Realizable design of field taper via coordinate transformation

Jianjia Yi; Sawyer D. Campbell; Rui Feng; Shah Nawaz Burokur; Douglas H. Werner

doi:10.1364/OE.26.000505

1. Introduction

The transformation electromagnetics (also referred to as transformation optics) technique, which exploits the invariance of Maxwell’s equations under coordinate mappings, was formalized in 2006 by J. B. Pendry [1] and U. Leonhardt [2]. Since then, the concept of spatial coordinate transformations (CT), a powerful and effective method to control electromagnetic (EM) fields, has been widely used to design new classes of optical and electromagnetic devices for cloaking [1,3,4], illusion [5–10], waveguide transitions [11–15], microwave antennas [16–21], and lenses [22–25]. Theoretically, CT are used to establish a mapping between a transformed space and the initial space by applying a transformation between the initial and transformed coordinate systems. Typically, such a transformation consists of compression, rotation, or expansion of the initial space or geometry. The resulting transformation results in a new space that recreates the desired functionality in the initial space through an appropriate mapping of the material tensors. The new calculated tensors are viewed as a medium that mimics the defined topological space, and contain non-diagonal components due to non-linear mappings. Generally, the permittivity and permeability values resulting from CT-based mappings are both inhomogeneous and anisotropic.

A waveguide taper designed with CT-based mappings has been previously discussed [14], but required complex material parameters in order to be physically realized. In this work, using the CT concept, a field tapering device intended to guide electromagnetic fields between two waveguides of different cross sections is designed. Four different transformation procedures are presented to achieve a low reflection taper between the two waveguides. The media obtained from these transformations present complex anisotropic permittivities and permeabilities. In order to attain physically achievable material parameters, simplifications and reductions of the constituent material tensors are performed and the resulting material values are presented. Moreover, a 3D potentially realizable discrete model is developed based on a selected transformation technique. Both two- and three-dimensional full wave finite element method (FEM) numerical simulations are performed over a frequency range spanning from 9 to 11 GHz to validate the non-resonant performances and functionality of the proposed device.

2. Transformation formulations

The field tapering functionality of the proposed device is achieved by adding a transformation region between two waveguides of different cross-sections. We assume an electromagnetic wave propagating from the wide waveguide, which is referred to as the input waveguide in the remainder of the paper. It transmits through a tapered region where the coordinate transformation technique is applied. Within the tapered region, the electromagnetic field is compressed along the path defined by the transformation and then propagates into a narrow waveguide defined as the output waveguide.

Four formulations, corresponding to four different transformation procedures, are proposed to achieve a low reflection taper between two waveguides of different cross sections. The input and output waveguides are represented by the gray areas in their respective space shown in Cartesian coordinates as depicted in Fig. 1.

Fig. 1 Illustrations depicting of four different types of transformations. (a) cosinusoidal, (b) parabolic, (c) logarithmic and (d) reciprocal transformation.

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The widths of the input and output waveguides are denoted as a and b, respectively. The tapered region is considered to have a length l. Hence, after transmitting through the l-long tapered region, the electromagnetic field with waist a is narrowed into one with waist b. The streamlines in the tapered region show the reshaping behavior of the four different transformations. The aim is to connect each horizontal line of the input and output regions to ensure near-perfect transmission of electromagnetic waves. Thus, to a geometric approximation, each ray of an electromagnetic wave from the input waveguide propagates into the waveguide taper such that it is reshaped according to the transformation procedure applied in the tapering region. At the output end of the tapering region, the rays are compressed and guided into the output waveguide with low reflection at the different boundaries in the system.

