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

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 [8–10], 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 [21–23].

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 𝜙_𝑖 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 𝜃_𝑖 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.

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 [8–10] 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 𝐄^inc 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; 𝐯_𝑖̂‌ 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 𝜃_𝑖 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 “𝜀_r,” and relative permeability “𝜇_r,” can be expanded in “the individual 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 as 𝐄^inc_𝑣𝑅_𝑠,𝜙_𝑠,𝑧=𝐯_𝑖̂‌𝐸_0𝑣𝑒^{𝑖𝑘𝑟𝑠𝑑sin𝜓𝑖}^∞_𝑛=−∞𝑒^{−𝑖𝑛𝜓𝑖}𝐽_𝑛𝑘_𝑟𝑅_𝑠𝑒^{𝑖𝑛(𝜙𝑠+𝜋/2)}𝑒^{−𝑖𝑘𝑧𝑧}.(2.1)

The origin of each individual cylindrical coordinate system, namely, (𝑅_𝑠,𝜙_𝑠,𝑧), is located at the center of the corresponding cylinder. In the above description of the incident field, 𝐯_𝑖̂‌ is a unit vector that denotes the vertical polarization having a component parallel to all the cylinders, 𝜙_𝑖 is the angle of incidence in 𝑥-𝑦 plane measured from the 𝑥-axis in such a way that 𝜓_𝑖=𝜋+𝜙_𝑖, implying that the wave is obliquely incident in the first quadrant of the coordinate system, and “𝐽_𝑛(𝑥)” stands for a Bessel function of order n. In addition, we have the following definitions: 𝑘_𝑟=𝑘₀sin𝜃_𝑖,𝑘_𝑧=𝑘₀cos𝜃_𝑖,𝑘₀=𝜔𝑐.(2.2) “𝑒^{−𝑖𝜔𝑡}” 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 𝐫_𝑠, plus a summation of cylindrical waves outgoing from each individual 𝑚th cylinder located at 𝐫_𝑚, as |𝐫−𝐫_𝑚|→∞, that is,𝐸^(ext)_𝑧𝑅_𝑠,𝜙_𝑠,𝑧=𝐸^inc_𝑧𝑅_𝑠,𝜙_𝑠,𝑧+^+∞_𝑚=−∞𝐸^(𝑚)_𝑧𝑅_𝑚,𝜙_𝑚,𝑧.(2.3)

Lemma 2.2 (expressions for the 𝑧-components of the exterior fields [8]). Let {𝐴_𝑛,𝐴^𝐻_𝑛}^∞_𝑛=−∞ for all 𝑛∈𝐙, 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)_𝑧𝑅_𝑠,𝜙_𝑠,𝑧=⎧⎨⎩𝑒^{𝑖𝑘𝑟𝑠𝑑sin𝜓𝑖+∞}_𝑛=−∞𝐸^𝑖_𝑛+^∞_𝑚=−∞𝐴_𝑚ℐ_𝑛−𝑚𝑘_𝑟𝑑𝐽_𝑛𝑘_𝑟𝑅_𝑠+𝐴_𝑛𝐻⁽¹⁾_𝑛𝑘_𝑟𝑅_𝑠𝑒^{𝑖𝑛(𝜙𝑠+𝜋/2)}⎫⎬⎭𝑒^{−𝑖𝑘𝑧𝑧},(2.4a)𝐻^(ext)_𝑧𝑅_𝑠,𝜙_𝑠,𝑧=⎧⎨⎩𝑒^{𝑖𝑘𝑟𝑠𝑑sin𝜓𝑖+∞}_𝑛=−∞^∞_𝑚=−∞𝐴^𝐻_𝑚ℐ_𝑛−𝑚𝑘_𝑟𝑑𝐽_𝑛𝑘_𝑟𝑅_𝑠+𝐴^𝐻_𝑛𝐻⁽¹⁾_𝑛𝑘_𝑟𝑅_𝑠𝑒^{𝑖𝑛(𝜙𝑠+𝜋/2)}⎫⎬⎭𝑒^{−𝑖𝑘𝑧𝑧}.(2.4b)

In this representation, {𝐴_𝑛}^∞_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 {𝐴^𝐻_𝑛}^∞_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 𝐸^𝑖_𝑛=sin𝜃_𝑖𝐸_0𝑣𝑒^{−𝑖𝑛𝜓𝑖},(2.5a)ℐ_𝑛(2𝜋Δ)=^+∞_𝑝=1𝐻⁽¹⁾_𝑛(2𝜋𝑝Δ)𝑒^{2𝜋𝑖𝑝Δsin𝜓𝑖}(−1)^𝑛+𝑒^{−2𝜋𝑖𝑝Δsin𝜓𝑖},(2.5b)where  Δ≡𝑘_𝑟𝑑/2𝜋 and “𝐻⁽¹⁾_𝑛(𝑥)” denotes the 𝑛th order Hankel function of first kind, for all 𝑛∈𝐙. The series ℐ_𝑛−𝑚(𝑘_𝑟𝑑) in expression (2.4b) is the generalization of the “Schlömilch series for obliquely incident electromagnetic waves” [10, 17] and converges provided that 𝑘_𝑟𝑑(1±sin𝜓_𝑖)/2𝜋 does not equal integers. The integral values of 𝑘_𝑟𝑑(1±sin𝜓_𝑖)/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{(𝑘_𝑟𝑑/2𝜋)(1±sin𝜓_𝑖)}≡𝑘_𝑟𝑑/𝜋<𝑘_𝑟𝑑≪1 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 {𝐴_𝑛;𝐴^𝐻_𝑛}^+∞_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 𝑏^𝜇_𝑛⎧⎨⎩𝐴_𝑛+𝑐_𝑛⎡⎢⎣𝐸^𝑖_𝑛+^+∞_𝑚=−∞𝐴_𝑚ℐ_𝑛−𝑚𝑘_𝑟𝑑⎤⎥⎦⎫⎬⎭=−⎡⎢⎣𝐴^𝐻_𝑛+𝑎^𝜇𝑛+∞_𝑚=−∞𝐴^𝐻_𝑚ℐ_𝑛−𝑚𝑘_𝑟𝑑⎤⎥⎦,∀𝑛∈𝐙,𝑏^𝜀_𝑛⎡⎢⎣𝐴^𝐻_𝑛+𝑐^𝑛+∞_𝑚=−∞𝐴^𝐻_𝑚ℐ_𝑛−𝑚𝑘_𝑟𝑑⎤⎥⎦=𝐴_𝑛+𝑎^𝜀_𝑛⎡⎢⎣𝐸^𝑖_𝑛+^+∞_𝑚=−∞𝐴_𝑚ℐ_𝑛−𝑚𝑘_𝑟𝑑⎤⎥⎦,∀𝑛∈𝐙.(3.1) The coefficients arising in this infinite set of linear algebraic equations are defined as 𝑐_𝑛∶=𝐽_𝑛𝑘_𝑟𝑎𝐻⁽¹⁾_𝑛𝑘_𝑟𝑎,∀𝑛∈𝐙.(3.2) Two sets of constants 𝑎^𝜁_𝑛 and 𝑏^𝜁_𝑛, in which 𝜁_𝑟∈{𝜀_𝑟,𝜇_𝑟} stands for the relative permittivity and permeability of the dielectric cylinders, respectively, are given as 𝑎^𝜁_𝑛=⎡⎢⎣𝐽_𝑛𝑘₁𝑎𝐽^_𝑛𝑘_𝑟𝑎−𝜁_𝑟𝑘_𝑟/𝑘₁𝐽_𝑛𝑘_𝑟𝑎𝐽^_𝑛𝑘₁𝑎𝐽_𝑛𝑘₁𝑎𝐻⁽¹⁾_𝑛^𝑘_𝑟𝑎−𝜁_𝑟𝑘_𝑟/𝑘₁𝐻⁽¹⁾_𝑛𝑘_𝑟𝑎𝐽^_𝑛𝑘₁𝑎⎤⎥⎦(3.3) for 𝜁∈{𝜀,𝜇}, and for all 𝑛∈𝐙; where 𝑘₁ is defined as 𝑘₁=𝑘₀√𝜀_𝑟𝜇_𝑟−cos²𝜃_𝑖, and 𝑏^𝜁_𝑛=⎷𝜀₀𝜇₀𝜁²₀⎡⎢⎣𝐽_𝑛𝑘₁𝑎𝐻⁽¹⁾_𝑛𝑘_𝑟𝑎𝐽_𝑛𝑘₁𝑎𝐻⁽¹⁾_𝑛^𝑘_𝑟𝑎−𝜁_𝑟𝑘_𝑟/𝑘₁𝐻⁽¹⁾_𝑛𝑘_𝑟𝑎𝐽^_𝑛𝑘₁𝑎⎤⎥⎦𝑖𝑛𝐹𝑘_𝑟𝑎(3.4) for 𝜁∈{𝜀,𝜇} and for all 𝑛∈𝐙, where𝐹 in the expression above is a constant and given as 𝐹=𝜇_𝑟𝜀_𝑟−1cos𝜃_𝑖𝜇_𝑟𝜀_𝑟−cos²𝜃_𝑖,∀𝑛∈𝐙.(3.5) In these equations 𝜀_𝑟 and 𝜇_𝑟 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, 𝐽^_𝑛, and 𝐻⁽¹⁾_𝑛^ in expressions (3.2) and (3.3) are defined as 𝐽^_𝑛(𝜍)≡𝑑𝑑𝜍𝐽_𝑛(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 𝑘_𝑟𝑑≪1). 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_{±(2𝑛−1)}≅𝑘_𝑟𝑎^4𝑛−2𝐷𝑠^𝜀𝜇_2𝑛−1𝐸^𝑖_{±(2𝑛−1)}+^∞_m=−∞ℋ_{±(2𝑛−1)−𝑚}𝐴_𝑚+𝑠^𝜉_{±(2𝑛−1)}^∞_m=−∞ℋ_{±(2𝑛−1)−𝑚}𝐴^𝐻_𝑚,𝐴^𝐻_{±(2𝑛−1)}≅𝑘_𝑟𝑎^4𝑛−2𝐷𝑠^𝑛_{±(2𝑛−1)}𝐸^𝑖_{±(2𝑛−1)}+^∞_m=−∞ℋ_{±(2𝑛−1)−𝑚}𝐴_𝑚+𝑠^𝜇𝜀_2𝑛−1^∞_m=−∞ℋ_{±(2𝑛−1)−𝑚}𝐴^𝐻_𝑚.(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_±2𝑛≅𝑘_𝑟𝑎^4𝑛𝐷𝑠^𝜀𝜇_2𝑛𝐸^𝑖_±2𝑛+^∞_𝑚=−∞ℋ_{±2𝑛−𝑚}𝐴_𝑚+𝑠^𝜉_±2𝑛^∞_𝑚=−∞ℋ_{±2𝑛−𝑚}𝐴^𝐻_𝑚,𝐴^𝐻_±2𝑛≅𝑘_𝑟𝑎^4𝑛𝐷𝑠^𝜂_±2𝑛𝐸^𝑖_±2𝑛+^∞_𝑚=−∞ℋ_{±2𝑛−𝑚}𝐴_𝑚+𝑠^𝜇𝜀_2𝑛^∞_𝑚=−∞ℋ_{±2𝑛−𝑚}𝐴^𝐻_𝑚,∀𝑛∈𝐍,(3.8) where 𝐍 denotes the set of all natural numbers.

