Diffraction of plane waves incident on a grated dielectric slab: An entire domain integral equation analysis

Authors

  • Nikolaos L. Tsitsas,

    1. Microwaves and Fiber Optics Laboratory, School of Electrical and Computer Engineering, National Technical University of Athens, Heroon Polytechniou 9, Zografou, Athens, Greece
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  • Nikolaos K. Uzunoglu,

    1. Microwaves and Fiber Optics Laboratory, School of Electrical and Computer Engineering, National Technical University of Athens, Heroon Polytechniou 9, Zografou, Athens, Greece
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  • Dimitra I. Kaklamani

    1. Microwaves and Fiber Optics Laboratory, School of Electrical and Computer Engineering, National Technical University of Athens, Heroon Polytechniou 9, Zografou, Athens, Greece
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Abstract

[1] Diffraction of electromagnetic waves by periodic grating waveguides is investigated by using a rigorous integral equation method, which combines semianalytical techniques and the Method of Moments with entire domain basis functions. The electric field integral equation is employed with unknown function the electric field on the grooves. This equation is subsequently solved by applying an entire domain Galerkin's technique. The proposed analysis provides high numerical stability and controllable accuracy. All the involved computations are analytically carried out, leading to an analytic solution with the sole approximation of the final truncation of the expansion functions sets. The computed results exhibit superior accuracy and numerical efficiency compared with those already derived by applying different methods. The effect of the incident field's and grating's characteristics on the diffraction process as well as the grating structure's efficient operation as a narrow band reflection filter are thoroughly investigated. The numerical results obtained provide design guidelines, which may be exploited appropriately in the development of millimeter and optical waveguide structures.

1. Introduction

[2] Specific wave propagation and radiation properties of certain diffraction phenomena by periodic structures motivate the use of such structures in several areas of optics, acoustics, electromagnetics and integrated optoelectronics. A dielectric waveguide with periodically varying refractive index is mainly utilized to obtain a frequency selective surface. In the microwave region, dielectric waveguide gratings are used as dichroic subreflectors in large reflecting antennas [Agrawal and Imbriale, 1979] and reflection and transmission filters [Tibuleac et al., 2000]. Thin-film periodic dielectric structures are important in many integrated optics devices such as beam-to-surface-wave couplers, distributed feedback amplifiers and lasers, multiplexers, and demultiplexers [Wang and Magnusson, 1994]. Also, dielectric gratings play a significant role in resonant narrow-band plane-wave reflection filters. These devices operate by exciting a resonance (leaky wave mode) in a grating-waveguide structure and yield highly efficient filtering at the desired frequency with relatively small sidebands. Therefore, resonant grating filters are suitable for many applications such as laser-cavity mode selectors [Norton et al., 1997] as well as security and anti-counterfeit devices, used in credit and identification cards [Turunen and Wyrowski, 1997].

[3] The diffraction phenomena by dielectric grating waveguides have already been analyzed by perturbation techniques and rigorous differential-equation methods. Specifically, the perturbation techniques provide physically intuitive but approximate results (valid only for weak grating perturbations, e.g. small grating thickness) [Miyanaga and Akasura, 1982]. The most widely used rigorous differential-equation methods are the rigorous coupled-wave analysis [Moharam et al., 1995] and the modal method [Chu and Kong, 1977]. Both are vector diffraction methods that provide reflected and transmitted diffraction efficiencies for given sets of structural and incident field parameters. Their disadvantages consist in the facts that they are computationally intensive, require time-consuming root searches, large numerical matrices, and multiple time-intensive steps to determine the resonance's width [Norton et al., 1997]. Besides, the finite difference time domain method [Zunoubi and Kalhor, 2006] and the boundary element method [Nakata and Koshiba, 1990] have also been applied for the investigation of the diffraction by grating waveguides. Subdomain methods, like for example the boundary element method, possess drawbacks concerning the creation of fictitious charges and currents on the boundaries of the subdomains due to the discontinuous variation of the electromagnetic field as well as the large number of unknowns, leading to large required memory and CPU time and possible numerical instabilities [Athanasoulias and Uzunoglu, 1995]. Almost all the above mentioned algorithms solve the diffraction problem following the three basic steps: (i) the unknown fields outside the grating region are expressed by appropriate series of plane waves with under determination coefficients (ii) a general solution of Maxwell's equations in the grating region is determined by using an appropriate method (iii) the boundary conditions at the diffraction grating and homogeneous grating interfaces are matched (see the related detailed discussion in Jarem and Banerjee [2000]). In addition, it is worth noting that layered periodic waveguides and antennas have also been analyzed by applying two-dimensional fast spectral domain algorithms for volume integral equations and using layered media Green's functions [Eibert and Volakis, 2000; Eibert et al., 2003]. Besides, the propagation characteristics of coupled nonsymmetric grating slab waveguides have been investigated in [Tsitsas et al., 2006].

[4] In this paper, we investigate the diffraction behavior of an infinite periodic grating slab waveguide subject to plane wave incidence with arbitrary incidence angle. The grating region consists of a periodic layer containing rectangular grooves. The developed methodology combines semianalytical integral equation techniques and is essentially based on a Method of Moments technique using entire domain basis functions. The standard electric field integral equation is employed for the electric field on the grooves. By applying Sommerfeld's method, a Fourier integral representation of the Green's function of the nongrating problem is derived. The integral equation is subsequently solved by applying an entire domain Galerkin's technique, based on a Fourier series expansion of the electric field on the grooves, leading to a nonhomogeneous linear system. The solution of this system provides the field on the grooves, by means of which the reflected and transmitted field's distributions are determined.

