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Reflection and Transmission from Biaxially Anisotropic - Isotropic Interfaces Syracuse University SURFACE Electrical Engineering and Computer Science College of Engineering and Computer Science 2013 Reflection and transmission from biaxially anisotropic - Isotropic interfaces Jennifer W. Graham Syracuse University, [email protected] Jay K. Lee Syracuse University, [email protected] Follow this and additional works at: https://surface.syr.edu/eecs Part of the Electrical and Computer Engineering Commons Recommended Citation J. W. Graham and J. K. Lee, "Reflection and transmission from biaxially anisotropic - Isotropic interfaces," Progress in Electromagnetics Research, vol. 136, pp. 681-702, 2013. This Article is brought to you for free and open access by the College of Engineering and Computer Science at SURFACE. It has been accepted for inclusion in Electrical Engineering and Computer Science by an authorized administrator of SURFACE. For more information, please contact [email protected]. Progress In Electromagnetics Research, Vol. 136, 681{702, 2013 REFLECTION AND TRANSMISSION FROM BIAXIALLY ANISOTROPIC | ISOTROPIC INTERFACES Jennifer W. Graham* and Jay K. Lee Department of Electrical Engineering and Computer Science, Syracuse University, Syracuse, NY 13244, USA Abstract|In this paper we explore electromagnetic behavior of arbitrarily oriented biaxially anisotropic media; speci¯cally with respect to reection and transmission. The reection and transmission of electromagnetic waves incident upon half-space and two-layer interfaces are investigated. The waves may be incident from either the isotropic region or the biaxial region. The biaxial medium considered may be aligned with a principal coordinate system or may be arbitrarily oriented. Critical angle and Brewster angle e®ects are also analyzed. 1. INTRODUCTION The study of biaxially anisotropic (or biaxial) materials is of increasing interest. There are several naturally occurring materials with biaxial properties. When we ignore this biaxial nature, we are unable to accurately predict the behavior of circuits using these materials. However, more interesting is the current research in material science. Much of this research has been fueled by electromagnetic interests in a variety of metamaterials. Some studies have shown these materials to have biaxial properties [1, 2]. Despite these interests, theoretical analysis of electromagnetic behavior in biaxial media has not been rigorously studied. Many authors have studied reection and transmission in complex materials. Bianisotropic media (in which there is cross-coupling between electric and magnetic ¯elds [3]) has garnered particular attention [4, 5]. In [4] Tsalamengas provides a formulation to compute the reection and transmission of an arbitrarily polarized wave incident upon a general bianisotropic slab. This slab is described by four tensors, with no limitations on the tensors Received 8 November 2012, Accepted 7 January 2013, Scheduled 1 February 2013 * Corresponding author: Jennifer W. Graham ([email protected]). 682 Graham and Lee themselves. Therefore, this formulation could be used to evaluate reection and transmission coe±cients of an arbitrarily oriented biaxial slab. However, we have only one tensor and this formulation is unnecessary. Further, Tsalamengas does not analyze the results or provide numerical examples. In [5] Semchenko and Khakhomov derive and compute reection and transmission coe±cients for unrotated uniaxial bianisotropic material and explore the varying incident wave polarizations. Lee [6] studied wave behavior in tilted and untilted uniaxial media including a detailed study of reection and transmission from an isotropic-uniaxial interface. Metamaterials, recently a hot research area, have also been studied for their reective and refractive characteristics. Grzegorczyk et al. [1, 2] provide an extensive study of the behavior of waves incident upon metamaterial layers. Their work is particularly relevant because they ¯rst consider a general bianisotropic medium (with biaxial permittivity and permeability tensors), and then apply the properties of left-handed materials. Therefore, their formulation is general but their results are speci¯c to negative epsilon materials. In fact, the inclusions used to create negative epsilon (or mu) materials make the material anisotropic in general so it is important to understand the anisotropic behaviors. A few researchers have considered reection and transmission from biaxial boundaries. Stamnes and Sithambaranathan [7] considered reection and refraction from a plane interface separating an isotropic and a biaxial medium. In their paper, they consider only the unrotated biaxial medium with a diagonal permittivity tensor. Further, they do not present numerical results but rather the formulation of the resulting ¯elds when a plane TE (transverse electric) or TM (transverse magnetic) wave is incident on the interface. Abdulhalim [8] presents a 2 £ 2 matrix approach to solving for reection and transmission coe±cients from biaxial boundaries but does not