compact set in the parameter space R” reflecting some natural constraints on the design of the profile. Note that the choice cf = —l (resp. cf 2 1) in (6.1) amounts to maximizing (resp. minimizing ) the efficiency of the corresponding reflected or transmitted propagating mode of order n. 13 To find local minimal of problem (6.1), gradient-based minimization methods can be applied. Thus, we must calculate the gradient of J, VJ (I‘) = (DJ-J (I‘))'{‘, where e.g. for j = 1 0.16“) = (ah-1W.) — m» (6.3) = tsh‘kml+h,---,d)-J(n.---,d)) (6.4) Here I‘h denotes the binary profile with the height d and the transition points t1+h,t2,---,d. We obviously have, for j 2: 1, - - - ,m DjJ(F) = 22w: MHCZERB (A: (0)0in (1‘)} +32%; /fi){CI Re (A; (F ”01A; (P)} (6-5) Therefore , we have to calculate the partial derivatives DjAflI‘) of the reflection and transmission coefficients. We fix n E P+ and derive a formula for the partial derivative DIA;r (I‘) of the Rayleigh coefficient of the n-th reflected mode in the TE case. Let u be the solution of the TE transmission problem and let uh denote the solution of the corresponding problem for the profile I‘h = I‘(t1 + h, t2, ~ - - ,d): Bh(uh, 90) := f v. -—Vaso — wet/Lune — / (1::th — [(71:th (6.6) fl I‘+ I“ = — /2iBe$p(—ifib)¢ V90 6 HIKQ) (6-7) I‘+ where 80, h > 0 Eh = in Hh, 8+, h < O 5;. = e in Q\II,, By the reflection and transmission coefficients and the definition of DIA: (I‘), we have 14 $2 d / / 21 II). h>0 / / / 42: 0 t1 t1+h 1 Figure 1.3. Geometry for the calculation of gradients _- + DlA:(I‘) = I’im 63:“ w" b) [(uh — u)e:cp(—in:c1)d:r1 I‘+ —> 27th We define the adjoint TE problem, seeks v 6 H1161) such that Bop. v) = (v. f“) + (so, f') for all so 6 H362) where fit 6 Hp‘1/2(I‘i). Let to be the solution of the adjoint transmission problem B(<,o,w) = fcpexp(—inx1)d:r1\7’cp 6 H1162). I‘+ Then obviously h‘1 [(uh — u)exp(—inx1)d:r1 = h’lB(uh — u,w) I‘+ = h_1(B(uh,W) - Bh(uh,W))) = h‘1/w2u(€h — €)uhu’1 n = one. — 5+)|h|’1 / We nu 15 (6.8) (6.9) (6.10) (6.11) (6.12) (6.13) (6.14) we can prove lim |h|'1 fuhu‘) = /U1I}d$2. h—iO 11,. 21 Together , we have the formulas ([17]): Zn (6.15) _1 j-1 _ Din:(P) = ( ) e‘flibwzmeo — 5+) fuwidxgj = 1, - - - ,m — 1 (6.16) E. J 1 . DmA: : geflflguflpko — 5+) juwidml 2m 16 (6.17) CHAPTER 2 Guided Mode Grating Resonance Filters (GMGRF) 2.1 Introduction The anomalies of optical diffraction gratings have been of interest since they were discovered by Wood in 1902. They manifest themselves as rapid variation in the intensity of the various diffracted spectral orders in certain narrow frequency hands. They were termed anomalies because the effects could not be explained by ordinary grating theory. There are two principal types of anomalous effects; the Rayleigh type, which is the classical Wood’s anomaly, and the less common resonant type. The Rayleigh type is due to one of the spectral orders appearing (or disappearing) at the grazing angle ( propagating along the surface). Note that for fixed 6* and incidence angle 0, condition (2.7) is violated for a discrete set of frequencies (12,, w,- —> oo, referred to as Rayleigh frequencies and corresponding to physically anomalous behavior first observed by Wood. me the efficiency formula, it is natural to expect that efficiencies will be redistributed when the new propagating mode appears or disappears. The resonant type anomaly is due to possible guided modes supportable by the 17 waveguide grating. One of the most interesting new deveIOpments in diffractive optics involves the integration of a zero-order grating with a planar waveguide to create a resonance. Taking the form of a planar dielectric layer with the grating providing a periodic modulation of the dielectric constant in one or more of the layers, such structures have been demonstrated to yield ultra narrow bandwidth filters for a selected center wavelength. I V. l Region 1 surrounding 53."... Region 2 core EH EL 611 EL Region 3 _ A _. substrate as“), Figure 2.1. Waveguide (Single layer) A conceptual structure representing a GMGRF is illustrated in Figure 2.1. Region 2 consisting of a planar thin film, separates two homogeneous half-space. The upper half-space is the “surrounding” and designated region 1; The lower half-space is the “substrate”, called region 3. Electromagnetic radiation (“light”) in the form of a polarized plane wave can be incident on region from either half-space. Region 2 is constrained to have two Special properties: 1. it must satisfy the requirement of a planar waveguide and have an average refractive index greater than the refractive indices of both half-spaces. 2. it must have a periodically modulated dielectric function 18 Thus, in addition to being the core of a waveguide that supports guided modes, region 2 is also a grating. For a given incident plane wave of wavelength A, incident angle, and polarization, it is possible to find a grating period A such that a first diffractive order of the grating couples to a guided mode of the waveguide. By arranging the grating to support only the zero propagating order, energy of the guided mode diffracted out of the core can only lie along the direction of the incident wave, and through this coupling a resonance is established which can lead in principle to 100% reflectance in a very narrow spectral bandwidth, as illustrated in Figure 2.2. 1 I I I I I I l I 1 0.9 - _ A = 0.3 0.8 - d = 0.131 " _. f = 0.5 _ 0.7 0 z 0 0.6 " €H=4.4 " - - EL = 3.6 _ REff 0.5 53“":231 0.4 ‘ 5311b — 2.31 ‘ 0.3 L _ 0.2 r- _ 0.1 e A O 1 l 1 1 M 1 1 % J 0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.6 Wavelength Figure 2.2. Resonance example The resonant wavelength is determined primarily by the grating period, and the bandwidth primarily by the modulation of refractive index in the grating. Furthermore, for wavelengths outside the resonance region, the structure appears 19 “homogenized” in its dielectric properties. Thus it may be considered approximately as a simple thin film structure with reflectance properties described by well-known thin film expressions ([12], section 1.6). In particular, it is possible to achieve antire- flection conditions in the thin film structure away from the resonant wavelength. With such extraordinary potential performance, these “resonant reflectors” have attracted attention for many applications, such as lossless spectral filters with arbi- trarily narrow, controllable line width, efficient and low-power optical switch elements, polarization control etc. 2.2 The reflectance in thin film For wavelengths outside the resonance region, the structure appear “ homogenized ” in its dielectric properties, and thus it may be considered approximately as a sim- ple thin film structure with reflectance properties described by well-known thin film expressions. Region 1 n1 2 fl Region 3 n3 2 \/€_3 V J. J Region 2 n2 2 $53 02 Figure 2.3. Propagation of an electromagnetic wave through a homogeneous film. 20 Define fl = 2—7rn1dcos 01 (2.1) /\o p.