Metamaterials with custom emissivity polarization in the near-infrared

Jeremy A. Bossard; Douglas H. Werner

doi:10.1364/OE.21.003872

1. Introduction

It has been recognized in recent years that metamaterial coatings offer great potential for tailoring the polarization and wavelength dependence of the infrared (IR) emission from an object [1,2]. Spectral emissivity control is useful for thermal management as the emissivity can be tailored to either heat or cool an object [3]. Previously, emissivity control has been achieved through material selection, applying paint or other coatings, or more recently by engineered metamaterial coatings [1–3]. Surfaces based on resonant, periodic metallic screens have recently been studied for IR emissivity polarization control and have been used to achieve linear polarization [1] as well as circular polarization with multi-layer structures [4]. Here, we investigate the use of thin electromagnetic band-gap (EBG) metamaterial coatings for achieving arbitrary emissivity polarization from a surface, without the need to consider multi-layer structures.

Emissivity is the fraction of blackbody radiation that is emitted from a surface and is dependent on wavelength as well as angle and polarization. It can be shown from Kirchoff’s law that the emissivity ε of a metamaterial is related to its absorptivity A according to

ε = A = 1 - R - T,

where R and T are the reflection and transmission magnitudes, respectively [1]. The Stokes-vector formalism is used in this work to describe the polarization of the emitted wave {ε = [ε_I, ε_Q, ε_U, ε_V]}, where ε_I is the emissivity intensity, ε_Q and ε_U are linear components, and ε_V is the circularly polarized component of the emissivity [5]. For the case of an opaque metamaterial, the polarized emissivity can be determined from the reflectivity of the surface [6]. As all of the metamaterials considered here are backed by a thick metallic ground layer, this method will be used to calculate the emissivity Stokes vector.

In this paper, a metamaterial structure based on an EBG is considered for use as a polarized emitter. This structure is composed of a periodic metal screen on top of a thin dielectric layer with a metal backing. EBG absorbers [7] have been explored in the infrared for controlling the spectral dependence of the absorption but have yet to be investigated for achieving custom arbitrary emission polarization. This paper first examines an EBG with an arbitrary, non-symmetric unit cell for achieving custom linear, circular, and elliptically polarized emission at the near-IR wavelength of 1.55 µm. Then, in order to better understand the geometry requirements for circularly polarized emissivity, a study is performed on a PEC backed bi-anisotropic slab in which it is determined that anisotropic chirality is necessary for achieving circularly polarized emissivity. Finally, a bi-anisotropic metamaterial with a two-fold rotationally symmetric screen is synthesized for circularly polarized emissivity. A robust genetic algorithm (GA) optimization technique [8] is employed to synthesize the EBG metamaterial coatings that control the emissivity polarization.

2. Stokes-vector representation of the emissivity polarization

The Stokes-vector formalism [5] provides a convenient mechanism for representing the polarization of a wave in a way that is easily understood and useful for optimization. The Stokes parameters [I, Q, U, V] represent the intensity of the wave as well as orthogonal linear and circular polarization components as illustrated in Fig. 1 . These parameters can be calculated from the complex electric field values in Cartesian or circular basis as follows:

I = {| E_{x} |}^{2} + {| E_{y} |}^{2}

Q = {| E_{x} |}^{2} - {| E_{y} |}^{2}

U = {| E_{a} |}^{2} - {| E_{b} |}^{2}

V = {| E_{l} |}^{2} - {| E_{r} |}^{2}

where E_x and E_y represent the electric field along the x and y axes, E_a and E_b represent the electric field along Cartesian coordinates _$\hat{a}$ and _$\hat{b}$ that are rotated 45° from _$\hat{x}$ and _$\hat{y}$, and E_l and E_r are the electric fields corresponding to the left- and right-handed circular bases _$\hat{l}$ and _$\hat{r}$. The Q, U, and V parameters are orthogonal to one other, while the total intensity I includes the contributions of both polarized and unpolarized portions of the wave. A wave that contains only a non-zero value for the I parameter is considered to be unpolarized, whereas a completely polarized wave will have an intensity I equivalent to the intensity contributed by all of the polarized components. Polarized waves that possess non-zero linear (Q or U) and circular (V) components are considered to be elliptically polarized. Figure 1(b) illustrates the polarization traces of elliptically and circularly polarized waves.

