# Exact Matrix Representation of the Transverse Magnetic Multiple Scattering of Obliquely Incident Plane Waves by the Diffraction Grating of Penetrable Cylinders

Authors: Ömer Kavaklıoğlu; Roger Henry Lang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

“An exact matrix conformation model” associated with the equations describing the exact behavior of the Fourier-Bessel multiple scattering coefficients of the diffraction grating consisting of an infinite number of infinitely long parallel penetrable circular cylinders, corresponding to the obliquely incident transverse-magnetic plane waves in “Twersky-Wait-Kavaklıoğlu representation,” originally excogitated in (Kavaklıoğlu, 2000), is acquired, and the exact solution for “the Fourier-Bessel multiple scattering coefficients of the diffraction grating at oblique incidence” is obtained by a matrix inversion procedure.


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## Body

## 1. Introduction

Twersky [1] solved the problem of multiple scattering of radiation by an arbitrary configuration of parallel cylinders by the infinite grating of insulating dielectric circular cylinders at normal incidence in terms of cylindrical wave functions as long ago as 1952, considering all possible contributions to the excitation of a particular cylinder by the radiation scattered by the remaining cylinders. He later extended his solution for the case where all axes of cylinders lie in the same plane [2] by expressing the scattered wave as an infinite sum of orders of scattering and derived the solution for the scattering of waves by the finite grating of cylinders [3] at normal incidence. In addition, Twersky introduced a “scalar functional equation” [4] for the grating in order to describe the relationship between the multiple scattered amplitude of the grating in terms of the “multiple scattered amplitude of an isolated cylinder within the grating.” He [5] then utilized the separation-of-variables technique to derive a set of algebraic equations for the “multiple scattering coefficients of the infinite grating” in terms of his famous elementary function representations of “Schlömilch series” [6], and in terms of the “well-known scattering coefficients of an isolated cylinder at normal incidence,” which was originally derived by Rayleigh [7, 8]. His results since then have been reiterated and extensively used by many other authors.For the more generalized case ofconical incidence, Sivov [9] first studied the problem of determining the coefficients of reflection and transmission by an infinite plane grating of parallel conductors assuming the period small in comparison with the wavelength. Lee [10] treated the multiple scattering by an arbitrary configuration of parallel, nonoverlapping infinite cylinders, and provided the solution for the “scattering by closely spaced radially-stratified parallel cylinders with an arbitrary number of stratified layers” [11]. Furthermore, Smith et al. [12] developed a formulation for cylinder gratings in conical incidence using a multipole method in modeling photonic crystal structures and study scattering matrices and “Bloch modes” in order to investigate the photonic band gap properties of woodpile structures [13]. In a more recent study, Henin et al. [14] presented a semianalytical solution by an array of circular dielectric cylinders and parallel-coated circular cylinders of arbitrary radii and positions [15].The exact equations describing the behavior of theFourier-Bessel multiple scattering coefficientsof an infinite grating of dielectric circular cylinders, which are aligned along the y-axis and parallel to the z-axis, for obliquely incident plane electromagnetic waves, θi being the obliquity angle made with z-axis, were first derived by Kavaklıoğlu [16–18] for both TM and TE polarizations. The generalized representations associated with the reflected and transmitted fields were formulated in terms of these Fourier-Bessel multiple scattering coefficients for obliquely incident plane H-polarized waves in Kavaklıoğlu and Schneider [19]. In addition, they [20] derived the generalized form of Twersky’s functional equation[4]for the infinite grating at oblique incidence in matrix form in terms of the “Fourier-Bessel scattering coefficients of an isolated dielectric circular cylinder at oblique incidence”originally derived by Wait [21]. Moreover, Kavaklıoğlu and Schneider [22] acquired the asymptotic solutionfor the Fourier-Bessel multiple scattering coefficients of the infinite grating associated with obliquely incident and vertically polarized waves up to and including third order as a function of the cylinder radius to grating spacing when the grating spacing “d” is small compare to a wavelength. Recently, Kavaklıoğlu and Lang [23] demonstrated a rigorous proof for the validity of this asymptotic solution derived in [22].The purpose of this investigation is to acquirethe exact conformation model for the Fourier-Bessel multiple scattering coefficients of an infinite array of penetrable circular cylinders associated with obliquely incident vertically polarized plane electromagnetic waves and to capture the exact representation of the aforementioned coefficients of the diffraction grating at oblique incidence in “matrix form.” In the generalized oblique incidence solution presented in this article, the direction of the incident plane wave makes an arbitrary oblique angle of arrival θi with the positive z-axis as depicted in Figure 1.Figure 1
The configuration of the problem of scattering of waves by an infinite array of penetrable circular cylinders at oblique incidence (side view).In addition, we have explicated and connoted that our proposed method of solution for the scattering coefficients is a new technique and an exact representation. Besides, the paper clearly describes how one can generate a numerical algorithm from this exact solution by the truncation of the system matrices associated with the undetermined multiple scattering coefficients. On the other hand, the objective of this paper is not to discover some numerical algorithm as its title designates, which would be completely approximate and requiring an error analysis.