- = meow.- (j=1.2.3) (2.2) According to the Fresnel formulae, we have, for a TE wave, n2 cos 02 — n1 cos 01 p2 — p1 7. = _ 2.3 21 n1 cos 61 + n; cos 02 191 + 192 ( ) 2712 COS 92 2112 t21 = = (2-4) 71.1 c0301 +n200862 191 +132 n c030 — n cos0 — 7'13 : 1 l 3 3 = pl 1’3 (25) n1 cos 01 + 71;, cos 93 111 + 1):; 2n c030 2p t13 = 1 1 = 1 (26) n1 COS 91 + 71.3 COS 63 [)1 + 193 (2.7) The formula for r and t become : 7‘21 + T1382”, (2 8) 1 + 7'217‘13C2ifi . : t21t13€2w (2 9) 1 + T21T1382i'6 . The reflectivity and transmissivity are therefore given by R: |T|2 : T21 +27%; +27217‘13C082fi (210) 1 + r21r13 + 2r21r13 cos 23 T: Ell 27l3COS€3 tgltIB (2.11) {)2 n2 COS 62 1 + 7317?;3 + 27‘217‘13 COS 23 We first note (2.10) and (2.11) remain unchanged when 6 is replaced by [3 + 7r, i.e., when d is replaced by d + Ad, where Ao Ad = ———. 271.1 COS 61 Hence the reflectivity and transmissivity of dielectric films which differ in thickness A by an integral multiple of —0— are the same. 2721 COS 01 21 If we set on : 4cos01’ (m =0,1,2,---) (2.12) we find from (2.10) that R’ = 0 (2.13) We must distinguish two cases: 1. When m is odd, _ A0 3A0 5A0 .. _4cosi91’4cosi91’4cosdl’ (2.14) then cos 25 = —1 and (2.10) reduces to 7‘21 — 7‘13 2 R: -———-—— . 2.15 (1 _ 7,217.13) ( ) In particular for normal incidence, one has 2 mm — n1 2 (112713 + n?) ( ) 2. When m is even, i.e. when the Optical thickness has any of the values _ A0 210 3A0 — 2cosOl’ 200361’ 2cost91’ then cosfi = 1 and (2.10) reduces to _ 21.211: 2 1 + 7'217‘13 R (2.17) In particular, for normal incidence, this becomes 712—713 2 2.18 7124.713) ( ) R=< 22 Next we must determine the nature of these extreme values. After a straightforward calculation we have that maximum, 1f (—1)"‘(n2 — n1)(n1 — 12.3) > O. (2.19) minimum, if (—1)'"(n2 — n1)(n1 - n3) < 0. (2.20) Example In Figure 2.2, thickness d = 0.131pm is the half-wavelength and since the waveguide is symmetric (53m. = am), the almost zero reflectance is obtained. When the grating thickness is not close to a multiple of a half-wavelength deter- mined by the resonance wavelength, a filter response with an asymmetrical line is ob- tained. For parameters EH = 405;, = 361,581"... 2 10,53”, 2 2.31 and Are, = 0.609, Figure 2.4 illustrates a calculated asymmetrical line shape corresponding to thick- ness d = 0.2,um while the thickness determined by the resonance wavelength is d 2 0.16pm. 0.9-~- .. 0.8- ., 0.6-~ 0'2-.. . .. f, 0.1- 0 1 1 1 1 i i 1 0.602 0.604 0.6% 0.608 0.61 0.612 0.614 0.616 0.618 0.62 Figure 2.4. Asymmetric reflectance curve 23 2.3 Guided Modes for the Slab Waveguide n3=\/E§ 1 ”IZEAA? n2=\/52- Figure 2.5. Guided wave in the slab waveguide. We simplify the description of the slab waveguide by assuming that there is no variation in z direction, which we express symbolically by the equation 6 57: = o (3.1) Maxwell’s equations can be written in the form BE 2 = — .2 V X H 507?. 3 (3 ) and 8H E' = — — . V x at (3 3) TE modes have only three field components : Ez, Hz and Hg. Since we are interested in obtaining the normal modes of the slab waveguide, we assume that the :1: dependence of the mode fields is given by the function 6—”: (3.4) 24 By combining the two factors, we obtain e—M‘fl” (3.5) With E1. = 0, By 2 0 and Hz = 0, we obtain from Maxwell’s equations: -z',8Hy — 601:]; = iweoanz (3.6) z'flE'z = -iwpoHy (3.7) 0B,, . = — Hz . 6y wuo (3 8) We thus obtain the H components in terms of the E7, component . 0E2. Hy = —zme (3.9) 2' 6E; 35 = — 3.10 W) 6y ( ) Substitution of these two equations into (3.6) yields the one-dimensional reduced wave equation for the Ez component: 62 Ez 0:132 + (We2 — [32)Ez = 0 (3.11) with k2 = «2260110 2 (27“)2. Solve the ordinary differential equation, we must require that E; are continuous at y==0, y: ——d and vanish at y: ioo. E; = Ae‘éy for y __>_ 0 (3.12) = Acos Icy + Bsin my for 0 2 y 2 —d (3.13) 2 (A cos nd - B sin nd)e7(y+d) for y g —d (3.14) The H1. component is obtained Hz 2 (—i6/wp0)Ae'6y for :1: Z 0 (3.15) = (—in/wpo)(A sin my — B cos 53/) for 0 2 y 2 —d (3.16) = (iv/wuoXA cos nd — B sin nd)e7(y+d) for y S —d (3-17) 25 The Hz component does not immediately satisfy the boundary conditions. The requirement of continuity of Hz at y = 0 and y = —d leads to the following system of equations: 6A+ch = 0 (3.18) (rs sin red — 7cos nd)A + (It cos red + 'ysin nd)B = 0 (3.19) This homogeneous equation system has a solution only if the system determinant vanishes. We thus obtain the eigenvalue equation 6(n cos nd + 'y sin red) - K:(Ic sin red — '7 cos red) 2 0 (3.20) The eigenvalue equation can be written in a different form: rah + 6) n2 — 76 tan red 2 (3.21) where K 2 (50k? _ [32)1/2, ’Y Z (32 " €1k2)1/2, 6 = (52 _ €3k2)1/2. 26 2.4 Design Methodology Design requirements are specified in terms of the resonant wavelength, Am, at which maximum reflectance occurs, the spectral bandwidth of the resonance AA, as well as the polarization and angle of the incident wave, and possibly the “out-of-band” reflectance away from resonance. Given refractive indices for the surrounding (region 1) and the substrate (region 3), a thin film material must be chosen with refractive index greater than both. If AR (antireflection) prOperties are desired, then a multiple thin film structure is generally required for the waveguide. Since region 2 is also a grating, it must have a periodically modulated dielectric function. With these modulated dielectric functions, we need to find the period A, thickness d of core, and fill factor f. 5H 5L <—;——> Figure 2.6. Design parameters If f represents the fill-factor, the effective dielectric constant for the grating film is given by (for TB modes ) €eff=f*€H+(1-f)*€1. This value of Eeff is then used in ordinary thin film design expressions for the AR stack and in the calculation of the waveguide eigenvalues. 