Proof. The exact equations in (3.1) can be solved for 𝐴_𝑛, and 𝐴^𝐻_𝑛 when the distance between the cylinders of the infinite grating is smaller than the wavelength of the incident wave, that is, for 𝑘_𝑟𝑑≪1 the exact equations take the following form: ⎛⎜⎜⎝𝐴_±𝑛𝐴^𝐻_±𝑛⎞⎟⎟⎠≅𝑆_𝑛⎛⎜⎜⎜⎜⎝𝐸^𝑖_±𝑛+^∞_𝑚=−∞𝐴_𝑚ℋ_{±𝑛−𝑚}𝑘_𝑟𝑑_∞𝑚_=−∞𝐴^𝐻_𝑚ℋ_{±𝑛−𝑚}𝑘_𝑟𝑑⎞⎟⎟⎟⎟⎠,(3.9) where 𝑆_𝑛 is a (2×2) matrix defined as 𝑆_𝑛∶=⎛⎜⎜⎝𝑠^𝜀𝜇_𝑛𝑠^𝜉_±𝑛𝑠^𝜂_±𝑛𝑠^𝜇𝜀_𝑛⎞⎟⎟⎠𝑘_𝑟𝑎^2𝑛𝐷,(3.10) and “ℋ_𝑛(𝑘_𝑟𝑑)” connotes the approximation to the “exact form of the Schlömilch series ℐ_𝑛(𝑘_𝑟𝑑)” in the limiting case when for 𝑘_𝑟𝑑≪1. 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_±𝑛𝐴^𝐻_±𝑛⎞⎟⎟⎠≅𝑘_𝑟𝑎^2𝑛𝐷⎛⎜⎜⎝𝑠^𝜀𝜇_𝑛𝑠^𝜉_±𝑛𝑠^𝜂_±𝑛𝑠^𝜇𝜀_𝑛⎞⎟⎟⎠⎛⎜⎜⎜⎜⎝𝐸^𝑖_±𝑛+^∞_𝑚=−∞𝐴_𝑚ℋ_{±𝑛−𝑚}𝑘_𝑟𝑑_∞𝑚_=−∞𝐴^𝐻_𝑚ℋ_{±𝑛−𝑚}𝑘_𝑟𝑑⎞⎟⎟⎟⎟⎠.(3.11) In the above, we have 𝐷=1+𝜀_𝑟𝑘_𝑟𝑘₁²1+𝜇_𝑟𝑘_𝑟𝑘₁²−𝐹².(3.12) The 𝑛-dependent constants appearing in (3.10) and (3.11) are defined as𝑠^𝜀𝜇_𝑛∶=𝑖𝑛𝜋(2^𝑛𝑛!)²𝑠_𝜀𝜇,(3.13a)𝑠^𝜉_±𝑛∶=𝑖𝑛𝜋(2^𝑛𝑛!)²𝑠_±𝜉,(3.13b)𝑠^𝜂_±𝑛∶=𝑖𝑛𝜋(2^𝑛𝑛!)²𝑠_±𝜂,(3.13c)𝑠^𝜇𝜀_𝑛∶=𝑖𝑛𝜋(2^𝑛𝑛!)²𝑠_𝜇𝜀.(3.13d)The various constants appearing in the definitions (3.13a)–(3.13d) are expressed as 𝑠_𝜀𝜇=1−𝜀_𝑟𝑘_𝑟𝑘₁²1+𝜇_𝑟𝑘_𝑟𝑘₁²+𝐹²,𝑠_𝜇𝜀=1−𝜇_𝑟𝑘_𝑟𝑘₁²1+𝜀_𝑟𝑘_𝑟𝑘₁²+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 (𝑠_𝜀𝜇/𝐷) and (𝑠_𝜇𝜀/𝐷) terms can be written as 𝑠_𝜀𝜇𝐷≡1−𝜀_𝑟sin²𝜃_𝑖𝜇_𝑟𝜀_𝑟−cos²𝜃_𝑖1+𝜇_𝑟sin²𝜃_𝑖𝜇_𝑟𝜀_𝑟−cos²𝜃_𝑖+⎡⎢⎣𝜇_𝑟𝜀_𝑟−1cos𝜃_𝑖𝜇_𝑟𝜀_𝑟−cos²𝜃_𝑖⎤⎥⎦²1+𝜀_𝑟sin²𝜃_𝑖𝜇_𝑟𝜀_𝑟−cos²𝜃_𝑖1+𝜇_𝑟sin²𝜃_𝑖𝜇_𝑟𝜀_𝑟−cos²𝜃_𝑖−⎡⎢⎣𝜇_𝑟𝜀_𝑟−1cos𝜃_𝑖𝜇_𝑟𝜀_𝑟−cos²𝜃_𝑖⎤⎥⎦²,𝑠_𝜇𝜀𝐷≡1−𝜇_𝑟sin²𝜃_𝑖𝜇_𝑟𝜀_𝑟−cos²𝜃_𝑖1+𝜀_𝑟sin²𝜃_𝑖𝜇_𝑟𝜀_𝑟−cos²𝜃_𝑖+⎡⎢⎣𝜇_𝑟𝜀_𝑟−1cos𝜃_𝑖𝜇_𝑟𝜀_𝑟−cos²𝜃_𝑖⎤⎥⎦²1+𝜇_𝑟sin²𝜃_𝑖𝜇_𝑟𝜀_𝑟−cos²𝜃_𝑖1+𝜀_𝑟sin²𝜃_𝑖𝜇_𝑟𝜀_𝑟−cos²𝜃_𝑖−⎡⎢⎣𝜇_𝑟𝜀_𝑟−1cos𝜃_𝑖𝜇_𝑟𝜀_𝑟−cos²𝜃_𝑖⎤⎥⎦².(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_±𝑛𝐴^𝐻_±𝑛⎞⎟⎟⎠≅1𝐷⎛⎜⎜⎝𝑠_𝜀𝜇𝑠_±𝜉𝑠_±𝜂𝑠_𝜇𝜀⎞⎟⎟⎠⎛⎜⎜⎜⎜⎝𝐸^𝑖_±𝑛+^∞_𝑚=−∞𝐴_𝑚ℋ_{±𝑛−𝑚}𝑘_𝑟𝑑_∞𝑚_=−∞𝐴^𝐻_𝑚ℋ_{±𝑛−𝑚}𝑘_𝑟𝑑⎞⎟⎟⎟⎟⎠𝑖𝑛𝜋(2^𝑛𝑛!)²𝑘_𝑟𝑎^2𝑛,∀𝑛∈𝐍.(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 ℐ_𝑛(𝑘_𝑟𝑑) 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 ℋ_𝑛=𝒥_𝑛+𝑖𝒩_𝑛 in the limit of 𝑘_𝑟𝑑≪1. 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 ℋ_𝑛=𝒥_𝑛+𝑖𝒩_𝑛” in the limit of 𝑘_𝑟𝑑≪1 [10, 17]). We have obtained ℋ₀ for the special case of 𝑛=0 as ℋ₀=−1+1𝜋Δ⎛⎜⎝^𝜇+_𝜇=_−𝜇−1cos𝜙_𝜇⎞⎟⎠+2𝑖𝜋ln𝛾Δ2+𝑖𝜋⎛⎜⎝^𝜇+_𝜇=1+^𝜇−_𝜇=1⎞⎟⎠1𝜇+1𝑖𝜋^∞_𝜇=𝜇++11Δsinh𝜂⁺_𝜇−1𝜇+1𝑖𝜋^∞_{𝜇=𝜇−+1}1Δsinh𝜂⁻_𝜇−1𝜇,(3.17) where 𝛾=1.781…. In (3.17), cos𝜙_𝜇 is defined by the following relationship: sin𝜙_𝜇∶=sin𝜓_𝑖+𝜇2𝜋𝑘_𝑟𝑑.(3.18) The angles 𝜙_𝜇 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 |sin𝜙_𝜇|<1, and they correspond to |𝜇|≤𝜇_±, the 𝜇_±’s being the closest integers to the 𝑚_±’s for which |sin𝜙_𝜇|<1 should be satisfied, that is, 𝜇_±<𝑚_±, such that 𝑚_±=1∓sin𝜙_𝑖𝑘_𝑟𝑑2𝜋.(3.19) “Evanescent modes” are determined by |sin𝜙_𝜇|>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/2∓𝑖|𝜂^±_𝜇|. For this case the “grating equation” takes the form of cosh||𝜂^±_𝜇||=±sin𝜙_𝑖+𝜇2𝜋𝑘_𝑟𝑑>1,∀𝜇∈𝐙∣±𝜇≥𝜇_±+1.(3.20) For the general case, we have ℋ_𝑛, for all 𝑛∈𝐍as ℋ_2𝑛=1𝜋Δ^𝜇+_{𝜇=−𝜇−}cos2𝑛𝜙_𝜇cos𝜙_𝜇+𝑖𝜋⎡⎢⎣1𝑛+^𝑛_𝑚=1(−1)^𝑚2^2𝑚(𝑛+𝑚−1)!(2𝑚)!(𝑛−𝑚)!𝐵_2𝑚Δsin𝜓_𝑖Δ^2𝑚⎤⎥⎦+1𝑖𝜋Δ⎡⎢⎣⎛⎜⎝^𝜇+_𝜇=0−^−𝜇−_𝜇=−1⎞⎟⎠sin2𝑛𝜙_𝜇cos𝜙_𝜇+(−1)^𝑛⎛⎜⎝^∞_𝜇=_𝜇++1𝑒^{−2𝑛𝜂+𝜇}sinh𝜂⁺_𝜇+^∞_{𝜇=𝜇−+1}𝑒^{−2𝑛𝜂−𝜇}sinh𝜂⁻_𝜇⎞⎟⎠⎤⎥⎦,ℋ_2𝑛+1=1𝑖𝜋Δ^𝜇+_{𝜇=−𝜇−}sin(2𝑛+1)𝜙_𝜇cos𝜙_𝜇+2𝜋^𝑛_𝑚=0(−1)^𝑚2^2𝑚(𝑛+𝑚)!(2𝑚+1)!(𝑛−𝑚)!𝐵_2𝑚+1Δsin𝜓_𝑖Δ^2𝑚+1+1𝜋Δ⎡⎢⎣⎛⎜⎝^𝜇+_𝜇=0−^−𝜇−_𝜇=−1⎞⎟⎠cos(2𝑛+1)𝜙_𝜇cos𝜙_𝜇+(−1)^𝑛+1⎛⎜⎝^∞_𝜇=_𝜇++1𝑒^{−(2𝑛+1)𝜂+𝜇}sinh𝜂⁺_𝜇−^∞_{𝜇=𝜇−+1}𝑒^{−(2𝑛+1)𝜂−𝜇}sinh𝜂⁻_𝜇⎞⎟⎠⎤⎥⎦.(3.21) Finally, B_𝑛(𝑥) is the Bernoulli polynomial of argument “𝑥” and power “𝑛”, in ((3.21).

