Study Of Some Electromagnetic Scattering Problem Of Diffractive Grating For Homogeneous Media

It is well-known that the study of electromagnetic scattering problems is an important topic in mathematical physics.There has been a lot of work in this region.For the references,we refer to [1 ].In homogeneous medium, time-harmonic electromagnetic fields are governed by the Maxwell equations▽×E-ikH = 0,▽×E-ikH = 0,where E,H are the electric field and the magnetic field,respectively; k is the wave number.In this paper, theoretical analysis of electromagnetic scattering problems for homogeneous media are discussed.Firstly,we study electromagnetic scattering from diffractive grating.Given incident plane waves Eⁱ = se^ig.x and Hⁱ = pe^ip.x such thats=1/k(p×q), p.q = 0, q.q = k2.to find solutions E and H of the following Helmholtz equations ( using cylinder coordinate, derived from equations ( 0.1 )),1 d . dE, 1 d2E 82E , ,? tn .r or Or r2 862 oz2() +r dr dr r2In this paper , we derive a variational formulation of scattering problem. To do so requires the derivation of appropriate boundary conditions; this is done by establishing artificial boundaries.1. Because of singularity of the coefficient of Helmholtz equation on r = 0,wc supply the condition ,lim r—- 0, lim r—— — 0.r-?o+ or r^o+ or2.To reduce the domain from infinite extent to a bounded "box"with finite extent in Z-direction, we establish artificial plane boundary F:T = {(r, 6, z)- z = 6}, (b > f(r), 0 < r < +oo).DefiningOo = {(r,8, z);b < z} and using the method of separation of variables, in O0 we express u(r, 0, z) as :+ /- + OOu{r, 0,z) = J2 il>(h)Jt(y/k2 - h2r)elhz+iWdh,where+ OO(0-4)We drop the superscript s from us, thus we can calculate the derivative of u(r,8,z) with respect to v, the unit normal, on <9Qo-(0.5)where v = (0,0,1). LEMMA 1. There exist quasi-differential operator T, such that(T - — )u = +i{3ei(^rcowhere fordefine the operator T by(Tu)(r,6,z)= 2^ / ih^{h)Ji{^/k2-h2r)elhz+lWdh.3.We establish artificial plane boundary P': V = {(r,9,z);r — R, 1}. Then u satisfy on I"Thus,the scattering problem is : to find u G H1(Q), such that fc2?^ = 0,—)u\r = lQei(°t'-r (0.6)i- du n lim r— = 0,r--.o+ orThus, we study the variational form of the scattering problem. Applying some simple vector identities along with integration by parts formulae, it follows that the scattering problem has an equivalent variational form: find u e /^(Q), such thata(u,v) = {f,v), for all ueff'(fi) (0.7)wheref* /* j> ft f\ fta(u,v) = - / Vu-Wvdv+ / T(u)vds+ik / vuds+ / ^vds + k2 / u-vdv Jn Jv Jr> A=o or Ja= / fvdx. Inand(/. v)= I fvdx (0.8)Jv,Next, we study the solutions of two special scattering problem. l.If the grating iss = {(r, d, z); z = 0,0 < 6 < 2n, 0 < r < +00} and the incident wave is parallel to z axis,El = {e-^.e-^.O}, H* = {e^lk\ -e^lk\ 0}. we have the solution of the scattering problem:E = {-elk\ -eikz, 0}, H = {-elkz, elkz, 0}.2. For diffractive grating , the simplest structure is called Lamellar grating, we mark the unit of E, H with u, and discretize the document with a rectangle...