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Journal of Applied Mathematics
Volume 2011, Article ID 715087, 30 pages
http://dx.doi.org/10.1155/2011/715087
Research Article

Asymptotic Analysis of Transverse Magnetic Multiple Scattering by the Diffraction Grating of Penetrable Cylinders at Oblique Incidence

Department of Electrical and Computer Engineering, School of Engineering and Applied Science, The George Washington University, Washington, DC 20052, USA

Received 17 June 2011; Accepted 13 September 2011

Academic Editor: Yongkun Li

Copyright © 2011 Ömer Kavaklıoğlu and Roger Henry Lang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We have presented a derivation of the asymptotic equations for transverse magnetic multiple scattering coefficients of an infinite grating of penetrable circular cylinders for obliquely incident plane electromagnetic waves. We have first deducted an “Ansatz” delineating the asymptotic behavior of the transverse magnetic multiple scattering coefficients associated with the most generalized condition of oblique incidence (Kavaklıoğlu, 2000) by exploiting Schlömilch series corresponding to the special circumstance that the grating spacing is much smaller than the wavelength of the incident electromagnetic radiation. The validity of the asymptotic equations for the aforementioned scattering coefficients has been verified by collating them with the Twersky's asymptotic equations at normal incidence. Besides, we have deduced the consequences that the asymptotic forms of the equations at oblique incidence acquired in this paper reduce to Twersky's asymptotic forms at normal incidence by expanding the generalized scattering coefficients at oblique incidence into an asymptotic series as a function of the ratio of the cylinder radius to the grating spacing.

1. Introduction

Rayleigh [1] first treated the problem of the incidence of plane electric waves on an insulating dielectric cylinder as long ago as 1881. He published the classical electromagnetic problem of the diffraction of a plane wave at normal incidence by a homogeneous dielectric cylinder [2]. His solution was later generalized for obliquely incident plane waves when the magnetic vector of the incident wave is transverse to the axis of the cylinder by Wait [3]. Moreover, Rayleigh [4, 5] adduced the first theoretical investigation for the problem of diffraction by gratings. His results have been extended by Wait [6] for the treatment of scattering of plane waves by parallel-wire grids with arbitrary angle of incidence. Wait [6] developed the solution of the problem of the scattering of plane electromagnetic waves incident upon a parallel-wire grid that was backed by a plane-conducting surface. He generalized this result subsequently to a plane wave, incident obliquely with arbitrary polarization on a planar grid [7]. Wait did not treat the scattering of obliquely incident plane electromagnetic waves by the infinite array of thick dielectric cylinders. This configuration has recently been studied by Kavaklıoğlu [810], and an analytic expression for the generalized multiple scattering coefficients of the infinite grating at oblique incidence was captured in the form of a convergent infinite series [11].

The formal analytical solution for the scattering of a plane acoustic or electromagnetic wave by an arbitrary configuration of parallel cylinders of different radii and physical parameters in terms of cylindrical wave functions was obtained by Twersky [12] who considered all possible contributions to the excitation of a particular cylinder by the radiation scattered by the remaining cylinders in the grating and extended this solution to expound the case where all the axes of cylinders lie in the same plane [13]. Twersky [14] subsequently introduced the formal multiple scattering solution of a plane wave by an arbitrary configuration of parallel cylinders to the finite grating of cylinders. He later employed Green’s function methods to represent the multiple scattering amplitude of one cylinder within the grating in terms of the functional equation and the single-scattering amplitude of an isolated cylinder [15]. Furthermore, Twersky [16] acquired a set of algebraic equations for the multiple scattering coefficients of the infinite grating in terms of the elementary function representations of Schlömilch series [17] and the well-known scattering coefficients of an isolated cylinder.

In the area of acoustics, Millar [18] studied the problem of scattering of a plane wave by finite number of cylinders equispaced in a row that are associated with scatterers both “soft” and “hard” in the acoustical sense. The solutions in the form of series in powers of a small parameter, essentially the ratio of cylinder dimension to wavelength, were obtained. Besides, Millar [19] investigated the scattering by an infinite grating of identical cylinders. In a more recent investigation, Linton and Thompson [20] formulated the diffracted acoustic field by an infinite periodic array of circles and determined the conditions for resonance by employing the expressions which enable Schlömilch series to be computed accurately and efficiently [2123].

Previous investigations mentioned above do not include the most general case of oblique incidence although the grating is illuminated by an incident plane 𝐸-polarized electromagnetic wave at an arbitrary angle 𝜙𝑖 to the 𝑥-axis, whereas in the generalized oblique incidence solution presented in this investigation, the direction of the incident plane wave makes an arbitrary oblique angle 𝜃𝑖 with the positive 𝑧-axis as indicated in Figure 1. As far as can be ascertained by the writers, Sivov [24, 25] first treated the diffraction by an infinite periodic array of perfectly conducting cylindrical columns for the most generalized case of obliquely incident plane-polarized electromagnetic waves in order to determine the reflection and transmission coefficients of the infinite grating of perfectly conducting cylinders in free space under the assumption that the period of the grating spacing was small compared to a wavelength. The configuration of a greater relevance to the problem has recently been investigated by many other researchers. For instance, Lee [26] studied the scattering of an obliquely incident electromagnetic wave by an arbitrary configuration of parallel, nonoverlapping infinite cylinders and acquired the solution for the scattering of an obliquely incident plane wave by a collection of closely spaced radially stratified parallel cylinders that can have an arbitrary number of stratified layers [27]. Moreover, Lee [28] presented a general treatment of scattering of arbitrarily polarized incident light by a collection of radially stratified circular cylinders at oblique incidence, described the solution to the problem of scattering of obliquely incident light by a closely spaced parallel radially stratified cylinders embedded in a semi-infinite dielectric medium [29], and developed a general scattering theory for obliquely incident plane-polarized monochromatic waves on a finite slab containing closely spaced radially stratified circular cylinders [30]. In addition, the formulation for the extinction and scattering cross-sections of closely spaced parallel infinite cylinders in a dielectric medium of finite thickness is presented [31]. In the area of modeling photonic crystal structures, Smith et al. [32] developed a formulation for cylinder gratings in conical incidence using a multipole method and studied scattering matrices and Bloch modes in order to investigate the photonic band gap properties of woodpile structures [33]. This area of research has recently received a lot of attention due to potential applications to microcircuitry, nanotechnology, and optical waveguides.

715087.fig.001
Figure 1: The schematic of the scattering by an infinite grating at oblique incidence.

Three-dimensional generalization of Twersky’s solution [15, 16] for scattering of waves by the infinite grating of dielectric circular cylinders was originally developed by Kavaklıoğlu [810] by employing the separation-of-variables method for both TM and TE polarizations, and the reflected and transmitted fields were derived for obliquely incident plane 𝐻-polarized waves in [34]. Kavaklıoğlu and Schneider [35] presented the asymptotic solution of the multiple scattering coefficients for obliquely incident and vertically polarized plane waves as a function of the ratio of the cylinder radius to grating spacing when the grating spacing, 𝑑, is small compared to a wavelength.

Furthermore, Kavaklıoğlu and Schneider [11] acquired the exact analytical solution for the multiple scattering coefficients of the infinite grating for obliquely incident plane electromagnetic waves by the application of the direct Neumann iteration technique to two infinite sets of equations describing the exact behavior of the multiple scattering coefficients, which was originally published in [8, 10], in the form of a convergent infinite series and obtained the generalized form of Twersky’s functional equation for the infinite grating in matrix form for obliquely incident waves [11].