Remark 3.4 (Bessel series 𝒥₀, 𝒥_2𝑛, and 𝒥_2𝑛+1). The propagating range of the Schlömilch series ℋ_𝑛, for all 𝑛∈𝐙₊, where 𝐙₊={0,1,2,3,…} in (3.17) and ((3.21), is described by “𝒥_𝑛” Bessel series, which can explicitly be written as 𝒥_2𝑛=⎡⎢⎣2𝑘_𝑟𝑑^𝜇+_{𝜇=−𝜇−}cos2𝑛𝜙_𝜇𝑘_𝑟𝑑cos𝜙_𝜇−𝛿_𝑛0⎤⎥⎦;∀𝑛∈𝐙₊,𝒥_2𝑛+1=⎡⎢⎣2𝑖𝑘_𝑟𝑑^𝜇+_{𝜇=−𝜇−}sin(2𝑛+1)𝜙_𝜇cos𝜙_𝜇⎤⎥⎦;∀𝑛∈𝐙₊.(3.22)

Remark 3.5 (Neumann series 𝒩₀, 𝒩_2𝑛, and 𝒩_2𝑛+1). The evanescent range of the Schlömilch series ℋ_𝑛, for all 𝑛∈𝐙₊ in (3.17) and (3.21), is described by “𝑖𝒩_𝑛” where 𝒩_𝑛 is known as the Neumann series. 𝒩_𝑛 in (3.17) and ((3.21) can be put into the following form for this limiting case (𝑘_𝑟𝑑≪1) as 𝒩₀≅−2𝜋ln𝛾Δ2+1𝜋⎛⎜⎝^𝜇+_𝜇=1+^𝜇−_𝜇=1⎞⎟⎠1𝜇−1𝜋Δ^∞_𝜇=𝜇++1(1/2)(Δ/𝜇)−sin𝜓_𝑖(𝜇/Δ)𝜇/Δ+sin𝜓_𝑖−(1/2)(Δ/𝜇)−1𝜋Δ^∞_{𝜇=𝜇−+1}(1/2)(Δ/𝜇)+sin𝜓_𝑖(𝜇/Δ)𝜇/Δ−sin𝜓_𝑖−(1/2)(Δ/𝜇).(3.23a) In addition, we can obtain the simplified expressions for 𝒩_2𝑛 and 𝒩_2𝑛+1 as 𝒩_2𝑛≅1𝑛𝜋+1𝜋^𝑛_𝑚=1(−1)^𝑚2^2𝑚(𝑛+𝑚−1)!(2𝑚)!(𝑛−𝑚)!𝐵_2𝑚Δsin𝜓_𝑖Δ^2𝑚−1𝜋⎛⎜⎝⁻¹_𝜇=_−𝜇−−^𝜇+_𝜇=0⎞⎟⎠^𝑛_𝑚₌₁(−1)^𝑚2^2𝑚−1(𝑛+𝑚−1)!(2𝑚−1)!(𝑛−𝑚)!Δ^2𝑚𝜇+Δsin𝜓_𝑖^2𝑚−1−(−1)^𝑛𝜋Δ⎧⎨⎩^∞_𝜇=_𝜇++1(𝜇/2Δ)^2𝑛+𝑂(Δ/𝜇)²(𝜇/Δ)+sin𝜓_𝑖−(1/2)(Δ/𝜇)+𝑂(Δ/𝜇)²+^∞_{𝜇=𝜇−+1}(𝜇/2Δ)^2𝑛+𝑂(Δ/𝜇)²(𝜇/Δ)−sin𝜓_𝑖−(1/2)(Δ/𝜇)+𝑂(Δ/𝜇)²⎫⎬⎭,∀𝑛∈𝐍,(3.23b)𝒩_2𝑛+1≅2𝑖𝜋^𝑛_𝑚=0(−1)^𝑚2^2𝑚(𝑛+𝑚)!(2𝑚+1)!(𝑛−𝑚)!𝐵_2𝑚+1Δsin𝜓_𝑖Δ^2𝑚+1−1𝑖𝜋⎛⎜⎝⁻¹_𝜇=_−𝜇−−^𝜇+_𝜇=0⎞⎟⎠^𝑛_𝑚₌₀(−1)^𝑚2^2𝑚(𝑛+𝑚)!(2𝑚)!(𝑛−𝑚)!Δ^2𝑚+1𝜇+Δsin𝜓_𝑖^2𝑚−(−1)^𝑛𝑖𝜋Δ⎧⎨⎩^∞_𝜇=_𝜇++1(𝜇/2Δ)^2𝑛+𝑂(Δ/𝜇)²(𝜇/Δ)+sin𝜓_𝑖−(1/2)(Δ/𝜇)+𝑂(Δ/𝜇)²−^∞_{𝜇=𝜇−+1}(𝜇/2Δ)^2𝑛+𝑂(Δ/𝜇)²(𝜇/Δ)−sin𝜓_𝑖−(1/2)(Δ/𝜇)+𝑂(Δ/𝜇)²⎫⎬⎭,∀𝑛∈𝐙₊.(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, (𝑘_𝑟𝑑/2𝜋)(1±sin𝜓_𝑖)<1. Then the Bessel series for 𝜙₀=𝜋+𝜓_𝑖, which implies that the plane wave is incident onto the grating in the first quadrant, for all 𝑛∈𝐙₊, reduces to 𝒥_2𝑛=2cos2𝑛𝜙₀𝑘_𝑟𝑑cos𝜙₀−𝛿_𝑛0,𝒥_2𝑛+1=−2𝑖sin(2𝑛+1)𝜙₀𝑘_𝑟𝑑cos𝜙₀,(3.24) where 𝛿_𝑛𝑚 stands for the Kronecker delta function.

