An iterative semi-analytical technique solving the boundary value problem of transmission through an anisotropic wedge


Corresponding author: C. A. Valagiannopoulos, Department of Radio Science and Engineering, School of Electrical Engineering, Aalto University, PO Box 13000, FI-00076 Helsinki, Finland. (


[1] An iterative technique imitating the physical mechanism of transmission through an anisotropic prism is introduced, analyzed and tested. The solution is obtained by treating the boundary value problem along one boundary of the device at a time, as if the other is extended to infinite. This approximation makes the proposed method more suitable for small wedge angles. However, comprehensive numerical checks concerning the accuracy and the convergence speed of the method are performed with satisfying results. Several geometrical configurations with various constituent materials have been considered and certain conclusions for the operational features of the corresponding devices have been obtained.

1. Introduction

[2] Anisotropic media are extensively utilized in electromagnetic modeling and design due to their directionally dependent properties. In Luo et al. [2006], a modulated Gaussian beam transmitted into an anisotropic metamaterial with special dispersion relation exhibits a superluminal group velocity for the peak of the wave packet. In addition, anisotropic substances, comprised of periodic lattices and wires, that demonstrate negative refraction and focusing have been reviewed and presented [Fang et al., 2009]. An optical backward-wave bianisotropic composite medium matched to free space is also suggested inTretyakov et al. [2007] suitable for operation in the optical range where artificial magnetism is not easily achieved. Furthermore, interesting properties of an infinite homogeneous circular cylinder with full permittivity tensor, excited by a straight strip of arbitrary axial magnetic current, have been examined [Valagiannopoulos, 2007b]. It has been finally shown that a metal film with an one-dimensional array of subwavelength slits can be accurately designed with use of anisotropic and non-dispersive substances [Shin et al., 2006].

[3] On the other hand, structures with nonparallel boundaries such as wedges, junctions, and tapered waveguides are employed in numerous theoretical considerations and actual experiments because of their receptivity to semi-analytical treatment and their simplicity in construction. InNefedov and Tretyakov [2011], the wedge of an ultra-broadband electromagnetically indefinite medium, formed by aligned carbon nanotubes is employed to transform incident evanescent waves into propagating transmitted modes. The wavefront-tilt effect in nonparallel optical waveguides, caused by junctions in the input and output of the filter, is also treated with help from mode-matching methods [Huang and Lessard, 1992]. Furthermore, the operation of conical horn antennas has been investigated in Hamid [2003], where edge modification and wall corrugation are utilized to improve its radiation characteristics. Corners owned by devices with polygonal shape existing in any laboratory of electromagnetics, are modeled with two nonparallel metallic planes to proposed ways of reducing the singular field concentration along the edges [Valagiannopoulos, 2009]. Surface waves transmitted through prismatic structures composed of birefringent media, have been also theoretically explored in Takayama et al. [2011], where unexpected high transmission above the critical angle is predicted.

[4] In this work, we combine the two aforementioned topics (anisotropic materials, nonparallel boundaries) to formulate a simple, iterative method that solves the scattering problem defined by the plane wave excitation of an anisotropic prism. The structure is comprised by a front and a rear planar surface, nonparallel each other, and therefore is not possible for a single wave to satisfy simultaneously the necessary boundary conditions along both of them. Accordingly, we solve each time a partial, simplified boundary value problem along one surface by ignoring the presence of the other. In fact, we repeatedly treat each incidence-reflection-transmission effect as if the prism was infinitely extended toward the one or the other horizontal direction. The solution to the total problem is given as the sum of the corresponding partial solutions. In this way, the method does not take into account the effect of the edge and therefore is better applicable for points far from the crossing point of two surfaces. The average error at the boundaries of the prism, indicating the performance and the effectiveness of the technique, is represented with respect to geometrical and excitation parameters. The transmission through the device is investigated via a newly defined ratio whose variations with respect to the features of the structure are shown and discussed.