Fig. 1 Illustration of the wave polarization traces for the Stokes-vector components. (a) Trace illustrations for the linear components, Q and U, and the circular component V. (b) Wave traces illustrating right-handed elliptical (top) and left-handed circular (bottom) polarization with Stokes vectors of [I, Q, U, V] = [1, 0, −0.5, −0.5] and [I, Q, U, V] = [1, 0, 0, 1], respectively.

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Once the Stokes parameters for the emissivity {ε = [ε_I, ε_Q, ε_U, ε_V]} are calculated as described in [6], these components can be directly optimized to target linear, circular, or elliptical polarizations. The emissivity intensity ε_I can further be balanced with the polarized components to optimize the portion of the emitted wave that is polarized or unpolarized or to control the overall emission intensity. In the following sections, metamaterial examples are provided that explore all cases of linear, circular, and elliptical emission polarization.

3. EBG metamaterial synthesis with arbitrary emissivity polarization

The metamaterial structure considered here is based on the EBG surface originally introduced for radiofrequency (RF) applications. This EBG structure consists of a patterned Au layer, which is backed by a thin polyimide dielectric layer and a Au ground plane. The pattern for the top Au layer is periodic in two dimensions and defined by a pixellized unit cell that specifies which regions are filled in by Au or air. EBG structures act as resonant cavities for incident plane waves and exhibit enhanced fields within the structure at resonance. By adding loss into the dielectric or metallic layers, EBGs have been exploited for use as absorbers at both RF [9] and optical [7] wavelengths. Here, the intrinsic Au loss at 1.55µm is exploited for emissivity control.

The design parameters that need to be optimized include the pixellated screen geometry, the unit cell dimension, and the thicknesses of each Au and dielectric layer. A robust GA is employed to optimize these metamaterial design parameters for a targeted emissivity polarization at a 1.55 µm wavelength in the near-IR. The flowchart in Fig. 2 illustrates the operation of the GA, which is based on the principles of natural selection. The GA evolves optimum designs from a pool of randomized candidate designs. The design parameters, including the layer thicknesses, the unit cell dimension, and the screen geometry, are encoded into a binary string called a chromosome, where the unit cell dimension and thicknesses are 8-bit binary numbers. The screen geometry is pixellized and allowed to be arbitrary with no symmetry condition applied as shown in Fig. 3 . The screen geometry is encoded into the chromosome with each pixel being either “0” (contains air) or “1” (contains metal). In the first generation of the optimization, the initial population is filled with randomized chromosomes, so as to sample the design parameter space. Each member is evaluated by a fitness function, and then the population is ranked according to performance. Next, tournament selection is utilized to select pairs of parents to mate with single-point crossover to generate new offspring to fill the population for the next generation. Single-bit mutations are applied randomly to the new generation so that the algorithm continues exploring new areas of the parameter space. Elitism is also enforced, copying the previous best population member into the new generation, so that the global best fitness is always maintained or improved. Typically, the GA is run for a pre-determined number of generations, after which convergence is determined if the fitness has not improved over multiple generations. In all of the optimizations presented here, convergence was determined after the designs had not improved for more than 100 generations.

Fig. 2 Flowchart showing the genetic algorithm synthesis procedure used to evolve polarization-selective emitter designs.

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Fig. 3 Illustration of a pixellized EBG unit cell with no symmetry constraint. The chromosome encoding for the unit cell overlays the geometry.