Mathematical expressions defining each transformation formulation are provided in Table 1. x’, y’ and z’ represent the coordinates in the transformed (physical) space while x, y and z are those in the initial (virtual) space. Each of the four proposed transformations leads to a material with specific properties that achieves the desired taper functionality. The transformation approach can be summarized in two main steps. First, the Jacobian matrix of each transformation formulation is determined so as to obtain the properties of the “taper space”. Thus, the transformation expressed in Eq. (1) is used to obtain the permittivity and permeability tensors in the initial space which are x and y dependent:

ε^{i' j'} = \frac{J_{i}^{i'} J_{j}^{j'} ε_{0} δ^{i j}}{\det (J)} and μ^{i' j'} = \frac{J_{i}^{i'} J_{j}^{j'} μ_{0} δ^{i j}}{\det (J)} with J_{α}^{α'} = \frac{\partial x'^{α}}{\partial x^{α}}

where

J_{α}^{α'}

and

δ^{i j}

are, respectively, the Jacobian transformation matrix of the expressions and the Kronecker delta function.

Table 1. Mathematical Expressions Defining the Transformation Techniques

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The second step consists of calculating the new permittivity and permeability tensors in the coordinate system (x’,y’) so as to mimic the transformed space. At this stage, one arrives at a material with the specific desired physical properties that can then be described by the permeability and permittivity tensors $\bar{\bar{ε}} = \bar{\bar{θ}} ε_{0}$ and $\bar{\bar{μ}} = \bar{\bar{θ}} μ_{0}$ . In order to introduce a more compact notation to describe this two-step process, we take $J_{i}^{i'} J_{j}^{j'} δ^{i j} / \det (J) = θ^{i' j'}$ with

\bar{\bar{θ}} = [\begin{matrix} θ_{x x} (x', y') & θ_{x y} (x', y') & 0 \\ θ_{x y} (x', y') & θ_{y y} (x', y') & 0 \\ 0 & 0 & θ_{z z} (x', y') \end{matrix}] .

The component values of the $\bar{\bar{θ}}$ tensor are presented in Fig. 2, where a non-diagonal term (θ_xy) appears. Commonly known metamaterials [26–28] are anisotropic and respond to a specific polarization of the incident wave. As such, it is quite difficult to engineer off-diagonal components in the material parameter tensor. This non-diagonal term, while leading to challenges for realization of the device, is, however, required to guide the electromagnetic waves in the xy plane and is necessary for the taper.

Fig. 2 Components of the permittivity and permeability tensors $\bar{\bar{θ}}$ for the four different transformation techniques: (a) cosinusoidal, (b) parabolic, (c) logarithmic, and (d) reciprocal transformation.

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3. Two-dimensional simulations and parameter reduction

To verify the formulations expressed in the previous section, the finite element method based full-wave commercial software COMSOL Multiphysics is employed to model the described waveguide taper in the microwave domain.

Simulations are performed in a semi-infinite two-dimensional mode for rapid validation of the proposed material parameters. Waveports are used to excite the first transverse electric (TE₁) mode of the input waveguide with the E-field directed along the z-axis to verify the conservation of modes through the taper. The waveguide boundaries are assumed as perfect electric conductors (PECs) and matched boundary conditions are applied to the taper.

The waveguides widths are fixed to a = 55 mm and b = 20 mm while the length of the taper is l = 70 mm. With these dimensions fixed, the spatially-dependent $\bar{\bar{θ}}$ tensor components can be calculated and are evaluated in Fig. 2. The material parameter distributions presented in Fig. 2 are obtained from the expressions given in Table 1. The spatially-dependent permittivity and permeability values enable control of the electromagnetic field in the taper and conservation of the propagating mode from the input to output waveguides. Simulated E-field distributions for the four different transformation procedures are presented in Fig. 3. The distributions in the tapered waveguides are compared to the configuration without any taper at 10 GHz for the fundamental (TE₁) excitation mode.

Fig. 3 Electric field distribution for the four different types of transformation techniques: (a) cosinusoidal, (b) parabolic, (c) logarithmic, and (d) reciprocal transformation. (e) Electric field distribution for transmission without taper between the two waveguides. (f) Transmission (S₂₁) of the four tapering structures compared to the case without taper.

