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

Abstract

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

1. Introduction

Twersky [1] solved the problem of multiple scattering of radiation by an arbitrary configuration of parallel cylinders by the infinite grating of insulating dielectric circular cylinders at normal incidence in terms of cylindrical wave functions as long ago as 1952, considering all possible contributions to the excitation of a particular cylinder by the radiation scattered by the remaining cylinders. He later extended his solution for the case where all axes of cylinders lie in the same plane [2] by expressing the scattered wave as an infinite sum of orders of scattering and derived the solution for the scattering of waves by the finite grating of cylinders [3] at normal incidence. In addition, Twersky introduced a “scalar functional equation” [4] for the grating in order to describe the relationship between the multiple scattered amplitude of the grating in terms of the “multiple scattered amplitude of an isolated cylinder within the grating.” He [5] then utilized the separation-of-variables technique to derive a set of algebraic equations for the “multiple scattering coefficients of the infinite grating” in terms of his famous elementary function representations of “Schlömilch series” [6], and in terms of the “well-known scattering coefficients of an isolated cylinder at normal incidence,” which was originally derived by Rayleigh [7, 8]. His results since then have been reiterated and extensively used by many other authors.

For the more generalized case 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 radially-stratified parallel cylinders with an arbitrary number of stratified layers” [11]. Furthermore, Smith et al. [12] developed a formulation for cylinder gratings in conical incidence using a multipole method in modeling photonic crystal structures and study scattering matrices and “Bloch modes” in order to investigate the photonic band gap properties of woodpile structures [13]. In a more recent study, Henin et al. [14] presented a semianalytical solution by an array of circular dielectric cylinders and parallel-coated circular cylinders of arbitrary radii and positions [15].

The exact equations describing the behavior of 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 Kavaklıoğlu [16–18] for both TM and TE polarizations. The generalized representations associated with the reflected and transmitted fields were formulated in terms of these Fourier-Bessel multiple scattering coefficients for obliquely incident plane H-polarized waves in Kavaklıoğlu and Schneider [19]. In addition, they [20] derived the generalized form of Twersky’s functional equation [4] for the infinite grating at oblique incidence in matrix form in terms of the “Fourier-Bessel scattering coefficients of an isolated dielectric circular cylinder at oblique incidence” originally derived by Wait [21]. Moreover, Kavaklıoğlu and Schneider [22] acquired the asymptotic 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, Kavaklıoğlu and Lang [23] demonstrated a rigorous proof for the validity of this asymptotic solution derived in [22].

The purpose of this investigation is to 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.

Figure 1 

The configuration of the problem of scattering of waves by an infinite array of penetrable circular cylinders at oblique incidence (side view).

In addition, we have explicated and connoted that our proposed method of solution for the scattering coefficients is a new technique and an exact representation. Besides, the paper clearly describes how one can generate a numerical algorithm from this exact solution by the truncation of the system matrices associated with the undetermined multiple scattering coefficients. On the other hand, the objective of this paper is not to discover some numerical algorithm as its title designates, which would be completely approximate and requiring an error analysis.

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

We consider a vertically polarized obliquely incident plane electromagnetic wave upon an infinite array of insulating circular dielectric cylinders having infinite length with 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 𝐫₀, 𝐫₁, 𝐫₂,…, and so forth, and separated by a distance of 𝑑.

For this configuration, the incident wave can be written in the cylindrical coordinate system (𝑅_𝑠,𝜙_𝑠,𝑧) of the 𝑠th cylinder in terms of the cylindrical waves referred to the axis of 𝑠th cylinder [16] as𝐄^incident_𝑣𝑅_𝑠,𝜙_𝑠,𝑧=𝐯_𝑖𝐸_0𝑣𝑒^{𝑖𝑘𝑟𝑠𝑑sin𝜓𝑖∞}_𝑛=−∞𝑒^{−𝑖𝑛𝜓𝑖}𝐽_𝑛𝑘_𝑟𝑅_𝑠𝑒^{𝑖𝑛(𝜙𝑠+𝜋/2)}𝑒^{−𝑖𝑘𝑧𝑧}. (2.1) 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 have𝑘_𝑟=𝑘₀sin𝜃_𝑖,𝑘_𝑧=𝑘₀cos𝜃_𝑖, (2.2) where 𝑘₀ denotes the free space wave number with 𝑘₀∶=2𝜋/𝜆₀ and where 𝜆₀ 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).

Figure 2 

The geometry of the infinite grating problem (top view).

