TM Electromagnetic Scattering from 2D Multilayered Dielectric Bodies Numerical Solution

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1 TM Eectromagnetic Scattering from D Mutiayered Dieectric Bodies Numerica Soution F. Seydou,, R. Duraiswami, N.A. Gumerov & T. Seppänen. Department of Eectrica and Information Engineering University of Ouu, P.O. Box 3, 9 Finand. Institute for Advanced Computer Studies University of Maryand, Coege Park, MD Abstract An integra equation approach is derived for an eectromagnetic scattering from an M arbitrary mutiayered dieectric domain. The integra equation is vaid for the D and 3D Hemhotz equation. Here we show the numerica soution for the D case by using the Nyström method. For vaidating the method we deveop a mode matching method for the case when the domains are mutiayered circuar cyinders and give numerica resuts for iustrating the agorithm. Introduction Probems of eectromagnetic scattering in ayered media are of significant importance in many areas of technoogy such as optics, geophysica probing, communication, etc. see [6] and the references therein. In this paper we discuss some anaytica and computationa resuts for the probem of approximating the scattered eectromagnetic fied from M ayered two-dimensiona scatterer. The scatterer is a nested body consisting of a finite number of homogeneous ayers annuar regions with boundary conditions on the interfaces. For the case when the boundaries are circuar, cosed form soutions can be obtained via a mode matching approach see [9], [6] and [6], Chapter 6. For boundaries of arbitrary shapes, one of the most efficient techniques to tacke the probem is using voume or surface integra equation methods. There are aso other type of methods such as the domain decomposition methods [] and k-space methods Cf. [3] and []. In this paper we choose the surface integra equation method since the inhomogeneities are piecewise constants in each region. The probem can thus be soved via a boundary eement method on surfaces. It has an advantage over the voume integra equation method, where the whoe mutiayered domain has to be discretized and the unknowns are in a voume rather than on a surface see [3]. The straightforward way for soving this type of probems via boundary eement methods is by using Green s theorem in each domain [6]. Another aternative is to consider the use of singe and/or doube ayer potentias [7]. In the case of one interface, both methods yied a singe integra equation for a singe unknown if the interface is impenetrabe e.g., impedance core. However, when the body is penetrabe with one interface e.g., dieectric core, they ead to a pair of integra equations for a pair of unknowns [7]. We deduce that, by using these approaches in the mutiayered dieectric domain, for N interfaces we have N unknown functions to determine. From a computationa point of view, it is highy desirabe to obtain ess equations and ess unknowns. In the case of one interface, the so caed transmission probem, one integra equation invoving one unknown was obtained by a

2 few authors see [], [] and the references therein. In [] the singe integra equation for one unknown was obtained for the transmission probem by using a hybrid of Green s theorem and ayer potentias. In [6], Chapter.3, singe integra equations are obtained for mutiayered domains by using the extended boundary condition method. But this method suffers from iconditioned equations and is mainy convenient for a scatterer where the fieds around it are expandabe to cyindrica harmonics. The purpose of this paper is to obtain Fredhom type singe integra equations on each interface for the mutiayered domain case. To this end, we aternate the ayer potentias and Green s theorems in the mutiayered domain and impement numerica computations using the Nyström method. For a theoretica study of the probem, see [] and []. Our resuts are vaidated by deveoping a mode matching approach for the case of a mutiayered circuar cyinder and comparing the two agorithms. 3 5 be the C boundaries of D, =,,M. Now et Ω = D,Ω = D \D, =,,M, and Ω M = R \D M. We assume that Ω M is simpy connected. See Figure for Ω, =,,, 5. This is a specia case of the genera geometry where we have the cross section of M = 5 concentric cyinders that are infinite in ength and their axes are parae to the z direction. Each of the regions Ω is a dieectric materia of constant compex permittivity and permeabiity ɛ and µ =,, M, respectivey. This geometry is iuminated by an incident fied which is a pane wave with direction d = cos φ, sin φ. It can be shown that we have to sove the foowing type of boundary vaue probem for the Hemhotz equation. + κ u = in Ω, =,,M, where the wave numbers κ are given by κ = ω ɛ µ, ω is the frequency, with the foowing continuity conditions on the interna interfaces: ν u = ρ ν u on Γ, =,,M, u = u on Γ, =,,M, ˆρ with ρ = ˆρ, =,,,M, where ˆρ = intrinsic impedance. On the outermost interface we have u ν = ρ M ν u M on Γ M, µ ɛ is the u = u M on Γ M, d Figure : The geometry for the case of five concentric ayered cyinder. The incident fied is a pane wave propagating in a direction d. The mathematica formuation of the probem Let D, =,, M bem bounded domains in R such that D D, =,,,M. Let Γ where, u = u M + u i in Ω M and the given incident fied, u i, satisfies u i + κ M ui = everywhere. In addition, u M must satisfy the Sommerfed radiation condition, i.e., um im x x / x iκ Mu M =.