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The performance of each population member is evaluated by first building the metamaterial geometry from the design parameters encoded in the chromosome. The metamaterial unit cell is simulated using a periodic finite element boundary integral (FE-BI) full-wave electromagnetic solver, which calculates the scattering from the metamaterial structure when illuminated by normally or obliquely incident plane waves [10]. Measured dispersive properties for Au, and polyimide at 1.55 µm are incorporated into the simulation [11,12]. Once the scattering parameters are obtained for both orthogonal TE and TM impinging waves at normal incidence, the emissivity Stokes vector is calculated according to the procedure in [6] and used to determine the cost according to

C o s t = {| ε_{t a r g e t} - ε |}^{2}

where ε_target and ε are the desired and predicted emissivities, respectively.

In order to explore the effectiveness of controlling the emissivity polarization, three EBG metamaterials were optimized targeting linear, circular, and elliptical polarization. The first design example exhibiting linear emissivity polarization has a specified target emissivity of ε_target = [0.2, 0.0, 0.2, 0.0]. Because ε_I = ε_U, the target emissivity would be completely polarized with the polarization wave trace lying along the diagonal in the x-y plane. The GA evolved a population of 32 over 556 generations to arrive at the design shown in Fig. 4(a) with a unit cell dimension of 965 nm on a side and thicknesses of 19 nm, 478 nm, and 52 nm for the top Au, polyimide, and bottom Au layers, respectively. The predicted emissivity for this design is ε = [0.210, 0.006, 0.184, −0.008] at 1.55 µm in the near-IR, indicating that the diagonal linear component of the emissivity is much larger than the other polarization components and that the emissivity is mostly polarized. Because the Au ground plane is several skin depths thick, the transmittance for this design and for subsequent designs in this paper is on the order of 0.1% or less. Hence, the designs are considered to be opaque. As an alternative to optimizing the ground plane thickness, this design parameter could also be fixed at a suitably large value. The reflectance and emittance for this design are 79% and 21%, respectively, at the optimized wavelength. A trace of the targeted and optimized polarization for this design is shown in Fig. 5(a) demonstrating visually that the optimized polarization is close to the design goal.

Fig. 4 Synthesized EBG structures used to achieve custom polarized emissivity targeting (a) linear, (b) circular, and (c) elliptical polarization.

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Fig. 5 Polarization traces for the EBG emitter designs shown in Fig. 4. The targeted polarization trace is shown by a dashed curve, and the predicted metamaterial emissivity is shown by the solid trace. These curves illustrate (a) linear, (b) circular, and (c) elliptical polarization.

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The second design targeted a purely circular polarized emissivity, which is particularly challenging because the linearly polarized components must be eliminated. Since one of the circularly polarized wave-fields is eliminated from the emitted wave, a fully-polarized emitter can have a maximum intensity of 0.5, yielding an emissivity of ε = [0.5, 0.0, 0.0, ± 0.5], where ‘ + ’ and ‘–’ indicate left- and right-handed circular polarization, respectively. Hence, the target polarization for this design is chosen to be ε_target = [0.5, 0.0, 0.0, 0.5], representing the maximum possible left-handed circular polarization for the emissivity. A population of 32 was evolved by the GA over 600 generations to synthesize the design shown in Fig. 4(b), which has a unit cell size of 858 nm and layer thicknesses of 32 nm, 180 nm, and 51 nm for the top Au, polyimide, and bottom Au layers, respectively. This design has a predicted emissivity of ε = [0.563, 0.050, −0.014, 0.418], indicating that a circular polarization was achieved that is 83.6% of the maximum possible value. The reflectance and emittance for this design are 44% and 56%, respectively, at 1.55 µm. The polarization trace shown in Fig. 5(b) illustrates that the emitted wave is mostly circularly polarized with a small amount of linear polarization that slightly elongates the circle in the x-direction.