[5] The proposed method provides semianalytic solutions with high numerical stability, controllable accuracy, and high efficiency, since accurate results are obtained by using only a few expansion terms [Jones, 1964]. Therefore, this method achieves also economy in computer memory and CPU time. Convergence of the reflected and transmitted fields, which is checked by increasing successively the number of expansion functions, demonstrates the superior numerical efficiency with respect to the methods of [Pai and Awada, 1991; Morf, 1995]. This efficiency is mainly justified by the fact that both the unknown electric field and the entire domain expansion functions satisfy Helmholtz equation. A detailed analysis concerning convergence and numerical efficiency aspects is included in the numerical results and discussion section below. Furthermore, the Green's function is analytically expressed and all the involved integrations are analytically carried out. Thus, the computational cost is reduced, the accuracy increases and the sole approximation in the solution is the truncation of the expansion functions sets. Besides, the present formulation requires no discretization of the integral equation involved, while the boundary element method requires and depends strongly on the discretization in boundary elements [Nakata and Koshiba, 1990]. In addition, the use of the Green's function of the slab geometry provides a more compact formulation inherently satisfying the boundary conditions in the nongrating structure. This situation does not occur either in the above mentioned rigorous differential-equation methods (where the respective boundary conditions are imposed at the final step (iii)) or in the boundary element method (utilizing the free space Green's function).

[6] The numerical results of this paper, apart from the convergence mentioned above, mainly exhibit the optimal incident field's and grating's characteristics for the structure's efficient operation as a narrow band reflection or transmission filter. Specifically, Wood's anomalies are investigated by applying the developed integral equation method and its results are compared with those of previously established techniques. Moreover, the characteristics of the reflection and transmission resonances in the case of plane wave diffraction by arrays of dielectric rectangular cylinders are discussed. Finally, we investigate the dependence of the grating waveguide's reflectance spectrum from the grooves characteristics inside each grating's unit cell. It is demonstrated that the grooves modulation provides flexibility in the design of reflection grating filters and leads to very narrow resonances for specific choices of the grooves parameters.

[7] In the following analysis an exp(jωt) time dependence of the field quantities is assumed and suppressed.

2. Mathematical Formulation

2.1. Geometrical Configuration and Incident Field

[8] The configuration of the grating dielectric slab waveguide under investigation is depicted in Figure 1. The slab has refractive index n1 and thickness 2d. The semi-infinite cover and substrate plane regions, lying above and below the slab, have respective refractive indices n0 and n2. On the slab has been etched a periodic rectangular grating (composed of q rectangles per unit cell with refractive index n3, thickness w, lengths si and ai the distances from z = 0; i = 1,…,q) forming a Λ-periodic structure along the z-axis. The entire structure has constant magnetic permeability μ0 and is assumed uniform along the direction equation image. Thus, the refractive index distribution of the structure is determined by the periodic extension with respect to z of the step function n(x, z), defined on (−∞, +∞) × [0, Λ] by

display math

So far have been analyzed only binary gratings (with one rectangle per unit cell, i.e. q = 1); see for example [Moharam et al., 1995]. Here, the number q of the rectangles and their geometrical characteristics ai, si may be arbitrarily chosen.

Figure 1.

Geometrical configuration of the cross-section of the grating slab waveguide. The rectangular periodic grating's parameters are as follows: period Λ, thickness w and refractive index n3. A unit amplitude plane wave impinges on the structure at an angle equation imageinc.

[9] A TE-polarized plane wave of unit amplitude impinges on the periodic structure from the cover region at an angle θinc (see Figure 1). The incident electric field is given by

display math

where kinc is the incident wave vector

display math

and k0 = ω/c the free space wave number.

[10] The aspects of the associated theory and the proposed integral equation method concern transverse electric (TE) polarized waves. However, the method can be extended to transverse magnetic (TM) waves, by considering as the unknown quantity the magnetic field on the grating region.

2.2. Fields in the Nongrating Structure

[11] The investigation of the diffraction phenomena by the grating slab waveguide of Figure 1, by applying an entire domain method of moments, requires appropriate expressions of the fields induced on the nongrating structure of Figure 2a, due to the incident plane wave (2).

Figure 2.

Geometry of the nongrating problem, utilized for the computation of (a) the fields due to plane wave incidence from the cover region and (b) the Green's function due to an infinite current line source, located inside the slab waveguide.

[12] The total electric field induced on the nongrating structure

display math

is computed by expressing the transmitted and reflected fields on the regions of constant refractive index as linear superpositions of appropriate plane waves of the form

display math

The coefficients R, B, C, T and the relations of the angles θ0, θ1, θ′1, θ2 with the angle of incidence θinc will be determined by imposing the boundary conditions on x = ±d. In particular, the above angles are related by the well-known Snell's laws as

display math

Then, by using (4) and (5), we find that

display math
display math

Thus, by combining (4)–(6) the field on the nongrating structure is expressed as

display math

where

display math

2.3. Green's Function of the Nongrating Structure

[13] The application of an entire domain integral equation method requires also an appropriate analytic expression of the Green's function of the nongrating structure. For this purpose, we consider the nongrating structure of Figure 2b, excited by a 2-D infinite along the y-axis line source, located at an arbitrary point (x′, z′) inside the slab, with normalized current density

display math

where δ(·) is the Dirac function.

[14] The Green's function G is the electric field induced to the structure by this source and is expressed hereafter as Fourier integral by applying Sommerfeld's method [Sommerfeld, 1949], also known as the method of scattering superposition in the terminology of Tai [1994]. More precisely, the primary field Gp, generated by the source (8) radiating inside the space equation image2 filled by the material of the slab, is expressed as [Balanis, 1989; Collin, 1991]

display math

where

display math

with Re{g1} > 0 and Im{g1} > 0, so that Gp is outgoing and decreasing to zero for ∣x∣ → + ∞. On the other hand, the unknown secondary field Gs, induced on the nongrating structure, is also expressed as

display math

The spectral function γ under determination is required to satisfy the wave equation

display math

where the refractive index distribution nh of the nongrating structure is given by

display math

Moreover, γ is expressed as a linear combination of the fundamental solutions of (12)

display math

where the signs of g0 and g2 are selected as that of g1. The unknown coefficients A1A4 are determined by imposing the continuity of γ and equation image on x = ±d.

