Finite-Difference Frequency-Domain Algorithm for Modeling Electromagnetic Scattering from General Anisotropic Objects.
1. INTRODUCTIONThe finite-difference frequency-domain (FDFD) method is a simple and powerful numerical method for solving Maxwell's equations [1-4]. It is rigorous, excellent for visualizing the fields, and able to model structures with complex geometries. It is accurate, stable, and the sources of error are well understood. Lastly, it lends itself very well to parallel processing on graphical processing units (GPUs) [5,6]. The method uses central finite-difference approximations to transform Maxwell's equations into a large set of linear algebraic equations that can be written in matrix form as Ax = b. The field stored in the column vector x is calculated by solving x = [A.sup.-1]b. A complete discussion of the basic FDFD algorithm for ordinary materials can be found in Ref. [7].
This paper outlines how to implement an anisotropic FDFD (AFDFD) method that can handle fully anisotropic materials. An anisotropic medium is one where the permeability and/or permittivity depend on the direction of the electromagnetic fields. Anisotropy provides additional design freedom that can be used to produce a wide array of useful phenomena including surface waves [8], slow waves [9], invisibility and cloaking [10], double refraction, polarization control [11], and more. Further, very often devices composed of metamaterials can be modeled more efficiently using effective medium theory. "Homogeneous" materials with the effective properties of the metamaterials can be meshed more coarsely compared to the volumetrically complex structures of the metamaterial. In this regard, a method capable of modeling devices with arbitrary dielectric and magnetic anisotropy is very useful.
This paper discusses in detail how the three-dimensional (3D) FDFD method described in Ref. [7] can be modified to incorporate anisotropic materials with arbitrary nine-element tensors for both permittivity and permeability. It also discusses some of the subtleties encountered when dealing with tensor quantities like converting them between coordinate systems, rotating them to an arbitrary orientation, combining them with absorbing boundary conditions, and assigning them to points on a grid. Unfortunately, in general anisotropic media Maxwell's equations do not simply for two-dimensional or even one-dimensional analysis due to the complete coupling of the fields. For this reason, it is most straight forward to develop just a 3D FDFD code and use that to model lower dimensional problems. This can be done with virtually no decrease in efficiency if the appropriate precautions are taken. Using the 3D algorithm this way allows the grid to lie in any plane that is desired. This is advantageous when the algorithm is being used in conjunction with other numerical methods and one is trying to maintain consistent coordinates.
While some papers can be found on FDFD analysis of anisotropic structures [12-18], none of these outline a general purpose method for 3D scattering analysis of devices containing arbitrary permittivity [[epsilon]] and permeability [[micro]] tensors. The majority of these papers [12,14,15,18-21] describe waveguide analysis and not scattering. The papers covering scattering [13,16,17,22] are all restricted to two-dimensions and all but Ref. [22] place restrictions on the tensor quantities. This paper describes a general purpose FDFD technique that is rigorous, fully 3D, and incorporates arbitrary material tensors for both permittivity and permeability. The paper is organized as follows. First, the standard formulation of the anisotropic 3D-FDFD method is presented in order to calculate the wave matrix A. Second, it is shown how to apply the powerful total-field/scattered-field (TF/SF) technique [22-24] to incorporate sources. Finally, several practical examples are given before the paper is concluded.
2. FORMULATION OF THE METHOD
The formulation and implementation of the basic FDFD method is described in detail in Ref. [7], but only diagonal material tensors [ [micro]] and [ [epsilon]] were considered. The following text outlines how the formulation is generalized to incorporate arbitrary material tensors. After normalizing the magnetic field according to H = -j[[eta].sub.0]H, Maxwell's curl equations with a uniaxial PML (UPML) [25-27] can be written as
[Please download the PDF to view the formula] (1)
[Please download the PDF to view the formula] (2)
For a UPML, the tensor [s] is expressed as
[Please download the PDF to view the mathematical expression] (3)
Without any approximation to the material tensors, Eqs. (1) and (2) can be expanded into the following set of six coupled partial differential equations. The [Please download the PDF to view the formula] and [Please download the PDF to view the formula] terms are relative coefficients because the free space constants [[micro].sub.0] and [[epsilon].sub.0] have been factored out and absorbed into the normalized grid coordinates [7]. In addition, the PML parameters [s.sub.x], [s.sub.y], and [s.sub.z] are multiplied into the constitutive parameters such that [Please download the PDF to view the formula].