2. Primary Definitions

2.1. Structural Configuration

[5] The physical configuration of the investigated structure is depicted in Figure 1, where the used Cartesian, small-lettered coordinate system (x, y, z) is also defined. Note that a secondary, capital-lettered, coordinate system (X, Y, Z), which is derived via counterclockwise rotation by angle ξ of (x, y, z) with respect to the common axis z = Z, can be utilized interchangeably. We consider a two-dimensional anisotropic slab with non-parallel boundaries forming an angle equal toξ. In particular, the front surface of this prism coincides with the plane x = 0, while the rear one is defined by the equation X = W.The magnetically inert anisotropic substance, filling the region 1, separates the vacuum background into regions 0, 2 and possesses the following relative permittivity matrix expressed in the small-lettered coordinate system:

display math

With respect to the capital-lettered coordinate system, the same permittivity tensor takes the form:

display math

obtained from [ϵr], multiplied with the necessary rotation matrices. Different approach on anisotropic or isotropic wedges are presented in: Osipov and Senior [2008], Bernard [1998], and Yuan and Zhu [2005].

Figure 1.

The physical configuration of the examined device.

[6] The considered device is excited by a plane electromagnetic wave existing into region 0 (with z-polarized magnetic field), given by:

display math

where math formula is the wave number into vacuum and ω the operating circular frequency. The parameter math formula is real and corresponds to either a propagating math formula or an evanescent wave math formula, while q0 is the amplitude of the incident field measured in V/m. Note that the axial permittivity ϵrZZ = ϵrzz does not participate in the computations, due to the nature of the excitation. A harmonic time dependence of the type e+jωt is suppressed throughout the analysis, while the square roots within the entire manuscript are evaluated with a positive real part and, in case it is null, with a positive imaginary part.

[7] The scope of the present work is to propose a simple, semi-analytical technique to determine the solution for the defined nontrivial boundary value problem, namely the electromagnetic field within each region of the regarded configuration.

2.2. Plane Waves

[8] The proposed technique utilizes exclusively plane waves and thus the expressions of the related fields for each region should be derived. Any reflected wave back to region 0, is naturally written as:

display math

where math formula ( math formula can be any real parameter) and math formula denotes the corresponding reflection coefficient. The use of the superscript (n) would become evident in the next section, where the method is thoroughly described. Similarly, an arbitrary transmitting ray of region 2 (with spectral constant math formula and amplitude math formula), as function of the capital coordinates (X, Y, Z) is given by:

display math

with math formula.

[9] As far as the anisotropic region 1 is concerned, the vector of the electric field with respect to small-lettered variables takes the form:

display math

In the same way, the equivalent expression using the capital-lettered coordinates reads:

display math

where the two auxiliary quantities are defined as follows:

display math
display math

in proportion to which coordinate system we are referring to (χ = xX, ψ = yY). Mind that both the equivalent expressions are comprised of two modes (with amplitudes math formula and math formula respectively) corresponding to the two opposite signs of the quantities math formula. In particular, the waves math formula are related to the positive sign, while the waves math formula are related to the negative sign. We introduce the notation math formulawhich is double-valued and would be properly evaluated to obey the imposed physical constraints.

[10] It should be noticed that a positive/negative-sign wave math formula of (6), is not necessarily transformed into the same sign wave math formula of (7); it could correspond to the opposite-sign expression. In other words, the transformation from the small-lettered coordinate system into the capital-lettered one and vice versa, does not preserve the sign in theequations (6) and (7). The small-lettered parameters ( math formula) are connected to the capital-lettered ones ( math formula), via the following expressions:

display math
display math
display math

The inverse relations are very similar:

display math
display math
display math

Note that math formula is directly determined by k1ψ, through the relations (8a) and (8b).

3. Iterative Solution

3.1. Method Concept

[11] The simultaneous phase matching along the front and the rear surface of the prism, is not easily achieved using rigorous techniques; therefore, we propose a semi-analytical approach to treat the defined problem. First of all, we assume that the bounds of the wedge-shaped structure ofFigure 1 are infinite from both sides and do not cross each other. In other words, we ignore the effect of the edge defined by the intersection of the planes x = 0 and X = W, unlike in the well-known concepts of fractional-ordered Bessel functions solving wedge configurations [Valagiannopoulos, 2009]. Therefore, our approach would be more successful for large thicknesses W and relatively small angles ξ. It is indisputable that the edge condition plays a significant role when solving wedge configurations but for points far from the corner, the field radiated from the edge gets significantly attenuated: these are the objective of the present study. In particular, the boundary conditions in the vicinity of the edge cannot be satisfied as happens in other works that incorporate edge component [Meixner, 1972; Beker, 1991; Budaev and Bogy, 2007]. However, even if the no-crossing infinite boundary assumption has been made, the concurrent fulfillment of the boundary conditions across both surfaces (x = 0 and X = W) is not possible by one single wave. In fact, there are multiple reflecting optical rays into region 0 and multiple transmitting ones within region 2. To imitate that mechanism, we ignore the presence of the one surface when solving the boundary value problem along the other. Therefore, three simpler problems of transmission through a single interface, should be analyzed instead, whose configurations are shown in Figure 2. Once the explicit formulas solving the aforementioned triad of problems are determined, they will be used as modules for obtaining an iterative approximate solution of the total problem of Figure 1.