[15] Now, by applying Sommerfeld's method and using the techniques of Tsitsas et al. [2006], we obtain the integral representation

display math

where the kernel μ is given by

display math

and the functions Po, Pe and Λ in (16) are defined by (i = 0, 2)

display math

[16] The poles of the integrant function in (15) are the roots of the algebraic equation

display math

and are of the following two kinds [Collin, 1991; Chew, 1995]:

[17] 1. The roots λ = ±βs, that satisfy the radiation condition Re(g0) > 0 and Re(g2) > 0, and constitute a finite set of real numbers with

display math

correspond to the propagation constants of the surface wave modes of the slab waveguide.

[18] 2. The roots, not satisfying the radiation condition, are infinite in number, complex, and are known as leaky wave poles.

3. Integral Representation of the Unknown Electric Field

[19] In the case of TE polarized waves the electric field is of the form

display math

and the problem is reduced to the determination of the unknown scalar electric field factor Ψ. In the absence of external sources, time-harmonic solutions of Maxwell's equations in the structure of Figure 1 are sought by considering the fields on the grating domain as equivalent polarization currents [Athanasoulias and Uzunoglu, 1995]. Therefore, the factor Ψ is written in integral form with kernel the Green's function G of the nongrating structure, determined in section 2.3

display math

where Sd is the total transverse cross-section of the rectangles and Ψ0 the field induced on the nongrating structure by the plane wave (1) (determined in section 2.2). A rigorous mathematical proof of the integral representation (19) is given in Appendix A.

[20] Now, since the infinitely periodic structure of Figure 1 is invariant under translations zz + mΛ (mequation imageequation image), so are also the solutions of the Maxwell's equations [Agassi and George, 1986]. Thus, according to the Floquet-Bloch theorem, the electric field factor satisfies the Bloch property [Weber and Mills, 1983] (pseudoperiodicity condition)

display math

which implies that

display math

with u(x, z) a Λ-periodic function of z. Hence, by combining equations (19) and (20) we get

display math

Next, we consider the expression

display math

of the total cross section Sd as countable union of the cross sections of the grating's rth unit cell

display math

Expression (22) and the derived Fourier integral (15) of the Green's function are used to reformulate representation (21)

display math

Furthermore, the transformation ζ′ = z′ − rΛ reduces the double integrals of equation (23) to integrals on the basic unit cell S0 (the grating's cross section in [0, Λ]). Then, by utilizing (23), the Poisson's summation formula [Morse and Feshbach, 1953] for the Dirac function

display math

and taking into account the basic property-definition of the Dirac function, we obtain

display math

4. Solution of the Integral Equation

[21] The specific frame, where the integral equation (24) is established, encourages its solution by applying a highly efficient entire domain Galerkin technique, analyzed below.

4.1. Application of an Entire-Domain Galerkin Technique

[22] The electric field's factor u(x, z) on the grating's basic unit cell S0 is expanded in the Fourier series with respect to z

display math

where the Fourier coefficients ϕn are the transverse components of the spatial harmonics [Collin, 1991], satisfying here the ordinary differential equation

display math

where

display math

Next, the functions ϕn are expressed as linear combinations of the fundamental solutions of (26)

display math

with cn± under determination coefficients.

[23] Note that the fundamental solutions in the linear combinations (28) are centered at the grating's symmetry axis x = d − (w/2), leading to smaller arguments of the exponential functions and hence assuring numerical stability of the results. In the terminology of the Method of Moments the entire domain functions equation imagen(x)exp[−j(2πn/Λ)z] are referred to as the electric field expansion (basis) functions. The physical meaning of equation (25) in conjunction with (28) is that the Fourier coefficients of the field on the grating region are expressed by means of the transverse eigenmodes exp{±g3,n[xd + (w/2)]}, representing waves traveling in the equation image directions.

[24] Now, substitution of equation (25) into (24) yields

display math

where the functions Jpn and Qnp± are defined in Appendix B. Function u(x, z) of (29) is expressed as a double series of products of two functions of one variable, namely Jpnexp[−j(2πp/Λ)z] and cn+Qnp+(x) + cnQnp(x), depending on the grating's geometrical characteristics along the z-axis (lengths si and distances ai) and the x-axis (thickness w) respectively.

[25] Furthermore, for the determination of the coefficients cn± in equation (29) by applying a Galerkin technique, we need to restrict the observation vector (x, z) in (29) on the grating's domain. So, for (x, z) ∈ Sd the function u in the left hand side of (29) may be expanded in the Fourier series (25). In addition, by considering the inner products of both sides of (29) with the test functions (conjugates of the expansion functions of (25))

display math

and carrying out the resulting integrations, we get

display math

The infinite matrices [Kmn±±] and [Qmnp±±] and vectors [Vm±] are defined in Appendix B.

4.2. Formulation of the Linear System

[26] Consider the algebraic infinite nonhomogeneous square linear system of the equations (30) with respect to the unknown coefficients cn± (nequation imageequation image). This infinite system is solved numerically by truncation. More precisely, by taking into account the terms of the expansion in equation (25) and the test functions in the inner products with maximum absolute order N, the infinite system reduces to the (4N + 2) × (4N + 2) linear system

display math

where the (2N + 1) × (2N + 1) matrices A±± are given by

display math

(Rmnp±± are defined in Appendix B), c± are 2N + 1 column vectors of the coefficients cn± and b± the 2N + 1 column vectors

display math

t and B, C are defined by the equations (5) and (6)) depending on the action of the test functions on the field inside the slab in the nongrating structure.