3 The unit outward norma ν to Γ is assumed to be directed towards the exterior domain. The above probem is known as the TM mode. The TE mode is obtained by repacing ρ by ρ. We denote the fundamenta soution to the Hemhotz equations the freespace source by Φ k x, y = i H κ k x y, k =,,,M, where H is the Hanke function of the first kind and order zero. Throughout this paper i wi denote the compex constant satisfying i =. The integra equation approach First, for non-zero functions φ,=,,,m, define the singe and doube ayer potentias as Sk φ x = Φ k x, yφ y dsy, x R \Γ, Γ and Dkφ x = Γ ν y Φ kx, yφ y dsy, x R \Γ, respectivey, for k =,,, M. Their norma derivatives are denoted by Pk and Q k, respectivey, for k =,,,M. We have the continuity reations and the jump reations S k = Ŝ k, Q k = ˆQ k, D k = I + ˆD k and P k = ±I + ˆP k, where, the upper ower sign corresponds to the imit when x approaches Γ from the outside inside. The hat on each operator represents it on the boundary Γ. To arrive at the desired integra equation we define a ayer ansatz by Ek := D k iη Sk for and Ek = with norma derivative H k := E k / ν in Ω k, where η s are nonzero compex constants chosen to obtain we-posedness, k =,,,, and Green s theorem in Ω k, k =, 3, 5,. In particuar, et us assume that M is odd. Then, in the core region Ω we define u x =E φ x, x Ω.. In the outermost domain, we use Green s theorem [7] pp. 6-7 to obtain { u M x =SM M ν ux DM M ux, x Ω M, u i x =SM M ν ux DM M ux, x R \Ω M.. In the other domains, for =,,,M, we define u x =E φ x+e + φ + x, x Ω,.3 and, using Green s theorem for =, 3,,M we have u x =S ν u x S + ν u x D D+ u x, x Ω, =S ν u x S + ν u. x D D+ u x, x R \Ω Now, using the jump and continuity reations we obtain the second equation in. on Γ M and the second equation in. on Γ and Γ + =, 3, 5, M. Using the boundary conditions, jump properties for the singe and doube ayer potentias together with their derivatives, and repacing u given in. and u given in.3 into these equations we arrive at a set of M integra equations with M unknowns φ on Γ, =,,M. In particuar, on Γ M we have u i = ρ M ŜM M ĤM M ˆD M M + IÊM M φ M + ρ M ŜM MHM,M M ˆD M M + IEM,M M φ M, and for =, 3, 5, M, we have = ρ Ŝ H, ˆD + IE, φ + ρ Ŝ Ĥ ˆD + IÊ φ ρ + S +, Ĥ + + D+, Ê + + φ + H +,+ + D +, E +,+ + φ + ρ + S +,

4 on Γ and = ρ + Ŝ + Ĥ + + ρ + Ŝ + H +,+ ρ S,+ H, ρ S,+ ˆD + + Ĥ D,+ ˆD + D,+ IÊ+ + φ + + IE +,+ φ Ê + E, φ + φ on Γ +, where Êm k = I + ˆD k m iηmŝm k, Ĥm k = ˆQ m k iη m±i + ˆP k m m,n, and by Tk T is for S, D, E or H we mean that Tk m is evauated on Γ n when n m. Numericay the above system has to be discretized and soved to obtain an approximation of the unknowns φ, =,,M. Then the soution of the ayered probem can be constructed for the discretized forms of.-.. Remark: The above system is aso vaid for the 3D Hemhotz equation. The ony difference is that the fundamenta soution is Φ k x, y = eiκ k x y π x y. 3 Numerica vaidation and resuts This section is devoted to the numerica soution of the above system and its vaidation for the D case. To this end, we use the Nyström method for the numerica soution and Besse function expansion for the vaidation. Then we show the numerica resuts. 3. Discretization and numerica soution The system is discretized using the Nyström method []. The resuting matrix equation, that invoves matrix mutipications resuted from the mutipications of ayer potentias and/or their derivatives, is soved by a standard LU decomposition approach. Let us note that the assumption that M is odd is not a oss of generaity. In fact, for an even M we can use the same method, but for M + regions, Γ M+ encoses the scatterer, with κ M+ = κ M and ρ M+ =. This way has the advantage of keeping the same system of equations and the disadvantage of adding another equation and an unknown function φ M+. This may be overcome by starting with Green s theorem in the core region, aternate with ayer ansatz and obtain the Green s theorem in Ω M, which gives a different system than the previous argument. 