[Please download the PDF to view the formula] (4)
[Please download the PDF to view the formula] (5)
[Please download the PDF to view the formula] (6)
[Please download the PDF to view the formula] (7)
[Please download the PDF to view the formula] (8)
[Please download the PDF to view the formula] (9)
[Please download the PDF to view the formula] (10)
[Please download the PDF to view the mathematical expression] (11)
Here, the grid coordinates have been normalized according to
[Please download the PDF to view the formula] (12)
2.1. Finite-difference Approximation of Maxwell's Equations
Following the procedure outlined in [7], the fields and materials are assigned to discrete points on a Yee grid [28] and the derivatives in Eqs. (4)-(9) are approximated using central finite-differences [29]. The integer variables i, j, and k are array indices.
[Please download the PDF to view the mathematical expression] (13)
[Please download the PDF to view the mathematical expression] (14)
[Please download the PDF to view the mathematical expression] (15)
[Please download the PDF to view the mathematical expression] (16)
[Please download the PDF to view the mathematical expression] (17)
[Please download the PDF to view the mathematical expression] (18)
In these equations, the terms containing off diagonal tensor elements require special treatment because each term in a finite-difference equation must exist at the same point in space. In Eq. (13), for example, each term must exist at the same point as [H.sub.x]. The central finite-difference ensures that the derivative terms on the left exist at this point. The last two terms on the right side of this equation are averaged at four points in the grid to interpolate the values at the same point as [H.sub.x]. The same interpolation operation is used on every term in Eqs. (13)-(18) containing an off diagonal tensor element. Figure 1 illustrates the Yee cell and summarizes where each of the tensor elements are defined to exist relative to the field components. The tensor elements [Please download the PDF to view the formula], and [Please download the PDF to view the formula] are placed on the Yee grid at the same points as [H.sub.x]. The tensor elements [Please download the PDF to view the formula], and [Please download the PDF to view the formula] are placed at the same points as [H.sub.y]. The tensor elements [Please download the PDF to view the formula], and [Please download the PDF to view the formula] are placed at the same points as [H.sub.z]. Similarly for the permittivity the tensor elements [Please download the PDF to view the formula], and [Please download the PDF to view the formula] are placed on the Yee grid at the same points as [E.sub.x]. The tensor elements [Please download the PDF to view the formula], and [Please download the PDF to view the formula] are placed at the same points as [E.sub.y]. The tensor elements [Please download the PDF to view the formula], and [Please download the PDF to view the formula]. are placed at the same points as [E.sub.z].
2.2. Maxwell's Equations in Matrix Form
Each of Eqs. (13)-(18) is written once for every cell in the grid so each finite-difference equation produces a large set of linear algebraic equations. Each set of equations can be written in matrix form. To do this, interpolation matrices [Please download the PDF to view the formula] are introduced to perform the four-point averaging. These will be discussed in more detail later.
[Please download the PDF to view the formula] (19)
[Please download the PDF to view the formula] (20)
[Please download the PDF to view the formula] (21)
[Please download the PDF to view the formula] (22)
[Please download the PDF to view the formula] (23)
[Please download the PDF to view the formula] (24)
The terms [e.sub.x], [e.sub.y], [e.sub.z], [h.sub.x], [h.sub.y] and [h.sub.z] are column vectors that contain all of the field components throughout the entire grid reshaped into linear arrays. The terms [Please download the PDF to view the formula] and [Please download the PDF to view the formula] are diagonal matrices containing the relative tensor elements throughout the grid along their diagonals with the PML incorporated. The terms [Please download the PDF to view the formula], [Please download the PDF to view the formula] and [Please download the PDF to view the formula] are banded matrices that calculate first-order spatial derivatives of the electric fields across the grid. Similarly, the terms [Please download the PDF to view the formula], [Please download the PDF to view the formula] and [Please download the PDF to view the formula] calculate first-order spatial derivatives of the magnetic fields across the grid. These are described in more detail in Ref. [7]. When the 3D code is being used to model 2D or 1D systems, the derivatives along the uniform directions take the special forms provided in Eqs. (25)-(27). The wave vector of the source is given by [Please download the PDF to view the formula] and I is the identity matrix.