The final design targeted the most general case of an arbitrary elliptical polarization. The targeted polarization ε_target = [0.28, 0.0, 0.2, 0.2] includes a combination of the diagonal, linear component and the circular component, while the intensity component forces the emissivity to be completely polarized. The GA evolved a population of 32 over 600 generations to arrive at the geometry shown in Fig. 4(c), which has a unit cell dimension of 524 nm and layer thicknesses of 30 nm, 243 nm, and 58 nm for the Au, polyimide, and Au layers, respectively. The final emissivity for this design is predicted to be ε = [0.324, 0.016, 0.191, 0.153], indicating that the U and V components dominate the emissivity polarization. The total reflectance and emittance for this design are 68% and 32%, respectively, at the optimized wavelength. The trace in Fig. 5(c) illustrates that the achieved polarization is elliptical but slightly smaller than the targeted ellipse.

4. Theoretical bi-anisotropic slab with circular emissivity polarization

While the three EBG designs in the previous section demonstrate the effectiveness of our approach at achieving metamaterials with arbitrary emissivity polarization, an important question remains as to what metamaterial requirements are necessary for achieving circular polarization. In order to better understand what metamaterial structures can produce circularly polarized emissivity, a homogeneous slab of theoretical material backed by a PEC ground was studied.

The GA was coupled with the bi-anisotropic FEBI solver in order to optimize the _{$\bar{\bar{ε}}$}, _{$\bar{\bar{μ}}$}, and _{$\bar{\bar{κ}}$} tensors of a lossy, bi-anisotropic slab for circular polarization. During this study, it was determined that anisotropic material paramaters (_{$\bar{\bar{κ}}$} = 0) could not produce emissivity with a non-zero circular polarization component. Furthermore, a bi-isotropic slab on PEC always produces unpolarized emissivity in the normal direction. However, bi-anisotropic slabs with anisotropic chirality in the plane of the slab (κ_x ≠ κ_y) can produce circularly polarized emission. As an example, the GA was used to synthesize the bi-anisotropic slab shown in Fig. 6 for circularly polarized emissivity. For this example, the slab thickness, the real and imaginary parts of isotropic ε and µ, and anisotropic κ_x and κ_y values were encoded into the chromosome, and a target emissivity of ε_target = [0.5, 0.0, 0.0, 0.5] was specified. After evolving a population of 32 for 600 generations, the GA produced a 1.99 µm thick bi-anisotropic slab on PEC with isotropic permittivity ε_r = 0.467–0.013j, isotropic permeability µ_r = 0.482–0.015j, and a chirality tensor given by

Fig. 6 Theoretical homogeneous bi-anisotropic slab backed by a PEC ground plane. This slab was optimized by a GA to achieve circularly polarized emissivity.

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\bar{\bar{κ}} = [\begin{matrix} - 0.357 & 0.0 & 0.0 \\ 0.0 & 0.992 & 0.0 \\ 0.0 & 0.0 & 0.0 \end{matrix}]

The polarized emissivity for this slab is calculated to be ε = [0.507, −0.003, −0.010, 0.466] at normal incidence, which represents 93.2% of the maximum possible circularly polarized component. The reflectance and emittance of this slab are predicted to be 49% and 51%, respectively, at 1.55 µm. The emissivity polarization components are plotted in Fig. 7(a) , indicating the linear components vary sinusoidally with azimuth angle as the incident wave orientation changes, but ε_V is azimuth independent. Figure 7(b) illustrates the predicted emissivity polarization trace, which is almost completely circularly polarized. Based on this result, we can deduce from the constitutive parameters for this slab that a metamaterial should have matched impedance with some intrinsic loss and chirality that is different along the two orthogonal in-plane axes. An analytical solution to this problem is also considered in Appendix A, confirming the fact that bi-anisotropy is required for a non-zero circular emissivity component at normal incidence.

Fig. 7 Near-IR emissivity polarization for a PEC-backed bi-anisotropic slab. (a) Azimuth dependence of the emissivity Stokes vector and (b) polarization trace showing near complete circular polarization.