3. The Mode Matching Approach This method is studied in detai in the iterature see e.g., [9]. Consider the case when the regions D s are circuar cyinders with radii r + and origins O +, =,,,,M ; then we have the foowing expansions [6]: For the outermost region u r M,φ M = n= b M n H n κ M r M +J n κ M r M and for other regions we have u r,φ = n= inφm φo e b nh n κ r +a nj n κ M r e inφ φo, =,,, M, where b n =. To enforce the boundary conditions we need the addition formua for u, =,,M which means that the fieds expressed in terms of X O Y be transated to X O Y coordinates. This yieds, by the addition theorem cf. [5] pp. 3-3, u r,φ = n= i= J i nκ d [ ] b n H i κ r +a n J iκ r e iφ i nφ, where d is the distance between O and O, and φ is the ange between O O and the x axis. The sums in the above equations have to be truncated, at some number, N, to obtain a finite system. Now we can use the expansions on the boundary together with their derivatives and the boundary conditions to obtain a inear system in the unknowns a n and b n. This system is aso soved via LU decomposition approach. 3.3 Numerica Resuts In this section, numerica soution obtained by using the integra equation IE and mode matching MM

5 methods are presented. We have conducted severa numerica experiments. First we try to vaidate the MM method by anayzing the physica properties of the waves, by potting the absoute vaue of the waves against the boundaries. To this end, we consider a cyinder with radius r = 3, so that we ony have the inner and outermost domains, and keep the ange of the incident fied φ = 9 deg and ρ = fixed. We woud expect, for rea κ and compex κ with negative imaginary part, the wave to diverge at the boundary. If, on the other hand, we have that the two wave numbers are rea and equa, the absoute vaue of the wave shoud be unity. Finay, for the case when κ is rea and κ is compex with a positive imaginary part, because of absorption, the absoute vaue of the wave must decay at the boundary and the bigger the imaginary part, the faster the wave shoud decay. Our numerica computations show that a theses properties are satisfied, and the resuts are summarized in Figure. Next we vaidated the IE method for one interface, centered at O =.,.7, by potting the absoute vaue of the far fied pattern measured at a fixed observation point ˆx against the incidence ange for two different wave numbers using the IE and MM methods. See [], page, for the definition of the far fied pattern f. The resut is given in Figure 3, which shows a very good agreement of the two methods. Uness otherwise stated we use 3 grid points for the Nyström sover. Our next exampes are for the two and three-ayered circuar cases. First we pot the absoute vaue of the far fied against the incidence ange for the two-ayered case and then for the three-ayered case. The resuts are shown in Figure and Figure 5, respectivey. In these cases as we we have very good agreements of the two methods. For the case of more circuar ayers we have the same concusions, except that more grid points are needed, which is due to the errors in the computation of the ayer potentias. Our ast exampe is for the case of three boundaries of kite type where MM method can not be performed Figure 6. Here we investigate the convergence as we as the boundary conditions. For the former we compute the far fied pattern for different wave numbers. absoute vaue of the fied u 5 5 absoute vaue of the fied u the radius r the radius r absoute vaue of the fied u absoute vaue of the fied u the radius r the radius r Figure : The case of one circuar boundary r = 3. The absoute vaue of the wave potted against the radius. We have used κ = and κ =.5i top eft κ =andκ = top right, κ = and κ = +.5i bottom eft, and κ = and κ =+.5i bottom right. absoute vaue of the far fied f φ rad Figure 3: The absoute vaue of the far fied pattern potted against the incidence ange using the MM o and IE soid ine methods. The case of one interface. We used κ =,κ = 3 and the radius is r =.