present any numerical results. The most extensive work to date on reection and transmission characteristics in biaxial media is presented by Landry and Maldonado [9, 10]; they study half space reection and transmission characteristics for biaxial-biaxial, isotropic-biaxial and biaxial- isotropic con¯gurations. They also study 2-layered and multi-layered problems. Their approach is considerably di®erent than the approach presented here. The direction and magnitude of the reected and refracted waves are treated separately in the half-space case and the 2-layered case is treated by formulating each bounce the incident wave undergoes to compute reection and transmission coe±cients. The multi-layered problem is treated similarly. In our approach, we expand the plane waves in each medium Progress In Electromagnetics Research, Vol. 136, 2013 683 then apply the boundary conditions. We use material parameters to determine the directions of each expansion wave and then apply boundary conditions to solve for the magnitude. We apply this treatment to both the half-space and 2-layered problems. This is a familiar and straightforward formulation. Another di®erence is that we de¯ne the electric ¯eld vectors in each medium based on the known material parameters (permittivity matrix and rotation matrix) while Landry uses the refractive index and a set of angles to de¯ne the relationship between the wave vector and ¯elds. Landry uses a formulation more commonly used in the physics and optics communities and not familiar to most electrical engineers. Our approach is also more general than the approach used in [9, 10]. Finally, we expand upon this research by analyzing the Brewster angle e®ect and critical angle as functions of permittivity and rotation angles. We begin by de¯ning the material parameters in Section 1.1. In Section 2, we de¯ne the half-space reection and transmission coe±cients of each incident wave on either side of an isotropic-biaxial boundary. We use these results to analyze the critical angle, an analysis that was not considered in [9, 10]. We then go on to formulate the 2- layer case in Section 3 with a wave incident from one isotropic layer onto the biaxial layer. We use these results to analyze the Brewster angle e®ect which was also not presented in [9, 10]. Conclusions are drawn in Section 4. 1.1. Biaxially Anisotropic Medium De¯nitions The de¯ning property of electrically biaxial media is the permittivity tensor. Unlike isotropic materials (with only one permittivity) and uniaxial materials (with 2 di®erent permittivities), biaxially anisotropic materials have three unique values in the permittivity tensor as in " # "x 0 0 "¹ = 0 "y 0 (1) 0 0 "z Equation (1) represents a biaxial medium whose principal axes are aligned with the laboratory coordinate system. If, however, the biaxial medium is not aligned with some reference coordinate system, the permittivity tensor would not be as simple. We can obtain the tensor for an arbitrarily oriented biaxial medium by applying rotations (Mudaliar and Lee [11]) to the tensor in Equation (1). If the permittivity tensor shown in (1) lies in the double-primed coordinate system, we can transform to the unprimed (laboratory) coordinate system through two rotations: the ¯rst about the x00 axis by an angle 684 Graham and Lee z' z'' z', z ψ1 ψ2 y' y y'' ψ2 ψ x' 1 y' x'',x' x (a) (b) Figure 1. Rotation diagrams. 0 Ã1, and the second about the z axis by an angle Ã2, as shown in Figures 1(a) and (b), resulting in a full permittivity matrix with all nine elements non-zero in the unprimed system. The permittivity tensor now is the full matrix where 2 ¡ 2 2 ¢ 2 "xx = "x cos Ã2 + "y cos Ã1 + "z sin Ã1 sin Ã2 ¡ 2 2 ¢ "xy = ¡"x + "y cos Ã1 + "z sin Ã1 sin Ã2 cos Ã2 "xz = ("z ¡ "y) sin Ã1 cos Ã1 sin Ã2 2 ¡ 2 2 ¢ 2 "yy = "x sin Ã2 + "y cos Ã1 + "z sin Ã1 cos Ã2 (2) "yz = ("z ¡ "y) sin Ã1 cos Ã1 cos Ã2 2 2 "zz = "y sin Ã1 + "z cos Ã1 "yx = "xy;"zx = "xz;"zy = "yz Another important electromagnetic property of biaxially anisotropic media is birefringence. If a wave is incident upon a biaxial medium, two characteristic waves will be refracted, the so called a-wave and b-wave [11]. 2. HALF SPACE REFLECTION AND TRANSMISSION COEFFICIENTS In general, the study of half-space (one interface) reection and trans- mission problems can be broken down into four main con¯gurations as noted by Pettis [12, Appendix G]. These con¯gurations are listed in Table 1. To derive the half-space reection and transmission coe±cients, we formulate the ¯elds in each region of interest, apply the boundary conditions and solve for the reection and transmission coe±cients. Progress In Electromagnetics Research, Vol. 136, 2013 685 Table 1. Half-space con¯gurations. Case Con¯guration horizontal or vertical wave downward 1 incident on isotropic-biaxial interface a-wave or b-wave upward incident 2 on biaxial-isotropic interface a-wave or b-wave downward incident 3 on biaxial-isotropic interface horizontal or vertical wave upward 4 incident on isotropic-biaxial interface Figure 2. A-wave incident upon biaxial-isotropic interface.
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