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-Müller radiation condition as |𝐫−𝐫_𝑛|→∞, that is,𝐸^{(exterior,TM)}_𝑧𝑅_𝑠,𝜙_𝑠,𝑧=𝐸^{(incident,TM)}_𝑧𝑅_𝑠,𝜙_𝑠,𝑧+^+∞_𝑛=−∞𝐸^{(𝑛thscatterer,TM)}_𝑧𝑅_𝑛,𝜙_𝑛,𝑧. (2.3) The generalized exact representation of the multiple scattered fields by an infinite grating of dielectric cylinders for obliquely incident plane waves has been rigorously treated by Kavaklıoğlu [16] for an arbitrary oblique angle of arrival 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 as𝐸^{(exterior,TM)}_𝑧𝑅_𝑠,𝜙_𝑠=𝑒,𝑧_{𝑖𝑘𝑟𝑠𝑑sin𝜓𝑖+∞𝑛=−∞}𝐸^𝑖_𝑛+𝑄_𝑛𝐽_𝑛𝑘_𝑟𝑅_𝑠+𝐴_𝑛𝐻_𝑛⁽¹⁾𝑘_𝑟𝑅_𝑠𝑒^{𝑖𝑛(𝜙𝑠+𝜋/2)}𝑒^{−𝑖𝑘𝑧𝑧},𝐻^{(exterior,TM)}_𝑧𝑅_𝑠,𝜙_𝑠=𝑒,𝑧_{𝑖𝑘𝑟𝑠𝑑sin𝜓𝑖+∞𝑛=−∞}𝑄^𝐻_𝑛𝐽_𝑛𝑘_𝑟𝑅_𝑠+𝐴^𝐻_𝑛𝐻_𝑛⁽¹⁾𝑘_𝑟𝑅_𝑠𝑒^{𝑖𝑛(𝜙𝑠+𝜋/2)}𝑒^{−𝑖𝑘𝑧𝑧}. (2.4) The infinite set of undetermined coefficients {𝐴_𝑛,𝐴^𝐻_𝑛}^∞_𝑛=−∞ arising in (2.4) denotes the Fourier-Bessel multiple scattering coefficients for the infinite array of insulating dielectric cylinders corresponding to the vertically polarized obliquely incident plane electromagnetic waves, forall𝑛∈𝐙 where 𝐙 represents the set of all integers, and 𝐸^𝑖_𝑛 in (2.4) is given as𝐸^𝑖_𝑛=sin𝜃_𝑖𝐸_0𝑣𝑒^{−𝑖𝑛𝜓𝑖}. (2.5) 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 as𝑄_𝑛=𝒥₀𝐴_𝑛+^∞_{𝑚=−∞𝑚≠𝑛}𝒥_𝑛−𝑚𝐴_𝑚,𝑄^𝐻_𝑛=𝒥₀𝐴^𝐻_𝑛+^∞_{𝑚=−∞𝑚≠𝑛}𝒥_𝑛−𝑚𝐴^𝐻_𝑚, (2.6) for all 𝑛∈𝐙. 𝒥_𝑛(𝑘_𝑟𝑑) is the generalized form of the Schlömilch series for obliquely incident waves (Twersky [4–6], Kavaklıoğlu [18]) explicitly given as𝒥_𝑛(2𝜋Δ)=^+∞_𝑝=1𝐻_𝑛⁽¹⁾2𝜋𝑝Δ𝑒^{2𝜋𝑖𝑝Δsin𝜓𝑖}(−1)^𝑛+𝑒^{−2𝜋𝑖𝑝Δsin𝜓𝑖}, (2.7) where, Δ≡𝑘_𝑟𝑑/2𝜋and 𝐻_𝑛⁽¹⁾(𝑥) denotes the 𝑛th order Hankel function of first kind, for all 𝑛∈𝐙. The series 𝒥_𝑛−𝑚(𝑘_𝑟𝑑) in expression (2.7) is the generalization of the “Schlömilch series for obliquely incident electromagnetic waves” (Twersky [6], Kavaklıoğlu [18]) 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” (Twersky [6]).