The purpose of this paper is to elucidate the derivation of the equations pertaining to the asymptotic behavior of the transverse magnetic multiple scattering coefficients of an infinite array of infinitely long circular dielectric cylinders illuminated by obliquely incident plane electromagnetic waves. The arbitrarily polarized obliquely incident plane wave depicted in Figure 1 can be decomposed into two different modes of polarization. The asymptotic representation associated with the transverse magnetic (TM) mode that is also defined as vertical polarization, for which the incident electric field 𝐄inc has a component parallel to the constituent cylinders of the grating, will be treated in this investigation.

2. Problem Formulation

“An infinite number of infinitely long identical dielectric circular cylinders,” which are separated by a distance “𝑑,” are placed parallel to each other in the 𝑦-𝑧 plane and positioned perpendicularly to the 𝑥-𝑦 plane as indicated in Figure 1. For TM mode; ̂𝐯𝑖 is the unit vector associated with the vertical polarization and has a component parallel to the cylinders of the grating. The fact that “the incident 𝐸-field has a component parallel to all the cylinders of the dielectric grating” does not mean that we deal with the TM mode as it does not exclude the existence of other components of 𝐸-field. The incident plane wave depicted in Figure 1 makes an angle of obliquity 𝜃𝑖 with the positive 𝑧-axis.

Lemma 2.1 (multiple scattering representation for an infinite grating of dielectric circular cylinders for obliquely incident 𝐸-polarized plane electromagnetic waves [3, 7, 8]). A vertically polarized plane electromagnetic wave, which is obliquely incident upon the infinite array of identical insulating dielectric circular cylinders with radius “a,” dielectric constant “𝜀r,” and relative permeability “𝜇r,” can be expanded in “the individual cylindrical coordinate system (R𝑠,𝜙𝑠,𝑧) of the 𝑠th cylinder” in terms of the cylindrical waves referred to the axis of 𝑠th cylinder as 𝐄inc𝑣𝑅𝑠,𝜙𝑠,𝑧=̂𝐯𝑖𝐸0𝑣𝑒𝑖𝑘𝑟𝑠𝑑sin𝜓𝑖𝑛=𝑒𝑖𝑛𝜓𝑖𝐽𝑛𝑘𝑟𝑅𝑠𝑒𝑖𝑛(𝜙𝑠+𝜋/2)𝑒𝑖𝑘𝑧𝑧.(2.1)

The origin of each individual cylindrical coordinate system, namely, (𝑅𝑠,𝜙𝑠,𝑧), is located at the center of the corresponding cylinder. In the above description of the incident field, ̂𝐯𝑖 is a unit vector that denotes the vertical polarization having a component parallel to all the cylinders, 𝜙𝑖 is the angle of incidence in 𝑥-𝑦 plane measured from the 𝑥-axis in such a way that 𝜓𝑖=𝜋+𝜙𝑖, implying that the wave is obliquely incident in the first quadrant of the coordinate system, and “𝐽𝑛(𝑥)” stands for a Bessel function of order n. In addition, we have the following definitions: 𝑘𝑟=𝑘0sin𝜃𝑖,𝑘𝑧=𝑘0cos𝜃𝑖,𝑘0=𝜔𝑐.(2.2)𝑒𝑖𝜔𝑡” time dependence is suppressed throughout the paper, where “𝜔” stands for the angular frequency of the incident wave in radians per second, “𝑘0” is the free-space wave number, “𝑐” denotes the speed of light in free space, and “𝑡” represents time in seconds. The centers of the cylinders in the infinite grating are located at the positions 𝐫0,𝐫1,𝐫2,, and so forth. The exact solution for the 𝑧-component of the electric field in the exterior of the grating belonging to this configuration 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 each individual 𝑚th cylinder located at 𝐫𝑚, as |𝐫𝐫𝑚|, that is,𝐸(ext)𝑧𝑅𝑠,𝜙𝑠,𝑧=𝐸inc𝑧𝑅𝑠,𝜙𝑠,𝑧++𝑚=𝐸(𝑚)𝑧𝑅𝑚,𝜙𝑚,𝑧.(2.3)

Lemma 2.2 (expressions for the 𝑧-components of the exterior fields [8]). Let {𝐴𝑛,𝐴𝐻𝑛}𝑛= for all 𝑛𝐙, where “𝐙” stands for the set of all integers, denote the set of all multiple scattering coefficients corresponding to the exterior electric and magnetic fields of the infinite grating associated with obliquely incident plane 𝐸-polarized electromagnetic waves, respectively. Then, the exterior electric and magnetic field intensities associated with vertically polarized obliquely incident plane electromagnetic waves are given as 𝐸(ext)𝑧𝑅𝑠,𝜙𝑠,𝑧=𝑒𝑖𝑘𝑟𝑠𝑑sin𝜓𝑖+𝑛=𝐸𝑖𝑛+𝑚=𝐴𝑚𝑛𝑚𝑘𝑟𝑑𝐽𝑛𝑘𝑟𝑅𝑠+𝐴𝑛𝐻(1)𝑛𝑘𝑟𝑅𝑠𝑒𝑖𝑛(𝜙𝑠+𝜋/2)𝑒𝑖𝑘𝑧𝑧,(2.4a)𝐻(ext)𝑧𝑅𝑠,𝜙𝑠,𝑧=𝑒𝑖𝑘𝑟𝑠𝑑sin𝜓𝑖+𝑛=𝑚=𝐴𝐻𝑚𝑛𝑚𝑘𝑟𝑑𝐽𝑛𝑘𝑟𝑅𝑠+𝐴𝐻𝑛𝐻(1)𝑛𝑘𝑟𝑅𝑠𝑒𝑖𝑛(𝜙𝑠+𝜋/2)𝑒𝑖𝑘𝑧𝑧.(2.4b)

In this representation, {𝐴𝑛}𝑛= depicts the set of all undetermined multiple scattering coefficients associated with exterior electric fields defined by the expressions (29) and (34)–(37) in [8], and {𝐴𝐻𝑛}𝑛= delineates the set of all undetermined multiple scattering coefficients associated with exterior magnetic fields defined by the expressions (40)–(42) in [8], respectively. In expressions (2.4a) and (2.4b), we have 𝐸𝑖𝑛=sin𝜃𝑖𝐸0𝑣𝑒𝑖𝑛𝜓𝑖,(2.5a)𝑛(2𝜋Δ)=+𝑝=1𝐻(1)𝑛(2𝜋𝑝Δ)𝑒2𝜋𝑖𝑝Δsin𝜓𝑖(1)𝑛+𝑒2𝜋𝑖𝑝Δsin𝜓𝑖,(2.5b)where  Δ𝑘𝑟𝑑/2𝜋 and “𝐻(1)𝑛(𝑥)” denotes the 𝑛th order Hankel function of first kind, for all 𝑛𝐙. The series 𝑛𝑚(𝑘𝑟𝑑) in expression (2.4b) is the generalization of the “Schlömilch series for obliquely incident electromagnetic waves” [10, 17] 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” [17]. The convergence of the series for the scattering coefficients can be found on page 342 in [11]. Moreover, the convergence of the Schlömilch series has been discussed by Twersky [17] in detail, who also gives additional references. The exact expressions corresponding to the radial and angular components of the electric and magnetic field intensities have already been obtained in [8] by employing the 𝑧-component of the external field in the expressions (2.4a) and (2.4b).