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5. Bi-anisotropic EBG emitters with circular polarization

In order to simplify the previous general EBG screen geometry shown in Fig. 3 while maintaining the capability for the structure to exhibit anisotropic chirality, two-fold 180° rotational symmetry is applied to the unit cell as shown in Fig. 8 . As illustrated in Fig. 8(b) the pixel values for the top half of the screen are encoded into the chromosome for optimization, while the bottom half of the unit cell is filled out by rotating the top half by 180°. Furthermore, the period of the unit cell in the x- and y-directions is allowed to differ during the optimization.

Fig. 8 Simplified EBG emitter screen geometry for circular polarization. (a) Two-fold rotational symmetry is enforced on the screen geometry. (b) The top half of the unit cell is encoded in the chromosome, while the bottom half is generated by rotating the top half by 180°.

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Using a target emissivity of ε_target = [0.5, 0.0, 0.0, 0.5], which represents the maximum possible circularly polarized emissivity, the GA evolved a population of 32 over 600 generations to produce the design shown in Fig. 9 . This design has unit cell dimensions of 1.287 μm and 0.535 μm in the x and y directions, respectively, and the top Au, polyimide, and bottom Au layer thicknesses are 28 nm, 200 nm, and 55 nm, respectively. Interestingly, the 180° rotationally symmetric unit cell geometry seen in Fig. 9 contains a two-arm spiral feature that is expected to give rise to anisotropic chirality. The polarized emissivity for this metamaterial surface is calculated to be ε = [0.532, 0.002, 0.017, 0.465] at 1.55 µm, which represents 93.0% of the maximum possible circularly polarized component. The total reflectance and emittance at the optimized wavelength are predicted to be 47% and 53%, respectively. Figure 10(a) shows that the circular polarized emissivity is azimuth independent. The polarization trace of the emitted wave in Fig. 10(b) is nearly circular and close to the trace for an ideal, fully circularly polarized emitter.

Fig. 9 GA optimized Au and polyimide EBG emitter with two-fold rotationally symmetric screen geometry that is optimized to have high circularly polarized emission. (a) 3D unit cell and (b) 3x3 tiling.

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Fig. 10 Near-IR emissivity polarization at λ = 1.55µm for the EBG emitter design in Fig. 9. (a) Azimuth dependence of the emissivity Stokes vector and (b) polarization trace showing near complete circular polarization.

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In order to better understand the physics behind the emissivity/absorption mechanism, the relative fields were plotted in Fig. 11 for the design in Fig. 9 under illumination by a normally incident TE wave at λ = 1.55 µm. These plots show that the field in the structure is concentrated around the patterned top Au layer. This enhanced field results in high absorption in the top Au layer, which is primarily responsible for the high emissivity at resonance. At λ = 1.55 µm the polyimide dielectric layer is lossless and does not contribute directly to the absorption. However, the dielectric layer forms a resonant cavity with the top and bottom Au layers and therefore influences the resonance wavelength. By contrast, the theoretical bi-anisotropic slab shown in Fig. 6 possesses dielectric and magnetic loss, so the absorption is distributed throughout the slab. The optimized EBG designs in Fig. 4 also have concentrated fields in the top Au layer, which is primarily responsible for absorption. Each of these EBG designs exhibits the strongest field enhancement and absorption when illuminated by a polarized wave corresponding to the respective polarization trace shown in Fig. 5.

Fig. 11 Field plots for the EBG emitter design in Fig. 9 when illuminated by a TE wave at λ = 1.55µm. (a) x-y plane cut through the center of the top Au layer. (b) x-z plane cut through the center of the unit cell.