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

The equations describing the scattering coefficients for vertically polarized and obliquely incident plane electromagnetic waves are first derived in Kavaklıoğlu [16], which can be described as𝐴_𝑛+𝑎^𝜀_𝑛𝐸^𝑖_𝑛+^+∞_𝑚=−∞𝐴_𝑚𝒥_𝑛−𝑚𝑘_𝑟𝑑−𝑏^𝜀_𝑛𝐴^𝐻_𝑛+𝑐_𝑛^+∞_𝑚=−∞𝐴^𝐻_𝑚𝒥_𝑛−𝑚𝑘_𝑟𝑑𝐴=0,_𝐻𝑛+𝑎^𝜇_𝑛^+∞_𝑚=−∞𝐴^𝐻_𝑚𝒥_𝑛−𝑚𝑘_𝑟𝑑+𝑏^𝜇_𝑛𝐴_𝑛+𝑐_𝑛𝐸^𝑖_𝑛+^+∞_𝑚=−∞𝐴_𝑚𝒥_𝑛−𝑚𝑘_𝑟𝑑=0, (3.1)forall𝑛∈𝐙. In the equations above, we have𝑐_𝑛=𝐽_𝑛𝑘_𝑟𝑎𝐻_𝑛⁽¹⁾𝑘_𝑟𝑎, (3.2)𝑎^𝜁_𝑛=𝐽_𝑛𝑘₁𝑎𝐽^′_𝑛𝑘_𝑟𝑎−𝜁_𝑟𝑘_𝑟/𝑘₁𝐽_𝑛𝑘_𝑟𝑎𝐽^′_𝑛𝑘₁𝑎𝐽_𝑛𝑘₁𝑎𝐻_𝑛^(1)′𝑘_𝑟𝑎−𝜁_𝑟𝑘_𝑟/𝑘₁𝐻_𝑛⁽¹⁾𝑘_𝑟𝑎𝐽^′_𝑛𝑘_𝑟𝑎, (3.3)𝑏^𝜁_𝑛=𝑖𝐹𝑐𝜁₀𝐽_𝑛𝑘₁𝑎𝐻_𝑛⁽¹⁾𝑘_𝑟𝑎𝐽_𝑛𝑘₁𝑎𝐻_𝑛^(1)′𝑘_𝑟𝑎−𝜁_𝑟𝑘_𝑟/𝑘₁𝐻_𝑛⁽¹⁾𝑘_𝑟𝑎𝐽^′_𝑛𝑘_𝑟𝑎𝑛𝑘_𝑟𝑎 (3.4) for all 𝑛∈𝐙 for𝜁∈{𝜀,𝜇}, where 𝑘₁ is defined as 𝑘₁=𝑘₀𝜀_𝑟𝜇_𝑟−cos²𝜃_𝑖, F is defined as𝜇𝐹=^𝑟𝜀_𝑟−1cos𝜃_𝑖𝜇_𝑟𝜀_𝑟−cos²𝜃_𝑖, (3.5) and c stands for the speed of light in free space as1𝑐=√𝜇₀𝜀₀, (3.6)𝐽^′_𝑛(𝜍) and 𝐻_𝑛^′(1)(𝜍) in expression (3.3) and (3.4) are described as the first derivatives of these functions with respect to their arguments, that is, 𝐽^′_𝑛(𝜍)≡(𝑑/𝑑𝜍)𝐽_𝑛(𝜍) and 𝐻_𝑛^′(1)(𝜍)≡(𝑑/𝑑𝜍)𝐻_𝑛⁽¹⁾(𝜍).