## 2. Description of the Transverse Magnetic Multiple Scattering Coefficients of the Infinite Array at Oblique Incidence

We consider a vertically polarized obliquely incident plane electromagnetic wave upon an infinite array of insulating circular dielectric cylinders having infinite length with radiia, dielectric constant εr, and relative permeability μr. The constituent cylinders of the infinite array are placed periodically along the y-axis, with their axis parallel to the z-axis, located at positions r0, r1, r2,…, and so forth, and separated by a distance of d.For this configuration, the incident wave can be written in the cylindrical coordinate system(Rs,ϕs,z) of the sth cylinder in terms of the cylindrical waves referred to the axis of sth cylinder [16] as(2.1)Evincident(Rs,ϕs,z)=v̂iE0veikrsdsin⁡⁡ψi{∑n=-∞∞e-inψiJn(krRs)ein(ϕs+π/2)}e-ikzz.
In the representation above, v̂i denotes the vertical polarization vector associated with a unit vector having a component parallel to all the cylinders, ϕi is the angle of incidence in x-y plane measured from x-axis in such a way that ψi=π+ϕi as it is delineated in Figure 1, implying that the wave is arbitrarily incident in the first quadrant of the coordinate system, and Jn(x) denotes Bessel function of order n. In expression (2.1), we have(2.2)kr=k0sin⁡θi,kz=k0cos⁡θi,
where k0 denotes the free space wave number with k0:=2π/λ0 and where λ0 is the wavelength of the incident radiation; θi is the obliquity angle made with z-axis. e-iωt time dependence is suppressed throughout the article, where ω represents the angular frequency of the incident wave in radians per second and t stands for time in seconds (Figure 2).Figure 2
The geometry of the infinite grating problem (top view).The exact solution associated with thez-component of the electric field intensity in the exterior of the infinite grating can be expressed in terms of the incident electric field in the coordinate system of the sth cylinder located at rs, plus a summation of cylindrical waves outgoing from the individual nth cylinder located at rn, which satisfies Silver-Müller radiation condition as |r-rn|→∞, that is,(2.3)Ez(exterior,TM)(Rs,ϕs,z)=Ez(incident,TM)(Rs,ϕs,z)+∑n=-∞+∞Ez(nthscatterer,TM)(Rn,ϕn,z).
The generalized exact representation of the multiple scattered fields by an infinite grating of dielectric cylinders for obliquely incident plane waves has been rigorously treated by Kavaklıoğlu [16] for an arbitrary oblique angle of arrival θi with the positive z-axis, and it has been proved that the z-component of the total electric and magnetic fields in the exterior of the infinite grating for obliquely incident and vertically polarized plane electromagnetic wavescan be written as(2.4)Ez(exterior,TM)(Rs,ϕs,z)={eikrsdsin⁡⁡ψi∑n=-∞+∞[(Eni+Qn)Jn(krRs)+AnHn(1)(krRs)]ein(ϕs+π/2)}e-ikzz,Hz(exterior,TM)(Rs,ϕs,z)={eikrsdsin⁡ψi∑n=-∞+∞[QnHJn(krRs)+AnHHn(1)(krRs)]ein(ϕs+π/2)}e-ikzz.