Remark 3.7 (approximations for Neumann series𝒩₀, 𝒩_2𝑛, and 𝒩_2𝑛+1 in the limit of Δ≪1). Inserting 𝜇₊=𝜇₋=0 in (3.23a), (3.23b), and (3.23c), the expression for 𝒩₀ in (3.23a)) reduces to 𝒩₀≅−2𝜋ln𝛾Δ2−1𝜋Δ^∞_𝜇=1⎧⎨⎩1+2sin²𝜓_𝑖−(1/2)(Δ/𝜇)²(𝜇/Δ)³1−1+2sin²𝜓_𝑖(Δ/𝜇)²+(1/4)(Δ/𝜇)⁴⎫⎬⎭.(3.25a) The approximation of the Neumann series 𝒩₀, for 𝜙₀=𝜋+𝜓_𝑖, up to terms of the order (𝑘_𝑟𝑑)² can be obtained from (3.25a) as 𝒩₀≅−2𝜋ln𝛾Δ2−1+2sin²𝜓_𝑖Δ²𝜋𝜁(3),(3.25b)where 𝜁(𝑠), for all 𝑠∈ℜ, denotes the Riemann zeta function. In the same range, the Neumann series 𝒩_𝑛 reduces to 𝒩_2𝑛=1𝑛𝜋+1𝜋^𝑛_𝑚=1(−1)^𝑚2^2𝑚−1(𝑛+𝑚−1)!(2𝑚−1)!(𝑛−𝑚)!Δ^2𝑚⎡⎢⎣𝐵_2𝑚Δsin𝜓_𝑖𝑚+Δsin𝜓_𝑖^2𝑚−1⎤⎥⎦+ℱ_2𝑛,∀𝑛∈𝐍,𝒩_2𝑛+1=1𝑖𝜋^𝑛_𝑚=0(−1)^𝑚2^2𝑚(𝑛+𝑚)!(2𝑚)!(𝑛−𝑚)!Δ^2𝑚+1⎡⎢⎣𝐵_2𝑚+1Δsin𝜓_𝑖𝑚+1/2+Δsin𝜓_𝑖^2𝑚⎤⎥⎦+ℱ_2𝑛+1,∀𝑛∈𝐙₊,(3.26) where ℱ’s in (3.26) are given as ℱ_2𝑛≅(−1)^𝑛+1𝜋Δ^∞_𝜇=112^2𝑛−1Δ𝜇^2𝑛+1,ℱ_2𝑛+1≅𝑖(−1)^𝑛+1𝜋Δsin𝜓^𝑖∞_𝜇=112^2𝑛Δ𝜇^2𝑛+3.(3.27)

Remark 3.8 (approximations for Schlömilch series, ℋ_𝑛=𝒥_𝑛+𝑖𝒩_𝑛 in the limit of Δ≪1). If𝑘_𝑟𝑑 is small, that is to say if (𝑘_𝑟𝑑/2𝜋)(1±sin𝜓_𝑖)<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+𝜓_𝑖; the Schlömilch Series in this range can be expressed as ℋ₀≅2𝑘_𝑟𝑑cos𝜙₀−2𝑖𝜋ln𝛾𝑘_𝑟𝑑4𝜋−1−𝑖𝑘_𝑟𝑑²2𝜋³12+sin²𝜙₀𝜁(3)+𝑂𝑘_𝑟𝑑³,ℋ₁≅−2𝑖sin𝜙₀𝑘_𝑟𝑑cos𝜙₀+2sin𝜙₀𝜋+𝑘_𝑟𝑑²sin𝜙₀4𝜋³𝜁(3)+𝑂𝑘_𝑟𝑑³,ℋ₂≅4𝜋3𝑖𝑘_𝑟𝑑²+2cos2𝜙₀𝑘_𝑟𝑑cos𝜙₀+𝑖𝜋1−2sin²𝜙₀+𝑖𝑘_𝑟𝑑²(2𝜋)³𝜁(3)+𝑂𝑘_𝑟𝑑³,ℋ₃≅−16𝜋sin𝜙₀3𝑘_𝑟𝑑²−𝑖2sin3𝜙₀𝑘_𝑟𝑑cos𝜙₀+2sin𝜙₀𝜋1−43sin²𝜙₀−sin𝜙₀2𝑘_𝑟𝑑⁴(2𝜋)⁵𝜁(5)+𝑂𝑘_𝑟𝑑⁵,ℋ₄≅2⁵𝜋³15𝑖𝑘_𝑟𝑑⁴−𝑖16𝜋𝑘_𝑟𝑑²16−sin²𝜙₀+2cos4𝜙₀𝑘_𝑟𝑑cos𝜙₀+𝑖2𝜋1−8sin²𝜙₀+8sin⁴𝜙₀−𝑖𝑘_𝑟𝑑⁴4(2𝜋)⁵𝜁(5)+𝑂𝑘_𝑟𝑑⁵.(3.28)