Introducing the intrinsic impedance of free space as𝜉₀=𝜇₀𝜀₀∶=𝜂₀⁻¹≅377Ω, (3.7) 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} as𝑏^𝜀_𝑛=𝐽_𝑛𝑘₁𝑎𝐻_𝑛⁽¹⁾𝑘_𝑟𝑎𝐽_𝑛𝑘₁𝑎𝐻_𝑛^′(1)𝑘_𝑟𝑎−𝜀_𝑟𝑘_𝑟/𝑘₁𝐻_𝑛⁽¹⁾𝑘_𝑟𝑎𝐽^′_𝑛𝑘₁𝑎𝑖𝑛𝐹𝜉₀𝑘_𝑟𝑎,𝑏^𝜇_𝑛=𝐽_𝑛𝑘₁𝑎𝐻_𝑛⁽¹⁾𝑘_𝑟𝑎𝐽_𝑛𝑘₁𝑎𝐻_𝑛^′(1)𝑘_𝑟𝑎−𝜇_𝑟𝑘_𝑟/𝑘₁𝐻_𝑛⁽¹⁾𝑘_𝑟𝑎𝐽^′_𝑛𝑘₁𝑎𝑖𝑛𝐹𝜂₀𝑘_𝑟𝑎, (3.8) for all 𝑛∈𝐙. Rearranging the equations in (3.1), we have modified the equations for the Fourier-Bessel multiple scattering coefficients as1+𝑎^𝜀_𝑛𝒥₀𝐴_𝑛+𝑎^𝜀_𝑛^+∞_{𝑚=−∞𝑚≠𝑛}𝐴_𝑚𝒥_𝑛−𝑚−𝑏^𝜀_𝑛1+𝑐_𝑛𝒥₀𝐴^𝐻_𝑛+𝑐_𝑛^+∞_{𝑚=−∞𝑚≠𝑛}𝐴^𝐻_𝑚𝒥_𝑛−𝑚=−𝑎^𝜀_𝑛𝐸^𝑖_𝑛,1+𝑎^𝜇_𝑛𝒥₀𝐴^𝐻_𝑛+𝑎^𝜇_𝑛^+∞_{𝑚=−∞𝑚≠𝑛}𝐴^𝐻_𝑚𝒥_𝑛−𝑚+𝑏^𝜇_𝑛1+𝑐_𝑛𝒥₀𝐴_𝑛+𝑐_𝑛^+∞_{𝑚=−∞𝑚≠𝑛}𝐴_𝑚𝒥_𝑛−𝑚=−𝑏^𝜇_𝑛𝑐_𝑛𝐸^𝑖_𝑛, (3.9) for all 𝑛∈𝐙. Equations (3.9) for the special case of 𝑛=0 can be simplified as 𝐴₀𝑎=−^𝜀01+𝑎^𝜀₀𝒥₀𝐸^𝑖₀+^+∞_{𝑚=−∞𝑚≠𝑛}𝐴_𝑚𝒥_−𝑚,𝐴^𝐻₀𝑎=−^𝜇01+𝑎^𝜇₀𝒥₀^+∞_{𝑚=−∞𝑚≠0}𝐴^𝐻_𝑚𝒥_−𝑚. (3.10) Or equivalently, (3.10) can be written as𝐴₀𝑎=−^𝜀01+𝑎^𝜀₀𝒥₀𝐸^𝑖₀+𝐴_𝑛𝒥_−𝑛+^+∞_{𝑚=−∞𝑚≠0,𝑛}𝐴_𝑚𝒥_−𝑚,𝐴^𝐻₀𝑎=−^𝜇01+𝑎^𝜇₀𝒥₀𝐴^𝐻_𝑛𝒥_−𝑛+^+∞_{𝑚=−∞𝑚≠0,𝑛}𝐴^𝐻_𝑚𝒥_−𝑚, (3.11) for the Fourier-Bessel multiple scattering associated with the scattered electric and magnetic fields, respectively. Indicating those scattering coefficients corresponding to 𝐴₀ and 𝐴^𝐻₀ explicitly in (3.9), we have1+𝑎^𝜀_𝑛𝒥₀𝐴_𝑛+𝑎^𝜀_𝑛𝐴₀𝒥_𝑛+∑^+∞_{𝑚=−∞𝑚≠0,𝑛}𝐴_𝑚𝒥_𝑛−𝑚−𝑏^𝜀_𝑛1+𝑐_𝑛𝒥₀𝐴^𝐻_𝑛+𝑐_𝑛𝐴^𝐻₀𝒥_𝑛+^+∞_{𝑚=−∞𝑚≠0,𝑛}𝐴^𝐻_𝑚𝒥_𝑛−𝑚=−𝑎^𝜀_𝑛𝐸^𝑖_𝑛,1+𝑎^𝜇_𝑛𝒥₀𝐴^𝐻_𝑛+𝑎^𝜇_𝑛𝐴^𝐻₀𝒥_𝑛+^+∞_{𝑚=−∞𝑚≠0,𝑛}𝐴^𝐻_𝑚𝒥_𝑛−𝑚+𝑏^𝜇_𝑛1+𝑐_𝑛𝒥₀𝐴_𝑛+𝑐_𝑛𝐴₀𝒥_𝑛+^+∞_{𝑚=−∞𝑚≠0,𝑛}𝐴_𝑚𝒥_𝑛−𝑚=−𝑏^𝜇_𝑛𝑐_𝑛𝐸^𝑖_𝑛 (3.12){∀𝑛∈𝐙∣𝑛≠0}. We have noticed that the equations (3.11) represent the solutions for 𝐴₀ and 𝐴^𝐻₀ 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 𝐴₀ and 𝐴^𝐻₀ into (3.11) to eliminate 𝐴₀ and 𝐴^𝐻₀ terms from the equations of the scattering coefficients. For this purpose, we have first used (3.11) in the evaluation of the following identities as𝑎^𝜀_𝑛𝐴₀𝒥_𝑛+^+∞_{𝑚=−∞𝑚≠0,𝑛}𝐴_𝑚𝒥_𝑛−𝑚𝑎≡−^𝜀01+𝑎^𝜀₀𝒥₀𝑎^𝜀_𝑛𝒥_𝑛𝐸^𝑖₀−𝑎^𝜀₀1+𝑎^𝜀₀𝒥₀𝒥_𝑛𝒥_−𝑛𝑎^𝜀_𝑛𝐴_𝑛+𝑎^𝜀_𝑛^+∞_{𝑚=−∞𝑚≠0,𝑛}𝐴_𝑚𝒥_𝑛−𝑚−𝑎^𝜀₀1+𝑎^𝜀₀𝒥₀𝒥_𝑛𝒥_−𝑚,𝑎^𝜇_𝑛𝐴^𝐻₀𝒥_𝑛+^+∞_{𝑚=−∞𝑚≠0,𝑛}𝐴^𝐻_𝑚𝒥_𝑛−𝑚𝑎≡−^𝜇01+𝑎^𝜇₀𝒥₀𝒥_𝑛𝒥_−𝑛𝑎^𝜇_𝑛𝐴^𝐻_𝑛+𝑎^𝜇_𝑛^+∞_{𝑚=−∞𝑚≠0,𝑛}𝐴^𝐻_𝑚𝒥_𝑛−𝑚−𝑎^𝜇₀1+𝑎^𝜇₀𝒥₀𝒥_𝑛𝒥_−𝑚 (3.13){∀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 as1+𝑎^𝜀_𝑛𝒥₀−𝑎^𝜀₀1+𝑎^𝜀₀𝒥₀𝒥_𝑛𝒥_−𝑛𝐴_𝑛+𝑎^𝜀_𝑛^+∞_{𝑚=−∞𝑚≠0,𝑛}𝐴_𝑚𝒥_𝑛−𝑚−𝑎^𝜀₀1+𝑎^𝜀₀𝒥₀𝒥_𝑛𝒥_−𝑚−𝑏^𝜀_𝑛𝐴^𝐻_𝑛1+𝑐_𝑛𝒥₀−𝑎^𝜇₀1+𝑎^𝜇₀𝒥₀𝒥_𝑛𝒥_−𝑛+𝑐_𝑛^+∞_{𝑚=−∞𝑚≠0}𝐴^𝐻_𝑚𝒥_𝑛−𝑚−𝑎^𝜇₀1+𝑎^𝜇₀𝒥₀𝒥_𝑛𝒥_−𝑚=−𝑎^𝜀_𝑛𝐸^𝑖_𝑛−𝑎^𝜀₀1+𝑎^𝜀₀𝒥₀𝒥_𝑛𝐸^𝑖₀,1+𝑎^𝜇_𝑛𝒥₀−𝑎^𝜇₀1+𝑎^𝜇₀𝒥₀𝒥_𝑛𝒥_−𝑛𝐴^𝐻_𝑛+𝑎^𝜇_𝑛^+∞_{𝑚=−∞𝑚≠0,𝑛}𝐴^𝐻_𝑚𝒥_𝑛−𝑚−𝑎^𝜇₀1+𝑎^𝜇₀𝒥₀𝒥_𝑛𝒥_−𝑚+𝑏^𝜇_𝑛𝐴_𝑛1+𝑐_𝑛𝒥₀−𝑎^𝜀₀1+𝑎^𝜀₀𝒥₀𝒥_𝑛𝒥_−𝑛+𝑐_𝑛^+∞_{𝑚=−∞𝑚≠0}𝐴_𝑚−𝑎^𝜀₀1+𝑎^𝜀₀𝒥₀𝒥_𝑛𝒥_−𝑚=−𝑏^𝜇_𝑛𝑐_𝑛𝐸^𝑖_𝑛 (3.14){∀𝑛∈𝐙∣𝑛≠0}. The equations in (3.14) together represent the complete set of equations for the Fourier-Bessel multiple scattering coefficients of the infinite