3. Derivation of the Asymptotic Equations for the Multiple Scattering Coefficients of the Infinite Grating at Oblique Incidence

This section is devoted to the formal derivation of the asymptotic equations for the exterior electric and magnetic multiple scattering coefficients of the infinite grating of dielectric cylinders for obliquely incident vertically polarized plane waves. Since the wavelength of the incident radiation is much larger than the grating spacing, the condition max{(𝑘𝑟𝑑/2𝜋)(1±sin𝜓𝑖)}𝑘𝑟𝑑/𝜋<𝑘𝑟𝑑1 is automatically satisfied thereby excluding any special case associated with the grazing modes. In order to demonstrate the procedure of obtaining the asymptotic equations for the TM multiple scattering coefficients of the infinite grating at oblique incidence, we will first introduce the exact equations corresponding to the transverse magnetic multiple scattering coefficients {𝐴𝑛;𝐴𝐻𝑛}+𝑛= associated with the exterior electric and magnetic fields of the infinite grating of dielectric circular cylinders at oblique incidence by asserting the following lemma.

Lemma 3.1 (exact equations of the transverse magnetic multiple scattering coefficients of the infinite grating of insulating dielectric cylinders at oblique incidence [8]). Exact equations corresponding to the transverse magnetic multiple scattering coefficients of an infinite grating of insulating dielectric cylinders associated with obliquely incident plane electromagnetic waves are first presented by the equations (85a) and (85b) in [8] as 𝑏𝜇𝑛𝐴𝑛+𝑐𝑛𝐸𝑖𝑛++𝑚=𝐴𝑚𝑛𝑚𝑘𝑟𝑑=𝐴𝐻𝑛+𝑎𝜇𝑛+𝑚=𝐴𝐻𝑚𝑛𝑚𝑘𝑟𝑑,𝑛𝐙,𝑏𝜀𝑛𝐴𝐻𝑛+𝑐𝑛+𝑚=𝐴𝐻𝑚𝑛𝑚𝑘𝑟𝑑=𝐴𝑛+𝑎𝜀𝑛𝐸𝑖𝑛++𝑚=𝐴𝑚𝑛𝑚𝑘𝑟𝑑,𝑛𝐙.(3.1) The coefficients arising in this infinite set of linear algebraic equations are defined as 𝑐𝑛=𝐽𝑛𝑘𝑟𝑎𝐻(1)𝑛𝑘𝑟𝑎,𝑛𝐙.(3.2) Two sets of constants 𝑎𝜁𝑛 and 𝑏𝜁𝑛, in which 𝜁𝑟{𝜀𝑟,𝜇𝑟} stands for the relative permittivity and permeability of the dielectric cylinders, respectively, are given as 𝑎𝜁𝑛=𝐽𝑛𝑘1𝑎𝐽𝑛𝑘𝑟𝑎𝜁𝑟𝑘𝑟/𝑘1𝐽𝑛𝑘𝑟𝑎𝐽𝑛𝑘1𝑎𝐽𝑛𝑘1𝑎𝐻(1)𝑛𝑘𝑟𝑎𝜁𝑟𝑘𝑟/𝑘1𝐻(1)𝑛𝑘𝑟𝑎𝐽𝑛𝑘1𝑎(3.3) for 𝜁{𝜀,𝜇}, and for all 𝑛𝐙; where 𝑘1 is defined as 𝑘1=𝑘0𝜀𝑟𝜇𝑟cos2𝜃𝑖, and 𝑏𝜁𝑛=𝜀0𝜇0𝜁20𝐽𝑛𝑘1𝑎𝐻(1)𝑛𝑘𝑟𝑎𝐽𝑛𝑘1𝑎𝐻(1)𝑛𝑘𝑟𝑎𝜁𝑟𝑘𝑟/𝑘1𝐻(1)𝑛𝑘𝑟𝑎𝐽𝑛𝑘1𝑎𝑖𝑛𝐹𝑘𝑟𝑎(3.4) for 𝜁{𝜀,𝜇} and for all 𝑛𝐙, where𝐹 in the expression above is a constant and given as 𝐹=𝜇𝑟𝜀𝑟1cos𝜃𝑖𝜇𝑟𝜀𝑟cos2𝜃𝑖,𝑛𝐙.(3.5) In these equations 𝜀𝑟 and 𝜇𝑟 denote the relative dielectric constant and the relative permeability of the insulating dielectric cylinders; 𝜀0 and 𝜇0 stand for the permittivity and permeability of the free space, respectively. In addition, 𝐽𝑛, and 𝐻(1)𝑛 in expressions (3.2) and (3.3) are defined as 𝐽𝑛(𝜍)𝑑𝑑𝜍𝐽𝑛(𝜍),𝐻(1)𝑛(𝜍)𝑑𝑑𝜍𝐻(1)𝑛(𝜍),(3.6) which imply the first derivatives of the Bessel and Hankel functions of first kind and of order 𝑛 with respect to their arguments.

Theorem 3.2 (approximate equations for the scattering coefficients of the infinite grating at oblique incidence when 𝑘𝑟𝑑1). The asymptotic form of the exact equations for the transverse magnetic multiple scattering coefficients of an infinite grating at oblique incidence can be inferred by two different sets, in which the first one contains only the odd coefficients and the second set contains only the even coefficients. Odd multiple scattering coefficients associated with the infinite grating of dielectric circular cylinders at oblique incidence satisfy the following two sets of asymptotic equations: 𝐴±(2𝑛1)𝑘𝑟𝑎4𝑛2𝐷𝑠𝜀𝜇2𝑛1𝐸𝑖±(2𝑛1)+m=±(2𝑛1)𝑚𝐴𝑚+𝑠𝜉±(2𝑛1)m=±(2𝑛1)𝑚𝐴𝐻𝑚,𝐴𝐻±(2𝑛1)𝑘𝑟𝑎4𝑛2𝐷𝑠𝑛±(2𝑛1)𝐸𝑖±(2𝑛1)+m=±(2𝑛1)𝑚𝐴𝑚+𝑠𝜇𝜀2𝑛1m=±(2𝑛1)𝑚𝐴𝐻𝑚.(3.7) Similarly, the even multiple scattering coefficients satisfy the following two infinite sets of asymptotic equations associated with the transverse magnetic multiple scattering coefficients of the infinite grating of dielectric circular cylinders at oblique incidence as 𝐴±2𝑛𝑘𝑟𝑎4𝑛𝐷𝑠𝜀𝜇2𝑛𝐸𝑖±2𝑛+𝑚=±2𝑛𝑚𝐴𝑚+𝑠𝜉±2𝑛𝑚=±2𝑛𝑚𝐴𝐻𝑚,𝐴𝐻±2𝑛𝑘𝑟𝑎4𝑛𝐷𝑠𝜂±2𝑛𝐸𝑖±2𝑛+𝑚=±2𝑛𝑚𝐴𝑚+𝑠𝜇𝜀2𝑛𝑚=±2𝑛𝑚𝐴𝐻𝑚,𝑛𝐍,(3.8) where 𝐍 denotes the set of all natural numbers.