[27] The required truncation order is determined by applying a convergence control to the solutions for increasing N. A basic advantage of the proposed method is that small values of N provide sufficient convergence (see section 6.1 below for more details).

5. Computation of the Reflected and Transmitted Fields

[28] Once the solutions c± of the nonhomogeneous linear system (31) are determined, the field's distribution in the structure of Figure 1 is computed by means of the basic representation (29) of the electric field factor. In this way, the reflected Ψr and transmitted Ψt electric fields are given by

display math
display math

where the wavevectors' components are expressed

display math

The complex coefficients rp and tp in equations (34) and (35), designating the amplitudes of the diffracted orders in the reflection and transmission regions respectively, are determined by

display math
display math

where R and T are given by equations (6), functions ρnp± and τnp± are defined in Appendix B, and δp0 is the Kronecker symbol (:δ00 = 1 and δp0 = 0 ∣ p ≠ 0).

[29] Equations (34) and (35) express the reflected and transmitted fields as infinite weighted sums of plane waves with common propagation constants in the z-axis (specified by kz,p) and different propagation constants in the x-axis (specified by kx,pr and kx,pt). In all the following, a specific field component indexed by p will be referred to as the p- reflected or transmitted order. It is interesting to investigate which field orders are propagating along the x-axis. First, the 0-order fields are always propagating. For orders p ≠ 0, we define the thresholds

display math

and remark that: for p < pr+ or p > pr and p < pt+ or p > pt the constants kx,pr and kx,pt are real and thus the p-order reflected and transmitted fields are propagating along the x-axis. On the other hand, for p > pr+ or p < pr and p > pt+ or p < pt the constants kx,pr and kx,pt become purely imaginary and the p-order fields are evanescent. In other words, pr± and pt± are the thresholds, for which the fields switch from propagation to cutoff. Moreover, the angles between the wavevectors of the p-order propagating fields and the z-axis are determined by

display math

The situation described is clarified by a demonstrative example, concerning a grating structure supporting three propagating orders, as depicted in Figure 3.

Figure 3.

Demonstrative interpretation of a grating structure, supporting three reflected and transmitted propagating orders. The angles between the wavevectors of the p-order propagating fields and the z-axis are also indicated.

[30] Besides, the additional terms in the coefficients (37) and (38), corresponding to the 0-diffracted order, represent the reflected and transmitted fields on the nongrating structure (given by equations (6) and (7)). The distinction of these terms is actually due to the utilized integral equation formulation and its physical importance in the investigation of the resonant phenomena in grating waveguides will be discussed in section 6.2 below.

6. Numerical Results and Discussion

[31] The effect of the wavelength and angle of incidence and the grating's characteristics on the transmission and reflection coefficients, given by (37) and (38), is demonstrated by the numerical results established hereafter. For each set of parameters, the required number of expansion functions in the Fourier series (25) is determined by applying the convergence check, discussed in section 6.1.

6.1. Convergence and Numerical Efficiency

[32] The method's convergence and numerical efficiency are tested against two examples, investigated previously in the literature. The first example concerns the structure with n0 = n1 = n2 = 1, n32 = 3, d = 0.2λ, w = 2d, Λ = 0.6λ, q = 1, a1 = 0, s1 = 0.5Λ, proposed in the work of Pai and Awada [1991], and the second one concerns the “metallic” grating structure with n0 = n1 = n2 = 1, n3 = 1.8 + 7.12i, d = 0.05, w = 2d, Λ = 1, q = 1, a1 = 0, s1 = 0.1Λ, θinc = 85°, proposed in the work of Morf [1995]. The 0-order coefficients convergence patterns for the above two examples are shown in Figures 4a and 5a. The curves of Figures 4a and 5a, for which convergence is achieved by the present method, are in excellent agreement with the corresponding ones of Figures 4 and 5 in Pai and Awada [1991] and Figure 15 in Morf [1995]. In addition, Figures 4b and 5b depict the variation of ∣r02 and ∣t02 near the resonances, as obtained by the present method and by those of Pai and Awada [1991] and Morf [1995], respectively. High accuracy of the present method is achieved by considering only 2N + 1 = 7 expansion functions in expression (25) for the first example and 2N + 1 = 5 for the second one. These numbers of required coefficients are by far smaller than the respective ones (NT = 31, N = 21) of the multiple reflection series method [Pai and Awada, 1991] and (N = 59 modes) of the method of eigenfunction expansion in polynomial basis functions [Morf, 1995]. Moreover, as it is shown in Figure 5a, the resonant wavelength λ = 0.9132 in the metallic grating is calculated accurately by using only 2N+ 1 = 3 expansion functions, while the method of Morf [1995] requires 59 Legendre basis functions to achieve the respective convergence. This high numerical efficiency of the present method is justified by the facts that the unknown electric field factor and the entire domain expansion functions satisfy the same physical laws (not satisfied in the considerations of Morf [1995]) and that the Green's function, utilized herein, satisfies inherently the boundary conditions in the nongrating structure.

Figure 4.

(a) Convergence patterns of ∣r02 and ∣t02 with respect to the number N of expansion functions for n0 = n1 = n2 = 1, n32 = 3, d = 0.2λ, w = 2d, Λ = 0.6λ, q = 1, a1 = 0, s1 = 0.5Λ. (b) Variation of ∣r02 near the resonance with respect to θinc, as obtained by the present method for N = 3 and by that of Pai and Awada [1991].