grating of circular dielectric cylinders at oblique incidence. Incorporating the first terms inside the brackets into the infinite summation, (3.14) can be written more compactly as𝐴_𝑛+𝑎^𝜀_𝑛^+∞_{𝑚=−∞𝑚≠0}𝐴_𝑚𝒥_𝑛−𝑚−𝑎^𝜀₀1+𝑎^𝜀₀𝒥₀𝒥_𝑛𝒥_−𝑚−𝑏^𝜀_𝑛𝐴^𝐻_𝑛+𝑐_𝑛^+∞_{𝑚=−∞𝑚≠0}𝐴^𝐻_𝑚𝒥_𝑛−𝑚−𝑎^𝜇₀1+𝑎^𝜇₀𝒥₀𝒥_𝑛𝒥_−𝑚=−𝑎^𝜀_𝑛𝐸^𝑖_𝑛−𝑎^𝜀₀1+𝑎^𝜀₀𝒥₀𝒥_𝑛𝐸^𝑖₀,𝐴^𝐻_𝑛+𝑎^𝜇_𝑛^+∞_{𝑚=−∞𝑚≠0}𝐴^𝐻_𝑚𝒥_𝑛−𝑚−𝑎^𝜇₀1+𝑎^𝜇₀𝒥₀𝒥_𝑛𝒥_−𝑚+𝑏^𝜇_𝑛𝐴_𝑛+𝑐_𝑛^+∞_{𝑚=−∞𝑚≠0}𝐴_𝑚𝒥_𝑛−𝑚−𝑎^𝜀₀1+𝑎^𝜀₀𝒥₀𝒥_𝑛𝒥_−𝑚=−𝑏^𝜇_𝑛𝑐_𝑛𝐸^𝑖_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, 𝐴₀ and 𝐴^𝐻₀ 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^(𝑣,𝜀) and 𝑓_𝑛^(𝑣,𝜇), {∀𝑛∈𝐙∣𝑛≠0}, and {∀𝑚∈𝐙∣𝑚≠0} as𝑑^𝜁_𝑛,𝑚=𝒥_𝑛−𝑚−𝑎^𝜁₀1+𝑎^𝜁₀𝒥₀𝒥_𝑛𝒥_−𝑚𝑒;∀𝜁∈𝜀,𝜇,_{𝑛(𝑣,𝜀)}=−𝑎^𝜀_𝑛𝐸^𝑖_𝑛−𝑎^𝜀₀1+𝑎^𝜀₀𝒥₀𝒥_𝑛𝐸^𝑖₀𝑓_{𝑛(𝑣,𝜇)},=−𝑏^𝜇_𝑛𝑐_𝑛𝐸^𝑖_𝑛, (3.16)(3.17)(3.18) 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 as𝐴_𝑛+𝑎^𝜀_𝑛^+∞_{𝑚=−∞𝑚≠0}𝑑^𝜀_𝑛,𝑚𝐴_𝑚−𝑏^𝜀_𝑛𝐴^𝐻_𝑛+𝑐_𝑛^+∞_{𝑚=−∞𝑚≠0}𝑑^𝜇_𝑛,𝑚𝐴^𝐻_𝑚=𝑒_𝑛^(𝑣,𝜀),𝐴^𝐻_𝑛+𝑎^𝜇_𝑛^+∞_{𝑚=−∞𝑚≠0}𝑑^𝜇_𝑛,𝑚𝐴^𝐻_𝑚+𝑏^𝜇_𝑛𝐴_𝑛+𝑐_𝑛^+∞_{𝑚=−∞𝑚≠0}𝑑^𝜀_𝑛,𝑚𝐴_𝑚=𝑓_𝑛^(𝑣,𝜇) (3.19){∀𝑛∈𝐙∣𝑛≠0}. We have obtained the following matrix equations from (3.19) as𝐈+𝐃𝐢𝐚𝐠𝐚^𝜀_𝐧⋅𝐃^𝜀⋅𝐀−𝐃𝐢𝐚𝐠𝐛^𝜀_𝐧⋅𝐈+𝐃𝐢𝐚𝐠𝐜_𝐧⋅𝐃^𝜇⋅𝐀^𝐇=𝐞^(𝐯,𝜀),𝐈+𝐃𝐢𝐚𝐠𝐚^𝜇_𝐧⋅𝐃^𝜇⋅𝐀^𝐇+𝐃𝐢𝐚𝐠𝐛^𝜇_𝐧⋅𝐈+𝐃𝐢𝐚𝐠𝐜_𝐧⋅𝐃^𝜀⋅𝐀=𝐟^(𝐯,𝜇). (3.20) In (3.20), we have defined𝐃𝐢𝐚𝐠𝐚^𝜁_𝐧⋅𝑎∶=^𝜁+2𝑎^𝜁₊₁𝟎𝟎𝑎^𝜁₋₁𝑎^𝜁₋₂⋅≡𝚲^𝜁 (3.21){∀𝑛∈𝐙∣𝑛≠0}, where Λ^𝜁 is an (∞×∞) diagonal matrix, defined for 𝜁∈{𝜀,𝜇}, and