Figure 2.

The configurations of (a) Problem I, (b) Problem II and (c) Problem III.

3.2. Problem I

[12] In this subsection, we use the small-lettered coordinate system. Suppose that the rear surface of the prism does not exist, namely the whole semi-infinite areax > 0 (extended region 1) is filled by the anisotropic material with permittivity tensor (1) and the considered structure is excited by the primary incident wave (3) that possesses amplitude equal to q0 measured in V/m. As shown in Figure 2a, the magnitude of the reflecting wave into region 0 is denoted by math formula (n = 1 in (4)), while the notation math formula is used for the transmitting amplitude into region 1 (n = 1 in (6)). Mind that we do not know a priori which of the two terms ( math formula or math formula) represents the outgoing wave from the boundary x = 0, as it depends on the arbitrary tensor [ϵr]. Consequently, a primitive sign check is required in order to employ the correct expression for the field into region 1. The rigorous expressions are given as follows:

display math
display math

where math formula.

3.3. Problem II

[13] In this subsection, we use the capital-lettered coordinate system. Assume that the front surface of the prism does not exist, namely the whole semi-infinite areaX < W (extended region 1) is filled with the constructing substance of the prism. Now, the permittivity tensor of the anisotropic material is given by (2), due to the rotation of the coordinate system. The plane X = W is illuminated by the wave (7) with amplitude math formula existing into region 1; thus, the other wave math formula and a transmitting ray (5) of magnitude math formula are developed into regions 1,2 respectively (as depicted in Figure 2b). It should be remarked that the excitation field does not necessarily propagates along the positive Xsemi-axis; it could be outgoing from the separating surface. By imposing the necessary boundary conditions, one obtains:

display math
display math

with math formula.

3.4. Problem III

[14] In this subsection, we are referred to the same configuration as in Problem I with the difference that the excitation wave travels into (extended) region 1 with amplitude math formula which gives rise to the other wave with magnitude math formula and a reflecting ray math formula into regions 1 and 0 respectively, as indicated in Figure 2c. In such a consideration, one does not know which of the two supported waves into the anisotropic region 1 is the cause (index (n)), and which the response (index (n + 1)); consequently, we give the solution for both cases in mutual notation:

display math
display math

where math formula.

3.5. Algorithmic Pseudocode

[15] It should be noted that each of the aforementioned three problems concerns the fulfillment of the boundary conditions along one only planar surface of the considered structure (x = 0, X = W). We are going to incorporate the solutions of them in an iterative procedure during which one boundary value problem is solved at a time, by using as excitation the field into region 1 computed via the previous boundary value problem. The proposed technique can be summarized in the following steps, where the waves are represented by their amplitudes.

[16] Step 1: Solve Problem I of Figure 2a and determine the modes math formula and math formula from (11a) and (11b); Set index n equal to one.

[17] Step 2: Transform the wave math formula into math formula through (10a)(10c); Solve Problem II of Figure 2b and determine the modes math formula and math formula from (12a) and (12b).

[18] Step 3: Transform the wave math formula into math formula from (9a)(9c); Solve Problem III of Figure 2c and determine the modes math formula and math formula from (13a) and (13b); Increase index n by one; Go to Step 2.

[19] To obtain the solution in each region, we apply the operator math formula on the expressions (4)(7) by using the evaluated magnitudes and propagation constants. The number of iterations N is related to the stoppage criterion of the algorithm which could demand the computed amplitudes math formula of Step 3 (which are proportional to the average error of the method), to fall below a certain threshold. Only then, the corresponding boundary conditions would be approximately satisfied.

4. Indicative Numerical Results

4.1. Input and Output Quantities

[20] Prior to presenting the numerical results, we should first clarify the value intervals into which the input parameters belong and define the necessary output quantities. We consider propagating and evanescent modes with a spectrum extended within math formula, while the inclination angle is moderate (5° < ξ < 25°), as required by the adopted approximate technique. The diagonal elements of the permittivity tensor for the anisotropic prism are chosen close to those used in Liu et al. [2008], namely −3 < ϵrxxϵryy < 3, while the off-diagonal elements are of lower magnitude. The free-space wave numberk0 and the width W are selected so that the structure has considerable electrical thickness.