The infinite set of undetermined coefficients {An,AnH}n=-∞∞ arising in (2.4) denotes the Fourier-Bessel multiple scattering coefficients for the infinite array of insulating dielectric cylinders corresponding to the vertically polarized obliquely incident plane electromagnetic waves, foralln∈Z where Z represents the set of all integers, and Eni in (2.4) is given as(2.5)Eni=sin⁡⁡θiE0ve-inψi.
In addition, Qn and QnH represent the multiple scattering effects, which are expressed as a linear combination of the undetermined Fourier-Bessel multiple scattering coefficients of the infinite grating at oblique incidence as(2.6)Qn=J0An+∑m=-∞m≠n∞Jn-mAm,QnH=J0AnH+∑m=-∞m≠n∞Jn-mAmH,
for all n∈Z. 𝒥n(krd) is the generalized form of the Schlömilch series for obliquely incident waves(Twersky [4–6], Kavaklıoğlu [18]) explicitly given as(2.7)Jn(2πΔ)=∑p=1+∞Hn(1)(2πpΔ)[e2πipΔsin⁡⁡ψi(-1)n+e-2πipΔsin⁡⁡ψi],
where, Δ≡krd/2πand Hn(1)(x) denotes the nth order Hankel function of first kind, for all n∈Z. The series 𝒥n-m(krd) in expression (2.7) is the generalization of the “Schlömilch series for obliquely incident electromagnetic waves” (Twersky [6], Kavaklıoğlu [18]) and converges provided that krd(1±sinψi)/2π does not equal integers. The integral values of krd(1±sinψi)/2π are known as the “grazing modes” or “Rayleigh values” (Twersky [6]).


## 3. The Generalized Fourier-Bessel Multiple Scattering Coefficients of the Infinite Grating for Vertically Polarized Obliquely Incident Plane Electromagnetic Waves in Matrix Form

The equations describing the scattering coefficients forvertically polarized and obliquely incident plane electromagnetic wavesare first derived in Kavaklıoğlu [16], which can be described as(3.1)An+anε[Eni+∑m=-∞+∞AmJn-m(krd)]-bnε[AnH+cn∑m=-∞+∞AmHJn-m(krd)]=0,[AnH+anμ∑m=-∞+∞AmHJn-m(krd)]+bnμ{An+cn[Eni+∑m=-∞+∞AmJn-m(krd)]}=0,foralln∈Z. In the equations above, we have(3.2)cn=Jn(kra)Hn(1)(kra),(3.3)anζ=[Jn(k1a)Jn'(kra)-ζr(kr/k1)Jn(kra)Jn'(k1a)Jn(k1a)Hn(1)′(kra)-ζr(kr/k1)Hn(1)(kra)Jn'(kra)],(3.4)bnζ=iFcζ0[Jn(k1a)Hn(1)(kra)Jn(k1a)Hn(1)′(kra)-ζr(kr/k1)Hn(1)(kra)Jn'(kra)](nkra[Jn(k1a)Hn(1)(kra)Jn(k1a)Hn(1)'(kra)-ζr(kr/k1)Hn(1)(kra)Jn'(kra)])
for all n∈Z forζ∈{ε,μ}, where k1 is defined as k1=k0εrμr-cos2θi, F is defined as(3.5)F=(μrεr-1)cos⁡⁡θiμrεr-cos⁡⁡2θi,