[Please download the PDF to view the formula] (25)
[Please download the PDF to view the formula] (26)
[Please download the PDF to view the formula] (27)
It is possible to "absorb" both the interpolation matrices and PML terms into the permittivity and permeability tensors. When this is done, the tensor elements are written with a double-prime superscript to remind us that the PML and interpolations have been absorbed into them.
[Please download the PDF to view the formula] (28)
[Please download the PDF to view the formula] (29)
The interpolation matrices [Please download the PDF to view the formula] are banded matrices that interpolate quantities across the grid [30]. The sign of the superscript indicates whether the interpolation is calculated by averaging a point with the next or previous point on the grid. In this sense, the subscript indicates in which direction the interpolation is happening. The interpolation matrices are very similar in form to the derivative operators and differ only in signs and scaling. It is important to premultiply the tensor elements by the interpolation matrices because it is the [epsilon]E and [micro]H products that are being interpolated. As a quick example of what the interpolation matrices look like, they were computed for a two-dimensional grid composed of only 4 X 4 cells. Using Dirichlet boundary conditions, the interpolation matrices [Please download the PDF to view the formula] and [Please download the PDF to view the formula] for this simple case are
[Please download the PDF to view the mathematical expression] (30)
[Please download the PDF to view the mathematical expression] (31)
For this simple case, the interpolation matrices are composed of two dominant diagonals. Some off diagonal terms can arise when periodic boundaries or other boundary conditions are incorporated. Additional bands will appear when higher-order interpolation is used. The positive and negative interpolation matrices are related through the complex Hermitian operation as [Please download the PDF to view the formula]. The interpolation matrices can have complex numbers when there are periodic boundary conditions. Using interpolation matrices in this manner allows straightforward implementation of higher-order interpolation schemes without having to modify the rest of the AFDFD code.
Maxwell's curl equations can now be written in block matrix form as
[Please download the PDF to view the formula] (32)
[Please download the PDF to view the formula] (33)
[Please download the PDF to view the formula] (34)
[Please download the PDF to view the formula] (35)
2.3. Matrix Wave Equation
A matrix wave equation can be derived for the electric field by solving Eq. (32) for h and substituting that expression into Eq. (33). A similar matrix wave equation can be derived for the magnetic field by solving Eq. (33) for e and substituting that expression into Eq. (32). It is useful to write the matrix
wave equations in terms of a wave matrix A and the unknown fields, e or h.
[Please download the PDF to view the formula] (36)
[Please download the PDF to view the formula] (37)
These equations cannot yet be solved because a source has not yet been incorporated.
3. ROTATION OF THE TENSORS
The material tensors at each point in the grid can be completely unique so the anisotropy can be varied in any manner across the grid. Physical tensors have only three degrees of freedom, where three numbers represent the material response along the principle axes (a, b, and c) of the anisotropic material. When the tensors are expressed in this diagonalized system, the three degrees of freedom are explicit. Most often, anisotropic materials are specified in this manner. The convention is to list the constitutive values along the diagonal in order of increasing magnitude.
[Please download the PDF to view the mathematical expression] (38)
To incorporate a tensor into a Cartesian grid, it is first necessary to convert the tensor given along the principle axes [[[epsilon].sup.abc]] to an equivalent tensor in the coordinates of the model [[[epsilon].sup.xyz]]. In the present work, Cartesian coordinates are used. The transformation is accomplished using Eq. (39) where P is a rotation matrix.