Figure 5.

(a) Convergence patterns of ∣t02 with respect to the number N of expansion functions for n0 = n1 = n2 = 1, n3 = 1.8 + 7.12i, d = 0.05, w = 2d, Λ = 1, q = 1, a1 = 0, s1 = 0.1Λ, θinc = 85°. (b) Variation of ∣t02 near the resonance with respect to λ/Λ, as obtained by the present method for N = 1 and by that of Morf [1995].

[33] Also, the present semi analytical method is very efficient in terms of CPU time, due to the fact that all the computations and integrations are analytically carried out. For the calculation of a specific order's reflectivity or transmittivity 0.1 seconds are sufficient (Pentium IV, 2.80 GHz with 1 GByte of RAM) per wavelength or incident angle sampling point.

[34] In the numerical results of sections 6.2–6.4, for the determination of the required number N of expansion functions in (25) the following two criteria, in consistency with literature, are used: (a) an energy conservation condition [Bertoni et al., 1989] and (b) convergence to the solution with increasing N for all the grating and the incident wave parameters [Moharam et al., 1995]. Criterion (a) is implemented by demanding the reflected and transmitted fields to conserve power within 1 part in 108; for example in the case of the 0-order propagating ∣r02 + ∣t02 differs from unity by less than 1 × 10−8. In all the following results the choice N = 7 is sufficient in order for both criteria to be satisfied, a fact that validates the method's numerical efficiency.

6.2. Wood's Anomalies

[35] Recent applications of resonant integrated optics frequency-selective devices such as dielectric filters, spectrally selective polarized laser mirrors, and beam samplers [Wang and Magnusson, 1994] have revived the interest in the investigation of anomalous resonance effects, commonly known as Wood's anomalies (first observed by Wood [1902] in metallic gratings). Such effects occur when under special illumination conditions (with respect to the wavelength or angle of incidence) the intensity of the grating diffracted spectral orders drops from maximum to minimum (approximate ratio 10 to 1) within a very narrow range [Nevière et al., 1995; Tamir and Zhang, 1997]. Fano [1941] was the first suggesting that these anomalies could be associated with the excitation of a surface wave along the grating waveguide. This explanation has been fully confirmed by Hessel and Oliner [1965], who demonstrated that the stronger anomalies express a forced resonance associated with leaky waves of the grating structure. Furthermore, Nevière et al. [1995] have asserted that Wood's anomalies can be predicted and studied from the complex poles and zeros of the scattering matrix.

[36] In order to investigate the Wood's anomalies with the developed integral equation methodology and compare its results with previously established techniques, we study the proposed by Tamir and Zhang [1997] grating configuration with constant parameters n0 = 1, n1 = 2, n22 = 2.31, n32 = 3.61, d = 0.1 μm, w = 0.15 μm, Λ = 0.39 μm, q = 1, a1 = 0.3Λ, s1 = 0.4Λ. The analysis concerns the rapid changes in the reflection and transmission coefficients, due to the interaction between the incident field and the grating structure. These changes occur for wavelengths λ or incident angles θinc lying in regions where the diffracted orders of the incident wave are phase matched to leaky wave space harmonics. The amplitudes of the 0- and −1-orders of the reflected and transmitted fields as functions of λ for fixed θinc = 52° are depicted in Figures 6a and 6b. Furthermore, the respective amplitudes as functions of θinc for fixed λ = 0.47736 μm are shown in Figures 7a and 7b. Figures 6a and 7a also indicate the variations of the term ∣t0T∣ (i.e. the 0-order transmitted field without the contribution of the respective field in the nongrating structure; see equation (38)).

Figure 6.

Variations of (a) ∣r0∣, ∣t0∣ and ∣t0T∣ and (b) ∣r−1∣ and ∣t−1∣ as functions of the wavelength λ, for fixed θinc = 52° with n0 = 1, n1 = 2, n22 = 2.31, n32 = 3.61, d = 0.1 μm, w = 0.15 μm, Λ = 0.39 μm, q = 1, a1 = 0.3Λ, s1 = 0.4Λ.

Figure 7.

Variations of (a) ∣r0∣, ∣t0∣ and ∣t0T∣ and (b) ∣r−1∣ and ∣t−1∣ as functions of the incidence angle θinc, for fixed λ = 0.47736 μm and the other parameters that of Figure 6.

[37] The curves of Figures 6a and 7a and Figures 6b and 7b for ∣r0∣, ∣t0∣ and ∣r−1∣, ∣t−1∣ essentially coincide with the respective ones of Figures 4a and 10a and Figures 9a and 11a in the work of Tamir and Zhang [1997], where the modal method is utilized. Abrupt changes (namely the Wood's anomalies mentioned above) are observed in the 0-order reflectivity and transmittivity. On the other hand, the −1-order diffraction curves are smoother than the 0-order ones. Besides, narrow peaks appear in the curves of Figures 6 and 7 centered at λ1 = 0.4686 μm, λ2 = 0.4743 μm and θinc,1 = 50.7°, θinc,2 = 53.8°. The locations of these resonant wavelengths and angles are independent of the order p. The width of the narrow peaks is smaller for p = 0 than for p = −1, due to the rapid changes of the coefficients amplitude for p = 0.

[38] Also, it is worth to emphasize the Lorentzian behavior (in the terminology of Norton et al. [1997]) of the curve of ∣t0T∣. As reported in the statements of Figure 8a of Tamir and Zhang [1997], an almost identical Lorentzian behavior of the 0-order reflection and transmission coefficients is due to the ignorance of a complex zero in the fractional expressions of the coefficients. Here, by the statements of Figures 6a and 7a we conclude that this Lorentzian behavior can also be justified by the subtraction of the field component T due to the nongrating structure from the 0-order transmitted field. This conclusion follows by the use of the integral equation methodology, which separates in the analysis the fields in the nongrating structure (function Ψ0 in the mathematical formulation) from the fields due to the presence of the grating. On the contrary, the methodology of Tamir and Zhang [1997] involves uniform field expansions in the different layers, without making the above mentioned separation. Moreover, the −1-order exhibits a similar type of Lorentzian behavior. This order is not affected by the field on the nongrating structure, because this field contributes only in the 0-order.