27 The calculation of the waveguide eigenvalues makes use of standard expressions for planar waveguide and can be found for a simple structure in section 2.3 and for a multilayer structure in [34]. Zero-order gratings are most effective in coupling incident radiation to a guided mode. Although it is possible to achieve a resonance with a higher order grating, the existence of more than one channel for light to pr0pagate makes it diffith to get high reflectance, and in practice we constrain the design only to zero—order gratings. Intuitively, the coupling between the grating and waveguide is realized by equation the first-order wave vector of the grating to a wave vector of a guided mode, viz., A . [3 = k(n0 sinfliiX) mode matching condition, (4.1) where = guided mode eigenvalue 2 axial component of guided mode wavevector A = wavelength of incident wave in vacuum k = incident wavenumber = 3f 0 2 angle of incident plane wave in region 1 no 2 refractive index of region 1 A :2 grating period In the case that 0 = 0, zero-mode corresponding to z' = :tl, and the mode matching condition becomes: A = 7 (4.2) For unmodulated (5(at) = 56”) slab waveguide the eigenvalue equation for guided waves is rah + 6) n2 _ 75 (4.3) tan(K.d) = 28 where d is the thickness of the slab waveguide, K : (€0,172 _ ,62)1/2 ,7 = (62 _ 51k2)1/2 (5 = (,82 _ 6.3k2)1/2 The dielectric modulation is adjusted most easily by the fill factor of the grating. Since the grating supports only zero order, the period is smaller than the wave- length, A < Am. With the eigenvalues calculated for the effective structure, one can then estimate the required grating period using (4.1). The design is not complete at this point because the expressions are only approx- imately true being rigorous in the limit EH — 5;, —> 0 [24, 33, 34]. To complete the design, one must calculate the performance of the structure using a rigorous Maxwell solver code to get accurate values for the resonant wavelength, spectral bandwidth and spectral reflectance. We will study the resonant wavelength in the following sections. These values are then compared with the desired values, and if they exceed spec- ified tolerance, the entire procedure is iterated until a satisfactory structure is found. 29 Input Design Values Polar. w 0 1 Are... AA Multilayer Waveguide Solver - homogenized grating index - estimate A with mode matching 1 Rigorous Maxwell Solver - input actual structure with A - calculate resonant A, AA Figure 2.7. Design methodology for GMGRF 30 2.5 The dissipative diffraction problem The direct determination of the resonant frequencies appears to be difficult. However, the study of the singularities of the scattering matrix is easier and provides valuable information about the resonant frequencies which stay along the real axis in the vicinity of these scattering frequencies (see Figure 2.8). 1 0.9 L 0.8 0.7 L 0.6 0.5 0.4 0.3 r 0.2 l I T I I L l l .L l l 0.1 .. 0.522 0.523 0.524 0.525 ° 0.526 0.527 0.528 O: singular wavelength *: Resonant wavelength Figure 2.8. Resonance wavelength and singular wavelength For sake of simplicity, we assume that only the zero order propagates, with am- plitude A0, which is dependent on w, and the grating shape. If we assume the shape is fixed, A0 is dependent on 02 only, and we denote it by A0 (0:). We consider the ex- tension of the problem to the complex plane. The research of poles for Ao(w) implies that we try to see if it possible to find such solutions of Maxwell equations without any incident wave (homogeneous problem). If we find a value 0), for which the prob- lem has a non trivial solution, it means that there exist finite coefficient A0 without any incident wave, which has the consequence that, for a given incident wave with 31 a complex frequency 02, A0 is infinite. For 0).. lies not far from the real axis in the complex plane, when 02 passes near the real part of 0)., which is closed to the singular frequency 02., the coefficient A0 takes very high values and gives rise to the resonance phenomenon. For 5(3) 2 n2(a:) E R, consider the so-called dissipative problem obtained by extending the physical diffraction problem to complex values of w E C, let u = 0.12 E C. The variational formulation becomes: au(u,v) = — [2264/3511 V1) 6 H1162) (5.1) [‘+ We rewrite it as: (I + T(u))u = L; in H1162) (5.2) where a..(u,v) = [(Vau-Vafi—Vepufi)— /T+(u)u17— [T’(u)u1‘) o H I._ = /[Vu ' V17 + (Ial2 — 115101117 — 210011117] — /T+(V)m7 — /T'(u)uf) N I‘- n (T(V)u, v) = [(Vau - V5.17 — Vu - V17 — (V511 + 1)u17) — /T+(u)m7 — /T“(u)u17 n r+ I‘- 2 [Hal2 — (us/1 + 1)u17 — 2150105] — /T+(u)u17 - [T‘(u)u17 (5.3) a ‘ r+ I‘- (Lav) = - / 2ifi'im’v P'f' Let A(V) : H1} (9) —> Hgm) be the continuous operator associated to the sesqulin- ear form a” (A(u)u,v) = au(u,v) for u, v E H;(Q) 32 Proposition 2.5.1 For any complex number 11, the operator A(V) admits a Fredholm decomposition, i. e. we have A(u) = 80/) + C(11), where B(u) is an automorphism of H; (9) and C(11) is a compact operator on H; (9). Proof: Set a” = by + cu, where the sesqulinear forms by and c., are defined on H; ((2) x 115(9) by: b,,(u,v) = [(vu . V27 + u . 5) — / T+(u)m'2 — / T“(V)m7 n r+ r- cy(u,v) = — [(ueu +1 — |a|2)u-17 + ia(u0117 — 811127) a Let B (V) and C(11) be the continuous operator of H1162) associated to the sesqulin- ear forms bV(-, ) and c,,(-, ). Then it is easy to check that B (V) is an automorphism and C(11) is compact operator. | Proposition 2.5.2 ([9]) T(l/) is holomorphic in the domain C\R‘. A(V) is an analytic perturbation T(u) of I. From now on we shall denote (I + T(1/))’1 by R(u). 2.6 Scattering frequency V. is scattering frequency, if there exists u 75 0 such that A(V.)u = 0 (6.1) i.e., ay,(u, v) = [{Vau - V617 - menufi} — /+ T+(V)m7 — [T' (u)m7 = 0 o F r- for V2) 6 H1(Q). (6.2) 33 (I + T(V))u = 0 in H;(Q). (6.3) Proposition 2.6.1 The scattering frequencies V. of the diffraction problem are the solutions of the nonlinear eigenvalue problem (6. 