[Please download the PDF to view the formula] (39)
[Please download the PDF to view the formula] (40)
The Cartesian tensor can be rotated into any arbitrary orientation using rotation matrices [31]. Equation (41) rotates the tensor [[epsilon]] by angle [theta] about the ith axis. Rotation matrices are both real and unitary so [[P].sup.T] = [[P].sup.-1]
[Please download the PDF to view the formula] (41)
Rotation matrices that rotate about the x, y, and z axes by an angle [theta] are given by Eq. (42), Eq. (43), and Eq. (44) respectively.
[Please download the PDF to view the mathematical expression] (42)
[Please download the PDF to view the mathematical expression] (43)
[Please download the PDF to view the mathematical expression] (44)
Rotation matrices can also be used in any combination. For example, to rotate a tensor by 30[degrees] about the z-axis and then 120[degrees] about the x-axis, the following sequence of multiplications should be used.
[Please download the PDF to view the formula] (45)
4. TOTAL-FIELD/SCATTERED-FIELD FORMULATION
The powerful total-field/scattered-field (TF/SF) technique described in Ref. [7] for incorporating a source can still be applied, but three field components are needed to describe the source. The source field [e.sub.src] is constructed according to Eq. (46). It is a column vector composed of three smaller column vectors [e.sub.x,src], [e.sub.y,src], and [e.sub.z,src] that each contain the different field components of the source throughout the Yee grid, but reshaped into 1D arrays.
[Please download the PDF to view the mathematical expression] (46)
[Please download the PDF to view the mathematical expression] (47)
The masking matrix Q presented in Ref. [7] must be slightly modified and is constructed in the form of Eq. (48). It is a block diagonal matrix composed of three matrices along its diagonal. [Q.sub.x] is the scattered-field masking matrix for the [E.sub.x] field, [Q.sub.y] is the scattered-field masking matrix for the [E.sub.y] field, and [Q.sub.z] is the scattered-field masking matrix for the [E.sub.z] field.
[Please download the PDF to view the mathematical expression] (48)
Given the wave matrix A, source field [f.sub.src], and the masking matrix Q, the source vector b is calculated according to the equation presented in Ref. [7].
b = (QA-AQ) [e.sub.src] (49)
Now that a source has been incorporated, the field is calculated according to Eq. (50). Note that the solution e must be parsed to extract the individual field components [e.sub.x], [e.sub.y], and [e.sub.z]. These three terms can then then be reshaped from column vectors back to the original grid.
[Please download the PDF to view the formula] (50)
If needed, the magnetic field can be calculated from the electric field using Eq. (32). This is
[Please download the PDF to view the formula] (51)
5. BENCHMARK SIMULATIONS
5.1. Scattering from an Anisotropic Guided-Mode Resonance Filter
Guided-mode resonance (GMR) filters consist of two electromagnetically coupled devices; a grating and a slab waveguide [32]. When an external electromagnetic wave is incident on the device and a precise phase matching condition is met, the external wave is diffracted by the grating and coupled into a guided mode within the slab. Due to reciprocity, the guided waves slowly leak back out of the slab waveguide. The leaked wave interferes with the incident wave to produce the overall frequency response.
GMR devices using ruled gratings are inherently sensitive to polarization due to the birefringence imposed by the grating [32-36]. By incorporating anisotropy into the design, it becomes possible to introduce birefringence that counteracts the grating to realize polarization independent designs. Using the AFDFD code, an anisotropic ruled-grating GMR filter was designed to be polarization insensitive. The results were verified by simulating the same design with Ansys HFSS. Both HFSS and AFDFD used a plane wave as a source and periodic boundaries on the sides. AFDFD used UPML's on the top and bottom, but HFSS used Floquet ports.
Figure 2 depicts the spectral response of the anisotropic GMR filter obtained using both HFSS and AFDFD and the results matched almost exactly. Both the transvers electric (TE) and transvers magnetic (TM) modes were made resonant at 24.94 GHz by adjusting the anisotropy of the material. The TE mode exhibited a 0.2% fractional bandwidth while the TM mode exhibited 0.16% fractional bandwidth. Using the method presented in reference [37], a concept of how this device could be realized is illustrated in Figure 3. In this diagram, anisotropy is introduced artificially using an array of air holes formed through the otherwise ordinary dielectric. The array must be highly subwavelength to ensure that it is not resonant.