6.3. Array of Dielectric Rectangular Cylinders

[39] Plane wave diffraction by an array of dielectric rectangular cylinders, which has been analyzed separately in the literature [see, e.g., Zunoubi and Kalhor, 2006], is treated here as a special case of the developed method. Specifically, in this section by selecting n0 = n2 = 1, w = 2d, q = 1 and a1 = 0, we study the structure of an infinite Λ-periodic array composed of two adjacent rectangular cylinders per unit cell with indices n3 and n1 and lengths s1 and Λ − s1 (see Figure 8).

Figure 8.

Geometry of the infinite Λ-periodic array of dielectric rectangular cylinders with n0 = n2 = 1, w = 2d, q = 1 and a1 = 0, studied in section 6.3.

[40] Figures 9a and 9b depict the dependence of the amplitude of the 0-reflection order with respect to the normalized frequency Λ/λ for n32 = 2.44 and n32 = 5, corresponding to three different values of s1 with n1 = 1, Λ = 2d, θinc = 90°. For n32 = 2.44 occur two wavelengths of total reflection for each value of s1. These reflection peaks become wider as s1 increases. The reflection curves become smoother as s1 approaches Λ (mainly demonstrated by the behavior for s1 = 0.7Λ); i.e. as the array approaches the homogeneous dielectric slab. On the other hand, for n32 = 5 occur two wavelengths of total reflection for s1 = 0.1Λ and 0.3Λ and four for s1 = 0.5Λ. For both values of n3 one and two narrow total transmission wavelengths occur for s1 = 0.1Λ and s1 = 0.5Λ. Also, for fixed s1, the reflection peaks are wider for n32 = 5 than n32 = 2.44.

Figure 9.

Variation of ∣r0∣ with respect to the normalized wavelength Λ/λ for (a) n32 = 2.44, s1 = 0.1Λ, 0.5Λ and 0.7Λ and (b) n32 = 5, s1 = 0.1Λ, 0.3Λ and 0.5Λ, with n1 = 1, Λ = 2d, θinc = 90°, n0 = n2 = 1, w = 2d, q = 1 and a1 = 0.

[41] Furthermore, Figures 10a and 10b show the variations of ∣r0∣ and ∣r02 with respect to the normalized frequency 2k0d for Λ = 2d/1.713 and Λ = 2d/2.037, with n12 = 2.56, n32 = 1.44, θinc = 45°. In each figure, three curves are plotted corresponding to s1 = 0.1Λ, 0.3Λ and 0.5Λ. We have plotted up to the value 2k0d = 6.3 for Figure 10a and 2k0d = 7.5 for Figure 10b, since at these points pr = −1 and hence the reflected −1-order switches from cutoff to propagation in the air region along the x-axis. Note the agreement between the curves of Figures 10a and 10b for s1 = 0.5Λ and the corresponding ones in Figures 3 and 8 of Bertoni et al. [1989]. For Λ = 2d/1.713 occur five total transmission and three total reflection frequencies for lengths s1 = 0.1Λ, 0.3Λ and four total transmission and two total reflection frequencies for s1 = 0.5Λ. On the other hand, for Λ = 2d/2.037 occur two total transmission frequencies for each value of s1 and two and three total reflection frequencies for s1 = 0.1Λ and s1 = 0.3Λ, 0.5Λ. The physical mechanism generating these total reflection and transmission resonances is strongly connected to the excitation of the waveguide modes and is discussed thoroughly in the work of Bertoni et al. [1989]. For example, in the case of Λ = 2d/1.713 the first two total transmission frequencies for each s1 occur when the 0-order mode propagates inside the periodic layer. For higher frequencies, when the −1-order mode also propagates inside the layer, total reflection occurs at three frequencies for s1 = 0.1Λ, 0.3Λ and two for s1 = 0.5Λ, in the vicinity of which exist the same number of frequencies with zero reflection (total transmission).

Figure 10.

Variation of (a) ∣r0∣ and (b) ∣r02 with respect to the normalized frequency 2k0d for (a) Λ = 2d/1.713 and (b) Λ = 2d/2.037, both with n12 = 2.56, n32 = 1.44, θinc = 45°, n0 = n2 = 1, w = 2d, q = 1 and a1 = 0. Three curves are plotted corresponding to s1 = 0.1Λ, 0.3Λ and 0.5Λ.

[42] Furthermore, we define the fractional bandwidth of the above described resonant frequencies as the ratio of the difference between the frequencies at which ∣r02 = 0.9 (reflection) and ∣r02 = 0.1 (transmission) over the total reflection and transmission frequency respectively. This bandwidth constitutes a measure of the narrowness of a reflection or transmission peak. The values of the total reflection and transmission frequencies and the respective fractional bandwidths for the cases of Figures 10a and 10b are summarized in Table 1. For Λ = 2d/1.713 the total reflection and transmission frequencies and the fractional bandwidths increase with s1. It is worth to emphasize the very narrow reflection peaks at 2k0d = 4.899 and 5.085 for s1 = 0.1Λ and 0.3Λ with bandwidths 0.002 and 0.016%. On the other hand, the reflection peak at 2k0d = 5.311 of the structure proposed by Bertoni et al. [1989], corresponding also to that of the curve of Figure 10a with s1 = 0.5Λ, is wider (bandwidth 0.043%) than the above mentioned for s1 = 0.1Λ and 0.3Λ. Moreover, for Λ = 2d/2.037 the bandwidth of the reflection and transmission peaks increases with the normalized frequency 2k0d and the length s1. Very narrow resonances occur for s1 = 0.3Λ at 2k0d = 5.9941 and 5.9944 with bandwidths 0.003 and 0.008% and for s1 = 0.5Λ at 2k0d = 6.2315 and 6.2316 both with bandwidths 0.002%.