3). Proposition 2.6.2 If Im (V) > 0, the problem (5.1) has a unique solution. Proof: Suppose A(V)u = 0, then au(u, v) = 0 for V1) 6 H162). au(u, u) = [(Vau - V017 — Veuui‘J) — /T+(V)ui‘) — /T"(V)u17 = 0 o 1"+ P- Im (au(u,u)) = _ I{1m (14mm? — Im (I); T+(V)uu) — Im (I) T‘(V)uu) = -{f21m (VIENUIZ -F( ZRBWDUIBMW -rf_ ZRe(fi;)u;e‘"“fi = —({Im (Vue)]ul2 — Re Zflilu: 2 - Re 23.7%? Im (V115) > 0 and Ree,i 2 o All terms of this expression vanish since they are non-negative. We have u(:1:) = 0 for a: E (2. Corollary 2.6.1 The resolvent operator R(V) = A‘1(V) is holomorphic in the half plane Im (V) > 0. i.e., The scattering frequency is the half plane Im (V) S 0. 2.7 Numerical solution for scattering frequency By V), we denote a finite~dimensional subspace of H1} (Q). We will find the scattering frequency V approximation V), in Vh. Let X denote a complex Banach space and L(X ) denote the set of bounded, linear operators on X. For pencil [A, B] of operators in L(X) let p[A, B], o[A, B], and PUMB] denote the resolvent set, the spectrum, and the point spectrum defined by 34 p[A, B] = {A E C : A — AB is boundedly invertible } 01A. Bl = C\p[A. 3] P,[A, B] = {A E C : A — AB is not one-to—one } Q..[A, B] = {A e P,[A, B] : dim Ker[A — AB] < +50} For each A in some domain D in C, let F(A) be a linear, compact operator on X and let F be holomorphic in A. We call 5 E D a nonlinear eigenvalue if A(5) 2 I —F(£) is not one—to—one. Theorem 2.7.1 ([16, 25]) Define a(A) to be the dimension of Ker[I — F(A)]. Then a(A) is constant on D except at a countable number of isolated points. If a(A) = 0 for at least one point of D, then I — F (A) is bounded invertible in D except at a countable number of isolated points. Let g be an isolated nonlinear eigenvalue of I — F (A). Let ll), be a bounded projection from X onto X", a finite dimensional subspace of X, and assume that {11),} converges pointwise to the identity I, II), ——> I. point wise as h —> 0 (7.1) It is obvious that E is an eigenvalue of I — F ({ ) if and only if 0 E Q0[I — F({), I]. Theorem 2.7.2 [16, 25] Let 'y > 0 and Let F (A) be an L(X) valued function which is holomorphic on D = {IAI < '7} in the complex plane C. For 5 E D assume that 0 E QalI -F(§), I] is an isolated point with algebraic multiplicity m. Let U be an open set which isolates 0 from the remainder of OH - F (E ), I]. Then there is a positive 6, a positive integer p, and a positive integer k _<_ m so that for IA — {I S 6, the following hold: 35 1. U flo[I — F (A), I] consists of m eigenvalues [11(A), - . ~ ,,u,,,(A) counted according to multiplicity, k of which are distinct; each function u,(A) is a holomorphic function of the principal value of the p-th root of A and satisfies u,(€) = 0; the average MA) = {”l"+";:"'"m} is holomorphic in A. 2. for h sufficiently small U no[I — HhF(A),I] consists of m eigenvalues u1(A;h),--- ,um(A; h) counted according to multiplicity; the average fih(A) of these eigenvalues is holomorphic in A; 3. u,(A; h) converges uniformly to u,(A) as h —> 0. With the above Theorem, we can find 6,, such that flh(£h) = 0. To compute it), the algebraic multiplicity m must be known. Given A, we first compute the m eigenvalues u1(A;h),-- . ,u,,,(A;n) of I -— IIhF(A) closest to 0, and (A;h)+---+pm (AV!) . then the arithmetic mean ah(A) = ”1 m Let us consider the homogeneous dissipative problem: [I + T(V)]u = 0 (T(V)u, v) = [(Vau ~ Vat") — Vu - V17 — (Veu + 1)uv) — /T+(V)uv — /T“(V)m7 r+ I‘- ll :3\ :3 [[01]2 — (Veu + 1)uv — 2iaaluv] — /T+(V)uv — /T_(V)uv I‘+ 1‘- (Ju, v) = Vqu + uv (7.2) l T(V.)U(V..) = A(V.)JU(V...) (7.3) 36 It is obvious that scattering frequency is the solution of A04) : —1 (7.4) The solution of (7.4) by Newton’s method requires the derivative A’ (V) of A(V). We assume that A(V) is a simple eigenvalue of A(V), Proposition 2.7.1 X0!) = (T'(V)U (10,901)) where g(V) is the associated eigenvector of the adjoint operator, i.e., satisfying T’(V)9(V) = A—(V—NQM and (JU(V).9(V)) = 1 proof: WV) = (T(V)U(V).9(V))’ = (T'(V)U(V).9(V)) +[(T(V)U'(V), g(V)) + (T(V)U(V). 9’(V))l (T(V)U’(V),9(V)) + (T(V)U(V),g’(V)) = (U ’(V).T‘(V)9(V)) + (A(VN U (V).9’(V)) = (U'(V)./\—(VlJ9(V)) + (A(V)JU(V),9’(V)) = A(V)(JU’(V),9(V)) + A(V)(JU(V),9’(V)) = A(VXI)’ = 0 We can rewrite the eigenvalue problem as: 37 Find V, such that there exists u 75 0, [Vu - V17 + Ia(V)|2uv — 2ia(V)01uv — /T+(V)u27 — [T—(V)’U.'l_) = V/euuv (7.5) n r+ r- n It is equivalent to [Vu - V27 + [alzuv — 2ia61uv - /T+uv — [T710 2 r(V) leuuv (7.6) n r+ r- n r(V) = V (7.7) It is a nonlinear eigenvalue problem A(A):z: = ABcz: (7.8) where B S.P.D. We can solve it by simple iteration, i.e., given we, r(wn) = can“ The numerical experiment shows that the method converges, but we could not prove the convergence. We can solve the nonlinear equation r(V) — V = 0 by secant method. It is faster than simple iteration, since the convergent rate 1.618. Example For parameters A = 0.3um ( period), d = 0.125um ( thickness of core), f = 0.5 (fill-factor), 5;; = 4.4,eL = 36,133,," = 1,531.1; = 2.31 and theta = 0, we found 2 eigenvalues in [05,06], A1 = 0.51152267, A2 = 0.51362098. At A1 = 0.51152267, the reflective efficiency is 0.99993363, it is a resonant wave- length, see Figure 2.7. At A2 = 0.51362098, the reflective efliciency 0.1445, it is not a resonant wavelength, see 2.9. We notice that there is a small “bump” near this eigenvalue. 38 0.6" .. . . L . ,.. .. .... 0,3,. . .' 01-» 0.44 0.46 0.48 0.5 0 52 0.54 0 56 0.58 0 6 0.24 v 7 I r 1 01 l 1 1 l I 0.513 0.5132 0.5134 0.5136 0.5138 0514 0.5142 Figure 2.10. eigenvalue but not resonant wavelength 39 2.8 Design of GMGRF With the function r( f, d, A) which correspond the given grating structure to the resonance frequency ( wavelength), we may consider the design problem. Given permittivity constants €3,5L,e,,m. and cm, at first, we use fill-factor as 0.5. The grating film thickness is quarter-wavelength, or a multiple thereof. Solve the waveguide eigenvalue equation, we can decide the period of grating. Next step, we need to compute the resonance wavelength and bandwidth. It is hard to get accurate values of the spectral bandwidth. But it is easy to know whether the bandwidth for the structure is greater or less than a given data. These values are then compared with the desired values, and if necessary, the fill-factor will adjusted to get the the specified wavelength. This is to find zero of a nonlinear equation. After adjusted the fill-factor, we consider the reflectance out of resonance region. If necessary, we can change the thickness of the grating film. This is an optimization problem. Given: ewrnesmemeL, 6 ( incident angle) 1. Waveguide solver Initial data : f = 0.5, fill-factor. mA 4v€eff A From mode-matching condition. thickness of core, m = 1, 2, - - - (a) Waveguide eigenvalue Solve (3.21) for B (for single layer). It tan d=-—,———— K 152—76 40 where I6 = (50162 _ fi2)1/2 7 = (,62 _ €1k2)1/2 (S ___ (:32 _ €3k2)1/2 (b) mode-matching condition Solve (4.1) for A. 2_1r 6 = 0, zero-mode —> A = fl If A is not in the interval (0, A), we need to change the material. 