its elements 𝑎^𝜁_𝑛, are given in (3.3),𝐃𝐢𝐚𝐠𝐛^𝜁_𝐧⋅𝑏∶=^𝜁+2𝑏^𝜁₊₁𝟎𝟎𝑏^𝜁₋₁𝑏^𝜁₋₂⋅≡𝐁^𝜁, (3.22) where 𝐁^𝜁 is an(∞×∞)diagonal matrix defined for 𝜁∈{𝜀,𝜇} and its elements, 𝑏^𝜁_𝑛, are given in (3.4),𝐃𝐢𝐚𝐠𝐜_𝐧⋅𝑐∶=⁺²𝑐₊₁𝟎𝟎𝑐₋₁𝑐₋₂⋅≡𝚪, (3.23) where Γ is an (∞×∞) diagonal matrix, and its elements, 𝑐_𝑛, are given in (3.2),𝐝^𝜉_𝐧,𝐦∶=⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅𝑑^𝜉_2,+2𝑑^𝜉_2,+1𝑑^𝜉_2,−1𝑑^𝜉_2,−2⋅⋅⋅⋅𝑑^𝜉_1,+2𝑑^𝜉_1,+1𝑑^𝜉_1,−1𝑑^𝜉_1,−2⋅⋅⋅⋅𝑑^𝜉_−1,2𝑑^𝜉_−1,1𝑑^𝜉_−1,−1𝑑^𝜉_−1,−2⋅⋅⋅⋅𝑑^𝜉_−2,2𝑑^𝜉_−2,1𝑑^𝜉_−2,−1𝑑^𝜉_−2,−2⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅=𝐃^𝜉 (3.24) where 𝐃^𝜉 is an (∞×∞) matrix, defined for𝜁∈{𝜀,𝜇}, and its elements, 𝑑^𝜉_𝑛,𝑚, are given in (3.16), and 𝐈 is an (∞×∞) identity matrix. Besides, we have𝐞^(𝐯,𝜺)≡⋅⋅𝑒₃^(𝑣,𝜀)𝑒₂^(𝑣,𝜀)𝑒₁^(𝑣,𝜀)𝑒^(𝑣,𝜀)₋₁𝑒^(𝑣,𝜀)₋₂𝑒^(𝑣,𝜀)₋₃⋅⋅,𝐟^(𝐯,𝝁)≡⋅⋅𝑓^(v₃^,𝜇)𝑓^(v₂^,𝜇)𝑓^(v₁^,𝜇)𝑓^(v,𝜇)₋₁𝑓^(v,𝜇)₋₂𝑓^(v,𝜇)₋₃⋅⋅, (3.25) where 𝐞^(𝐯,𝜺) and 𝐟^(𝐯,𝝁) are (∞×1) vectors, and their elements 𝑒_𝑛^(𝑣,𝜀), and 𝑓_𝑛^(𝑣,𝜀) are given in (3.17) and (3.18), respectively. On the other hand, the unknown scattering coefficients associated with the exterior electric and magnetic fields are defined as𝐀≡⋅⋅𝐴₃𝐴₂𝐴₁𝐴₋₁𝐴₋₂𝐴₋₃⋅⋅,𝐀^𝐇≡⋅⋅𝐴^𝐻₃𝐴^𝐻₂𝐴^𝐻₁𝐴^𝐻₋₁𝐴^𝐻₋₂𝐴^𝐻₋₃⋅⋅, (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 as𝐈+𝚲^𝜺⋅𝐃^𝜺−𝐁^𝜺⋅𝐈+𝚪⋅𝐃^𝝁𝐁^𝝁⋅𝐈+𝚪⋅𝐃^𝜺𝐈+𝚲^𝝁⋅𝐃^𝝁𝐀𝐀^𝐇=𝐞^(𝐯,𝜀)𝐟^(𝐯,𝝁). (3.27)