and c stands for the speed of light in free space as(3.6)c=1μ0ε0,Jn'(ς) and Hn(1)′(ς) in expression (3.3) and (3.4) are described as the first derivatives of these functions with respect to their arguments, that is, Jn'(ς)≡(d/dς)Jn(ς) and Hn(1)′(ς)≡(d/dς)Hn(1)(ς).Introducing the intrinsic impedance of free space as(3.7)ξ0=μ0ε0≔η0-1≅377Ω,
two parameters that appears in the equations of the Fourier-Bessel multiple scattering coefficients of the infinite grating in (3.1), namely, bnε and bnμ, can be acquired by employing (3.7) in (3.4) for ζ∈{ε,μ} as(3.8)bnε=[Jn(k1a)Hn(1)(kra)Jn(k1a)Hn(1)'(kra)-εr(kr/k1)Hn(1)(kra)Jn'(k1a)](inFξ0kra[Jn(k1a)Hn(1)(kra)Jn(k1a)Hn(1)'(kra)-εr(kr/k1)Jn'(k1a)]),bnμ=[Jn(k1a)Hn(1)(kra)Jn(k1a)Hn(1)'(kra)-μr(kr/k1)Hn(1)(kra)Jn'(k1a)](inFη0kra[Jn(k1a)Hn(1)(kra)Jn(k1a)Hn(1)'(kra)-εr(kr/k1)Jn'(k1a)]),
for all n∈Z. Rearranging the equations in (3.1), we have modified the equations for the Fourier-Bessel multiple scattering coefficients as(3.9)(1+anεJ0)An+anε∑m=-∞m≠n+∞AmJn-m-bnε[(1+cnJ0)AnH+cn∑m=-∞m≠n+∞AmHJn-m]=-anεEni,(1+anμJ0)AnH+anμ∑m=-∞m≠n+∞AmHJn-m+bnμ[(1+cnJ0)An+cn∑m=-∞m≠n+∞AmJn-m]=-bnμcnEni,
for all n∈Z. Equations (3.9) for the special case of n=0 can be simplified as (3.10)A0=-[a0ε(1+a0εJ0)∑m=-∞m≠n+∞][E0i+∑m=-∞m≠n+∞AmJ-m],A0H=-[a0μ(1+a0μJ0)∑m=-∞m≠n+∞][∑m=-∞m≠0+∞AmHJ-m].
Or equivalently, (3.10) can be written as(3.11)A0=-[a0ε(1+a0εJ0)∑m=-∞m≠n+∞(AnI-n+∑m=-∞m≠0,n+∞AmHI-m)][E0i+(AnJ-n+∑m=-∞m≠0,n+∞AmJ-m)],A0H=-[a0μ(1+a0μJ0)∑m=-∞m≠n+∞∑m=-∞m≠0,n+∞][AnHJ-n+∑m=-∞m≠0,n+∞AmHJ-m],
for the Fourier-Bessel multiple scattering associated with the scattered electric and magnetic fields, respectively. Indicating those scattering coefficients corresponding to A0 and A0H explicitly in (3.9), we have(3.12)(1+anεJ0)An+anε[A0Jn+∑m=-∞m≠0,n+∞AmJn-m]-bnε[(1+cnJ0)AnH+cn(A0HJn+∑m=-∞m≠0,n+∞AmHJn-m)]=-anεEni,(1+anμJ0)AnH+anμ[A0HJn+∑m=-∞m≠0,n+∞AmHJn-m]+bnμ[(1+cnJ0)An+cn(A0Jn+∑m=-∞m≠0,n+∞AmJn-m)]=-bnμcnEni{∀n∈Z∣n≠0}. We have noticed that the equations (3.11) represent the solutions for A0 and A0H which are expressed in terms of all the other Fourier-Bessel multiple scattering coefficients {An,AnH}, {∀n∈Z∣n≠0}. Therefore, we can insert these solutions of A0 and A0H into (3.11) to eliminate A0 and A0H terms from the equations of the scattering coefficients. For this purpose, we have first used (3.11) in the evaluation of the following identities as(3.13)anε[A0Jn+∑m=-∞m≠0,n+∞AmJn-m]≡-[a0ε(1+a0εJ0)∑m=-∞m≠0,n+∞]anεJnE0i-[a0ε(1+a0εJ0)∑m=-∞m≠0,n+∞]JnJ-nanεAn+anε∑m=-∞m≠0,n+∞Am{Jn-m-[a0ε(1+a0εJ0)∑m=-∞m≠0,n+∞]JnJ-m},anμ[A0HJn+∑m=-∞m≠0,n+∞AmHJn-m]≡-[a0μ(1+a0μJ0)∑m=-∞m≠0,n+∞]JnJ-nanμAnH+anμ∑m=-∞m≠0,n+∞AmH{Jn-m-[a0μ(1+a0μJ0)∑m=-∞m≠0,n+∞]JnJ-m}{∀n∈Z∣n≠0}. Using (3.13) in (3.12), we have acquired the equations for the scattering coefficients of the infinite grating of circular dielectric cylinders as(3.14){1+anε{J0-[a0ε(1+a0εJ0)]JnJ-n}}An+anε∑m=-∞m≠0,n+∞Am{Jn-m-[a0ε(1+a0εJ0)]JnJ-m}-bnε{AnH[1+cn{J0-[a0μ(1+a0μJ0)]JnJ-n}]+cn∑m=-∞m≠0+∞AmH{Jn-m-[a0μ(1+a0μJ0)]JnJ-m}}=-anε{Eni-[a0ε(1+a0εJ0)]JnE0i},{1+anμ{J0-[a0μ(1+a0μJ0)]JnJ-n}}AnH+anμ∑m=-∞m≠0,n+∞AmH{Jn-m-[a0μ(1+a0μJ0)]JnJ-m}+bnμ{An[1+cn{J0-[a0ε(1+a0εJ0)]JnJ-n}]+cn∑m=-∞m≠0+∞Am{-[a0ε(1+a0εJ0)]JnJ-m}}=-bnμcnEni{∀n∈Z∣n≠0}. The equations in (3.14) together represent the complete set of equations for the Fourier-Bessel multiple scattering coefficients of the infinite grating of circular dielectric cylinders at oblique incidence. Incorporating the first terms inside the brackets into the infinite summation, (3.14) can be written more compactly as(3.15)An+anε∑m=-∞m≠0+∞Am{Jn-m-[a0ε(1+a0εJ0)]JnJ-m}-bnε{AnH+cn∑m=-∞m≠0+∞AmH{Jn-m-[a0μ(1+a0μJ0)]JnJ-m}}=-anε{Eni-[a0ε(1+a0εJ0)]JnE0i∑m=-∞m≠0+∞},AnH+anμ∑m=-∞m≠0+∞AmH{Jn-m-[a0μ(1+a0μJ0)]JnJ-m}+bnμ{An+cn∑m=-∞m≠0+∞Am{Jn-m-[a0ε(1+a0εJ0)]JnJ-m}}=-bnμcnEni{∀n∈Z∣n≠0}. The equations in (3.15) represent the complete set of equations for the Fourier-Bessel multiple scattering coefficients An and AnH’s of the infinite grating at oblique incidence. On the other hand, A0 and A0H are obtained from (3.10) in terms of all the other scattering coefficients, i.e., An and AnH’s, {∀n∈Z∣n≠0}. Defining a new set of parameters dn,mζ, ∀ζ∈{ε,μ}; en(v,ε) and fn(v,μ), {∀n∈Z∣n≠0}, and {∀m∈Z∣m≠0} as(3.16)dn,mζ={Jn-m-[a0ζ(1+a0ζJ0)]JnJ-m};∀ζ∈{ε,μ},(3.17)en(v,ε)=-anε{Eni-[a0ε(1+a0εJ0)]JnE0i},(3.18)fn(v,μ)=-bnμcnEni,
and employing the expressions (3.16), (3.17), and (3.18) in the equations (3.15), we can express the equations for the generalized Fourier-Bessel multiple scattering coefficients of the infinite grating associated with obliquely incident and vertically polarized wavesmore compactly as(3.19)[An+anε∑m=-∞m≠0+∞dn,mεAm]-bnε[AnH+cn∑m=-∞m≠0+∞dn,mμAmH]=en(v,ε),[AnH+anμ∑m=-∞m≠0+∞dn,mμAmH]+bnμ[An+cn∑m=-∞m≠0+∞dn,mεAm]=fn(v,μ){∀n∈Z∣n≠0}. We have obtained the following matrix equations from (3.19) as(3.20){I̲̲+[Diaganε]⋅D̲̲ε}⋅A̲-[Diagbnε]⋅{I̲̲+[Diagcn]⋅D̲̲μ}⋅A̲H=e̲(v,ε),{I̲̲+[Diaganμ]⋅D̲̲μ}⋅A̲H+[Diagbnμ]⋅{I̲̲+[Diagcn]⋅D̲̲ε}⋅A̲=f̲(v,μ).