2. Rigorous Maxwell Solver Input period A, calculate resonant wavelength A 3. iteration modify fill-factor (first), thickness or period. From initial start value f0, do, A, we need solve the following two mathematical problems: 1. Solve nonlinear equation r(fj-l-l) dj, A) = Ares This function is a real valued function, we use Van Wijingaarden—Dekker-Brent method. It combines the bisection and secant methods, providing a synthesis of the advantages of both. 41 2. Optimal problem: It is hard to get the accurate value of bandwidth. In fact, we don’t need to know the accurate value of bandwidth. We can specified a desired bandwidth AA, compute the reflectance at wave- length A + AA and A — AA, then we know whether the actual bandwidth is satisfied or not by comparing it with AA. At wavelengths A + AA ( and/ or A — AA), we can compute the reflectance e0 by solving the direct diffraction problem. If it is not satisfied, then at wavelength A = Ac + AA (A = AC — AA ) solve the following optimal problem. Solve dj+1 80(fj+11 dj+la A) : mdin 60(fj-1-11 d) A) For this optimization problem, we can use the gradient formula described in section 1.6. The above procedures iterated until a satisfactory structure is found. In numerical experiment, this procedure is found to converge quite quickly. 42 2.9 Numerical example Given 5,, = 2.12 = 441,51, = 2.02 = 40,5..." = 1.02 = 1.0, a... = 1.522 = 2.3104 and 0 = 0. The specified resonant center is Am 2 0.54 . . . 0.54 , 1. First we need to estimate thickness, d — 2 * (2.1 * 0.5 + 2.0 * 0.5) —— 0.132 Solve (3.21) for B, we have B i 20.423, then A = ’2?“ é- 0.307 2. For this structure (A = 0.307, d = 0.132, f = 0.5), we found that the eigenvalue is 0.5385 ‘ 7 . ' ' r ' . r . 0.9- . O,8*-~: . . .... 0.6» 0,4. . .. .. .. . . .1. .. . .4. . .. ............. 0.1- )L _, 0 l 1 l l 1 l l 1 l 0.5 0 51 0 52 0 53 0.54 0 55 0.56 0.57 0.50 0.59 0.6 Figure 2.11. reflectance curve for initial data Solve nonlinear equation (with A = 0.307, d = 0.132 ) r( f ) = 0.54 The solution is f 2': 0.714. 43 With fill-factor f = 0.714, the reflectance curve is given by Figure 2.12: 1 l m _r r I 7 f 0.9” : " 03" .1 0,. x 0,5... . .. .;. . . _ .. . . . 0,4» . .1. 5, I .' . f. 1 0.2» , .. .. ‘, . o i . 1 . . 1 ' . 1 1 0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0 58 0.59 0.6 Figure 2.12. reflectance curve 3. Consider AA = 0.015, At A = Am, + AA = 0.5415, Reflectance is 0.0389. At A 2 Am — AA = 0.5385, Reflectance is 0.0591. We need to solve the minimization problem at wavelength A = 0.5385 (with A = 0.307, f = 0.714): min eo(d) We got d = 0.131 For this structure (A = 0.307, d = 0.131) , solve nonlinear equation r( f) = 0.54 again, the new fill-factor is f = 0.741. 44 At A = Am + AA = 0.5415, Reflectance = 0.041 At A 2 Am —- AA = 0.5385, Reflectance = 0.048 We may stop with the parameters as A = 0.307, d = 0.131, f = 0.741. 0.9» , q 04~ ,. ....... 0.3-. ..... .......... . - l l l l l l l 0.5 0.51 0.52 0.53 0.54 0.55 0.56 0 57 0.58 0.59 0.6 Figure 2.13. reflectance curve for the design example. 45 CHAPTER 3 Optimal design of nonlinear grating 3.1 Introduction Consider a plane wave of frequency 021 incident on a grating or periodic structure con- sisting of some nonlinear optical material. Because of the presence of the nonlinear material, the nonlinear Optical interaction gives rise to diffracted waves at frequencies 011 and 022 = 20.21. This process represents the simplest situation in nonlinear Optics - second harmonic generation (SHG). An exciting application of SHG is to obtain coherent radiation at a wavelength shorter than that of the available lasers. Unfortu- nately, it is well known that nonlinear optical effects from SHG are generally so weak that their observation requires extremely high intensity of laser beams. Effective en- hancement of nonlinear Optical effects presents one of the most challenging tasks in nonlinear optics. The present research is concerned with important aspects for systematically design of surface (grating) enhanced nonlinear Optical effects. Recently, in a sequence of papers [29], [30], [27], a PDE model based on Maxwell’s equations has been introduced to model nonlinear SHG in periodic structures. In particular, it has been announced in [29] and [30] that SHG can be greatly enhanced by using diffraction gratings or periodic structures and the PDE model can predict the field propagation accurately. 46 Our goal is to provide the mathematical foundation of Optimization methods for solving the Optimal design problem of nonlinear periodic gratings. By conducting a perturbation analysis Of the grating problems that arise from smooth variations of the interfaces, we derive explicit formulas for the partial derivatives of the reflection and transmission coeflicients. Such derivatives allow us to compute the gradients for a general class of functionals involving the Rayleigh coefficients. Optimal design of periodic grating has recently received much attention [1], [3], [6], [17], [18]. For linear grating structures, significant results have been Obtained by Dobson [15](weak convergence), Bao and Bonnetier [3](homogenization), and Eschner and Schmidt [17], [18] (optimization). To our best knowledge, the present work is the first attempt to solve the Optimal design problem of nonlinear gratings. Little is known concerning the questions of existence and uniqueness for nonlinear Maxwell’s equations in periodic structures. In two simple cases, where Maxwell’s equations can be reduced to a system of nonlinear Helmholtz equations, existence and uniqueness results have been Obtained recently in Bao and Dobson [6] and [7]. Computational results have also been obtained by using a combination Of the method of finite elements and the fixed point iteration algorithm. More recently, a more general model has been studied by Bao and Chen [4]. Their model supports a general class Of nonlinear Optical materials with cubic symmetry structures. Our present work is devoted to study the Optimal design problem for this model problem. A good background on the linear theory Of diffractive Optics in grating structures may be found in Petit [28] and Bao, Cowsar and Masters [5]. For the underlying physics of nonlinear Optics, we refer the reader to the classic books of Bloembergen [11] and Shen [31]. 