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

1 Division of Electromagnetics Research, Department of Electrical and Computer Engineering, School of Engineering and Applied Science, The George Washington University, Washington, DC 20052, USA 2 Division of Electromagnetic Fields and Microwave Techniques, Department of Electrical and Electronics Engineering, Faculty of Engineering and Architecture, İzmir Kâtip Çelebi University, 35620 Çiğli, İzmir, Turkey


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 "Schl ömilch 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 of conical 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 radiallystratified 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 the Fourier-Bessel multiple scattering coefficients of 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, θ i being the obliquity angle made with z-axis, were first derived by Kavaklıo glu 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 Kavaklıo glu 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, Kavaklıo glu and Schneider 22 acquired the asymptotic solution for 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, Kavaklıo glu and Lang 23 demonstrated a rigorous proof for the validity of this asymptotic solution derived in 22 .
The purpose of this investigation is to acquire the 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 θ i with the positive z-axis as depicted in Figure 1.
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 The configuration of the problem of scattering of waves by an infinite array of penetrable circular cylinders at oblique incidence side view .
numerical algorithm as its title designates, which would be completely approximate and requiring an error analysis.

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 radii a, dielectric constant ε r , and relative permeability μ r .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 r 0 , r 1 , r 2 ,. .., and so forth, and separated by a distance of d.For this configuration, the incident wave can be written in the cylindrical coordinate system R s , φ s , z of the sth cylinder in terms of the cylindrical waves referred to the axis of sth cylinder 16 as In the representation above, v i denotes the vertical polarization vector associated with a unit vector having a component parallel to all the cylinders, φ i is the angle of incidence in x-y plane measured from x-axis in such a way that ψ i π φ i as it is delineated in Figure 1, implying that the wave is arbitrarily incident in the first quadrant of the coordinate system, and J n x denotes Bessel function of order n.In expression 2.1 , we have where k 0 denotes the free space wave number with k 0 : 2π/λ 0 and where λ 0 is the wavelength of the incident radiation; θ i 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 .
The exact solution associated with the z-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 r s , plus a summation of cylindrical waves outgoing from the individual nth cylinder located at r n , which satisfies Silver-M üller radiation condition 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 Kavaklıo glu 16 for an arbitrary oblique angle of arrival θ i 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 waves can be written as

2.4
The infinite set of undetermined coefficients 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, for all n ∈ Z where Z represents the set of all integers, and E i n in 2.4 is given as In addition, Q n and Q H n 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 for all n ∈ Z. J n k r d is the generalized form of the Schlömilch series for obliquely incident waves Twersky 4-6 , Kavaklıo glu 18 explicitly given as Journal of Applied Mathematics where, Δ ≡ k r d/2π and H 1 n x denotes the nth order Hankel function of first kind, for all n ∈ Z.The series J n−m k r d in expression 2.7 is the generalization of the "Schlömilch series for obliquely incident electromagnetic waves" Twersky 6 , Kavaklıo glu 18 and converges provided that k r d 1 ± sin ψ i /2π does not equal integers.The integral values of k r d 1 ± sin ψ i /2π are known as the "grazing modes" or "Rayleigh values" Twersky 6 .

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 for vertically polarized and obliquely incident plane electromagnetic waves are first derived in Kavaklıo glu 16 , which can be described as 3.1 for all n ∈ Z.In the equations above, we have and c stands for the speed of light in free space as J n ς and H 1 n ς in expression 3.3 and 3.4 are described as the first derivatives of these functions with respect to their arguments, that is, J n ς ≡ d/dς J n ς and H Introducing the intrinsic impedance of free space as 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 for all n ∈ Z. Equations 3.9 for the special case of n 0 can be simplified as

Journal of Applied Mathematics
Or equivalently, 3.10 can be written as

3.11
for the Fourier-Bessel multiple scattering associated with the scattered electric and magnetic fields, respectively.Indicating those scattering coefficients corresponding to A 0 and A H 0 explicitly in 3.9 , we have {∀n ∈ Z | n / 0}.We have noticed that the equations 3.11 represent the solutions for A 0 and A H 0 which are expressed in terms of all the other Fourier-Bessel multiple scattering coefficients {A n , A H n }, {∀n ∈ Z | n / 0}.Therefore, we can insert these solutions of A 0 and A H 0 into 3.11 to eliminate A 0 and A H 0 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 {∀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 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 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 waves more compactly as {∀n ∈ Z | n / 0}.We have obtained the following matrix equations from 3.19 as

3.20
In 3.20 , we have defined

Diag a
ζ n : where Γ is an ∞ × ∞ diagonal matrix, and its elements, c n , are given in 3.2 , where D ξ is an ∞ × ∞ matrix, defined for ζ ∈ {ε, μ}, and its elements, d ξ n,m , are given in 3.16 , and I is an ∞ × ∞ identity matrix.Besides, we have where e v,ε and f v,μ are ∞×1 vectors, and their elements e v,ε n , and f v,ε n 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
A and A H 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

Exact Solution to the "Twersky-Wait-Kavaklıo glu 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.The matrix 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 Journal of Applied Mathematics From 4.1 we can solve for A as In a similar manner, we can solve from 4.2 for A H as Using the expression of 4.4 for A H in 4.3 , we have obtained the solution for A as or, equivalently

4.6
Defining two ∞ × ∞ matrices in terms of the previously defined matrices, namely, and inserting 4.7 into 4.5 , 4.6 , we can express A as 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, A as

4.9
Inserting the expression of A in 4.9 into A H 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

4.11
This is the complete solution vector for the whole Fourier-Bessel multiple scattering coefficients of the infinite grating at oblique incidence.

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 associated Fourier-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 coefficients associated with the obliquely incident and vertically polarized waves has been acquired as a function of cylinder radius to grating spacing up 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 the generalized Fourier-Bessel multiple scattering coefficients of an infinite array of penetrable circular cylinders associated with

Figure 2 :
Figure 2: The geometry of the infinite grating problem top view .

. 7 two
parameters that appears in the equations of the Fourier-Bessel multiple scattering coefficients of the infinite grating in 3.1 , namely, b ε n and b μ n , can be acquired by employing 3.7 in 3.4 for ζ ∈ {ε, μ} as b

n 3 . 15 {∀n∈,
Z | n / 0}.The equations in 3.15 represent the complete set of equations for the Fourier-Bessel multiple scattering coefficients A n and A H n 's of the infinite grating at oblique incidence.On the other hand, A 0 and A H 0 are obtained from 3.10 in terms of all the other scattering coefficients, i.e., A n and A H n 's, {∀n ∈ Z | n / 0}.Defining a new set of parameters d ζ n,m , ∀ ζ ∈ {ε, μ}; e {∀n ∈ Z | n / 0}, {∀m ∈ Z | m / 0} and as defined for ζ ∈ {ε, μ},