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The simulation results shown in Fig. 3 illustrate proper guidance of the incident wave from one waveguide to the other with only a slight impact on the guided mode when the transformed medium is embedded between the two waveguides. The difference in the transformation formulations indicates a change in the path of electromagnetic waves in the tapered section. The field in the output waveguide in the case of the reciprocal transformation is observed to be more concentrated, implying that less reflection occurs on the boundaries between adjacent regions of the system. Moreover, the field tapering device is still efficient for other excitation modes, as illustrated by the guidance of the third order (TE₃) mode from the input to the output waveguides corresponding to the reciprocal transformation in Fig. 4. The scattering or S-parameters showing reflection and transmission magnitudes are further presented at 25 GHz in Table 2 for the four transformations.

Fig. 4 Electric field distribution of the TE₃ mode in the case of the reciprocal transformation at 25 GHz.

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Table 2. S-parameters of the Transformation Techniques at 25 GHz for a TE₃ Excitation Mode

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The physical realization of such a taper can be facilitated by a slow variation of the material parameters. This can, in turn, be achieved through gradual variations of the geometrical parameters of the metamaterial inclusions. The permittivity and permeability tensors of the proposed reciprocal transformation are given as:

(\begin{matrix} \frac{m}{n} & - \frac{y}{n} & 0 \\ - \frac{y}{n} & \frac{n^{2} + y^{2}}{m n} & 0 \\ 0 & 0 & \frac{m}{n} \end{matrix})

where m = 1/(n + x).

Again, the off-diagonal components present a challenge for realizing the desired behavior with metamaterials. Therefore, proper parameter simplification is required for a possible realization of the field tapering device. The parameter simplification is performed in two steps as illustrated in Fig. 5.

Fig. 5 Diagram of the two-step parameter simplification procedure. (a) Original Cartesian coordinate. (b) Eigenvectors of material properties matrix from (a). (c) Coordinate system of simplified parameters: (1) Calculation of eigenvectors from (a) to (b) and (2) parameter reduction by dispersion relation from (b) to (c).

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The parameters distributions are expressed in Cartesian coordinates. By calculating the eigenvectors and eigenvalues of the parameter distribution matrix, new expressions of the parameter distributions are obtained. Thus, the (x, y, z) Cartesian coordinates in Fig. 5(a) are transformed into the (u, v, z) coordinate system shown in Fig. 5(b), where the z-axis remains the same while the u- and v-axes are along two eigenvectors defined in Eq. (5). The reduced parameter matrix consists of three diagonal components as the eigenvalues in Eq. (4). It should be noted that the eigenvalues and eigenvectors are dependent on the position of each point (x, y) in the physical domain, which means that the coordinate systems are different for different points. The eigenvalues and eigenvectors of the parameter matrix are calculated from Eq. (3):

α = [\begin{matrix} \frac{m^{2} + n^{2} + y^{2} + \sqrt{{(m^{2} + n^{2} + y^{2})}^{2} - 4 m^{2} n^{2}}}{2 m n} \\ \frac{m^{2} + n^{2} + y^{2} - \sqrt{{(m^{2} + n^{2} + y^{2})}^{2} - 4 m^{2} n^{2}}}{2 m n} \\ \frac{m}{n} \end{matrix}]

λ = [\begin{matrix} - \frac{m^{2} - n^{2} - y^{2} + \sqrt{{(m^{2} + n^{2} + y^{2})}^{2} - 4 m^{2} n^{2}}}{2 m y} & 1 & 0 \\ - \frac{m^{2} - n^{2} - y^{2} - \sqrt{{(m^{2} + n^{2} + y^{2})}^{2} - 4 m^{2} n^{2}}}{2 m y} & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]

The simplified permittivity components are then calculated in the new coordinate system and obey the following dispersion relation in the transverse electric (TE) mode:

ε_{z z} (μ_{u v}^{2} - μ_{u v} μ_{v v}) + μ_{u u} k_{u}^{2} + k_{v} (μ_{v v} k_{v} - 2 μ_{u v} k_{u}) = 0.