In (3.20), we have defined(3.21)[Diaganζ]:=(⋅a+2ζa+1ζ0̲̲0̲̲a-1ζa-2ζ⋅)≡Λ̲̲ζ{∀n∈Z∣n≠0}, where Λ̲̲ζ is an (∞×∞) diagonal matrix, defined for ζ∈{ε,μ}, and its elements anζ, are given in (3.3),(3.22)[Diagbnζ]:=(⋅b+2ζb+1ζ0̲̲0̲̲b-1ζb-2ζ⋅)≡B̲̲ζ,
where B̲̲ζ is an(∞×∞)diagonal matrix defined for ζ∈{ε,μ} and its elements, bnζ, are given in (3.4),(3.23)[Diagcn]:=(⋅c+2c+10̲̲0̲̲c-1c-2⋅)≡Γ̲̲,
where Γ̲̲ is an (∞×∞) diagonal matrix, and its elements, cn, are given in (3.2),(3.24)[dn,mξ]:=(⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅d2,+2ξd2,+1ξd2,-1ξd2,-2ξ⋅⋅⋅⋅d1,+2ξd1,+1ξd1,-1ξd1,-2ξ⋅⋅⋅⋅d-1,2ξd-1,1ξd-1,-1ξd-1,-2ξ⋅⋅⋅⋅d-2,2ξd-2,1ξd-2,-1ξd-2,-2ξ⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅)=Dξ
where Dξ is an (∞×∞) matrix, defined forζ∈{ε,μ}, and its elements, dn,mξ, are given in (3.16), and I̲̲ is an (∞×∞) identity matrix. Besides, we have(3.25)e̲(v,ε)≡(⋅⋅e3(v,ε)e2(v,ε)e1(v,ε)e-1(v,ε)e-2(v,ε)e-3(v,ε)⋅⋅),f̲(v,μ)≡(⋅⋅f3(v,μ)f2(v,μ)f1(v,μ)f-1(v,μ)f-2(v,μ)f-3(v,μ)⋅⋅),
where e̲(v,ε) and f̲(v,μ) are (∞×1) vectors, and their elements en(v,ε), and fn(v,ε) are given in (3.17) and (3.18), respectively. On the other hand, the unknown scattering coefficients associated with the exterior electric and magnetic fields are defined as(3.26)A̲≡(⋅⋅A3A2A1A-1A-2A-3⋅⋅),A̲H≡(⋅⋅A3HA2HA1HA-1HA-2HA-3H⋅⋅),A̲ and A̲H are (∞×1) unknown vectors for the Fourier-Bessel multiple scattering coefficients of the electric and magnetic fields of the infinite array of dielectric circular cylinders corresponding to the vertically polarized obliquely incident plane electromagnetic waves. We can use the definitions of (3.21)–(3.26) in combining (3.20) into a single matrix system of equations for the scattering coefficients of the infinite grating at oblique incidence, and write (3.20) as a unique matrix system of equation for all of the scattering coefficients as(3.27)[(I̲̲+Λ̲̲ε⋅D̲̲ε)-B̲̲ε⋅(I̲̲+Γ̲̲⋅D̲̲μ)B̲̲μ⋅(I̲̲+Γ̲̲⋅D̲̲ε)(I̲̲+Λ̲̲μ⋅D̲̲μ)](A̲A̲H)=(e̲(v,ε)f̲(v,μ)).