Table 1. Normalized Frequencies 2k0d and Bandwidths for Total 0-Order Reflection and Transmission, Corresponding to Three Different Values of s1 and Λ = 2d/1.713 and Λ = 2d/2.037a
 Total Reflection ResonanceTotal Transmission Resonance
2k0dBandwidth, %2k0dBandwidth, %
  • a

    The grating's parameters are those of Figure 10.

Λ = 2d/1.713
s1 = 0.1Λ  2.25029.95
  4.49514.91
4.8990.0024.8990.041
5.3660.0455.3610.093
6.0600.0746.0710.313
s1 = 0.3Λ  2.38833.42
  4.73314.98
5.0850.0165.0880.511
5.5770.3805.5341.066
6.2130.5426.2820.637
s1 = 0.5Λ  2.55638.38
  4.99514.37
5.3110.0435.3175.492
5.8300.9525.7304.589
    
Λ = 2d/2.037
s1 = 0.1Λ6.21410.0316.20810.203
6.88420.0446.76046.745
s1 = 0.3Λ5.99410.0035.99440.008
6.46400.2546.42111.055
7.11670.327  
s1 = 0.5Λ6.23150.0026.23160.002
6.76810.5876.67881.613
7.41540.859  

6.4. Effect of Modulation Inside the Grating's Unit Cell

[43] In this last section, we investigate the dependence of the grating waveguide's reflectance spectrum from the characteristics ai, si and q inside each grating's unit cell. So far in the literature has been investigated only the particular case of binary gratings (q = 1). The developed integral equation method admits the arbitrary selection of the number q and the geometrical characteristics ai, si of the rectangles, hence offering additional degrees of freedom to the designing of grating waveguides with the desirable operational standards.

[44] Two representative frequency selective structures, incorporating a grating waveguide and operating in the microwave region, are analyzed. The grating periods and the angles of incidence are chosen so that both grating structures admit only the 0-order propagating in the substrate and cover regions. First, we investigate a reflection filter under normal incidence (θinc = 90°) with air in the substrate and cover (n0 = n2 = 1) and grating waveguide characteristics n12 = 2.59 (Plexiglas), n32 = 2.05 (Teflon), w = 2d, Λ = 1.65 cm. Figures 11a and 11b depict the filter's spectral response ∣r02 for 2d = 0.5 cm and 2d = 0.7 cm. In each figure are plotted three curves corresponding to the sets of parameters (α) q = 1, a1 = 0, s1 = 0.5Λ, (β) q = 2, a1 = 0, a2 = 0.5Λ, s1 = 0.1Λ, s2 = 0.3Λ and (γ) q = 3, a1 = 0, a2 = 0.3Λ, a3 = 0.6Λ, s1 = 0.2Λ, s2 = 0.1Λ, s3 = 0.2Λ. The curves of Figure 11 for the set (α) are the same with that of Figure 2 in the work of Tibuleac et al. [2000], where the rigorous coupled wave analysis is utilized. The slab thickness 2d = 0.7 cm results in symmetric reflectance curves with small sideband reflection for the three sets of parameters, justified also by the fact that this thickness satisfies the half-wavelength condition (analyzed in Wang and Magnusson [1994]). On the other hand, the thickness 2d = 0.5 cm yields asymmetric reflectance curves with larger sideband reflection compared to 2d = 0.7 cm. For set (α) one reflection resonance occurs in the frequency range 13–15 GHz for both values of 2d. However, for sets (β) and (γ) two reflection resonances are generated in the same frequency range. Furthermore, the bandwidth of the resonant reflection frequencies (defined in section 6.3) decreases with increasing q. More precisely, for 2d = 0.5 cm the total reflection bandwidths are: 0.17% for set (α), 0.012% and 0.04% for (β) and 0.0007% and 0.0013% for (γ), while for 2d = 0.7 cm: 0.15% for (α), 0.011% and 0.032% for (β) and 0.0022% and 0.0014% for (γ). The slab's thickness does not significantly influence the distances between the two resonant frequencies, which are approximately 600 MHz for set (β) and 500 MHz for (γ) for both values of 2d.

Figure 11.

Reflection filter's spectral response ∣r02 for (a) 2d = 0.5cm and (b) 2d = 0.7cm, both with n0 = n2 = 1, n12 = 2.59, n32 = 2.05, w = 2d, Λ = 1.65 cm, θinc = 90°. Three curves are plotted corresponding to the sets of parameters: (α) q = 1, a1 = 0, s1 0.5Λ, (β) q = 2, a1 = 0, a2 = 0.5Λ, s1 = 0.1Λ, s2 = 0.3Λ, and (γ) q = 3, a1 = 0, a2 = 0.3Λ, a3 = 0.6Λ, s1 = 0.2Λ, s2 = 0.1Λ, s3 = 0.2Λ.