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In practical applications, it is important to know how the performance of a device is affected by a change in environment or fabrication inaccuracies. As a case study, the dependence of the emissivity Stokes vector on incidence angle is shown in Fig. 12 for the design in Fig. 9, and the wavelength dependence for this design is plotted in Fig. 13 . Because all of the designs were optimized for normal incidence and at a single wavelength, the performance away from those operation points cannot be guaranteed. However, from Fig. 12 it can be noted that in the φ = 90° plane the emissivity does not change much within |θ|≤10°, whereas the emissivity changes more rapidly in the φ = 0° plane with a 10% reduction in ɛ_V at θ = 5°. The unit cell dimension in the φ = 0° plane is much larger, which could contribute to the smaller field of view. The plot of wavelength dependence in Fig. 13 shows that the circularly polarized emissivity component ɛ_V peaks close to 1.55 µm and maintains 90% of its peak value over a line width of approximately 30 nm. As a point of reference, for optical infrared telecommunications at 1.55 µm, the conventional C band covers a 35 nm line width from 1530 nm to 1565 nm.

Fig. 12 Near-IR emissivity polarization at λ = 1.55µm for the EBG emitter design in Fig. 9 as a function of incidence angle for (a) φ = 0° and (b) φ = 90°.

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Fig. 13 Near-IR emissivity polarization for the EBG emitter design in Fig. 9 as a function of wavelength.

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In order to evaluate the sensitivity of the designs in Fig. 4 and Fig. 9 to fabrication tolerances, each design was simulated with layer thickness and unit cell dimension variations up to ± 10%. The emissivity Stokes vector elements for each design were then evaluated to see what % change in the design parameter values caused a variation of ± 0.05 or ± 0.10 in one or more element. These results were then tabulated to form Table 1 . A wide range in fabrication tolerance can be observed across the designs, but all of the designs share a low sensitivity to ground plane thickness variation. Future optimizations would benefit from fixing the ground plane thickness to a suitable value in order to reduce the load on the optimizer.

Table 1. Fabrication tolerances for each design showing what % change in a layer thickness or unit cell dimension will cause an element in the ɛ Stokes vector to change by ± 0.05 or ± 0.10

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6. Conclusion

Metamaterial coatings based on EBG structures were explored for emissivity polarization control at the near-IR wavelength of 1.55 µm. The Stokes-vector formalism was introduced to aid in the analysis and optimization of the emissivity polarization. A general, pixellized geometry without symmetry constraints was optimized by a robust GA to demonstrate custom linear, circular, and elliptical emissivity polarization. In order to determine a symmetry condition that would allow EBG emitters to achieve circular and elliptical polarization, a study was conducted on a homogeneous bi-anisotropic slab backed by a PEC ground, which indicated that anisotropic chirality was required. Hence, a two-fold rotational symmetry was applied to the EBG unit cell geometry. This simplified EBG structure was optimized to produce a highly circularly polarized emissivity. These examples demonstrate that EBG metamaterial coatings can be effectively synthesized to achieve custom polarization-selective emission.

7. Appendix A

Here the analytical formulation in [13] is utilized to calculate the normal incidence reflection matrix for a general bianisotropic slab with thickness d placed on a PEC ground plane, illustrating the need for anisotropic chirality to obtain cross-coupled reflection. The permittivity and permeability tensors are assumed to be anisotropic with only diagonal non-zero values, and the chirality parameter is anisotropic with non-zero values in x and y as follows:

\bar{\bar{ε}} = [\begin{matrix} ε_{x} & 0 & 0 \\ 0 & ε_{y} & 0 \\ 0 & 0 & ε_{z} \end{matrix}], \bar{\bar{μ}} = [\begin{matrix} μ_{x} & 0 & 0 \\ 0 & μ_{y} & 0 \\ 0 & 0 & μ_{z} \end{matrix}], and \bar{\bar{κ}} = [\begin{matrix} κ_{x} & 0 & 0 \\ 0 & κ_{y} & 0 \\ 0 & 0 & 0 \end{matrix}] .