Remark 3.9 (leading terms of the Schlömilch series,ℋ_𝑛=𝒥_𝑛+𝑖𝒩_𝑛 in the limit of  Δ≪1). The leading terms of ℋ’s for large “𝑛”, for all 𝑛∈𝐍 is given as ℋ_2𝑛≅2^4𝑛−1⎡⎢⎣(−1)^𝑛𝜋^2𝑛−1𝐵_2𝑛(0)𝑘_𝑟𝑑^2𝑛⎤⎥⎦𝑖𝑛,ℋ_2𝑛+1≅2^4𝑛+1⎡⎢⎣(−1)^𝑛𝜋^2𝑛−1𝐵_2𝑛(0)𝑘_𝑟𝑑^2𝑛⎤⎥⎦sin𝜙₀,(3.29) where 𝐵_𝑛(𝑥) corresponds to the Bernoulli Polynomial. From (3.28) and (3.29), we can determine the leading terms of the Schlömilch series as ℋ₀≈ℎ₀𝑘_𝑟𝑑,ℎ₀≡2sec𝜙₀,ℋ₁≈ℎ₁𝑘_𝑟𝑑,ℎ₁≡−2𝑖tan𝜙₀,ℋ₂≈ℎ₂𝑘_𝑟𝑑²,ℎ₂≡4𝜋3𝑖,ℋ₃≈ℎ₃𝑘_𝑟𝑑²,ℎ₃≡−16𝜋sin𝜙₀3,ℋ₄≈ℎ₄𝑘_𝑟𝑑⁴,ℎ₄≡2⁵𝜋³15𝑖,ℋ₅≈ℎ₅𝑘_𝑟𝑑⁴,ℎ₅≡−2⁸𝜋³sin𝜙₀15.(3.30) The leading terms of ℋ_𝑛 for large “𝑛” are given by ℋ_2𝑛≈ℎ_2𝑛𝑘_𝑟𝑑^2𝑛,ℋ_2𝑛+1≈ℎ_2𝑛+1𝑘_𝑟𝑑^2𝑛,(3.31) where  ℎ_2𝑛’s and  ℎ_2𝑛+1’s for large "𝑛" are given as ℎ_2𝑛⟶𝑖𝑛(−1)^𝑛2^4𝑛−1𝜋^2𝑛−1𝐵_2𝑛(0),(3.32a)ℎ_2𝑛+1⟶(−1)^𝑛2^4𝑛+1𝜋^2𝑛−1𝐵_2𝑛(0)sin𝜙₀≡−4𝑖𝑛ℎ_2𝑛sin𝜙₀.(3.32b) In the above expressions, 𝐵_𝜉’s are the Bernoulli numbers, and the relationship between Bernoulli polynomial and Bernoulli numbers is given as 𝐵_2𝜉(0)≡(−1)^𝜉−1𝐵_𝜉.(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 (𝑘_𝑟𝑎)≪1, and (𝑘_𝑟𝑎/𝑘_𝑟𝑑)≡𝜉<1/2, as𝐴_{±(2𝑛−1)}≅𝐴_{±(2𝑛−1),0}𝑘_𝑟𝑎^2𝑛,(4.1a)𝐴^𝐻_{±(2𝑛−1)}≅𝐴^𝐻_{±(2𝑛−1),0}𝑘_𝑟𝑎^2𝑛(4.1b) for all 𝑛∈𝐍, 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_±2𝑛≅𝐴_±2𝑛,0𝑘_𝑟𝑎^2𝑛+2,(4.1c)𝐴^𝐻_±2𝑛≅𝐴^𝐻_±2𝑛,0𝑘_𝑟𝑎^2𝑛+2(4.1d)for all 𝑛∈𝐙₊ 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,𝐴^𝐻_±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 by⎡⎢⎢⎣𝐴_{±(2𝑛−1),0}𝐴^𝐻_{±(2𝑛−1),0}⎤⎥⎥⎦=𝛿_𝑛1𝐷⎡⎢⎢⎣𝑠^𝜀𝜇_2𝑛−1,0𝑠^𝜂_{±(2𝑛−1),0}⎤⎥⎥⎦𝐸^𝑖_{±(2𝑛−1),0}+^∞_𝑚=1𝑎𝑑^{2(𝑚+𝑛−1)}ℎ_{±2(𝑚+𝑛−1)}𝑆_{±(2𝑛−1),0}⎡⎢⎢⎣𝐴_{∓(2𝑚−1),0}𝐴^𝐻_{∓(2𝑚−1),0}⎤⎥⎥⎦,∀𝑛∈𝐍(4.2a) for the odd multiple scattering coefficients, and ⎡⎢⎢⎣𝐴_±2𝑛,0𝐴^𝐻_±2𝑛,0⎤⎥⎥⎦=𝛿_𝑛1𝐷⎡⎢⎢⎣𝑠^𝜀𝜇_2𝑛,0𝑠^𝜂_±2𝑛,0⎤⎥⎥⎦𝐸^𝑖_±2𝑛,0+^∞_𝑚=1𝑎𝑑^{2(𝑚+𝑛−1)}×𝑆_±2𝑛,0⎧⎪⎨⎪⎩ℎ_{±2(𝑚+𝑛−1)}⎡⎢⎢⎣𝐴_{∓2(𝑚−1),0}𝐴^𝐻_{∓2(𝑚−1),0}⎤⎥⎥⎦+ℎ_{±(2𝑚+2𝑛−1)}⎡⎢⎢⎣𝐴_{∓2(𝑚−1),0}𝐴^𝐻_{∓2(𝑚−1),0}⎤⎥⎥⎦⎫⎪⎬⎪⎭,∀𝑛∈𝐍,(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, (𝑘_𝑟𝑑)≪1, and (𝑘_𝑟𝑎/𝑘_𝑟𝑑)≡𝜉<1/2 as𝐺_±𝑛≡^∞_𝑚=−∞ℋ_{±𝑛−𝑚}𝐴_𝑚,(4.3a) for the electric field coefficients, and 𝐺^𝐻_±𝑛≡^∞_𝑚=−∞ℋ_{±𝑛−𝑚}𝐴^𝐻_𝑚,(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, (𝑘_𝑟𝑑)≪1, and (𝑘_𝑟𝑎/𝑘_𝑟𝑑)≡𝜉<1/2 as𝐺_{±(2𝑛−1),0}=^∞_𝑚=1𝑎𝑑^2𝑚ℎ_{±2(𝑚+𝑛−1)}𝐴_{∓(2𝑚−1),0},(4.4a)𝐺^𝐻_{±(2𝑛−1),0}=^∞_𝑚=1𝑎𝑑^2𝑚ℎ_{±2(𝑚+𝑛−1)}𝐴^𝐻_{∓(2𝑚−1),0}(4.4b) for all 𝑛∈𝐍, for the odd coefficients, 𝐺_±2𝑛,0=^∞_𝑚=1𝑎𝑑^2𝑚ℎ_{±2(𝑚+𝑛−1)}𝐴_{∓(2𝑚−2),0}+ℎ_{±(2𝑚+2𝑛−1)}𝐴_{∓(2𝑚−1),0},(4.4c)𝐺^𝐻_±2𝑛,0=^∞_𝑚=1𝑎𝑑^2𝑚ℎ_{±2(𝑚+𝑛−1)}𝐴^𝐻_{∓(2𝑚−2),0}+ℎ_{±(2𝑚+2𝑛−1)}𝐴^𝐻_{∓(2𝑚−1),0}(4.4d)for all 𝑛∈𝐍, for the even coefficients; and the special case for 𝑛=0 is given by 𝐺_0,0=¹_𝑚=−1ℎ_𝑚𝐴_−𝑚,0,𝐺^𝐻_0,0=¹_𝑚=−1ℎ_𝑚𝐴^𝐻_−𝑚,0.(4.5) Defining the wavelength independent parts of the scattering matrices from (3.10)) as𝑆_𝑛∶=𝑆_𝑛,0𝑘_𝑟𝑎^2𝑛,(4.6a)𝑆_±𝑛,0∶=1𝐷⎡⎢⎢⎣𝑠^𝜀𝜇_𝑛𝑠^𝜉_±𝑛𝑠^𝜂_±𝑛𝑠^𝜇𝜀_𝑛⎤⎥⎥⎦,(4.6b)𝑆_{±(2𝑛−1),0}≡1𝐷⎡⎢⎢⎣𝑠^𝜀𝜇_2𝑛−1,0𝑠^𝜉_{±(2𝑛−1),0}𝑠^𝜂_{±(2𝑛−1),0}𝑠^𝜇𝜀_2𝑛−1,0⎤⎥⎥⎦(4.6c) for all 𝑛∈𝐍, corresponding to the odd, and 𝑆_±2𝑛,0≡1𝐷⎡⎢⎢⎣𝑠^𝜀𝜇_2𝑛,0𝑠^𝜉_±2𝑛,0𝑠^𝜂_±2𝑛,0𝑠^𝜇𝜀_2𝑛,0⎤⎥⎥⎦,∀𝑛∈𝐍,(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_{±(2𝑛−1),0}𝐴^𝐻_{±(2𝑛−1),0}⎤⎥⎥⎦=𝑆_{±(2𝑛−1),0}⎡⎢⎢⎣𝛿_𝑛1𝐸^𝑖_{±(2𝑛−1),0}+𝑎𝑑^2(𝑛−1)𝐺_{±(2𝑛−1),0}𝑎𝑑^2(𝑛−1)𝐺^𝐻_{±(2𝑛−1),0}⎤⎥⎥⎦,∀𝑛∈𝐍,(4.7a) corresponding to the odd scattering coefficients, and ⎡⎢⎢⎣𝐴_±2𝑛,0𝐴^𝐻_±2𝑛,0⎤⎥⎥⎦=𝑆_±2𝑛,0⎡⎢⎢⎣𝛿_𝑛1𝐸^𝑖_±2𝑛,0+𝑎𝑑^2(𝑛−1)𝐺_±2𝑛,0𝑎𝑑^2(𝑛−1)𝐺^𝐻_±2𝑛,0⎤⎥⎥⎦,∀𝑛∈𝐍,(4.7b)corresponding to the even scattering coefficients. Splitting the matrices in (4.7a) and (4.7b) into two parts, we have ⎡⎢⎢⎣𝐴_{±(2𝑛−1),0}𝐴^𝐻_{±(2𝑛−1),0}⎤⎥⎥⎦=𝛿_𝑛1𝐷⎡⎢⎢⎣𝑠^𝜀𝜇_2𝑛−1,0𝑠^𝜂_{±(2𝑛−1),0}⎤⎥⎥⎦𝐸^𝑖_{±(2𝑛−1),0}+𝑎𝑑^2(𝑛−1)𝑆_{±(2𝑛−1),0}⎡⎢⎢⎣𝐺_{±(2𝑛−1),0}𝐺^𝐻_{±(2𝑛−1),0}⎤⎥⎥⎦,∀𝑛∈𝐍,(4.8a) for the odd scattering coefficients, and ⎡⎢⎢⎣𝐴_±2𝑛,0𝐴^𝐻_±2𝑛,0⎤⎥⎥⎦=𝛿_𝑛1𝐷⎡⎢⎢⎣𝑠^𝜀𝜇_2𝑛,0𝑠^𝜂_±2𝑛,0⎤⎥⎥⎦𝐸^𝑖_±2𝑛,0+𝑎𝑑^2(𝑛−1)𝑆_±2𝑛,0⎡⎢⎢⎣𝐺_±2𝑛,0𝐺^𝐻_±2𝑛,0⎤⎥⎥⎦,∀𝑛∈𝐍(4.8b)for even scattering coefficients. From (4.4a)–(4.4d), we have established the following terms: 𝑎𝑑^2(𝑛−1)𝐺_{±(2𝑛−1),0}=^∞_𝑚=1𝑎𝑑^{2(𝑚+𝑛−1)}ℎ_{±2(𝑚+𝑛−1)}𝐴_{∓(2𝑚−1),0}(4.9a) for the multiple interactions corresponding to the scattering coefficients of the electric field, and 𝑎𝑑^2(𝑛−1)𝐺^𝐻_{±(2𝑛−1),0}=^∞_𝑚=1𝑎𝑑^{2(𝑚+𝑛−1)}ℎ_{±2(𝑚+𝑛−1)}𝐴^𝐻_{∓(2𝑚−1),0}(4.9b)for the multiple interactions corresponding to the scattering coefficients of the magnetic field, for all 𝑛∈𝐍, for the odd scattering coefficients, and 𝑎𝑑^2(𝑛−1)𝐺_±2𝑛,0=^∞_𝑚=1𝑎𝑑^{2(𝑚+𝑛−1)}ℎ_{±2(𝑚+𝑛−1)}𝐴_{∓(2𝑚−2),0}+ℎ_{±(2𝑚+2𝑛−1)}𝐴_{∓(2𝑚−1),0}(4.10a) for the multiple interactions corresponding to the scattering coefficients of the electric field, 𝑎𝑑^2(𝑛−1)𝐺^𝐻_±2𝑛,0=^∞_𝑚=1𝑎𝑑^{2(𝑚+𝑛−1)}ℎ_{±2(𝑚+𝑛−1)}𝐴^𝐻_{∓(2𝑚−2),0}+ℎ_{±(2𝑚+2𝑛−1)}𝐴^𝐻_{∓(2𝑚−1),0}(4.10b)for the multiple interactions corresponding to the scattering coefficients of the magnetic field, for all 𝑛∈𝐍, 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 𝐴₀≡𝑝₀𝑞_𝑒,𝐴₁≡𝑝₁𝑞_𝑜,𝐴₂≡𝑝₂𝑞_𝑒,𝐴₃≡𝑝₃𝑞_𝑜,(5.1a) where, the numerator terms are given as 𝑝₀=𝑏₀1+2𝑏₂ℋ₂,(5.1b)𝑝₁=𝑏₁1+𝑏₃ℋ₂+ℋ₄,(5.1c)𝑝₂=𝑏₂1+𝑏₀ℋ₂,(5.1d)𝑝₃=𝑏₃1+𝑏₁ℋ₂+ℋ₄,(5.1e) and the denominator terms are given as 𝑞_𝑒=1−2𝑏₀𝑏₂ℋ²₂,(5.1f)𝑞_𝑜=1−𝑏₁𝑏₃ℋ₂+ℋ₄².(5.1g)The 𝑏_𝑛’s, for all 𝑛∈𝐍, are given by 𝑏₀=𝑎₀1−𝑎₀ℋ₀,𝑏_𝑛=𝑎_𝑛1−𝑎_𝑛ℋ₀+ℋ_2𝑛.(5.2) Finally, 𝑎_𝑛’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 (𝑘_𝑟𝑎)≪1 as 𝑎₀≅𝑎_0,0𝑘_𝑟𝑎²,𝑎_𝑛≅𝑎_𝑛,0𝑘_𝑟𝑎^2𝑛,(5.3) for all 𝑛∈𝐍,where𝑎_0,0≡𝑖𝜋4𝜀_𝑟−1,(5.4a)𝑎_𝑛,0≡𝑖𝑛𝜋(2^𝑛𝑛!)²𝜇_𝑟−1𝜇_𝑟+1,(5.4b)for all 𝑛∈𝐍.  Inserting (5.3), (3.29), and (3.30) into (5.2), 𝑏_𝑛’s can be evaluated as𝑏₀≅⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑎_0,01−𝑎_0,0⎡⎢⎢⎢⎣ℎ₀(𝑎/𝑑)𝑘_𝑟𝑎_{negligiblefor𝑘𝑟𝑎≪1}⎤⎥⎥⎥⎦⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭𝑘_𝑟𝑎²,(5.5a)𝑏₁≅⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑎_1,01−𝑎_1,0⎡⎢⎢⎢⎣ℎ₂(𝑎/𝑑)²+ℎ₀(𝑎/𝑑)𝑘_𝑟𝑎_{negligiblefor𝑘𝑟𝑎≪1}⎤⎥⎥⎥⎦⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭𝑘_𝑟𝑎²,(5.5b)𝑏₂≅⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑎_2,01−𝑎_2,0⎡⎢⎢⎢⎣ℎ₄(𝑎/𝑑)⁴+ℎ₀(𝑎/𝑑)𝑘_𝑟𝑎³_{negligiblefor𝑘𝑟𝑎≪1}⎤⎥⎥⎥⎦⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭𝑘_𝑟𝑎⁴,(5.5c)𝑏₃≅⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑎_3,01−𝑎_3,0⎡⎢⎢⎢⎣ℎ₆(𝑎/𝑑)⁶+ℎ₀(𝑎/𝑑)𝑘_𝑟𝑎⁵_{negligiblefor𝑘𝑟𝑎≪1}⎤⎥⎥⎥⎦⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭𝑘_𝑟𝑎⁶.(5.5d)Expressions in (5.5a)–(5.5d) are valid for 𝑘_𝑟𝑎≪1, and 𝑘_𝑟𝑑≪1. In general, 𝑏_𝑛’s can be expressed as 𝑏_𝑛≅⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑎_𝑛,01−𝑎_𝑛,0⎡⎢⎢⎢⎣ℎ_2𝑛(𝑎/𝑑)^2𝑛+ℎ₀(𝑎/𝑑)𝑘_𝑟𝑎^2𝑛−1_{negligiblefor𝑘𝑟𝑎≪1}⎤⎥⎥⎥⎦⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭𝑘_𝑟𝑎^2𝑛,∀𝑛∈𝐍.