Proof. The exact equations in (3.1) can be solved for 𝐴𝑛, and 𝐴𝐻𝑛 when the distance between the cylinders of the infinite grating is smaller than the wavelength of the incident wave, that is, for 𝑘𝑟𝑑1 the exact equations take the following form: 𝐴±𝑛𝐴𝐻±𝑛𝑆𝑛𝐸𝑖±𝑛+𝑚=𝐴𝑚±𝑛𝑚𝑘𝑟𝑑𝑚=𝐴𝐻𝑚±𝑛𝑚𝑘𝑟𝑑,(3.9) where 𝑆𝑛 is a (2×2) matrix defined as 𝑆𝑛=𝑠𝜀𝜇𝑛𝑠𝜉±𝑛𝑠𝜂±𝑛𝑠𝜇𝜀𝑛𝑘𝑟𝑎2𝑛𝐷,(3.10) and “𝑛(𝑘𝑟𝑑)” connotes the approximation to the “exact form of the Schlömilch series 𝑛(𝑘𝑟𝑑)” in the limiting case when for 𝑘𝑟𝑑1. Introducing (3.10)) into (3.9), the approximate set of equations for the scattering coefficients of the infinite grating at oblique incidence can explicitly be written as 𝐴±𝑛𝐴𝐻±𝑛𝑘𝑟𝑎2𝑛𝐷𝑠𝜀𝜇𝑛𝑠𝜉±𝑛𝑠𝜂±𝑛𝑠𝜇𝜀𝑛𝐸𝑖±𝑛+𝑚=𝐴𝑚±𝑛𝑚𝑘𝑟𝑑𝑚=𝐴𝐻𝑚±𝑛𝑚𝑘𝑟𝑑.(3.11) In the above, we have 𝐷=1+𝜀𝑟𝑘𝑟𝑘121+𝜇𝑟𝑘𝑟𝑘12𝐹2.(3.12) The 𝑛-dependent constants appearing in (3.10) and (3.11) are defined as𝑠𝜀𝜇𝑛=𝑖𝑛𝜋(2𝑛𝑛!)2𝑠𝜀𝜇,(3.13a)𝑠𝜉±𝑛=𝑖𝑛𝜋(2𝑛𝑛!)2𝑠±𝜉,(3.13b)𝑠𝜂±𝑛=𝑖𝑛𝜋(2𝑛𝑛!)2𝑠±𝜂,(3.13c)𝑠𝜇𝜀𝑛=𝑖𝑛𝜋(2𝑛𝑛!)2𝑠𝜇𝜀.(3.13d)The various constants appearing in the definitions (3.13a)–(3.13d) are expressed as 𝑠𝜀𝜇=1𝜀𝑟𝑘𝑟𝑘121+𝜇𝑟𝑘𝑟𝑘12+𝐹2,𝑠𝜇𝜀=1𝜇𝑟𝑘𝑟𝑘121+𝜀𝑟𝑘𝑟𝑘12+𝐹2,𝑠±𝜉=±2𝑖𝜉0𝐹,𝑠±𝜂=2𝑖𝜂0𝐹.(3.14) The elements of the matrix of coefficients in ((3.11) can be calculated using the expressions (3.14), for instance (𝑠𝜀𝜇/𝐷) and (𝑠𝜇𝜀/𝐷) terms can be written as 𝑠𝜀𝜇𝐷1𝜀𝑟sin2𝜃𝑖𝜇𝑟𝜀𝑟cos2𝜃𝑖1+𝜇𝑟sin2𝜃𝑖𝜇𝑟𝜀𝑟cos2𝜃𝑖+𝜇𝑟𝜀𝑟1cos𝜃𝑖𝜇𝑟𝜀𝑟cos2𝜃𝑖21+𝜀𝑟sin2𝜃𝑖𝜇𝑟𝜀𝑟cos2𝜃𝑖1+𝜇𝑟sin2𝜃𝑖𝜇𝑟𝜀𝑟cos2𝜃𝑖𝜇𝑟𝜀𝑟1cos𝜃𝑖𝜇𝑟𝜀𝑟cos2𝜃𝑖2,𝑠𝜇𝜀𝐷1𝜇𝑟sin2𝜃𝑖𝜇𝑟𝜀𝑟cos2𝜃𝑖1+𝜀𝑟sin2𝜃𝑖𝜇𝑟𝜀𝑟cos2𝜃𝑖+𝜇𝑟𝜀𝑟1cos𝜃𝑖𝜇𝑟𝜀𝑟cos2𝜃𝑖21+𝜇𝑟sin2𝜃𝑖𝜇𝑟𝜀𝑟cos2𝜃𝑖1+𝜀𝑟sin2𝜃𝑖𝜇𝑟𝜀𝑟cos2𝜃𝑖𝜇𝑟𝜀𝑟1cos𝜃𝑖𝜇𝑟𝜀𝑟cos2𝜃𝑖2.(3.15) In terms of the definitions of (3.13a)–(3.13d), the approximate set of equations for the multiple scattering coefficients of the infinite grating at oblique incidence given in ((3.11) takes the following form: 𝐴±𝑛𝐴𝐻±𝑛1𝐷𝑠𝜀𝜇𝑠±𝜉𝑠±𝜂𝑠𝜇𝜀𝐸𝑖±𝑛+𝑚=𝐴𝑚±𝑛𝑚𝑘𝑟𝑑𝑚=𝐴𝐻𝑚±𝑛𝑚𝑘𝑟𝑑𝑖𝑛𝜋(2𝑛𝑛!)2𝑘𝑟𝑎2𝑛,𝑛𝐍.(3.16) Statement of Theorem 3.2 follows immediately upon decomposition of (3.16) into its odd and even components as it is designated by (3.7) and (3.8).

The elementary function representations of the Schlömilch series 𝑛(𝑘𝑟𝑑) in (2.5b) have originally been derived by Twersky [17] for the normal incidence and modified by Kavaklıoğlu [10] for the oblique incidence. We will employ these elementary function representations for the evaluation of the asymptotic forms of the Schlömilch series 𝑛=𝒥𝑛+𝑖𝒩𝑛 in the limit of 𝑘𝑟𝑑1. Twersky’s forms [16, 17] are still valid for the case of obliquely incident waves [10] with a slight modification in their arguments.