47 3.2 Modeling Of the nonlinear scattering problem Throughout, the media are assumed to be nonmagnetic with constant magnetic per- meability. For convenience, the magnetic permeability constant is assumed to be equal to unity everywhere. Assume also that no external charge or current is present. The time harmonic Maxwell equations that govern SHG then take the form: VXE = EEH, V-H=0, (2.1) VxH = —’—:’—D, v-D=0, (2.2) along with the constitutive equation: D = 6E + 4nx(2)(a:,w) : EE , (2.3) where E is electric field, H is magnetic field, D is electric displacement, e is dielectric coefficient, c is speed of the light, 02 is angular frequency, X0) is the second order nonlinear susceptibility tensor Of third rank, i.e., X0) : EE is a vector whose j-th component is 133:1 xgi]EkEz, j = 1, 2, 3. Remark 2.11—The medium is said to be linear if D = 6E or X0) vanishes. In principle, essentially all Optical media are nonlinear, i.e., D is a nonlinear function of E. The physics Of SHG may be described as follows: when a plane wave at frequency w = wl is incident on a nonlinear medium, because of the interaction Of the incident wave and nonlinear medium, diffracted waves at frequencies 0) = 021 and w = 2021 are generated. The fact that new frequency components are present is the most striking difference between nonlinear and linear Optics. However, for most media, nonlinear Optical effects are so weak that they may reasonably be ignored. In particular, the conversion of energy into the new frequency component is very small. The Observation 48 of nonlinear phenomena in the Optical region normally can only be made by using high intensity beams, say by application of a high intensity laser. Assume that the depletion of energy from the pump waves (at frequency w = 011) may be neglected, which is the well known undepleted pump approximation in the literature, see [29] and [30]. Under the approximation, equation (2.3) at frequencies 02 = w] and w = (.02 = 2021, respectively, may be written as D(z,w1) = c(m,w1)E($,w1), (2.4) D(a:,w2) = e(x,w2)E(a:,w2) + 47rx(2)(x,w2) : E(:I:,w1)E(a:,w1) . (2.5) We then reduce the nonlinear coupled system (2.1) and (2.2). Throughout the paper, all fields are assumed to be invariant in the 1:3 direction. Here, as in the linear case, in TE polarization the electric field is transversal to the (ml,zg)-plane, and in TM polarization the magnetic field is transversal to the (1:1, x2)-plane. In the non- linear case, however, the polarization is determined by group symmetry properties Of xlz). In this work, motivated by applications, we assume that the electromagnetic fields are TM polarized at frequency 021 and TE polarized at frequency 012. This po- larization assumption is known to support a large class Of nonlinear Optical materials, for example, crystals with cubic symmetry structures. See Appendix for additional discussion. Therefore Hlmlwll = H($1,$2,w1)53, (2-6) E(.’L‘,L¢)2) = E($1,$2,w2)f3. (2.7) Define for convenience e, = €($1,$2,w]'),j=1,2, (2.8) k.- = “’— 5. Imzo,j=1,2. (2.9) 49 The system (2.1), (2.2) at frequency 0.21 can be simplified to V-(E15VH)+H=O. (2.10) 1 Because of Equation (2.10), c E(a:,w1) — ”IEIV >< H(:1:,w1) (2.11) c _ Wm (3,211, ~611H. 0) . (2.12) Hence the second harmonic field satisfies 477012 [A+k%lE = “672- Z xii-km,w2> 0, I m nJ-g 2 0. The case I m njg > 0 accounts for materials which absorb energy. We assume that nj0(:c) are piecewise constant functions in $20 satisfying Re njo > 0, I m njo Z 0. We wish to solve the system (3.8) and (2.13) when an incoming plane wave u; = u,-e’°‘1$1—’l’“‘2 (2.15) is incident on SI from I)? where u,- is a real positive constant, (11 = kn sin 0, 51 = kn cos 0, kn = Eel-nu, and —§ < 0 < g is the angle Of incidence. We are interested in “quasi-periodic” solutions (H, E), that is, solutions (H, E) such that _ H —ia1:c1 d _ E -i021'1 _ k - 6 k _ “)2 u— e an v— e (02— 213m , 21——c—n21) are 277-periodic in the 231 direction. 51 It follows from the system (3.8) and (2.13) that Va, - (imp) +u = 0 , ki (A0, + k§)v = Z pgafm arm, j,l=l,2 where A02 = A + 22028;, — [C12]2 , Val = V + i(01,0) and a i2a—a :1: a _ 1 a _ pjlzpjle( 1 2)1) all—631+7’ali a2l—a$2° Define, for j = 1, 2, the coefficients 12 /. 62, 52%“) = ei‘m/2 [kg]. _ (n+ a1)2[ n , . 1/2 239(4) = .121/2[1.g,_(n+a,)2| , nez, where 71]. = arg(kfj — (n + al)2), 0 S 71,- < 27r , 721' = 31'8“; " (n ‘l' a2)2), 0 S 72,- < 27r . (2.16) (2.17) (2.18) (2.19) (2.20) (2.21) (2.22) Throughout, assume that 15% yé (n + (11)2 and 1:3, 75 (n + 012)2 for all n E Z, j = 1,2. This assumption excludes the “Rayleigh anomalous” cases where waves propagate along the 2.1-axis. For function f E H %(I‘j) (the Sobolev space of complex valued functions on the circle), define the Operator T3- by (Tf'ijxl) = Z -ifi§}’)(a)f(")e’"“. 7162 (2.23) 211' , for s, j = 1, 2, where f (n) = i f f 2:1 e“”"1 and the equality is taken in the sense Of 21r 0 distributions. 