(5.6) Obviously, (5.5a) and (5.6) will asymptotically be written as 𝑏₀≅𝑏_0,0𝑘_𝑟𝑎²,𝑏_𝑛≅𝑏_𝑛,0𝑘_𝑟𝑎^2𝑛,(5.7) for all𝑛∈𝐍,  where 𝑏_0,0≡𝑎_0,0,(5.8a)𝑏_𝑛,0≡𝑎_𝑛,01−𝑎_𝑛,0ℎ_2𝑛(𝑎/𝑑)^2𝑛(5.8b)represent 𝑘_𝑟𝑎-independent parts of 𝑏_𝑛’s for wavelengths larger than the radii, that is, (𝑘_𝑟𝑎)≪1. The numerator terms appearing in (5.1b)–(5.1e) can be approximated as 1+2𝑏₂ℋ₂≅1+2𝑏_2,0ℎ₂𝑎𝑑²𝑘_𝑟𝑎²_{negligiblefor𝑘𝑟𝑎≪1},1+𝑏₀ℋ₂≅1+𝑏_0,0ℎ₂𝑎𝑑²,1+𝑏₃ℋ₂+ℋ₄≅1+𝑏_3,0𝑎𝑑²ℎ₄𝑎𝑑²+ℎ₂𝑘_𝑟𝑎²𝑘_𝑟𝑎²_{negligiblefor𝑘𝑟𝑎≪1},1+𝑏₁ℋ₂+ℋ₄≅1+𝑏_1,0𝑎𝑑²⎡⎢⎣ℎ₂+ℎ₄𝑘_𝑟𝑑²⎤⎥⎦.(5.9) The denominator terms appearing in (5.1c) can be approximated as 𝑞_𝑒=1−2𝑏₀𝑏₂ℋ²₂≅1−2𝑏_0,0𝑏_2,0ℎ²₂𝑎𝑑⁴𝑘_𝑟𝑎²_{negligiblefor𝑘𝑟𝑎≪1},𝑞_𝑜=1−𝑏₁𝑏₃ℋ₂+ℋ₄²≅1−𝑏_1,0𝑏_3,0𝑎𝑑⁴⎡⎢⎢⎢⎣ℎ₄𝑎𝑑²+ℎ₂𝑘_𝑟𝑎²_{negligiblefor𝑘𝑟𝑎≪1}⎤⎥⎥⎥⎦².(5.10) Inserting (5.5a)–(5.5d) to (5.10) into (5.1a)–(5.1g), we have 𝐴₀≅1+2𝑏_2,0ℎ₂(𝑎/𝑑)²𝑘_𝑟𝑎^{negligiblefor𝑘𝑟𝑎≪12}1−2𝑏_0,0𝑏_2,0ℎ²₂(𝑎/𝑑)⁴𝑘_𝑟𝑎²_{negligiblefor𝑘𝑟𝑎≪1}⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑎_0,01−𝑎_0,0⎡⎢⎢⎢⎣ℎ₀(𝑎/𝑑)𝑘_𝑟𝑎_{negligiblefor𝑘𝑟𝑎≪1}⎤⎥⎥⎥⎦⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭𝑘_𝑟𝑎²,𝐴₁≅1+𝑏_3,0(𝑎/𝑑)²ℎ₄(𝑎/𝑑)²+ℎ₂𝑘_𝑟𝑎²𝑘_𝑟𝑎^{negligiblefor𝑘𝑟𝑎≪12}1−𝑏_1,0𝑏_3,0(𝑎/𝑑)⁴⎡⎢⎢⎢⎣ℎ₄(𝑎/𝑑)²+ℎ₂𝑘_𝑟𝑎²_{negligiblefor𝑘𝑟𝑎≪1}⎤⎥⎥⎥⎦²×⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑎_1,01−𝑎_1,0⎡⎢⎢⎢⎣ℎ₂(𝑎/𝑑)²+ℎ₀(𝑎/𝑑)𝑘_𝑟𝑎_{negligiblefor𝑘𝑟𝑎≪1}⎤⎥⎥⎥⎦⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭𝑘_𝑟𝑎²,𝐴₂≅1+𝑏_0,0ℎ₂(𝑎/𝑑)²1−2𝑏_0,0𝑏_2,0ℎ²₂(𝑎/𝑑)⁴𝑘_𝑟𝑎²_{negligiblefor𝑘𝑟𝑎≪1}⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑎_2,01−𝑎_2,0⎡⎢⎢⎢⎣ℎ₄(𝑎/𝑑)⁴+ℎ₀(𝑎/𝑑)𝑘_𝑟𝑎³_{negligiblefor𝑘𝑟𝑎≪1}⎤⎥⎥⎥⎦⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭𝑘_𝑟𝑎⁴,𝐴₃≅1+𝑏_1,0(𝑎/𝑑)²⎡⎢⎢⎢⎣ℎ₄/𝑘_𝑟𝑑²+ℎ^{negligiblefor𝑘𝑟𝑑≪1}₂⎤⎥⎥⎥⎦1−𝑏_1,0𝑏_3,0(𝑎/𝑑)⁴⎡⎢⎢⎢⎣ℎ₄(𝑎/𝑑)²+ℎ₂𝑘_𝑟𝑎²_{negligiblefor𝑘𝑟𝑎≪1}⎤⎥⎥⎥⎦²×⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑎_3,01−𝑎_3,0⎡⎢⎢⎢⎣ℎ₆(𝑎/𝑑)⁶+ℎ₀(𝑎/𝑑)𝑘_𝑟𝑎⁵_{negligiblefor𝑘𝑟𝑎≪1}⎤⎥⎥⎥⎦⎫⎪⎪⎪⎪⎬⎪⎪⎪⎪⎭𝑘_𝑟𝑎⁶.(5.11) From (5.11), we have deduced that when the grating spacing is much smaller than a wavelength, that is, for the range of 𝑘_𝑟𝑎≪1 and 𝑘_𝑟𝑑≪1, the asymptotic form of the transverse magnetic multiple scattering coefficients of the infinite grating associated with the exterior electric field, 𝐴_𝑛, can asymptotically be represented as 𝐴₀≅𝐴_0,0𝑘_𝑟𝑎²,𝐴₁≅𝐴_1,0𝑘_𝑟𝑎²,𝐴₂≅𝐴_2,0𝑘_𝑟𝑎⁴,𝐴₃≅𝐴_3,0𝑘_𝑟𝑎⁴,(5.12)𝐴_𝑛,0’s represent the wavelength-independent parts of the multiple scattering coefficient 𝐴_𝑛, and the first four of them are given in terms of previously defined constants as 𝐴_0,0≅𝑎_0,0≡𝑠^𝜀𝜇₀||_{𝜃𝑖=𝜋/2}=𝑖𝜋4𝜀_𝑟−1,(5.13a)𝐴_1,0≅𝑏_1,01−𝑏_1,0𝑏_3,0ℎ₄(𝑎/𝑑)⁴²,(5.13b)𝐴_2,0≅𝑏_2,01+𝑏_0,0ℎ₂𝑎𝑑²,(5.13c)𝐴_3,0≅ℎ₄𝑎𝑑⁴𝑏_3,0𝐴_1,0=𝑏_1,0𝑏_3,0ℎ₄(𝑎/𝑑)⁴1−𝑏_1,0𝑏_3,0ℎ₄(𝑎/𝑑)⁴².(5.13d)In the expressions (5.13a)–(5.13d), 𝑎_𝑛,0 denotes the wavelength-independent parts of the scattering coefficients associated with an isolated cylinder within the grating at normal incidence [16]. From the set of equations in (5.12), we have observed that they conform to the “Ansatz” statement of (3.7) and (3.8). Inserting (5.8b) into (5.13b), we have obtained 𝐴_1,0≅𝑎_1,0/1−𝑎_1,0ℎ₂(𝑎/𝑑)²1−𝑎_1,0/1−𝑎_1,0ℎ₂(𝑎/𝑑)²𝑎_3,0/1−𝑎_3,0ℎ₆(𝑎/𝑑)⁶ℎ₄(𝑎/𝑑)⁴²,(5.14a)𝐴_1,0≅𝑎_1,0−𝑎_1,0𝑎_3,0ℎ₆(𝑎/𝑑)⁶1−𝑎_1,0ℎ₂(𝑎/𝑑)²−𝑎_3,0ℎ₆(𝑎/𝑑)⁶+𝑎_1,0𝑎_3,0ℎ₂ℎ₆−ℎ₄²(𝑎/𝑑)⁸.(5.14b) We can equivalently deduct the same result from the “asymptotic equations of the infinite grating at oblique incidence” in ((3.11) for the special case of normal incidence as 𝐴_1,0≅𝑠^𝜀𝜇₁1−𝑠^𝜀𝜇₃ℎ₆(𝑎/𝑑)⁶1−𝑠^𝜀𝜇₁ℎ₂(𝑎/𝑑)²−𝑠^𝜀𝜇₃ℎ₆(𝑎/𝑑)⁶+𝑠^𝜀𝜇₁𝑠^𝜀𝜇₃ℎ₂ℎ₆−ℎ₄²(𝑎/𝑑)⁸|||||_{𝜃𝑖=𝜋/2},(5.14c)𝑠^𝜀𝜇_𝑛≡𝑖𝑛𝜋(2^𝑛𝑛!)²𝑠_𝜀𝜇𝐷||||_{𝜃𝑖=𝜋/2}=𝑖𝑛𝜋(2^𝑛𝑛!)²𝜇_𝑟−1𝜇_𝑟+1.(5.14d)Keeping those terms up to (𝑎/𝑑)⁶ in both numerator and denominator of (5.14b), we have obtained 𝐴_1,0≅𝑎_1,0+𝑂(𝑎/𝑑)⁶1−𝑎_1,0ℎ₂(𝑎/𝑑)²+𝑂(𝑎/𝑑)⁶,(5.15a)𝐴_1,0≅𝑠^𝜀𝜇₁+𝑂(𝑎/𝑑)⁶1−𝑠^𝜀𝜇₁ℎ₂(𝑎/𝑑)²+𝑂(𝑎/𝑑)⁶|||||_{𝜃𝑖=𝜋/2},(5.15b)and expanding the denominator of (5.15a) in the form of a geometric series, that is, 1/(1−𝑥)=1+𝑥+𝑥²+𝑥³+⋯, |𝑥|<1 we have derived an asymptotic expansion for 𝐴_1,0 as 𝐴_1,0≅𝑎_1,0+𝑎²_1,0ℎ₂𝑎𝑑²+𝑎³_1,0ℎ²₂𝑎𝑑⁴+𝑂𝑎𝑑⁶,(5.16a)𝑎_1,0≅𝑠^𝜀𝜇₁||_{𝜃𝑖=𝜋/2}=𝑖𝜋4𝜇_𝑟−1𝜇_𝑟+1,(5.16b)𝐴_1,0≅𝑖𝜋4𝜇_𝑟−1𝜇_𝑟+11+13𝜋𝑎𝑑²𝜇_𝑟−1𝜇_𝑟+1+13𝜋𝑎𝑑²𝜇_𝑟−1𝜇_𝑟+1²+𝑂𝑎𝑑⁶.(5.16c)Thereby, employing the definition of the first-order multiple scattering coefficient when the radius of the cylinders is small compared to a wavelength, we have obtained Twersky’s solution for the normal incidence. Similarly, inserting (5.8b) into (5.13c) for 𝑛=2, we have 𝐴_2,0≅𝑎_2,01−𝑎_2,0ℎ₄(𝑎/𝑑)⁴1+𝑎_0,0ℎ₂𝑎𝑑²,(5.17a)𝐴_2,0≅𝑎_2,0⎡⎢⎣1+𝑎_0,0ℎ₂(𝑎/𝑑)²1−𝑎_2,0ℎ₄(𝑎/𝑑)⁴⎤⎥⎦,(5.17b)𝐴_2,0≅𝑠^𝜀𝜇₂⎡⎢⎣1+𝑠^𝜀𝜇₀ℎ₂(𝑎/𝑑)²1−𝑠^𝜀𝜇₂ℎ₄(𝑎/𝑑)⁴⎤⎥⎦|||||_{𝜃𝑖=𝜋/2}.(5.17c)Expanding the denominator of (5.17b) in the form of a geometric series, we have obtained 𝐴_2,0≅𝑎_2,01+𝑎_0,0ℎ₂𝑎𝑑²1+𝑎_2,0ℎ₄𝑎𝑑⁴+𝑂𝑎𝑑⁸,(5.18a) or neglecting terms of the order of (𝑎/𝑑)⁶, we have 𝐴_2,0≅𝑎_2,01+𝑎_0,0ℎ₂𝑎𝑑²+𝑎_2,0ℎ₄𝑎𝑑⁴+𝑂𝑎𝑑⁶,(5.18b)𝑎_2,0≅𝑠^𝜀𝜇₂||_{𝜃𝑖=𝜋/2}=𝑖𝜋32𝜇_𝑟−1𝜇_𝑟+1,(5.18c)𝐴_2,0≅𝑖𝜋32𝜇_𝑟−1𝜇_𝑟+11+115𝜋𝑎𝑑⁴𝜇_𝑟−1𝜇_𝑟+1+𝑂𝑎𝑑⁶.(5.18d)This substantiates the validity of the second-order multiple scattering coefficients, which reduces to Twersky’s form for normal incidence. For the asymptotic expansion of 𝐴_3,0 in powers of (𝑎/𝑑), we have inserted (5.8b) into (5.13d) for 𝑛=3 and obtained 𝐴_3,0≅𝑎_1,0/1−𝑎_1,0ℎ₂(𝑎/𝑑)²𝑎_3,0/1−𝑎_3,0ℎ₆(𝑎/𝑑)⁶ℎ₄(𝑎/𝑑)⁴1−𝑎_1,0/1−𝑎_1,0ℎ₂(𝑎/𝑑)²𝑎_3,0/1−𝑎_3,0ℎ₆(𝑎/𝑑)⁶ℎ₄(𝑎/𝑑)⁴²,(5.19a)𝐴_3,0≅⎡⎢⎣𝑎_3,0ℎ₄(𝑎/𝑑)⁴1−𝑎_3,0ℎ₆(𝑎/𝑑)⁶⎤⎥⎦𝐴_1,0,(5.19b)𝐴_3,0≅𝐴_1,0⎡⎢⎣𝑠^𝜀𝜇₃ℎ₄(𝑎/𝑑)⁴1−𝑠^𝜀𝜇₃ℎ₆(𝑎/𝑑)⁶⎤⎥⎦|||||_{𝜃𝑖=𝜋/2}.(5.19c)Employing (5.15a) in (5.19b), we have 𝐴_3,0≅⎡⎢⎣𝑎_3,0ℎ₄(𝑎/𝑑)⁴1−𝑎_3,0ℎ₆(𝑎/𝑑)⁶⎤⎥⎦⎡⎢⎣𝑎_1,0+𝑂(𝑎/𝑑)⁶1−𝑎_1,0ℎ₂(𝑎/𝑑)²+𝑂(𝑎/𝑑)⁶⎤⎥⎦,(5.20a)𝐴_3,0≅𝑎_1,0𝑎_3,0ℎ₄(𝑎/𝑑)⁴+𝑂(𝑎/𝑑)¹⁰1−𝑎_1,0ℎ₂(𝑎/𝑑)²+𝑂(𝑎/𝑑)⁶.(5.20b) Equation (5.20b) can be deducted from the oblique coefficients as 𝐴_3,0≅𝑠^𝜀𝜇₁𝑠^𝜀𝜇₃ℎ₄(𝑎/𝑑)⁴+𝑂(𝑎/𝑑)¹⁰1−𝑠^𝜀𝜇₁ℎ₂(𝑎/𝑑)²+𝑂(𝑎/𝑑)⁶|||||_{𝜃𝑖=𝜋/2}.(5.20c)Expanding the denominator of (5.20b) in the form of a geometric series, we have obtained 𝐴_3,0≅𝑎_1,0𝑎_3,0ℎ₄𝑎𝑑⁴1+𝑎_1,0ℎ₂𝑎𝑑²+𝑎²_1,0ℎ²₂𝑎𝑑⁴+𝑂𝑎𝑑⁶.(5.21a) Keeping terms up to the order of (𝑎/𝑑)⁴, we have 𝐴_3,0≅𝑎_1,0𝑎_3,0ℎ₄𝑎𝑑⁴+𝑂𝑎𝑑⁶.(5.21b) Using the definition of the single-scattering coefficients 𝑠^𝜀𝜇_𝑛 in (3.13a) at oblique incidence, 𝑎_𝑛,0 at normal incidence [16] can be acquired as 𝑎_3,0≅𝑠^𝜀𝜇₃∣_{𝜃𝑖=𝜋/2}=𝑖𝜋768𝜇_𝑟−1𝜇_𝑟+1,(5.21c) and the third-order scattering coefficient at normal incidence can then be acquired as 𝐴_3,0≅𝑖𝜋96115𝜋𝑎𝑑⁴𝜇_𝑟−1𝜇_𝑟+1²+𝑂𝑎𝑑⁶.(5.21d)