Lemma 3.3 (approximate expressions for the “Schlömilch series 𝑛=𝒥𝑛+𝑖𝒩𝑛” in the limit of 𝑘𝑟𝑑1 [10, 17]). We have obtained 0 for the special case of 𝑛=0 as 0=1+1𝜋Δ𝜇+𝜇=𝜇1cos𝜙𝜇+2𝑖𝜋ln𝛾Δ2+𝑖𝜋𝜇+𝜇=1+𝜇𝜇=11𝜇+1𝑖𝜋𝜇=𝜇++11Δsinh𝜂+𝜇1𝜇+1𝑖𝜋𝜇=𝜇+11Δsinh𝜂𝜇1𝜇,(3.17) where 𝛾=1.781. In (3.17), cos𝜙𝜇 is defined by the following relationship: sin𝜙𝜇=sin𝜓𝑖+𝜇2𝜋𝑘𝑟𝑑.(3.18) The angles 𝜙𝜇 are the usual “diffraction angles” of  the grating, and (3.18) that provides these discrete angles is called the “grating equation”. “Propagating modes” are determined by |sin𝜙𝜇|<1, and they correspond to |𝜇|𝜇±, the 𝜇±’s being the closest integers to the 𝑚±’s for which |sin𝜙𝜇|<1 should be satisfied, that is, 𝜇±<𝑚±, such that 𝑚±=1sin𝜙𝑖𝑘𝑟𝑑2𝜋.(3.19) “Evanescent modes” are determined by |sin𝜙𝜇|>1, and they correspond to integer values of 𝜇 such that |𝜇|𝑚±+1, we have ±sin𝜙±𝜇>1, and 𝜙±𝜇 are determined by 𝜙±𝜇=±𝜋/2𝑖|𝜂±𝜇|. For this case the “grating equation” takes the form of cosh||𝜂±𝜇||=±sin𝜙𝑖+𝜇2𝜋𝑘𝑟𝑑>1,𝜇𝐙±𝜇𝜇±+1.(3.20) For the general case, we have 𝑛, for all 𝑛𝐍as 2𝑛=1𝜋Δ𝜇+𝜇=𝜇cos2𝑛𝜙𝜇cos𝜙𝜇+𝑖𝜋1𝑛+𝑛𝑚=1(1)𝑚22𝑚(𝑛+𝑚1)!(2𝑚)!(𝑛𝑚)!𝐵2𝑚Δsin𝜓𝑖Δ2𝑚+1𝑖𝜋Δ𝜇+𝜇=0𝜇𝜇=1sin2𝑛𝜙𝜇cos𝜙𝜇+(1)𝑛𝜇=𝜇++1𝑒2𝑛𝜂+𝜇sinh𝜂+𝜇+𝜇=𝜇+1𝑒2𝑛𝜂𝜇sinh𝜂𝜇,2𝑛+1=1𝑖𝜋Δ𝜇+𝜇=𝜇sin(2𝑛+1)𝜙𝜇cos𝜙𝜇+2𝜋𝑛𝑚=0(1)𝑚22𝑚(𝑛+𝑚)!(2𝑚+1)!(𝑛𝑚)!𝐵2𝑚+1Δsin𝜓𝑖Δ2𝑚+1+1𝜋Δ𝜇+𝜇=0𝜇𝜇=1cos(2𝑛+1)𝜙𝜇cos𝜙𝜇+(1)𝑛+1𝜇=𝜇++1𝑒(2𝑛+1)𝜂+𝜇sinh𝜂+𝜇𝜇=𝜇+1𝑒(2𝑛+1)𝜂𝜇sinh𝜂𝜇.(3.21) Finally, B𝑛(𝑥) is the Bernoulli polynomial of argument “𝑥” and power “𝑛”, in ((3.21).

Remark 3.4 (Bessel series 𝒥0, 𝒥2𝑛, and 𝒥2𝑛+1). The propagating range of the Schlömilch series 𝑛, for all 𝑛𝐙+, where 𝐙+={0,1,2,3,} in (3.17) and ((3.21), is described by “𝒥𝑛Bessel series, which can explicitly be written as 𝒥2𝑛=2𝑘𝑟𝑑𝜇+𝜇=𝜇cos2𝑛𝜙𝜇𝑘𝑟𝑑cos𝜙𝜇𝛿𝑛0;𝑛𝐙+,𝒥2𝑛+1=2𝑖𝑘𝑟𝑑𝜇+𝜇=𝜇sin(2𝑛+1)𝜙𝜇cos𝜙𝜇;𝑛𝐙+.(3.22)

Remark 3.5 (Neumann series 𝒩0, 𝒩2𝑛, and 𝒩2𝑛+1). The evanescent range of the Schlömilch series 𝑛, for all 𝑛𝐙+ in (3.17) and (3.21), is described by “𝑖𝒩𝑛” where 𝒩𝑛 is known as the Neumann series. 𝒩𝑛 in (3.17) and ((3.21) can be put into the following form for this limiting case (𝑘𝑟𝑑1) as 𝒩02𝜋ln𝛾Δ2+1𝜋𝜇+𝜇=1+𝜇𝜇=11𝜇1𝜋Δ𝜇=𝜇++1(1/2)(Δ/𝜇)sin𝜓𝑖(𝜇/Δ)𝜇/Δ+sin𝜓𝑖(1/2)(Δ/𝜇)1𝜋Δ𝜇=𝜇+1(1/2)(Δ/𝜇)+sin𝜓𝑖(𝜇/Δ)𝜇/Δsin𝜓𝑖(1/2)(Δ/𝜇).(3.23a) In addition, we can obtain the simplified expressions for 𝒩2𝑛 and 𝒩2𝑛+1 as 𝒩2𝑛1𝑛𝜋+1𝜋𝑛𝑚=1(1)𝑚22𝑚(𝑛+𝑚1)!(2𝑚)!(𝑛𝑚)!𝐵2𝑚Δsin𝜓𝑖Δ2𝑚1𝜋1𝜇=𝜇𝜇+𝜇=0𝑛𝑚=1(1)𝑚22𝑚1(𝑛+𝑚1)!(2𝑚1)!(𝑛𝑚)!Δ2𝑚𝜇+Δsin𝜓𝑖2𝑚1(1)𝑛𝜋Δ𝜇=𝜇++1(𝜇/2Δ)2𝑛+𝑂(Δ/𝜇)2(𝜇/Δ)+sin𝜓𝑖(1/2)(Δ/𝜇)+𝑂(Δ/𝜇)2+𝜇=𝜇+1(𝜇/2Δ)2𝑛+𝑂(Δ/𝜇)2(𝜇/Δ)sin𝜓𝑖(1/2)(Δ/𝜇)+𝑂(Δ/𝜇)2,𝑛𝐍,(3.23b)𝒩2𝑛+12𝑖𝜋𝑛𝑚=0(1)𝑚22𝑚(𝑛+𝑚)!(2𝑚+1)!(𝑛𝑚)!𝐵2𝑚+1Δsin𝜓𝑖Δ2𝑚+11𝑖𝜋1𝜇=𝜇𝜇+𝜇=0𝑛𝑚=0(1)𝑚22𝑚(𝑛+𝑚)!(2𝑚)!(𝑛𝑚)!Δ2𝑚+1𝜇+Δsin𝜓𝑖2𝑚(1)𝑛𝑖𝜋Δ𝜇=𝜇++1(𝜇/2Δ)2𝑛+𝑂(Δ/𝜇)2(𝜇/Δ)+sin𝜓𝑖(1/2)(Δ/𝜇)+𝑂(Δ/𝜇)2𝜇=𝜇+1(𝜇/2Δ)2𝑛+𝑂(Δ/𝜇)2(𝜇/Δ)sin𝜓𝑖(1/2)(Δ/𝜇)+𝑂(Δ/𝜇)2,𝑛𝐙+.(3.23c)

Remark 3.6 (special case when 𝜇+=𝜇=0). The physical problem under consideration corresponds to the special case for which there is only one propagating mode and the scattering of wavelengths is larger than the grating spacing, that is, (𝑘𝑟𝑑/2𝜋)(1±sin𝜓𝑖)<1. Then the Bessel series for 𝜙0=𝜋+𝜓𝑖, which implies that the plane wave is incident onto the grating in the first quadrant, for all 𝑛𝐙+, reduces to 𝒥2𝑛=2cos2𝑛𝜙0𝑘𝑟𝑑cos𝜙0𝛿𝑛0,𝒥2𝑛+1=2𝑖sin(2𝑛+1)𝜙0𝑘𝑟𝑑cos𝜙0,(3.24) where 𝛿𝑛𝑚 stands for the Kronecker delta function.