[45] Second, we investigate the reflection filter with n0 = n2 = 1, n12 = 6.13(E-glass), n32 = 3.7(silica), w = 2d = 4.37 mm, Λ = 11.28 mm. Figures 12a and 12b depict the filter's spectral response ∣r02 corresponding to (a) the sets of parameters (α), (β), (γ) of Figure 11 with θinc = 90° and (b) θinc = 75°, 80° and 85° with the parameters of set (γ). We also observe in Figure 12a (as in Figure 11) that an additional reflection resonance is generated for sets (β) and (γ) and the bandwidth of the resonant frequencies decreases with increasing q. In addition, Figure 12b indicates that for set (γ) the bandwidth of the two resonant frequencies remains very small independently of θinc. However, the distance between the two resonant frequencies decreases with increasing θinc, making hence the incident angle a control mechanism of the resonances distance. For frequencies different from the resonant ones, the reflectance has almost the same values for all θinc.

Figure 12.

Reflection filter's spectral response ∣r02 for n0 = n2 = 1, n12 = 6.13, n32 = 3.7, w = 2d = 4.37 mm, Λ = 11.28 mm. The three curves plotted correspond to (a) the sets of parameters (α), (β), (γ) of Figure 11 with θinc = 90° and (b) θinc = 75°, 80° and 85° with the parameters of set (γ).

[46] Besides, it is interesting to investigate the convergence for q > 1 of the present method, utilizing essentially the Fourier series expansion (25) with respect to z. Figure 13 depicts the 0-order convergence patterns near the resonances with respect to the number N of expansion functions for the sets of parameters of Figure 11b with (a) q = 1, (b) and (c) q = 2 and (d) and (e) q = 3. The sufficient order N for high accuracy of the present method increases with the number q of rectangles per grating unit cell; namely N = 2 for q = 1, N = 6 for q = 2 and N = 10 for q = 3. However, even for q>1 these numbers of required coefficients in the present method are by far smaller than the numbers of the multiple reflection series terms (NT = 31, N = 21) and the polynomial basis functions (N = 59), needed in the methods of Pai and Awada [1991] and Morf [1995], corresponding to the simple case of q = 1. Therefore, the basis functions (25) utilized herein exhibit superior convergence for q > 1. In addition, we note that for ∣r02 < 0.8 (i.e. relatively far from the resonance) only a small number of terms is required for convergence, and moreover the convergence is independent of q. More precisely, for the specific parameters of Figure 13 simulations have indicated that 2N + 1 = 5 terms are sufficient for all q.

Figure 13.

Convergence patterns of ∣r02 near the resonances with respect to the number N of expansion functions for the sets of parameters of Figure 11b with a1 = 0 and (a) q = 1, s1 = 0.5Λ, (b) and (c) q = 2, a2 = 0.5Λ, s1 = 0.1Λ, s2 = 0.3Λ, and (d) and (e) q = 3, a2 = 0.3Λ, a3 = 0.6Λ, s1 = 0.2Λ, s2 = 0.1Λ, s3 = 0.2Λ.

[47] We conclude this section by noting that the modulation inside each grating's unit cell provides flexibility in the design of reflection grating filters and leads to very narrow resonances for specific choices of the parameters ai, si, q. This offers an alternative way of generating narrow resonances to those proposed so far in the literature [Tibuleac et al., 2000; Coves et al., 2004], concerning mainly the addition of extra dielectric layers in the grating structure.

7. Concluding Remarks

[48] The diffraction phenomena by grating assisted waveguides have been investigated analytically by means of a rigorous integral equation method. The standard electric field integral equation was established for the electric field on the grating surface and was subsequently solved by applying a Galerkin's technique with entire domain expansion functions. The basic advantages of the proposed method, compared with already established methods, concern the high accuracy, the numerical efficiency and the analytic computation of the integrals involved. Several numerical results reveal the optimal grating's and incident field's characteristics, leading to efficient operation of the structure as a narrow band frequency filter. Besides, the strong dependence of the reflected and transmitted fields on the grating's physical and geometrical parameters makes the grating waveguide under consideration a highly flexible structure that can be tailored for a number of potential applications.

Appendix A

[49] This appendix outlines a rigorous proof of the electric field's integral representation (19), based on applications of Green's theorem. For this purpose, consider the rectangular partition Li = [x1, x2] × (iΛ, (i + 1)Λ]of equation image2 of Figure A1, containing the grating's ith equation imageequation image unit cell along the z-axis. In addition, dik(k = 1, 2, 3) and gij(j = 1,…,q) denote the plane regions of the intersection of Li with the grating waveguide structure.

Figure A1.

Rectangular partition of the grating waveguide, utilized for the proof of the integral representation (19).

[50] An application of second Green's theorem [Balanis, 1989] for the functions Ψ − Ψ0 and G onto the constant refractive index regions Mi = dik or gij results

display math

By the fact that the line integrals have opposite signs in the common parts of the boundaries of the regions, summation of equations (A1) gives

display math

where gi = equation imagegij and di = equation imagedik. Besides, since Ψ, Ψ0 and G are solutions of the Helmholtz equations

display math

(the refractive index distributions are defined by (1) and (13)) (A2) implies

display math

[51] Now, by summing equations (A3) we find

display math

If the observation vector r belongs to the rectangle Lk, then by the definition of the Dirac function, the double integral on Lk is equal to Ψ(r) − Ψ0(r), while on any other Li (ik) is equal to zero. Finally, for x1 → −∞ and x2 → +∞, by applying standard techniques (expansions of the fields in guided and radiated waves) we conclude that equation imageequation image vanishes, and hence equation (A4) implies

display math

which for equation imagegi = Sd leads to equation (19).

Appendix B

[52] The auxiliary functions, defined by (B1)–(B6), are utilized for the formulation of the method's linear system (31).

display math
display math
display math
display math
display math
display math

In (B7) and (B8) are defined auxiliary functions, used for the determination of the reflection and transmission coefficients (37) and (38).

display math
display math

Acknowledgments

[53] The authors would like to thank the reviewers and the Associate Editor for their valuable remarks and suggestions.

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