Under these assumptions, the matrix _{$\bar{\bar{P}}$} from Eq. 19 in [13] can be written as follows:

\bar{\bar{P}} = [\begin{matrix} 0 & k_{0} κ_{y} & 0 & - j ς_{0} k_{0} μ_{y} \\ - k_{0} κ_{x} & 0 & j ς_{0} k_{0} μ_{x} & 0 \\ 0 & j ς_{0}^{- 1} k_{0} ε_{y} & 0 & k_{0} κ_{y} \\ - j ς_{0}^{- 1} k_{0} ε_{x} & 0 & - k_{0} κ_{x} & 0 \end{matrix}],

where_{$k_{0} = ω \sqrt{μ_{0} ε_{0}}$} and _{$ς_{0} = \sqrt{μ_{0} / ε_{0}}$}. The 4x4 transition matrix _{$\bar{\bar{T}}$}, which is a tool used to propagate the field in the slab, is then obtained from Eq. 22 in [13] as follows:

\bar{\bar{T}} (z) = \exp (z \bar{\bar{P}}) = [\begin{matrix} \bar{\bar{T_{1}}} & \bar{\bar{T_{2}}} \\ \bar{\bar{T_{3}}} & \bar{\bar{T_{4}}} \end{matrix}],

where z is the depth in the slab, exp is the matrix exponent operation, and _{$\bar{\bar{T_{1}}}$},_{${\bar{\bar{T}}}_{2}$},_{${\bar{\bar{T}}}_{3}$}, and_{${\bar{\bar{T}}}_{4}$} are 2x2 sub-matrices of _{$\bar{\bar{T}}$}. Following the procedure outlined in [13], the eigenvalues of _{$\bar{\bar{P}}$} are first calculated and utilized to determine the scalar coefficients C₀(z), C₁(z), C₂(z), C₃(z) which are then employed to evaluate the transition matrix using only an identity matrix _{${\bar{\bar{I}}}_{4}$} and powers of _{$\bar{\bar{P}}$}.

\bar{\bar{T}} (z) = C_{0} (z) \bar{\bar{I_{4}}} + C_{1} (z) \bar{\bar{P}} + C_{2} (z) {\bar{\bar{P}}}^{2} + C_{3} (z) {\bar{\bar{P}}}^{3}

The 4x4 reflection matrix _{${\bar{\bar{R}}}_{m}$} for a grounded bianisotropic slab is derived in Eq. (52) of [13] to be

\bar{\bar{R_{m}}} = {[\bar{\bar{T_{4}}} (d) {\bar{\bar{T_{2}}}}^{- 1} (d) \bar{\bar{W}} - \bar{\bar{Q}}]}^{- 1} [\bar{\bar{T_{4}}} (d) {\bar{\bar{T_{2}}}}^{- 1} (d) \bar{\bar{U}} - \bar{\bar{V}}],

where for a normally incident wave _{$\bar{\bar{W}}$}, _{$\bar{\bar{Q}}$}, _{$\bar{\bar{U}}$}, and _{$\bar{\bar{V}}$} have the values

\bar{\bar{W}} = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}], \bar{\bar{Q}} = \frac{1}{ς_{0}} [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}], \bar{\bar{U}} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}], and \bar{\bar{V}} = \frac{1}{ς_{0}} [\begin{matrix} 0 & - 1 \\ 1 & 0 \end{matrix}] .