Lemma 5.2 (generalized asymptotic solution of the multiple scattering coefficients at oblique incidence [11, 34, 35]). The generalized asymptotic solution for the multiple scattering coefficients of an infinite grating of dielectric circular cylinders for obliquely incident vertically polarized waves has already been acquired in [35] by solving the asymptotic matrix equations of the infinite grating at oblique incidence as 𝐴_0,0≅sin𝜃_𝑖𝑖𝜋4𝜀_𝑟−1,(5.22a)𝐴^𝐻_0,0≡0,(5.22b)𝐴_±1,0≅sin𝜃_𝑖𝑖𝜋4𝐷𝑠_𝜀𝜇𝑒^{∓𝑖𝜓𝑖}+𝑎𝑑²ℎ₂𝑠²_𝜀𝜇−4𝐹²𝑖𝜋4𝐷𝑒^{±𝑖𝜓𝑖}+𝑎𝑑⁴ℎ²₂𝑠_𝜀𝜇𝑠²_𝜀𝜇−4𝐹²+8𝐹²𝜀_𝑟−𝜇_𝑟𝑘_𝑟𝑘₁²×𝑖𝜋4𝐷²𝑒^{∓𝑖𝜓𝑖}+𝑂𝑎𝑑⁶,(5.23a)𝐴^𝐻_±1,0≅∓2𝑖𝜂₀𝐹sin𝜃_𝑖𝑖𝜋4𝐷𝑒^{∓𝑖𝜓𝑖}+𝑎𝑑²ℎ₂2𝜇_𝑟−𝜀_𝑟𝑘_𝑟𝑘₁²𝑖𝜋4𝐷𝑒^{±𝑖𝜓𝑖}+𝑎𝑑⁴ℎ²₂𝑠²_𝜇𝜀−4𝐹²+2𝜇_𝑟−𝜀_𝑟𝑘_𝑟𝑘₁²𝑠_𝜀𝜇×𝑖𝜋4𝐷²𝑒^{∓𝑖𝜓𝑖}+𝑂𝑎𝑑⁶,(5.23b)𝐴_±2,0≅sin𝜃_𝑖𝑖𝜋32𝐷𝑠_𝜀𝜇𝑒^{∓2𝑖𝜓𝑖}+𝑎𝑑²ℎ₂𝑠^𝜀𝜇₀𝑠_𝜀𝜇±ℎ₃𝑠²_𝜀𝜇−4𝐹²𝑖𝜋4𝐷𝑒^{±𝑖𝜓𝑖}+𝑎𝑑⁴ℎ₄𝑖𝜋32𝐷𝑠²_𝜀𝜇−4𝐹²𝑒^{±2𝑖𝜓𝑖}±ℎ₅ℎ₂𝑖𝜋4𝐷²×𝑠_𝜀𝜇𝑠²_𝜀𝜇−4𝐹²+8𝐹²𝜀_𝑟−𝜇_𝑟𝑘_𝑟𝑘₁²𝑒^{∓𝑖𝜓𝑖}+𝑂𝑎𝑑⁶,(5.24a)𝐴^𝐻_±2,0≅∓2𝑖𝜂₀𝐹sin𝜃_𝑖𝑖𝜋32𝐷⋅𝑒^{∓2𝑖𝜓𝑖}+𝑎𝑑²ℎ₂𝑠^𝜀𝜇₀±ℎ₃𝑖𝜋4𝐷2𝜇_𝑟−𝜀_𝑟𝑘_𝑟𝑘₁²𝑒^{±𝑖𝜓𝑖}+𝑎𝑑⁴ℎ₄𝑖𝜋32𝐷2𝜇_𝑟−𝜀_𝑟𝑘_𝑟𝑘₁²𝑒^{±2𝑖𝜓𝑖}±ℎ₅ℎ₂𝑖𝜋4𝐷²×𝑠²_𝜀𝜇−4𝐹²+2𝜀_𝑟−𝜇_𝑟𝑘_𝑟𝑘₁²𝑠_𝜇𝜀𝑒^{∓𝑖𝜓𝑖}+O𝑎𝑑⁶,(5.24b)𝐴_±3,0≅sin𝜃_𝑖𝑖𝜋3.2⁸𝐷𝑎𝑑⁴ℎ₄𝑠²_𝜀𝜇−4𝐹²𝑖𝜋4𝐷𝑒^{±𝑖𝜓𝑖}+O𝑎𝑑⁶,(5.25a)𝐴^𝐻_±3,0≅∓2𝑖𝜂₀𝐹sin𝜃_𝑖𝑖𝜋3.2⁸𝐷𝑎𝑑⁴ℎ₄2𝜇_𝑟−𝜀_𝑟𝑘_𝑟𝑘₁²𝑖𝜋4𝐷𝑒^{±𝑖𝜓𝑖}+O𝑎𝑑⁶.(5.25b)