Therefore, permeability

μ_{u u}

can be normalized to 1. Finally, two components remain:

μ_{v' v'} = \frac{m^{2} + n^{2} + y^{2} - \sqrt{{(m^{2} + n^{2} + y^{2})}^{2} - 4 m^{2} n^{2}}}{2 n^{2}}

ε_{z' z'} = \frac{m^{2} + n^{2} + y^{2} + \sqrt{{(m^{2} + n^{2} + y^{2})}^{2} - 4 m^{2} n^{2}}}{m^{2} + n^{2} + y^{2} - \sqrt{{(m^{2} + n^{2} + y^{2})}^{2} - 4 m^{2} n^{2}}} .

The direction of the v’-axis is along the direction of the corresponding eigenvector. The angle that rotates around the z-axis from the x-axis is given as:

β = {Cos}^{- 1} [- \frac{m^{2} - n^{2} - y^{2} + \sqrt{{(m^{2} + n^{2} + y^{2})}^{2} - 4 m^{2} n^{2}}}{2 m y \sqrt{1 + \frac{{(m^{2} - n^{2} - y^{2} + \sqrt{{(m^{2} + n^{2} + y^{2})}^{2} - 4 m^{2} n^{2}})}^{2}}{4 m^{2} y^{2}}}}] .

Finally, after the reduction procedure, only two effective parameters remain, whose distributions are presented in Fig. 6.

Fig. 6 Effective parameter distributions: (a) permeability μ_v’v’ and (b) permittivity ε_z’z’.

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4. Metamaterial design and three-dimensional modeling

A 3D discrete lens is designed for further realistic numerical simulations as presented in Fig. 7(a). The taper is discretized into 76 unit cells, which are mapped to different permittivity and permeability values varying from 1 to 15 and 0.06 to 1, respectively. For ease of structuring, the lens is realized from three different types of unit cells containing both resonant and non-resonant elements. For permittivity values ranging from 1 to 4.4, we consider the use of a dielectric material with air holes where the volume fraction of air allows to tailor the desired value. Above the value of 4.4, the permittivity is achieved can be achieved using either cut wires [26] or electric-LC resonators [27]. In the same manner, permeability values can be tailored using split ring resonators [28]. The properties of the three types of unit cells are presented in Figs. 7(b)-7(d).

Fig. 7 (a) Design of 3D discrete taper composed of three different types of unit cells. (b) Effective permittivity of the cell composed of an air hole in a dielectric host medium. (c) Effective permittivity of an E-LC resonator embedded in a dielectric cube. (d) Effective permeability of a SRR resonator embedded in a dielectric cube.

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The resonant electric-LC resonators for permittivity engineering (ε > 4.4) and the split ring resonators for permeability engineering are simulated in a dielectric cube, such that all the resonant components are embedded in an object which can potentially be fabricated by a 3D printer. Size-varying air holes in a dielectric host medium of relative permittivity ε_h = 4.4 are constructed for the non-resonant cells. Next, binary mixing rules are assumed to calculate the effective material parameters by:

ε_{e} = ε_{a} f_{a} + ε_{h} f_{h}

where ε_a = 1 and f_a and f_h are the volume fraction of the air holes and the host material, respectively. Unit cells which are composed of three different types of structures are used to supply these particular values obtained by the parameter simplification procedures and presented in Fig. 6, where the period of the unit cell is chosen to be p = 5 mm. By adjusting the volume fraction of the air holes in the dielectric host medium and the structural dimensions of the printed metallic metamaterial resonators, the effective permittivity or permeability of the cell can then be modified, as shown in Figs. 7(b)-7(d).

Full-wave simulations using Ansys HFSS are performed on the 3D discrete taper to verify the field compressing functionality. The 3D lens model consists of 76 cubic cells containing the proposed resonant and non-resonant unit cells. A wave port is used as the source of the radiating emissions at 9.2 GHz, 10 GHz and 10.8 GHz as presented in Fig. 8. These frequencies were chosen because the resulting designs can potentially be fabricated by 3D polyjet printing and using printed circuit boards. As it can be observed, in the presence of the designed tapering device, the radiated electric field is compressed along the shape of the taper. These results further validate the 2D simulation predictions and once again show that the transformation-realized taper is able to reshape the field by virtue of a compression effect.