Remark 3.7 (approximations for Neumann series𝒩0, 𝒩2𝑛, and 𝒩2𝑛+1 in the limit of Δ1). Inserting 𝜇+=𝜇=0 in (3.23a), (3.23b), and (3.23c), the expression for 𝒩0 in (3.23a)) reduces to 𝒩02𝜋ln𝛾Δ21𝜋Δ𝜇=11+2sin2𝜓𝑖(1/2)(Δ/𝜇)2(𝜇/Δ)311+2sin2𝜓𝑖(Δ/𝜇)2+(1/4)(Δ/𝜇)4.(3.25a) The approximation of the Neumann series 𝒩0, for 𝜙0=𝜋+𝜓𝑖, up to terms of the order (𝑘𝑟𝑑)2 can be obtained from (3.25a) as 𝒩02𝜋ln𝛾Δ21+2sin2𝜓𝑖Δ2𝜋𝜁(3),(3.25b)where 𝜁(𝑠), for all 𝑠, denotes the Riemann zeta function. In the same range, the Neumann series 𝒩𝑛 reduces to 𝒩2𝑛=1𝑛𝜋+1𝜋𝑛𝑚=1(1)𝑚22𝑚1(𝑛+𝑚1)!(2𝑚1)!(𝑛𝑚)!Δ2𝑚𝐵2𝑚Δsin𝜓𝑖𝑚+Δsin𝜓𝑖2𝑚1+2𝑛,𝑛𝐍,𝒩2𝑛+1=1𝑖𝜋𝑛𝑚=0(1)𝑚22𝑚(𝑛+𝑚)!(2𝑚)!(𝑛𝑚)!Δ2𝑚+1𝐵2𝑚+1Δsin𝜓𝑖𝑚+1/2+Δsin𝜓𝑖2𝑚+2𝑛+1,𝑛𝐙+,(3.26) where ’s in (3.26) are given as 2𝑛(1)𝑛+1𝜋Δ𝜇=1122𝑛1Δ𝜇2𝑛+1,2𝑛+1𝑖(1)𝑛+1𝜋Δsin𝜓𝑖𝜇=1122𝑛Δ𝜇2𝑛+3.(3.27)

Remark 3.8 (approximations for Schlömilch series, 𝑛=𝒥𝑛+𝑖𝒩𝑛 in the limit of Δ1). If𝑘𝑟𝑑 is small, that is to say if (𝑘𝑟𝑑/2𝜋)(1±sin𝜓𝑖)<1, then there is only one discrete propagating mode. Employing the expansions for the Bessel and Neumann Series obtained in the previous sections for 𝜙0=𝜋+𝜓𝑖; the Schlömilch Series in this range can be expressed as 02𝑘𝑟𝑑cos𝜙02𝑖𝜋ln𝛾𝑘𝑟𝑑4𝜋1𝑖𝑘𝑟𝑑22𝜋312+sin2𝜙0𝜁(3)+𝑂𝑘𝑟𝑑3,12𝑖sin𝜙0𝑘𝑟𝑑cos𝜙0+2sin𝜙0𝜋+𝑘𝑟𝑑2sin𝜙04𝜋3𝜁(3)+𝑂𝑘𝑟𝑑3,24𝜋3𝑖𝑘𝑟𝑑2+2cos2𝜙0𝑘𝑟𝑑cos𝜙0+𝑖𝜋12sin2𝜙0+𝑖𝑘𝑟𝑑2(2𝜋)3𝜁(3)+𝑂𝑘𝑟𝑑3,316𝜋sin𝜙03𝑘𝑟𝑑2𝑖2sin3𝜙0𝑘𝑟𝑑cos𝜙0+2sin𝜙0𝜋143sin2𝜙0sin𝜙02𝑘𝑟𝑑4(2𝜋)5𝜁(5)+𝑂𝑘𝑟𝑑5,425𝜋315𝑖𝑘𝑟𝑑4𝑖16𝜋𝑘𝑟𝑑216sin2𝜙0+2cos4𝜙0𝑘𝑟𝑑cos𝜙0+𝑖2𝜋18sin2𝜙0+8sin4𝜙0𝑖𝑘𝑟𝑑44(2𝜋)5𝜁(5)+𝑂𝑘𝑟𝑑5.(3.28)

Remark 3.9 (leading terms of the Schlömilch series,𝑛=𝒥𝑛+𝑖𝒩𝑛 in the limit of  Δ1). The leading terms of ’s for large “𝑛”, for all 𝑛𝐍 is given as 2𝑛24𝑛1(1)𝑛𝜋2𝑛1𝐵2𝑛(0)𝑘𝑟𝑑2𝑛𝑖𝑛,2𝑛+124𝑛+1(1)𝑛𝜋2𝑛1𝐵2𝑛(0)𝑘𝑟𝑑2𝑛sin𝜙0,(3.29) where 𝐵𝑛(𝑥) corresponds to the Bernoulli Polynomial. From (3.28) and (3.29), we can determine the leading terms of the Schlömilch series as 00𝑘𝑟𝑑,02sec𝜙0,11𝑘𝑟𝑑,12𝑖tan𝜙0,22𝑘𝑟𝑑2,24𝜋3𝑖,33𝑘𝑟𝑑2,316𝜋sin𝜙03,44𝑘𝑟𝑑4,425𝜋315𝑖,55𝑘𝑟𝑑4,528𝜋3sin𝜙015.(3.30) The leading terms of 𝑛 for large “𝑛” are given by 2𝑛2𝑛𝑘𝑟𝑑2𝑛,2𝑛+12𝑛+1𝑘𝑟𝑑2𝑛,(3.31) where  2𝑛’s and  2𝑛+1’s for large "𝑛" are given as 2𝑛𝑖𝑛(1)𝑛24𝑛1𝜋2𝑛1𝐵2𝑛(0),(3.32a)2𝑛+1(1)𝑛24𝑛+1𝜋2𝑛1𝐵2𝑛(0)sin𝜙04𝑖𝑛2𝑛sin𝜙0.(3.32b) In the above expressions, 𝐵𝜉’s are the Bernoulli numbers, and the relationship between Bernoulli polynomial and Bernoulli numbers is given as 𝐵2𝜉(0)(1)𝜉1𝐵𝜉.(3.32c)

4. Asymptotic Expansions for the Scattering Coefficients of the Infinite Grating at Oblique Incidence in the Limiting Case of “(𝑎/𝑑)1

In order to find a solution for the set of equations given in (3.15) and (3.16), we have introduced an “Ansatz” [36] for the scattering coefficients of the electric and magnetic fields of the infinite grating assuming (𝑘𝑟𝑎)1, and (𝑘𝑟𝑎/𝑘𝑟𝑑)𝜉<1/2, as𝐴±(2𝑛1)𝐴±(2𝑛1),0𝑘𝑟𝑎2𝑛,(4.1a)𝐴𝐻±(2𝑛1)𝐴𝐻±(2𝑛1),0𝑘𝑟𝑎2𝑛(4.1b) for all 𝑛𝐍, for the odd multiple coefficients corresponding to the electric and magnetic field intensities of the infinite grating associated with obliquely incident plane electromagnetic waves, and𝐴±2𝑛𝐴±2𝑛,0𝑘𝑟𝑎2𝑛+2,(4.1c)𝐴𝐻±2𝑛𝐴𝐻±2𝑛,0𝑘𝑟𝑎2𝑛+2(4.1d)for all 𝑛𝐙+ for the even multiple coefficients. In the above expressions, we have delineated the wavelength-independent parts of the multiple scattering coefficients associated with the exterior electric and magnetic field intensities as {𝐴±𝑚,0,𝐴𝐻±𝑚,0}+𝑚=.