## 4. Exact Solution to the “Twersky-Wait-Kavaklıoğlu Equations” for an Infinite Grating of Circular Dielectric Cylinders at Oblique Incidence: Transverse-Magnetic Mode

The purpose of this section is to acquire the solution to the exact matrix system of equations for transverse magnetic Fourier-Bessel multiple scattering coefficients of the infinite grating at oblique incidence. Thematrix system of equations for the scattering coefficients of the exterior electric and magnetic fields corresponding to the vertically polarized obliquely incident plane electromagnetic waves expressed by (3.27) can be rewritten as two separate matrix equations as(4.1)(I̲̲+Λ̲̲ε⋅D̲̲ε)⋅A̲-B̲̲ε⋅(I̲̲+Γ̲̲⋅D̲̲μ)⋅A̲H=e̲(v,ε),(4.2)B̲̲μ⋅(I̲̲+Γ̲̲⋅D̲̲ε)⋅A̲+(I̲̲+Λ̲̲μ⋅D̲̲μ)⋅A̲H=f̲(v,μ).
From (4.1) we can solve for A̲ as(4.3)A̲=(I̲̲+Λ̲̲ε⋅D̲̲ε)-1⋅[e̲(v,ε)+B̲̲ε⋅(I̲̲+Γ̲̲⋅D̲̲μ)⋅A̲H].
In a similar manner, we can solve from (4.2) for A̲H as(4.4)A̲H=(I̲̲+Λ̲̲μ⋅D̲̲μ)-1⋅[f̲(v,μ)-B̲̲μ⋅(I̲̲+Γ̲̲⋅D̲̲ε)⋅A̲].
Using the expression of (4.4) for A̲H in (4.3), we have obtained the solution for A̲ as(4.5)A̲=[I̲̲+(I̲̲+Λ̲̲ε⋅D̲̲ε)-1⋅B̲̲μ⋅(I̲̲+Γ̲̲⋅D̲̲μ)⋅(I̲̲+Λ̲̲μ⋅D̲̲μ)-1⋅B̲̲ε⋅(I̲̲+Γ̲̲⋅D̲̲ε)]-1⋅(I̲̲+Λ̲̲ε⋅D̲̲ε)-1⋅[e̲(v,ε)+B̲̲ε⋅(I̲̲+Γ̲̲⋅D̲̲μ)⋅(I̲̲+Λ̲̲μ⋅D̲̲μ)-1⋅f̲(v,μ)],
or, equivalently(4.6)A̲=[(I̲̲+Λ̲̲ε⋅D̲̲ε)+B̲̲ε⋅(I̲̲+Γ̲̲⋅D̲̲μ)⋅(I̲̲+Λ̲̲μ⋅D̲̲μ)-1⋅B̲̲μ⋅(I̲̲+Γ̲̲⋅D̲̲ε)]-1⋅[e̲(v,ε)+B̲̲ε⋅(I̲̲+Γ̲̲⋅D̲̲μ)⋅(I̲̲+Λ̲̲μ⋅D̲̲μ)-1⋅f̲(v,μ)].
Defining two (∞×∞)matrices in terms of the previously defined matrices, namely,(4.7)Ω̲̲εμ∶=(I̲̲+Λ̲̲ε⋅D̲̲ε)-1⋅B̲̲ε⋅(I̲̲+Γ̲̲⋅D̲̲μ),Ω̲̲με∶=(I̲̲+Λ̲̲μ⋅D̲̲μ)-1⋅B̲̲μ⋅(I̲̲+Γ̲̲⋅D̲̲ε),
and inserting (4.7) into (4.5), (4.6), we can express A̲ as(4.8)A̲=[I̲̲+Ω̲̲εμ⋅Ω̲̲με]-1[(I̲̲+Λ̲̲ε⋅D̲̲ε)-1⋅e̲(v,ε)+Ω̲̲εμ⋅(I̲̲+Λ̲̲μ⋅D̲̲μ)-1⋅f̲(v,μ)].