6 absoute vaue of the far fied f φ rad The resuts are reported in the two tabes beow. We see cear convergence, and, as expected, it is fast. For the atter case we pot u and u on Γ, u and u on Γ, and u and u 3 + u i on Γ 3, against the incidence ange. From the boundary conditions we know that they must coincide. This is shown in Figures 7. One way we have used to compare the IE and MM methods in the case of 3 ayered kite is by encosing the tree ayers within a circuar domain and choose a the inner ayers to have the same wave numbers and the outer region to have a different wave number. Physicay, this is a one-ayered probem; but mathematicay the four ayers exist. By so doing we sti obtain a figure simiar to Figure 3. Figure : The absoute vaue of the far fied pattern potted against the incidence ange using the MM o and IE soid ine methods. The case of two-ayered circuar cyinders. Here κ =,κ =3,κ =,r = and r =. absoute vaue of the far fied f Figure 6: The geometry for the case of three boundaries of kite type. Tabe : Parameter vaues and description for the geometry in Figure 6. D, D and D3 are the first, the second and third data, respectivey, for the numerica computation Description Symbo D D D3 κ +i Wave numbers κ 3 5 κ i κ φ rad Figure 5: The absoute vaue of the far fied pattern potted against the incidence ange for the MM o and IE soid ine methods. The case of three-ayered circuar cyinder. Here κ =,κ =3,κ =,κ 3 = r =,r = and r 3 =3. Concusion and future work We have deveoped an integra equation approach for soving the M mutiayered eectromagnetic probem and used the Nyström method for the numerica computation. The agorithm was vaidated by a Fourier expansion method for circuar not necessariy concentric cyinders. One may think as a disadvantage for our

7 Tabe : The numerica resuts using the integra equation IE approach for the geometry in Figure 6. The data are given in Tabe. The number N is the number of Nystro m grid points. D D D3 N IE i i i i i i i i i i i i Dr. Seydou s research interests are finite and boundary eement methods appied to probems in computationa eectromagnetism, and inverse scattering probems. Ramani Duraiswami is a Research Associate Professor at the University of Maryand Institute for Advanced Computer Studies. He is aso Director of the Perceptua Interfaces and Reaity Laboratory. Dr. Duraiswami got his Ph.D. in 99 from The Johns φ rad c 6 6 absoute vaue of the fied u Dr. F. Seydou is a Docent Associate Professor in appied mathematics at the University of Ouu, Finand and a Visiting Researcher at the University of Maryand, Coege Park. He wrote his Ph.D. thesis at the University of Deaware and graduated at the University of Ouu in 997. b absoute vaue of the fied u absoute vaue of the fied u a method the numerous matrix vector mutipications. This probem can be overcome by using fast mutipoe methods see [5] where these operations are done very quicky. Our resuts aso show the expected fast convergence of the Nystro m method for anaytic boundaries. The natura expansion of our method is for the numerica soution of the three-dimensiona eectromagnetic probem, and to the case of mutipe scatterers. 