Theorem 4.1 (asymptotic equations for the multiple scattering coefficients corresponding to the exterior electric and magnetic field intensities associated with obliquely incident vertically polarized plane electromagnetic waves). The multiple scattering coefficients corresponding to the exterior electric and magnetic field intensities associated with obliquely incident vertically polarized plane electromagnetic waves satisfy two infinite sets of asymptotic equations described by𝐴±(2𝑛1),0𝐴𝐻±(2𝑛1),0=𝛿𝑛1𝐷𝑠𝜀𝜇2𝑛1,0𝑠𝜂±(2𝑛1),0𝐸𝑖±(2𝑛1),0+𝑚=1𝑎𝑑2(𝑚+𝑛1)±2(𝑚+𝑛1)𝑆±(2𝑛1),0𝐴(2𝑚1),0𝐴𝐻(2𝑚1),0,𝑛𝐍(4.2a) for the odd multiple scattering coefficients, and 𝐴±2𝑛,0𝐴𝐻±2𝑛,0=𝛿𝑛1𝐷𝑠𝜀𝜇2𝑛,0𝑠𝜂±2𝑛,0𝐸𝑖±2𝑛,0+𝑚=1𝑎𝑑2(𝑚+𝑛1)×𝑆±2𝑛,0±2(𝑚+𝑛1)𝐴2(𝑚1),0𝐴𝐻2(𝑚1),0+±(2𝑚+2𝑛1)𝐴2(𝑚1),0𝐴𝐻2(𝑚1),0,𝑛𝐍,(4.2b)for the even multiple scattering coefficients.

Proof. We have defined the overall effect of the multiple scattering terms when the wavelength is much larger than the grating spacing, that is, (𝑘𝑟𝑑)1, and (𝑘𝑟𝑎/𝑘𝑟𝑑)𝜉<1/2 as𝐺±𝑛𝑚=±𝑛𝑚𝐴𝑚,(4.3a) for the electric field coefficients, and 𝐺𝐻±𝑛𝑚=±𝑛𝑚𝐴𝐻𝑚,(4.3b)for the magnetic field coefficients. Employing the approximations of Schlömilch series given in (3.27) in the expressions (4.3a) and (4.3b), we can write the overall effect of the multiple scattering terms when the wavelength is much larger than the grating spacing, that is, (𝑘𝑟𝑑)1, and (𝑘𝑟𝑎/𝑘𝑟𝑑)𝜉<1/2 as𝐺±(2𝑛1),0=𝑚=1𝑎𝑑2𝑚±2(𝑚+𝑛1)𝐴(2𝑚1),0,(4.4a)𝐺𝐻±(2𝑛1),0=𝑚=1𝑎𝑑2𝑚±2(𝑚+𝑛1)𝐴𝐻(2𝑚1),0(4.4b) for all 𝑛𝐍, for the odd coefficients, 𝐺±2𝑛,0=𝑚=1𝑎𝑑2𝑚±2(𝑚+𝑛1)𝐴(2𝑚2),0+±(2𝑚+2𝑛1)𝐴(2𝑚1),0,(4.4c)𝐺𝐻±2𝑛,0=𝑚=1𝑎𝑑2𝑚±2(𝑚+𝑛1)𝐴𝐻(2𝑚2),0+±(2𝑚+2𝑛1)𝐴𝐻(2𝑚1),0(4.4d)for all 𝑛𝐍, for the even coefficients; and the special case for 𝑛=0 is given by 𝐺0,0=1𝑚=1𝑚𝐴𝑚,0,𝐺𝐻0,0=1𝑚=1𝑚𝐴𝐻𝑚,0.(4.5) Defining the wavelength independent parts of the scattering matrices from (3.10)) as𝑆𝑛=𝑆𝑛,0𝑘𝑟𝑎2𝑛,(4.6a)𝑆±𝑛,0=1𝐷𝑠𝜀𝜇𝑛𝑠𝜉±𝑛𝑠𝜂±𝑛𝑠𝜇𝜀𝑛,(4.6b)𝑆±(2𝑛1),01𝐷𝑠𝜀𝜇2𝑛1,0𝑠𝜉±(2𝑛1),0𝑠𝜂±(2𝑛1),0𝑠𝜇𝜀2𝑛1,0(4.6c) for all 𝑛𝐍, corresponding to the odd, and 𝑆±2𝑛,01𝐷𝑠𝜀𝜇2𝑛,0𝑠𝜉±2𝑛,0𝑠𝜂±2𝑛,0𝑠𝜇𝜀2𝑛,0,𝑛𝐍,(4.6d)corresponding to the even part. Using the definitions in (4.6a)–(4.6d), and introducing (4.4a)–(4.4d) into (3.7) and (3.8), we have obtained the following set of equations for the approximations of the scattering coefficients: 𝐴±(2𝑛1),0𝐴𝐻±(2𝑛1),0=𝑆±(2𝑛1),0𝛿𝑛1𝐸𝑖±(2𝑛1),0+𝑎𝑑2(𝑛1)𝐺±(2𝑛1),0𝑎𝑑2(𝑛1)𝐺𝐻±(2𝑛1),0,𝑛𝐍,(4.7a) corresponding to the odd scattering coefficients, and 𝐴±2𝑛,0𝐴𝐻±2𝑛,0=𝑆±2𝑛,0𝛿𝑛1𝐸𝑖±2𝑛,0+𝑎𝑑2(𝑛1)𝐺±2𝑛,0𝑎𝑑2(𝑛1)𝐺𝐻±2𝑛,0,𝑛𝐍,(4.7b)corresponding to the even scattering coefficients. Splitting the matrices in (4.7a) and (4.7b) into two parts, we have 𝐴±(2𝑛1),0𝐴𝐻±(2𝑛1),0=𝛿𝑛1𝐷𝑠𝜀𝜇2𝑛1,0𝑠𝜂±(2𝑛1),0𝐸𝑖±(2𝑛1),0+𝑎𝑑2(𝑛1)𝑆±(2𝑛1),0𝐺±(2𝑛1),0𝐺𝐻±(2𝑛1),0,𝑛𝐍,(4.8a) for the odd scattering coefficients, and 𝐴±2𝑛,0𝐴𝐻±2𝑛,0=𝛿𝑛1𝐷𝑠𝜀𝜇2𝑛,0𝑠𝜂±2𝑛,0𝐸𝑖±2𝑛,0+𝑎𝑑2(𝑛1)𝑆±2𝑛,0𝐺±2𝑛,0𝐺𝐻±2𝑛,0,𝑛𝐍(4.8b)for even scattering coefficients. From (4.4a)–(4.4d), we have established the following terms: 𝑎𝑑2(𝑛1)𝐺±(2𝑛1),0=𝑚=1𝑎𝑑2(𝑚+𝑛1)±2(𝑚+𝑛1)𝐴(2𝑚1),0(4.9a) for the multiple interactions corresponding to the scattering coefficients of the electric field, and 𝑎𝑑2(𝑛1)𝐺𝐻±(2𝑛1),0=𝑚=1𝑎𝑑2(𝑚+𝑛1)±2(𝑚+𝑛1)𝐴𝐻(2𝑚1),0(4.9b)for the multiple interactions corresponding to the scattering coefficients of the magnetic field, for all 𝑛𝐍, for the odd scattering coefficients, and 𝑎𝑑2(𝑛1)𝐺±2𝑛,0=𝑚=1𝑎𝑑2(𝑚+𝑛1)±2(𝑚+𝑛1)𝐴(2𝑚2),0+±(2𝑚+2𝑛1)𝐴(2𝑚1),0(4.10a) for the multiple interactions corresponding to the scattering coefficients of the electric field, 𝑎𝑑2(𝑛1)𝐺𝐻±2𝑛,0=𝑚=1𝑎𝑑2(𝑚+𝑛1)±2(𝑚+𝑛1)𝐴𝐻(2𝑚2),0+±(2𝑚+2𝑛1)𝐴𝐻(2𝑚1),0(4.10b)for the multiple interactions corresponding to the scattering coefficients of the magnetic field, for all 𝑛𝐍, for the even scattering coefficients. Inserting (4.9a)-(4.9b) and (4.10a)-(4.10b) into (4.8a)-(4.8b), we have finally obtained the infinite set of asymptotic equations for the multiple scattering coefficients corresponding to the exterior electric and magnetic field intensities of an infinite grating of dielectric circular cylinders associated with obliquely incident and vertically polarized electromagnetic waves as it is proposed by the statement of Theorem 4.1 introduced in (4.2a) and (4.2b). In addition, we have noticed that the scattering coefficients of the electric and magnetic fields appeared as coupled to each others.

