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The Theory of Planar Bragg Resonators P Technical Physics, Vol. 47, No. 2, 2002, pp. 141–147. Translated from Zhurnal TekhnicheskoÏ Fiziki, Vol. 72, No. 2, 2002, pp. 1–7. Original Russian Text Copyright © 2002 by Petrov. THEORETICAL AND MATHEMATICAL PHYSICS The Theory of Planar Bragg Resonators P. V. Petrov All-Russia Research Institute of Technical Physics, Russian Federal Nuclear Center, Snezhinsk, Chelyabinsk oblast, 456770 Russia e-mail: [email protected] Received February 12, 2001; in final form, June 8, 2001 Abstract—The reflecting properties of one-dimensional planar Bragg gratings are studied. A coupled resonator model for studying the diffraction of electromagnetic waves in an arbitrarily corrugated waveguide is sug- gested. It is based on exact relationships that follow from the two-dimensional boundary-value problem stated in terms of the Helmholtz equation. The specific relationships for the rectangular corrugation of the grating- forming plates are presented. The reflection coefficients of the Bragg gratings vs. corrugation length and inci- dent radiation frequency are calculated. An analytical solution for the “narrow” corrugation is obtained. © 2002 MAIK “Nauka/Interperiodica”. INTRODUCTION ematical model that adequately describes the diffrac- The use of Bragg resonators, which provide distrib- tion of electromagnetic waves by one- and two-dimen- uted feedback and the spatial coherency of radiation, sional Bragg gratings without invoking the perturbation seems to be a promising approach to implementing theory, has an accuracy close to that of direct numerical electrodynamic systems for free-electron masers [1–4]. calculations, and is free of awkward mathematics. Bragg resonators are waveguides with a single- or a In this work, we suggest a coupled resonator model double-periodic corrugation (Bragg gratings) [1, 5]. for the analysis of the electromagnetic fields in one- Mathematical models [1–5] used for describing the dimensional planar Bragg gratings and report their spectra of the modes and reflection coefficients in reflecting properties. Bragg gratings are based on the coupled mode theory, within which the mode coupling coefficients are found by the perturbation method [6, 7]. Namely, the EQUATION FOR MAGNETIC FIELD waveguide surface deformation (corrugation) defined by some function l(r) is replaced by the “equivalent” Consider a hollow planar waveguide with perfectly boundary condition on the undisturbed surface of a reg- conducting walls that is infinitely long in the 0Y direc- ular waveguide [7]: tion and symmetric about the plane 0X. Its undisturbed ± ω (regular) boundary is defined by the function x = Lx. Eτ = ∇()l()r En⋅ – i----l()r []nH, , (1) Let a finite number M of corrugations be patterned on c the side walls of the waveguide in the interval 0 < z < L. where n is the unit vector of the outer normal to the The corrugations are also symmetric about the plane waveguide surface; E, H ~ exp(–iωt) are the electric 0X. Let the contour of the qth corrugation be described and magnetic fields; ω is the radiation circular fre- by the function x = lq(z) (Fig. 1) quency; and ∇ is the Hamiltonian operator. The paren- () ∈ [], thesized and bracketed vectors mean the scalar product lq z , zaq, 1 aq, 2 and the vector product, respectively. x = (2) 0, za< , , za> , , q = 1, M. Approximation (1) applies [7] if (i) the deformation q 1 q 2 is small (the corrugation height is much smaller than the wavelength of the incident radiation and the corru- We assume that the Bragg grating is excited by a gation length) and (ii) the angle between the deformed natural mode of the regular waveguide from the side of and virgin surfaces is small. As a rule, waveguides used negative z (i.e., from left to right). The mode has a lon- in experiments with Bragg gratings have a rectangular gitudinal wave number hp and a transverse wave num- corrugation [1–5]. In this case, the deformation is usu- 2 2 ally small, while the angle is not. Even for a sinusoidal ber gp = k – h p . Let the TM modes of the waveguide corrugation on the surface, which is widely used in be excited (for them, only the components Ex, Ez, and numerical simulation, the angle between the surfaces is Hy are other than zero). Then, the initial problem of π/4 and can hardly be considered small. Because of finding the electromagnetic fields is reduced to the this, it would be appropriate to develop a physicomath- boundary-value problem