5.2. Scattering from a Flat Lens for Far-Zone Focusing
Transformation optics (TO) uses coordinate transforms to design electromagnetic devices that can control electromagnetic fields almost arbitrarily [38-40]. Since Maxwell's equations are form invariant [41, 42] the transform can be pulled out of the coordinates and absorbed into the constitutive parameters. When this is done, the electromagnetic fields in the new constitutive parameters behave the same as they did in the transformed coordinate system. Using TO, a flat lens for far-zone focusing was designed in Ref. [43]. The lens effectively converts cylindrical waves into plane waves. In this design, the permittivity and permeability were set equal. The resulting permittivity was
[Please download the PDF to view the formula] (52)
[Please download the PDF to view the formula] (53)
[Please download the PDF to view the formula] (54)
[Please download the PDF to view the formula] (55)
[Please download the PDF to view the formula] (56)
[Please download the PDF to view the formula] (57)
The dimensions in this design were w = 0.4 m, g = 0.05 m, and l = 0.1 m. The material parameters are plotted across a 2D grid in Figure 4.
The far zone lens was simulated using the AFDFD method presented here. The field calculated from the simulation is shown in Figure 5. There are UPML's on all four sides of the simulation space. The simulated results are extremely close to those shown in Figure 6 of Ref. [41]. There are slight reflections at the interface of the lens and the cylindrical wave because the materials properties were truncated to eliminate extreme values. Duplicating these results confirms the accuracy of the simple AFDFD method.
5.3. Scattering from a Cloak
Using TO, cloaks have been designed to work in the microwave region [10] and in the optical region [44]. Using the coordinate transform developed in Ref. [10], a cloak was made using the following coordinate transform in the cylindrical coordinate system. In these equations, the parameter a is the inner radius of the cloak and parameter b is the outer radius.
[Please download the PDF to view the formula] (58)
[Please download the PDF to view the formula] (59)
[Please download the PDF to view the formula] (60)
The dimensions used were a = 10 cm and b = 45 cm. For illustration purposes, Eqs. (58)-(60) were converted into Cartesian space and plotted in Figure 6.
The tensor elements not shown in Figure 6 were set to zero. A standard plane wave was simulated using AFDFD with UMPL's on the left and right and Dirichlet boundaries on the other sides. The same plane wave was simulated but with the cloak placed in the center of the simulation space. It can be seen in Figure 7 that the cloak was successfully simulated using AFDFD. The cloak was able to reconstruct a plane wave on the transmitted side of the device.
6. CONCLUSIONS
It was shown in detail how to modify the 3D FDFD method to be able to incorporate anisotropic materials with arbitrary tensors. It was also shown how to use the 3D code as a 2D code and a 1D code by special treatment of the derivative operators. The method was then successfully benchmarked by simulating an anisotropic GMR filter in HFSS and in AFDFD. Finally, AFDFD was used to simulate a far zone lens and a cloak generated with TO. All of the results obtained using AFDFD matched very closely to results obtained by other tools and other authors.
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Raymond C. Rumpf (*), Cesar R. Garcia, Eric A. Berry, and Jay H. Barton
The authors are with the EM Lab, W. M. Keck Center for 3D Innovation, University of Texas at El Paso, USA.
(*) Corresponding author: Raymond C. Rumpf (rcrumpf@utep.edu).
Received 16 July 2014, Accepted 10 September 2014, Scheduled 12 September 2014
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| Author: | Rumpf, Raymond C.; Garcia, Cesar R.; Berry, Eric A.; Barton, Jay H. |
|---|---|
| Publication: | Progress In Electromagnetics Research B |
| Article Type: | Report |
| Geographic Code: | 1USA |
| Date: | Sep 1, 2014 |
| Words: | 4878 |
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