52 From (2.23) and the definition of fig?) (0), it is clear that T8“;- is a standard pseudo- differential Operator (in fact, a convolution Operator) of order one. The scattering problem can be formulated as follows [4]: Va, - (évam) + u = 0 in Q, 1 (Am + k§)v = Z pfic’if‘u Bf‘u in $2, j,l=1,2 ‘ a a - —i[3 b (T11 + -—)u = —2zu,-Ble 1 on F1, 6V 0 6 ( 12 + 5;)u = 0 on 1‘2, 0 0 (T21+ 5;)v = 0 on P1, (T35 + 567% = 0 on F2. Integration by parts results in the variational relation: 1 —— _ 1 a _ 1 a _ BTM(U,80) =/—2' Val” ' Val? — [“90 + T/(Tnu) 90 + T/(T12U)cp 0 191 9 19111.1 1612,.2 _ 2211.961 8—w’b (€11 [95. VwEHflQl F1 8130,90) = / v..v - —va.so — / 13w + [(1‘sz + [(213072 (2 0 F1 P2 = _ Z pfl/ei(2a1-az)x1 aymaf’mo , V90 6 HIKQ). j,l=l,2 no (2.24) (2.25) (2.26) (2.27) (2.28) (2.29) (2.30) (2.31) Here H562) contains the functions of H 3 (0) that are 27r-periodic in the $1 direction. Note that usually the medium above the grating is air with optical index n1 ,- = l, which is independent of the wavelength. Thus 02 = 201 and p?) = pJ-z for all incidence angles, which simplify some of the formulas given below. In the following, assume that the functions nJ-o (:5) are constant on subdomains Q,- with piecewise smooth boundaries 60,-. The angles at the corners of Q,- are strictly between 0 and 277. Also, denote by AZUan\(F1UP2) 53 the set of interfaces between different materials. Assume further that the problems (2.30) and (2.31) with vanishing right—hand sides have only the trivial solution. Then it is well known [17] that the solution u of (2.30) belongs to the Sobolev space Hiram) for some 6 E (0, 1/2). Furthermore, we have 2 pgafm 6,021 e H-1(n) (2.32) j,l=1,2 by a direct application of the following regularity result of Beals [10]. Proposition 3.2.1 Iff E H’1(R”), g E H’2(R"), s,- _<_ n/2, 31 + 32 2 0, then the product fg E H31+32‘"/2“"(R") for arbitrary 6 > 0, and ||fg|[,,+,2_n/2_5 S C(6)llfll81llgll32' In view of (2.32), we Obtain the following result. Theorem 3.2.1 Under the assumptions made above, the problem (2.30), (2.31) has a unique solution v E H; ((2). Similar to the linear diffraction problem, the energy propagation of the diffracted fields is measured by the diffraction efliciencies. The efliciencies of the second har- monic fields are given by the formula: 6—2ffiéf)b . =(n)/,62 |E+|2 with E: = 2 ‘/ve""’“”1 dml for fig?) real. 7r 1‘, e—2ifi(")b . e; = flégl/flglE; [2 with E;= 2r / ve‘mfl‘1 dscl for fig) real. F2 3.3 Optimal design Our goal is to determine (or design) grating geometries that ensure maximal efficien- cies of the second harmonic fields. The Optimal design problem may be stated as follows: Find a grating profile A0 such that 54 mlax e:(A) = ej,‘(A°) . In order to apply certain gradient based Optimization methods, it is essential to study the differentiability of the efliciencies with respect to perturbations Of the interface A. Consider a family of perturbed interfaces A), given by A), = (1)},(A) , ‘13};(27): SE + hX(1L‘), (3.1) where x = (X11X2) is C1 continuous, 277~periodic in $1 and has compact support in [0,277] x (—b, b). Clearly, for sufficiently small [h] the mapping h is a C'1 diffeo— morphism Of 9 onto itself. Consequently, 4>h(fl) corresponds to a perturbed grating geometry which yields new piecewise constant functions 5;? as well as the perturbed bilinear forms B53," and 83E. Moreover, the nonlinear material is contained in the subdomain 0" = h(flo). It follows that (n) De7=,1,ing,h-1(e7(4.)— 27(4)) = 255; Re (12.713197) . Therefore, to compute Defi,‘ with respect to the perturbation (3.1), it suffices to cal- culate the derivatives DE;L defined by e—2ifién)b DEM =72. 2.. [(vh — v) e "“1 d231, (3.2) where v solves (2.30), (2.31) and v). is the solution of the perturbed problem 2iu,-B e431” _ BTM(uh1 (10) = _ [:2 [(10 7 11 F“ (3.3) Emma) 2 p,, / Wei-“2)“ 33771,, 0711,75, V90 6 11,;(12). To compute (3.2), it is useful to employ the concept of the material derivative [32]. Using the mapping h, we introduce the isomorphism 55 which maps u to u o (1),“;1. Since X is compactly supported in (2, it is easily seen that \Ilglulpj = ulpj , j = 1,2 , Vu E H1362). Hence —2ie§“)b 277h 6 1213700=-.’1li_1’1%J I[(111, v), —v)e ”1de. (3.4) Therefore, the derivative DE: (x) is a functional of the material derivative Of v with respect to the diffeomorphisms ‘Ilh, which is defined as - —1 —1 _ [12(1) h (‘11,, v), v) . The material derivative may be evaluated by introducing a change of the variables y = (1),,(32) in the bilinear forms B§~M and 8533. Note that k]? = thj and dy = J (m)d:1: with J(z) = 1 + h(a..x. + 8.2262) + 7176.17.79...» — 6.2.6.172) and = J(.)—1((1 + ha..X2)a.. — 126.226..) , 6112 = J($)_l( - ’10,;le 0x1 +(1'l' hax1X1)azz) - Applying the change of variables to the domain integrals of 37M, we Obtain __ 1 __ - n/(- \Ilhu‘ph‘P + W VafiI’hu . Valq’hcp) dy = —h/u(pJ($) dx +/((1 + h62X2)61 + ial']($) — h01X262)u ((1 + h52X2lal - 1011,05) — halx2632)¢ ,, Jk7 +/ ( — (1522051 +(1'l' h51X1)52)u( — (1322031 +(1+ (10127062)? J (3)1920?) _ —:/(— +k2(:1: -—)1Va,uV_O,—,h(:r) to the domain integrals of the form 8553, we obtain / ( — wow/hum + vazwhv - W) dy o = l ( — 1931)? + Vazvm)dat + hBTE,1(v, (,0) + thTE,2,h(v, (p), n with BTE,1(v, so) = - / k§(61xl + 62X2)v¢+ / 61xl(62v@ — <91va + 0322 o) +/82X2(8°’v8190— 62118—290) (3.9) _ / (61x2(0f7v— 02‘? + 02v all?) + 0200(0in + 021) m» n The remainder term satisfies IBTEHW ¢)| < C||v||1||xl(agm)2 :5 + h .711(u, 9p) + h2£u(u, 90) no with Junk 90) = /6i(2al-az)xl(01X1 + 02X2 + i(201 — 02)X1)(3?"u)2¢' no —2 / ei(2"1"°2)‘“0§"u (01X161u + 01X282‘U.) ¢ no 2. Forj =1,l=2: /ei(2°"°‘2)y1((6y, + ial)\Ilhu) 63,2th \Ilhcpdy 03 : /6i(2al _02)xlaiu’lj. 62%;? ‘l‘ h .7120”, (p) + h2£12(u2 (p) no with .712(u, cp) = — / ei<20r02>h2 + alxxaufw no +i01/€i(2al_02)xlu(51X1 52” — a2X1C91u)¢ no + 7:(2C!1 — ag)/ei(2“1'02)x‘xlaf‘u 6211,? no 59 3. Forj=l=2z / ei(201-az)yi (am wk“)? ‘I’hSO dy 93 = / ei<2m-ae>n(a.u)2¢ + h .7220». so) + W220”, S?) no with .722(u, 90) = / €i(2"""2)z‘((0ixl - 02X2 +i(201 — cr2)x1)(62u)2 - 232m 01110211)? . no Thus the right—hand side of (3.14) transforms to 1'20: -a y a E pfl/e( 1 2llr'il‘l’jluh(93,111”.\Ilh'wdy .7 as jal=192 a0 +h Z pfi~7fl(‘1’;1uh,w)+h2 Z szfiijI’EluhMl j,l=1,2 jrl=192 where .721 = .712. Note that due to (2.32) obviously lfiij(‘1’;1%w)| S Cll‘I’ZluhHilIWIh S CIIIUIIi- Using Theorem 3.3.1, we arrive at Z: lefei(2“1‘“2)y‘8;1uh6;‘uh\Ilhwdy = Z pjz/6i(2al‘“2)$16;71u6?1ufidx jrl=ly2 as j,l=l,2 no +2h Z pjz/ei(2a‘"'2)110?"uaf"u1fi + h 2 pij.7jz(u,w) + 0(h2) , 11:13 no j,l=1,2 which implies B%E(vh, \Ilhw) = BTE(v, w) +h Z p.,-(.7,,(u,w) + 2/e‘<201-02>eaymarml w) + 0(h2). no j,l=l,2 Thus from (3.13), we get BTEOIIKI’U)“ ’11)) + hBTE,1(\I’;l’Uh, 11)) + thTE’g’h(\IJ;IUh, U!) = BTE('U,’w) + h 2 sz(.7ij(u, w) + 2/e‘(2°'1‘°2)“‘6f‘u Bf‘ul E) + 0(h2) , j,l=1,2 no which together with (3.12) proves the following theorem. 