Further evaluation of the expression (4.8) yields the multiple scattering coefficients for the electric fields in the exterior region of the infinite grating associated with obliquely incident vertically polarized electromagnetic waves, namely, A̲ as(4.9)A̲=(I̲̲+Ω̲̲εμ⋅Ω̲̲με)-1⋅(I̲̲+Λ̲̲ε⋅D̲̲ε)-1⋅e̲(v,ε)+(Ω̲̲εμ-1+Ω̲̲με)-1⋅(I̲̲+Λ̲̲μ⋅D̲̲μ)-1⋅f̲(v,μ).
Inserting the expression of A̲ in (4.9) into A̲H in (4.4), we have finally obtained the multiple scattering coefficients for the magnetic fields in the exterior region of the infinite grating associated with obliquely incident vertically polarized electromagnetic waves as(4.10)A̲H=-(Ω̲̲εμ+Ω̲̲με-1)-1⋅(I̲̲+Λ̲̲ε⋅D̲̲ε)-1⋅e̲(v,ε)+[I̲̲-(I̲̲+Ω̲̲εμ-1⋅Ω̲̲με-1)-1]⋅(I̲̲+Λ̲̲μ⋅D̲̲μ)-1⋅f̲(v,μ).
Combining (4.9) and (4.10) into a single matrix expression, we can write(4.11)(A̲A̲H)=[(I̲̲+Ω̲̲εμ⋅Ω̲̲με)-1(Ω̲̲εμ-1+Ω̲̲με)-1-(Ω̲̲εμ+Ω̲̲με-1)-1I̲̲-(I̲̲+Ω̲̲εμ-1⋅Ω̲̲με-1)-1]×[(I̲̲+Λ̲̲ε⋅D̲̲ε)-10̲̲0̲̲(I̲̲+Λ̲̲μ⋅D̲̲μ)-1](e̲(v,ε)f̲(v,μ)).
This is the complete solution vector for the whole Fourier-Bessel multiple scattering coefficients of the infinite grating at oblique incidence.


## 5. Conclusion

In the closely related previous studies, the exact representations of the external fields corresponding to the obliquely incident plane electromagnetic waves and the exact equations describing the behavior of the associatedFourier-Bessel multiple scattering coefficients of an infinite array of penetrable circular cylinders, which are aligned along the y-axis and placed parallel to the z-axis, were derived [16–19] for both TM and TE polarizations. Furthermore, the generalized form of Twersky’s functional equation for the infinite grating at oblique incidence in matrix form was acquired [20] in terms of the Fourier-Bessel scattering coefficients of an isolated dielectric circular cylinder at oblique incidence,which was originally derived by Wait [21]. In a more recent treatment, by the implementation of the Ansatz proposed in [22], the asymptotic solution of the Fourier-Bessel multiple scattering coefficientsassociated with the obliquely incident and vertically polarized waves has been acquired as a function of cylinder radius to grating spacingup to and including the third order terms when the grating spacing, d,is small compare to a wavelength. In addition, a proof [23] for the validity of the asymptotic solution acquired in [22], have been provided.In this investigation, we have presented a procedure for acquiring the conformation of the exact equations describing the behavior of thegeneralized Fourier-Bessel multiple scattering coefficients of an infinite array of penetrable circular cylinders associated with obliquely incident transverse magnetic plane waves in matrix form.This exact representation is formulated in terms of the multiplication of the two (∞×∞)system matrices, and describes the exact solutions associated with the transverse magnetic multiple scattering coefficients at oblique incidence. Approximate expressions for the aforementioned multiple scattering coefficients can then easily be acquired by truncating these (∞×∞)system matrices.



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*Source: 371729-2012-07-05.xml*