6 5 φ rad 6 Hopkins University. His research interests incude computationa audio, scientific computing and computer vision. He is an associate editor of the ACM Transactions on Appied Perception Nai Gumerov is a Research Assistant Professor at the University of Mary5 6 and Institute for Advanced Computer Studies. He got his Ph.D. in 97 from The Lomonosov Moscow Figure 7: Here we pot uj o and uj+ + j j i State University and Sc.D. in 99 u soid ine on the boundary Γj against from the Tyumen University in Rusthe incidence ange. In a we have the case j=, in b sia. j= and c j=3. His research interests incude mathematica modeing, fuid mechanics, acoustics, and scientific computing. 5 φ rad

8 Tapio Seppänen is a professor of information engineering at the University of Ouu, Finand. Dr. Seppänen got his Ph.D. in 99 from the University of Ouu. His research interests incude speech and audio signa processing, mutimedia search engines, digita rights management of mutimedia, and processing of physioogica signas. References [] C. Athanastadis, On the acoustic scattering ampitude for a muti-ayered scatterer, J. Austra. Math. Soc. Ser. B 39, pp. 3, 99. [] C. Athanasiadis, A.G. Ramm and I.G. Stratis, Inverse acoustic scattering by a ayered obstace, in Inverse Probems, Tomography, and Image Processing ed. A.G. Ramm, Penum Press, New York, pp., 99. [3] N. N. Bojarski, The k-space formuation of the scattering probem in the time domain, J. Opt. Soc. Amer., Vo. 7, pp. 57 5, 9. [] O.P. Bruno and A. Sei, A fast high-order sover for EM scattering from compex penetrabe bodies: TE case, IEEE Transactions on Antennas and Propagation, Vo.,, pp. 6 6,. [5] K. Chadan, D. Coton, L. Paivarinta and W. Runde, An Introduction to Inverse Scattering and Inverse Spectra Probems, SIAM Monographs on Mathematica Modeing and Computation, 997. [6] W. C. Chew, Waves and Fieds in Inhomogeneous Media, IEEE Press, 995. [7] D. Coton and R. Kress, Integra Equation in Scattering Theory, John Wiey & Sons, 93. [9] A. Kishk, R. Parrikar and A. Esherbeni, Eectromagnetic Scattering from an eccentric mutiayered circuar cyinder, IEEE Trans. Antennas and Prop. Vo., No3 pp , 99. [] R.E. Keinmann and P.A. Martin, On singe integra equations for the transmission probem of acoustics, SIAM J. App. Math. Vo., No., pp , 9. [] R. Kress, On the numerica soution of a hypersinguar integra equation in scattering theory, J. Comp. App. Math. Vo. 6, pp , 995. [] E. Larsson, A domain decomposition method for the Hemhotz equation in a mutiayer domain, SIAM J. Sci. Comput., Vo., pp , 999. [3] Oivier J. F. Martin and Nicoas B. Pier, Eectromagnetic scattering in poarizabe backgrounds, Phys. Rev. E 5, No. 3, pp , 99. [] P.A. Martin and P. Oa, Boundary integra equations for the scattering of eectromagnetic waves by a homogeneous dieectric obstace, J. Proc. R. Soc. Edinb., Sect. A 3, No., pp. 5, 993. [5] J.M. Song and W.C. Chew, FMM and MLFMA in 3D and Fast Iinois Sover Code, Chapter 3 in Fast and Efficient Agorithms in Computationa Eectromagnetics, edited by Chew, Jin, Michiessen, and Song, Artech House,. [6] Z. Wu and L. Guo, Eectromagnetic scattering from a mutiayered cyinder arbitrariy ocated in a gaussian beam, a new recursive agorithms, Progress In Eectromagnetics Research, PIER, pp , 99. [] D. Coton and R. Kress, Inverse Acoustic and Eectromagnetic Scattering Theory, Springer Verag, 99.