60 Theorem 3.3.2 The derivative of the reflection coefficients E: with respect to the variations (3.1) of the interface A is given by the formula DEflX): —BTE,1(v,w)+ Z pfl(,7,-z(u,w)+2/ei(2°‘1’°2)310?’146?m1Tu'dxXfiJS) j,l=l,2 no where the bilinear form BT33 is defined by (3.9), u and v denote the solutions of the diffraction problems (2.30), (2.31), respectively, ul solves (3.8) and w is the solution of the adjoint TE problem (3.11). Following [18], the form BTE,1(v, w) given by (3.9) can be transformed to BTE,1(v,w) = — [kilA /(X,n)vfi A + [(Amv + kgv)(x1 61—212 + X20712) + / (X161 1) + X2 62 v)(Aa,'w' + kgm) ——[k§]A/(x,n)vw+ 2 P31] ei(2a1—02)xlamu 6011,00 .0—1w+x282w) j,l=1,2 no where we have used the equation (2.25) and Aaw+lc_§w = 0 in Q, for the solution w E H 2(9) of the adjoint problem (3.11). Here n denotes the normal to the interface A, and [k§]A stands for the jump of the function 19% when crossing A in the direction on n. Thus (3.15) takes the form DEf(X)-—— [k2]A [(x,n)vw+2 Z: [2,1]‘(Qal-amlaamoamlwdz j,l= 1,2 no + Z pfl(,7,-¢(u, w) — [e'(2"““2)116;“u6f“u(xl 61w + x2 62w)) . jrl=172 no Remark 3.1. To apply the above results to binary gratings by choosing different x, we can compute the derivative DjEf of the Rayleigh coefficients with respect to the transition points. For simplicity, consider a binary grating with two transition points 61 5132 Air is t1 t2 = 271' Figure 3.2. Cross section of a simple binary gratings t1, t2 = 27r and the height t3, as shown in Figure 2. Denote 01 = (t1, 0), 02 = (t1,t3), 03 = (0,t3), and 21 = 0102,22 = 0302, the fill factor FF 2 5%. To compute the derivative D1 E3 of the Rayleigh coefficients with respect to the variation of t1, the mapping (3.1) takes the form <1>h($) = m + hx($), X01?) = (x1(x),0) , where X1 5 l in a neighborhood of )31 and x 6 08° (U) for a bigger neighborhood U (not containing other corners of the profile curve A). 3.4 Numerical examples The above described approach has been numerically tested on a number of examples in the literature. The numerical solution of the model equations is based on our generalized finite element (GFEM) discretizations of the bilinear forms BTM and 62 BTE [17]. This finite element method avoids the pollution effects associated with usual domain—based methods for solving Helmholtz equations. Since the method is restricted to piecewise rectangular sub-partitioning of the integration domain, the numerical tests have been performed for binary gratings. Also, to obtain the starting values for the optimization procedure, we have determined the grating structure which yields minimal reflection in the TM case. This is done by using gradient based minimization algorithms [17]. We then proceed to compute the derivatives with respect to the grating depth and transition points as described in Remark 3.1 by the line search algorithm. In the following, we present results on specific examples. First, we consider an example introduced in [30]. It is concerned with the grat- ing enhancement of the second harmonic nonlinear optical effects for a silver layer. Obviously, the TE efficiency (the nonlinear effect) for the flat layer is small which is confirmed by our calculated the efficiency 1.2003160E04. The TE efficiency for the binary grating with the period 0.556um, the incidence angle 645°, and the wavelength 1.06pm is then computed. With the fill-factor 0.5, similar enhancement results are obtained as those reported in [30] concerning the efficiency dependence on the groove depth. In particular, the maximal enhancement is about 45 which occurs when the groove depth is close to 0.3,um. Our computation indicates in addition that by using the above algorithm, with the same data, a better enhancement for the fill-factor 0.834 may be achieved. In fact, at the groove depth 0.392um, the enhancement is more than 80. Figure 3.4 presents the enhancement of the efficiency of the second harmonic field at various groove depths. It is shown that around the optimal depth, the enhancement depends on the groove depth sharply. The second example is concerned with the grating enhancement of the second harmonic nonlinear optical effects for ZnS overcoated binary silver gratings. Once 63 l/ 1 l 1 L 0 0.1 0.2 0,3 0 4 0.5 0.6 0.7 Groove Depth (depth m) Figure 3.3. Numerical example 1. again, the enhancement of the second harmonic fields is computed with respect to the associated flat structure. The period of the grating is d = 0.4um, the incidence angle is 28.92° at the wavelength A = 1.06pm. The optimization parameters are the thickness of the ZnS coating, the fill factor, and the depth of the binary grating. Our computation indicates that optimal results are obtained at the thickness of 0.33pm of the coating layer, the fill factor 0.43, and the depth of 0.099um for the binary grating. Figure 3.4 illustrates the enhancement dependence on the grating depth. It should be pointed out that other thicknesses of the ZnS coating provide even higher enhancements for the second harmonic nonlinear optical effects compared to the flat structure. Figure 3.5 presents the corresponding enhancement factors for the thickness of 0.672um and a binary grating with the fill factor 0.505. Clearly, the maximum value is obtained at the depth 0.03pm. However this value only amounts to 17% of the maximum for the thickness 0.33pm. 64 700 600 500 400 300 200 100 l l 1 l l l 0 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 Figure 3.4. Numerical example 2. Appendix Recall that, for nonlinear material, the second order nonlinear susceptibility takes the form P‘<’~’>(2w) : x‘2’(2w) = was) , i.e., forj = 1,2,3, Wow) = 60 zx§?2,.<2w>E.E. . k,l According to the convention Xfil = 2d§2 and by the permutation symmetry: d§i>,(2w) = (mow), define J dj 2 $2? m=1a°”)62 where k, if k = l, 9-(k+l), “7&1. 65 45000 I 1 u 1 l I 40000 35000 30000 - -‘ 25000 20000 15000 10000 5000 T l I l l l T J l J l l l l l J 0 l l l l 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Figure 3.5. Numerical example 3. Thus 13(2)(2w)=€0 d2, (126 z (w). 6131 dss ( 2E... 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