Twersky’s asymptotic solution for the transverse magnetic multiple scattering coefficients of the infinite grating at normal incidence can then be acquired by exploiting the generalized asymptotic equations at oblique incidence derived in this investigation, thereby verifying the validity of the proposed “Ansatz” in Section 4.

Remark 5.3 (reduction of the generalized asymptotic solution at oblique incidence to Twersky’s asymptotic solution for the multiple scattering coefficients at normal incidence). In order to reduce Twersky’s results at normal incidence from the generalized multiple scattering coefficients at oblique incidence given in (5.22a)–(5.25b), we have used 𝜓_𝑖=0, 𝜃_𝑖=𝜋/2, sin𝜃_𝑖=1, and 𝐹=0 in (5.22a)–(5.25b) and acquired the following results: (a) a comparison of (5.22a), which determines the generalized scattering coefficient for 𝑛=0, with (5.4a) and (5.13a) proves that the generalized multiple scattering coefficient at oblique incidence reduces to the Twersky’s coefficient for the normal incidence case; (b) the following term, namely, (𝑖𝑛𝜋/(2^𝑛𝑛!)²)(𝑠_𝜀𝜇/𝐷), which represents the wavelength-independent part of the (1,1) element of the scattering matrix in (3.10)), reduces to the Twersky’s 𝑎_𝑛,0 for the normal incidence case as 𝑖𝑛𝜋(2^𝑛𝑛!)²𝑠_𝜀𝜇𝐷||||_{𝜃𝑖=𝜋/2}⟶𝑎_𝑛,0.(5.26) With this identification, the generalized scattering coefficient of (5.23a) reduces to Twersky’s 𝐴_1,0 normal incidence case as given in (5.16c). Similarly, the generalized scattering coefficient of (5.24a) becomes identical with the Twersky’s 𝐴_2,0 scattering coefficient of the normal incidence case as it is given in (5.18d); and finally, the generalized scattering coefficient of (5.25a) conforms to Twersky’s 𝐴_3,0 as it is given by (5.21d).

6. Conclusion

In this investigation, we have presented a rigorous derivation of the asymptotic equations associated with the multiple scattering coefficients of an infinite grating of dielectric circular cylinders for obliquely incident vertically polarized plane electromagnetic waves. We have predicted the asymptotic behavior of the multiple scattering coefficients of the infinite grating at oblique incidence when the wavelength of the incident radiation is much larger than the distance between the constituent cylinders of the grating, that is, (𝑑sin𝜃_𝑖)(1±sin𝜓_𝑖)≪𝜆₀≡2𝜋/𝑘₀. Furthermore, we have predicated that our results are nothing but the generalizations of those acquired by [16] for the nonoblique incidence case. We have inferred that these equations can be solved by a technique described by Kavaklıoğlu and Schneider [35], which reduces to Twersky’s asymptotic solution at normal incidence, as well.