[21] As far as the output quantities are concerned, the average errors of the boundary conditions along the front surface x = 0 are defined below:

display math
display math

where math formula is the magnetic field corresponding to the electric field E. These differences DEDM are measures of how well the method treats the formulated boundary value problem. There is no use to check the validity of the boundary conditions along the rear surface X = W, because they are directly imposed at Step 3 of the algorithm. To study the wave transmission through the anisotropic prism, we introduce the following ratio:

display math

where the summations are confined to those (n) that describe propagating modes math formula into region 2. This transmission ratio equals the power carried by the produced propagating waves over itself added to a quantity that expresses the intensity of the initial excitation either it is a propagating ( math formula) or an evanescent ( math formula) wave. In this sense, the parameter R expresses the “performance” of the device in transmitting (into region 2) the power of the impressed excitation.

4.2. Method Validation

[22] In Figure 3a, we represent the boundary conditions errors DEDM as functions of the inclination angle ξ for various ratios of the diagonal permittivities ϵrxx/ϵryy. First of all, the magnitudes of the errors are considerably low (less than 10−5) along the entire interval, which verifies the correctness of our approach. It is also sensible that DE and DM gets larger for increasing inclination angle, because our approximation is based on moderate ξ values. Mind additionally that when the diagonal permittivities into anisotropic region 1 differ substantially, the recorded error gets reinforced, which is explained by the inherent difficulty for any technique to manipulate strong material anisotropies. In Figure 3b, the quantities DEDM are shown with respect to angle ξ for several electrical widths k0W. It is clear that, in any case, the average error of the magnetic boundary condition DM is more significant than the corresponding one of the electric boundary condition DE; the same feature is also noticed in Figure 3a. Furthermore, a remarkable coincidence is exhibited for curves of different slab thicknesses, which indicates that the longitudinal dimension of the prism is not a critical factor for the effectiveness of the method, as long as it is not very tiny. Finally, the errors for small angles are equal to the computer precision, while the curves are again upward sloping with ξ as in Figure 3a.

Figure 3.

Logarithmic plot of boundary conditions errors as functions of inclination angle for (a) various diagonal permittivities ratios (k0W = 20π) and (b) various electrical widths of the slab (ϵrxx = 2.7). Plot parameters: q0 = 1 V/m, k0 = 10π Mrad/m, math formula, ϵryy = 1.1, ϵrxy = ϵryx = 0.1, ymax = −ymin = W.

[23] In Figure 4a, the variations of DEDM with respect to the propagation constant of the excitation wave math formula, are depicted for various ratios ϵrxx/ϵryy. Mind that the measured differences are smaller when the incident wave direction is “more oblique” to the front surface x = 0 ( math formula). Such a conclusion could be attributed to the finite (and non-constant) dimension of the device alongx axis, whose effect is reinforced in the event of normal incidence. Similarly to Figure 3a, the couple of red curves (solid and dashed) describing the ϵrxx = ϵryy case, are isolated at much smaller levels than the others; in addition, all the graphs exhibit periodically sharp minima at certain “resonant angles” for which our method works perfectly. In Figure 4b, the average errors of boundary conditions are represented as function of math formulafor several off-diagonal permittivitiesϵrxy/ϵryy = ϵryx/ϵryy. Surprisingly, the described methodology performs better when the cross elements of the permittivity matrix are larger, which adds to the overall powerfulness of the technique.

Figure 4.

Logarithmic plot of boundary conditions errors as functions of the propagation constant of the incident wave for (a) various diagonal permittivities ratios (ϵrxy = ϵryx = 0.1) and (b) various off-diagonal permittivities (ϵrxx = 2.5, ϵrxy = ϵryx). Plot parameters: q0 = 1 V/m, k0 = 10π Mrad/m, k0W = 20π Mrad/m, ξ = 15°, ϵryy = 1.1, ymax = −ymin = W.