that, in terms of the Helm- 1063-7842/02/4702-0141 $22.00 © 2002 MAIK “Nauka/Interperiodica” 142 PETROV Ω Ω X q (q = 1, M) (in 0, the magnetic field can be repre- x = lq(z); 1x = lq+1(z) sented as the sum of the incident and scattered fields); that is, Ω Ω 2 ∈ Ω a q a q+1 H0 + Hs, r 0 q1 q2 H = (8) n ∈ Ω Hq, r q, Eτ where H meets boundary conditions (4)–(6). Ω 0 0 L x At the boundary x = Lx with the regular waveguide, the tangential field components are continuous, 0 () () () Hq z = H0 z + Hs z (9) Z Y and the scattered field Hs meets homogeneous bound- ary conditions (4) and the condition that the modes Fig. 1. Planar Bragg grating: 1, contour of the qth corruga- tion; 2, contour of the undisturbed (regular) waveguide. leave the system at z = 0, L. At the interface between the corrugation and the regular waveguide, the electric field component along ≡ holtz scalar equation for the magnetic field H Hy(x, z), the 0Z axis is assumed to be some function Ez(x = Lx, y, Ω is stated as z) = Eτ(z). Then, in the region 0, Hs is obtained by 2 2 solving the boundary-value problem ∂ ∂ 2 -------- ++------- k H = 0 (3) 2 2 2 2 ∂ ∂ 2 ∂x ∂z -------- ++------- k H = 0, (10) ∂ 2 ∂ 2 s in the domain x z M ∂H ∈ Ω Ω Ω ± ---------s + ih H = 0 ()v = 012,,,… , (x, z) = ∪ q∪ 0 , v s ∂z , q = 1 z = 0 L (11) ∂ ∂ Ω ∞ ∞ Hs Hs where 0 = {0 < x < Lx, – < z < } is the region of the ==0, –ikEτ()z . Ω ---------∂ ---------∂ regular waveguide and q = {Lx < x < lq(z), 0 < x < L} x x = 0 x xL= x is the region of the qth corrugation (q = 1, …, M). The boundary conditions are as follows: the Neumann Its solution can be represented through the Green homogeneous condition function [8]: ∂ L H = 0 ik ------- H ()xz, = –------ dz'G ()xzx,,' = L , z' Eτ()z' , (12) ∂n ∪{}()∪{} (4) s π∫ k x S = xl= q z x = 0 4 q 0 ∞ on the metallic surface of the grating and on the plane π ε (),, , 2 i v ()() of symmetry 0Z Gk xzx' z' = -------- ∑ ----- cos gv x cos gv x' Lx hv ∂ v H () = 0 -------∂ +2ihp H = ihp cos g p x (5) z × exp()ihv zz– ' , (13) at z = 0 if the grating is excited by the TM mode, and π v ∂ v 2 2 2, = 0 H 2 2 gv ==------- , hv k –,gv εv = ------- –0,ihv H ==hv k –,gv L ≠ ∂z x 1, v 0. (6) π On the surface of the regular waveguide, expression g ==----- v ; v 01,,… v L s ≡ x (12) for Hs (z) Hs(x = Lx, z) can be written in the com- at z = L if the modes can freely leave the system. pact operator form The electric field components are given by the equa- s H = –ikGˆ Eτ, (14) tions s R 1 ∂H 1 ∂H ∞ ε L i v Ez ==–---- ------- , Ex ---- ------- . (7) ˆ () ik ∂x ik ∂z GR = -------- ∑ ----- ∫dz'exp ihv zz– ' . (15) 2Lx hv The solutions to Eqs. (7) under boundary conditions v = 0 0 (4)–(6) will be sought separately for the regular Let us find a relation between the electric field Eτ(z) Ω waveguide region 0 and for each of the corrugations and the magnetic field Hq at the interface between the TECHNICAL PHYSICS Vol. 47 No. 2 2002 THE THEORY OF PLANAR BRAGG RESONATORS 143 Ω qth corrugation region q and the regular waveguide: XX' ≤ ≤ x = Lx, aq, 1 z aq, 2. Irrespective of the corrugation Ω shape, the magnetic field Hq inside the region q is the d Z' solution of Helmholtz equation (3) with Neumann a homogeneous conditions (4) on the metallic surface x = 0' Ez ≤ ≤ lq(z), aq, 1 z aq, 2 of the corrugation and must obey b Lx relationship (9) at the boundary with the regular waveguide. Using the Green function [8] for determin- 0 Z ing Hq(x, z) through the field value at the boundary Sq = ∪ ≤ {x = lq(z)} {x = Lx}, aq, 1 aq, 2 between the qth cor- Fig. 2. Periodic rectangular corrugation in a planar Ω waveguide. rugation region q and the regular waveguide, we find Taking into account the corrugation geometry, one (), ()' (),,' ∈ Hq xz = °∫ dSq H rs Gq xzrs can express Eτ(z) on the regular waveguide surface z η Sq [0, L] through the Heaviside function (x) = {0, x < 0; (16) 1, x > 0} in the operator form aq, 2 = dz'()Hs ()z' + Hs()z' G ()xzz,,' , ∫ 0 s q () 1 ˆ ()s s Eτ z = –---- Gin H0 + Hs , aq, 1 ik M (20) q where Gq is the surface Green function for the Helm- Gˆ in = ∑ ()η()ηza– , – ()za– , Gˆ in. Ω q 1 q 2 holtz equation in the region q. Gq satisfies homoge- q = 1 neous Neumann conditions (4) on the metallic surface ≤ ≤ {x = lq(z), aq, 1 z aq, 2} and also the inhomogeneous Using relationships (14) and (20), we come to the Dirichlet conditions at the boundary x = Lx with the reg- s Fredholm integral equation of the first kind for Hs (z) ular waveguide.
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