5. Discussion and Comparison of the Generalized Transverse Magnetic Multiple Scattering Coefficients of the Infinite Grating with Twersky’s Normal Incidence Case

Remark 5.1 (Twersky’s asymptotic solution for the multiple scattering coefficients at normal incidence). The exact equations for the multiple scattering coefficients of the infinite grating associated with the vertically polarized normally incident waves [16] can be solved by truncation as 𝐴0𝑝0𝑞𝑒,𝐴1𝑝1𝑞𝑜,𝐴2𝑝2𝑞𝑒,𝐴3𝑝3𝑞𝑜,(5.1a) where, the numerator terms are given as 𝑝0=𝑏01+2𝑏22,(5.1b)𝑝1=𝑏11+𝑏32+4,(5.1c)𝑝2=𝑏21+𝑏02,(5.1d)𝑝3=𝑏31+𝑏12+4,(5.1e) and the denominator terms are given as 𝑞𝑒=12𝑏0𝑏222,(5.1f)𝑞𝑜=1𝑏1𝑏32+42.(5.1g)The 𝑏𝑛’s, for all 𝑛𝐍, are given by 𝑏0=𝑎01𝑎00,𝑏𝑛=𝑎𝑛1𝑎𝑛0+2𝑛.(5.2) Finally, 𝑎𝑛’s appearing in (5.2) represent the asymptotic forms of the single-scattering coefficients associated with an isolated cylinder within the grating at normal incidence and can be approximated for (𝑘𝑟𝑎)1 as 𝑎0𝑎0,0𝑘𝑟𝑎2,𝑎𝑛𝑎𝑛,0𝑘𝑟𝑎2𝑛,(5.3) for all 𝑛𝐍,where𝑎0,0𝑖𝜋4𝜀𝑟1,(5.4a)𝑎𝑛,0𝑖𝑛𝜋(2𝑛𝑛!)2𝜇𝑟1𝜇𝑟+1,(5.4b)for all 𝑛𝐍.  Inserting (5.3), (3.29), and (3.30) into (5.2), 𝑏𝑛’s can be evaluated as𝑏0𝑎0,01𝑎0,00(𝑎/𝑑)𝑘𝑟𝑎negligiblefor𝑘𝑟𝑎1𝑘𝑟𝑎2,(5.5a)𝑏1𝑎1,01𝑎1,02(𝑎/𝑑)2+0(𝑎/𝑑)𝑘𝑟𝑎negligiblefor𝑘𝑟𝑎1𝑘𝑟𝑎2,(5.5b)𝑏2𝑎2,01𝑎2,04(𝑎/𝑑)4+0(𝑎/𝑑)𝑘𝑟𝑎3negligiblefor𝑘𝑟𝑎1𝑘𝑟𝑎4,(5.5c)𝑏3𝑎3,01𝑎3,06(𝑎/𝑑)6+0(𝑎/𝑑)𝑘𝑟𝑎5negligiblefor𝑘𝑟𝑎1𝑘𝑟𝑎6.(5.5d)Expressions in (5.5a)–(5.5d) are valid for 𝑘𝑟𝑎1, and 𝑘𝑟𝑑1. In general, 𝑏𝑛’s can be expressed as 𝑏𝑛𝑎𝑛,01𝑎𝑛,02𝑛(𝑎/𝑑)2𝑛+0(𝑎/𝑑)𝑘𝑟𝑎2𝑛1negligiblefor𝑘𝑟𝑎1𝑘𝑟𝑎2𝑛,𝑛𝐍.(5.6) Obviously, (5.5a) and (5.6) will asymptotically be written as 𝑏0𝑏0,0𝑘𝑟𝑎2,𝑏𝑛𝑏𝑛,0𝑘𝑟𝑎2𝑛,(5.7) for all𝑛𝐍,  where 𝑏0,0𝑎0,0,(5.8a)𝑏𝑛,0𝑎𝑛,01𝑎𝑛,02𝑛(𝑎/𝑑)2𝑛(5.8b)represent 𝑘𝑟𝑎-independent parts of 𝑏𝑛’s for wavelengths larger than the radii, that is, (𝑘𝑟𝑎)1. The numerator terms appearing in (5.1b)–(5.1e) can be approximated as 1+2𝑏221+2𝑏2,02𝑎𝑑2𝑘𝑟𝑎2negligiblefor𝑘𝑟𝑎1,1+𝑏021+𝑏0,02𝑎𝑑2,1+𝑏32+41+𝑏3,0𝑎𝑑24𝑎𝑑2+2𝑘𝑟𝑎2𝑘𝑟𝑎2negligiblefor𝑘𝑟𝑎1,1+𝑏12+41+𝑏1,0𝑎𝑑22+4𝑘𝑟𝑑2.(5.9) The denominator terms appearing in (5.1c) can be approximated as 𝑞𝑒=12𝑏0𝑏22212𝑏0,0𝑏2,022𝑎𝑑4𝑘𝑟𝑎2negligiblefor𝑘𝑟𝑎1,𝑞𝑜=1𝑏1𝑏32+421𝑏1,0𝑏3,0𝑎𝑑44𝑎𝑑2+2𝑘𝑟𝑎2negligiblefor𝑘𝑟𝑎12.(5.10) Inserting (5.5a)–(5.5d) to (5.10) into (5.1a)–(5.1g), we have 𝐴01+negligiblefor𝑘𝑟𝑎12𝑏2,02(𝑎/𝑑)2𝑘𝑟𝑎212𝑏0,0𝑏2,022(𝑎/𝑑)4𝑘𝑟𝑎2