Fig. 8 Simulated electric near-field distribution. (a) Original space without the tapering device. 3D discrete tapering device in the xy plane at (b) 9.2 GHz, (c) 10 GHz, and (d) 10.8 GHz. The lower row shows the field distribution in the yz plane at the output waveguide.

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Calculations of the throughput for the four different transformed tapers, based on the 3D simulations, are analyzed respectively. The ratio of flux between the output and input ports presented in Table 3 demonstrates the field focusing functionality of the taper and the throughput enhancement compared to the corresponding configuration without the taper.

Table 3. Ratio of Throughput Between the Output and Input Ports

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5. Conclusion

In summary, a novel design procedure for a coordinate-transformation-based field tapering device has been presented and numerically validated. The taper is able to reshape the electric field distribution from a waveguide and compress and guide it into a narrower waveguide at microwave frequencies ranging from 9.2 GHz to 10.8 GHz. As shown by simulation results, such a device is able to narrow the radiated beam along the path defined by the proposed transformation technique. The method proposed in this paper can be easily implemented at microwave frequencies with potential airborne applications. Moreover, it is expected that the design methodology will also have applications to communication systems. Finally, the method is general and can be freely applied to arbitrary 2D and 3D shapes.

Funding

National Natural Science Foundation of China (No. 61601345); Fundamental Research Funds for the Central Universities (No. XJS16046, JB160109); Natural Science Foundation of Shaanxi Province of China (No. 2017JQ6025).

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Technique	Formulation	Annotation
Cosinusoidal	${\begin{cases} x' = x \\ y' = y \cos (k x) \\ z' = z \end{cases}$	$k = \arccos (b / a) / l$
Parabolic	${\begin{cases} x' = x \\ y' = y (p (^{x - l) 2} + q) \\ z' = z \end{cases}$	$p = \frac{a - b}{a l^{2}}$ , $q = \frac{b}{a}$
Logarithmic	${\begin{cases} x' = x \\ y' = \frac{2 y}{a} (\frac{a}{2} - \ln (s x + 1)) \\ z' = z \end{cases}$	$s = \frac{e^{\frac{a - b}{2}} - 1}{l}$
Reciprocal	${\begin{cases} x' = x \\ y' = y \frac{n}{x + n} \\ z' = z \end{cases}$	$n = \frac{b l}{a - b}$

Technique	S₁₁ (dB)	S₂₁ (dB)
Cosinusoidal	−24.48	−0.41
Parabolic	−28.65	−0.35
Logarithmic	−26.95	−0.37
Reciprocal	−29.53	−0.23

	*9.2 GHz*	*10 GHz*	*10.8 GHz*
Free Space	0.016%	0.537%	0.995%
Taper	8.739%	10.952%	14.736%

Technique	Formulation	Annotation
Cosinusoidal	${\begin{cases} x' = x \\ y' = y \cos (k x) \\ z' = z \end{cases}$	$k = \arccos (b / a) / l$
Parabolic	${\begin{cases} x' = x \\ y' = y (p (^{x - l) 2} + q) \\ z' = z \end{cases}$	$p = \frac{a - b}{a l^{2}}$ , $q = \frac{b}{a}$
Logarithmic	${\begin{cases} x' = x \\ y' = \frac{2 y}{a} (\frac{a}{2} - \ln (s x + 1)) \\ z' = z \end{cases}$	$s = \frac{e^{\frac{a - b}{2}} - 1}{l}$
Reciprocal	${\begin{cases} x' = x \\ y' = y \frac{n}{x + n} \\ z' = z \end{cases}$	$n = \frac{b l}{a - b}$

Technique	S₁₁ (dB)	S₂₁ (dB)
Cosinusoidal	−24.48	−0.41
Parabolic	−28.65	−0.35
Logarithmic	−26.95	−0.37
Reciprocal	−29.53	−0.23

Realizable design of field taper via coordinate transformation

Abstract

1. Introduction

2. Transformation formulations

3. Two-dimensional simulations and parameter reduction

4. Metamaterial design and three-dimensional modeling

5. Conclusion

Funding

References and links

Cited By

Figures (8)

Tables (3)

Equations (10)

Optics Express