[24] In Figure 5a, we represent the number Nτ of loop's iterations (of the algorithm stated in the previous section) as function of the inverse tolerance 1/τ for various angles ξ. In particular, we computed the minimum number required to repeat Step 2 and Step 3 of the iterative solution, in order to satisfy the inequality check: max{DEDM} < τ. Apparently, assigning smaller tolerances demands more iterations which is indicated by the increasing trend of the curves. It is also noted that, on average, the convergence is achieved for lower Nτ when the inclination angles are smaller. This correlation is not shown, in the strict sense, from the diagram; however, there is a clearly increasing trend between the group of small angles (ξ = 5°, 10°, 15°) and the group of larger angles (ξ = 20°, 25°). In Figure 5b, the variation of the same quantity is shown for several incident propagation constants math formula, where evanescent modes are also considered. Note that even when math formula, the behavior of the proposed method is very satisfying as only 3–4 terms are enough to obtain accurate results. Similarly to Valagiannopoulos [2007a], so few terms constitute practically a closed-form solution to a nontrivial problem.

Figure 5.

The sufficient number of iterations as function of the inverse tolerance of the algorithm for (a) various inclination angles ( math formula, ϵrxx = 2.5) and (b) various propagation constants of the incident wave (ξ = 15°, ϵrxx = −2.5). Plot parameters: q0 = 1 V/m, k0 = 10π Mrad/m, k0W = 20π, ϵryy = 1.5, ϵrxy = ϵryx = 0.3, ymax = −ymin = W.

4.3. Electromagnetic Transmission

[25] In Figure 6a, we show the behavior of the transmission ratio R as defined by (15) as function of the angle ξ for different excitation wave constants math formula. One can observe that the fluctuation exhibited by the measured quantity gets diminished for less significant propagation constant. On the contrary, for larger math formula, many abrupt boosts of R are recorded which correspond to successful transformation into propagating power within region 2. In Figure 6b, the ratio R is represented with respect to math formula for several opposite diagonal permittivities ϵrxx = − ϵryy with ϵrxy = ϵryx = 0. In other words, we examine the transmission via a hyperbolic metamaterial prism for variable spectral excitation [Valagiannopoulos and Simovski, 2011]. It should be noticed that R → 1 for math formula, while R vanishes when math formula. In the case of a propagating incident wave, the ratio is reduced with increasing math formula, while similar oscillations to Figure 6a are observed for evanescent excitation waves.

Figure 6.

The transmission ratio of the device as function of (a) the inclination angle for various propagation constants (k0W = 3, ϵrxx = −2.7, ϵryy = 1.1) and (b) the incident propagation constant for various diagonal permittivities (k0W = 20π, ξ = 15°). Plot parameters: q0 = 1 V/m, k0 = 10π Mrad/m, τ = 0.001, ϵrxy = ϵryx = 0, ymax = − ymin = W.

[26] In Figure 7a, the variation of the transmission ratio is shown in contour plot with respect to the diagonal permittivities of the anisotropic substance (ϵrxx, ϵryy). The quantity R becomes smaller when the contrast of the constituent material against the vacuum background gets higher (higher ϵrxxϵryy). Mind also that when ϵryy is larger, the same increment of ϵrxx leads to more substantial drops of R. In Figure 7b, we depict the variation of the same quantity on the two-dimensional map math formula. Needless to say that math formula due to the hermitian property of the permitivity tensor. We notice the rapid rise of R for increasing math formula and the even symmetry with respect to math formula.

Figure 7.

The transmission ratio of the device in contour plots with respect to (a) the diagonal permittivities (ϵrxy = ϵryx = 0) and (b) the real and imaginary part of the off-diagonal permittivity (ϵrxx = ϵryy = 2). Plot parameters: q0 = 1 V/m, k0 = 10π Mrad/m, k0W = 20π, ξ = 15°, τ = 0.001, ymax = −ymin = W.

5. Conclusion

[27] In the work at hand, we introduce a new and simple iterative technique which can approximately solve the scattering problem of a plane wave by a fully anisotropic slab with nonparallel surfaces. The method is based on the unbounded media transmission and its convergence rate and accuracy has been checked. Certain conclusions concerning the propagation through the prismatic device have been drawn and discussed.

[28] An expansion of the considered method could be employed in treating bent waveguides or layered anisotropic structures with slightly non-parallel boundaries. In addition, a challenging evolution of the described concept would be to take into account the effect of the edge by including correcting terms that allow us to evaluate properly the field in the vicinity of the axis formed by the two crossed planar surfaces. In this way, the transmission of the waves in the presence of a fully anisotropic wedge would be semi-analytically examined to reveal possible applicable features.


[29] C. Valagiannopoulos acknowledges the Academy of Finland for post-doctoral project funding.