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

The purpose of this section is to acquire the solution to the exact matrix system of equations for transverse magnetic Fourier-Bessel multiple scattering coefficients of the infinite grating at oblique incidence. 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 as𝐈+𝚲^𝜀⋅𝐃^𝜀⋅𝐀−𝐁^𝜀⋅𝐈+𝚪⋅𝐃^𝜇⋅𝐀^𝐇=𝐞^(𝐯,𝜀),𝐁^𝜇⋅𝐈+𝚪⋅𝐃^𝜀⋅𝐀+𝐈+𝚲^𝜇⋅𝐃^𝜇⋅𝐀^𝐇=𝐟^(𝐯,𝝁). (4.1)(4.2) From (4.1) we can solve for 𝐀 as𝐀=𝐈+𝚲^𝜀⋅𝐃^𝜀−𝟏⋅𝐞^(𝐯,𝜀)+𝐁^𝜀⋅𝐈+𝚪⋅𝐃^𝝁⋅𝐀^𝐇. (4.3) In a similar manner, we can solve from (4.2) for 𝐀^𝐇 as𝐀^𝐇=𝐈+𝚲^𝜇⋅𝐃^𝜇−𝟏⋅𝐟^(𝐯,𝝁)−𝐁^𝜇⋅𝐈+𝚪⋅𝐃^𝜀⋅𝐀. (4.4) Using the expression of (4.4) for 𝐀^𝐇 in (4.3), we have obtained the solution for 𝐀 as𝐀=𝐈+𝐈+𝚲^𝜀⋅𝐃^𝜀−𝟏⋅𝐁^𝜇⋅𝐈+𝚪⋅𝐃^𝜇⋅𝐈+𝚲^𝜇⋅𝐃^𝜇−𝟏⋅𝐁^𝜀⋅𝐈+𝚪⋅𝐃^𝜀−1⋅𝐈+𝚲^𝜀⋅𝐃^𝜀−𝟏⋅𝐞^(𝐯,𝜀)+𝐁^𝜀⋅𝐈+𝚪⋅𝐃^𝜇⋅𝐈+𝚲^𝜇⋅𝐃^𝜇−𝟏⋅𝐟^(𝐯,𝝁), (4.5) or, equivalently𝐀=𝐈+𝚲^𝜀⋅𝐃^𝜀+𝐁^𝜀⋅𝐈+𝚪⋅𝐃^𝜇⋅𝐈+𝚲^𝜇⋅𝐃^𝜇−𝟏⋅𝐁^𝜇⋅𝐈+𝚪⋅𝐃^𝜀−1⋅𝐞^(𝐯,𝜀)+𝐁^𝜀⋅𝐈+𝚪⋅𝐃^𝜇⋅𝐈+𝚲^𝜇⋅𝐃^𝜇−𝟏⋅𝐟^(𝐯,𝝁). (4.6) Defining two (∞×∞)matrices in terms of the previously defined matrices, namely,𝛀_𝜺𝝁∶=𝐈+𝚲^𝜀⋅𝐃^𝜀−1⋅𝐁^𝜀⋅𝐈+𝚪⋅𝐃^𝜇,𝛀_𝝁𝜺∶=𝐈+𝚲^𝜇⋅𝐃^𝜇−1⋅𝐁^𝜇⋅𝐈+𝚪⋅𝐃^𝜀, (4.7) and inserting (4.7) into (4.5), (4.6), we can express 𝐀 as𝐀=𝐈+𝛀_𝜺𝝁⋅𝛀_𝝁𝜺⁻¹𝐈+𝚲^𝜀⋅𝐃^𝜀−1⋅𝐞^(𝐯,𝜀)+𝛀_𝜺𝝁⋅𝐈+𝚲^𝜇⋅𝐃^𝜇−1⋅𝐟^(𝐯,𝝁). (4.8) 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 as𝐀=𝐈+𝛀_𝜺𝝁⋅𝛀_𝝁𝜺⁻¹⋅𝐈+𝚲^𝜀⋅𝐃^𝜀−1⋅𝐞^(𝐯,𝜀)+𝛀⁻¹_𝜺𝝁+𝛀_𝝁𝜺⁻¹⋅𝐈+𝚲^𝜇⋅𝐃^𝜇−1⋅𝐟^(𝐯,𝝁). (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 as𝐀^𝐇=−𝛀_𝜺𝝁+𝛀⁻¹_𝝁𝜺⁻¹⋅𝐈+𝚲^𝜀⋅𝐃^𝜀−1⋅𝐞^(𝐯,𝜀)+𝐈−𝐈+𝛀⁻¹_𝜺𝝁⋅𝛀⁻¹_𝝁𝜺⁻¹⋅𝐈+𝚲^𝜇⋅𝐃^𝜇−1⋅𝐟^(𝐯,𝜇). (4.10) Combining (4.9) and (4.10) into a single matrix expression, we can write𝐀𝐀^𝐇=𝐈+𝛀_𝜺𝝁⋅𝛀_𝝁𝜺⁻¹𝛀^−𝟏_𝜺𝝁+𝛀_𝝁𝜺⁻¹−𝛀_𝜺𝝁+𝛀^−𝟏_𝝁𝜺⁻¹𝐈−𝐈+𝛀⁻¹_𝜺𝝁⋅𝛀⁻¹_𝝁𝜺⁻¹×𝐈+𝚲^𝜀⋅𝐃^𝜀−1𝟎𝟎𝐈+𝚲^𝜇⋅𝐃^𝜇−1𝐞^(𝐯,𝜀)𝐟^(𝐯,𝜇). (4.11) This is the complete solution vector for the whole Fourier-Bessel multiple scattering coefficients of the infinite grating at oblique incidence.

5. Conclusion

In the closely related previous studies, the exact representations of the external fields corresponding to the obliquely incident plane electromagnetic waves and the exact equations describing the behavior of the 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 obliquely incident transverse magnetic plane waves in matrix form. This exact representation is formulated in terms of the multiplication of the two (∞×∞)system matrices, and describes the exact solutions associated with the transverse magnetic multiple scattering coefficients at oblique incidence. Approximate expressions for the aforementioned multiple scattering coefficients can then easily be acquired by truncating these (∞×∞)system matrices.