Only two of the sub-matrices of _{$\bar{\bar{T}}$} appear in Eq. (12) and are found to be:

{\bar{\bar{T_{2}}}}^{- 1} (d) = \bar{\bar{T_{2}}} (- d) = [\begin{matrix} C_{2} (- d) j k_{0}^{2} ς_{0} (κ_{x} μ_{y} + κ_{y} μ_{x}), \\ C_{1} (- d) j k_{0} ς_{0} μ_{x} - C_{3} (- d) j k_{0}^{3} ς_{0} (κ_{x}^{2} μ_{y} + ε_{y} μ_{x}^{2} + 2 κ_{x} κ_{y} μ_{x}), \end{matrix}

\begin{matrix} - C_{1} (- d) j k_{0} ς_{0} μ_{y} + C_{3} (- d) j k_{0}^{3} ς_{0} (κ_{y}^{2} μ_{x} + ε_{x} μ_{y}^{2} + 2 κ_{x} κ_{y} μ_{y}) \\ C_{2} (- d) j k_{0}^{2} ς_{0} (κ_{x} μ_{y} + κ_{y} μ_{x}) \end{matrix}]

\bar{\bar{T_{4}}} (d) = [\begin{matrix} C_{0} (d) - C_{2} (d) k_{0}^{2} (κ_{x} κ_{y} + ε_{y} μ_{x}), \\ - C_{1} (d) k_{0} κ_{x} + C_{3} (d) k_{0}^{3} (κ_{x}^{2} κ_{y} + κ_{x} ε_{x} μ_{y} + κ_{x} ε_{y} μ_{x} + κ_{y} ε_{x} μ_{x}), \end{matrix}

\begin{matrix} C_{1} (d) k_{0} κ_{y} - C_{3} (d) k_{0}^{3} (κ_{x} κ_{y}^{2} + κ_{x} ε_{y} μ_{y} + κ_{y} ε_{x} μ_{y} + κ_{y} ε_{y} μ_{x}) \\ C_{0} (d) - C_{2} (d) k_{0}^{2} (κ_{x} κ_{y} + ε_{y} μ_{x}) \end{matrix}]

These equations have been used to reproduce the results shown in Fig. 6 and Fig. 7 for the PEC backed bi-anisotropic slab. For this example, the cross-polarized reflection matrix terms are non-zero, allowing for circularly polarized reflection and emission values.

It can be observed from Eqs. (14) and (15) that the diagonal elements in _{${\bar{\bar{T}}}_{2}$}^-1 and _{${\bar{\bar{T}}}_{4}$} are equal for the case of a bi-anisotropic slab. Additionally, for a bi-isotropic slab, the off-diagonal terms satisfy the following relations:

{\bar{\bar{T_{2_{12}}}}}^{- 1} (d) = - {\bar{\bar{T_{2_{21}}}}}^{- 1} (d) and \bar{\bar{T_{4_{12}}}} (d) = - \bar{\bar{T_{4_{21}}}} (d)

Hence, the following substitutions will be performed to simplify the reflection matrix for a bi-isotropic slab:

{\bar{\bar{T_{2}}}}^{- 1} (d) = [\begin{matrix} f & g \\ - g & f \end{matrix}] and \bar{\bar{T_{4}}} (d) = [\begin{matrix} h & l \\ - l & h \end{matrix}]

Using these substitutions, the reflection matrix can be written as

\bar{\bar{R_{m}}} = \frac{- 1}{{(f h - g l)}^{2} + (f l + g h + ς_{0}^{- 1})} [\begin{matrix} - f^{2} h^{2} - g^{2} l^{2} - f^{2} l^{2} - g^{2} h^{2} + ς_{0}^{- 2} & 2 ς_{0}^{- 1} (f h - g l) \\ 2 ς_{0}^{- 1} (g l - f h) & f^{2} h^{2} + g^{2} l^{2} + f^{2} l^{2} + g^{2} h^{2} - ς_{0}^{- 2} \end{matrix}]

Combining this result with the fact that

{\bar{\bar{T_{2_{11}}}}}^{- 1} (d) \bar{\bar{T_{4_{11}}}} (d) = {\bar{\bar{T_{2_{12}}}}}^{- 1} (d) \bar{\bar{T_{4_{12}}}} (d)

leads to the conclusion that for a bi-isotropic slab the cross polarization terms are eliminated:

\bar{\bar{R_{m_{12}}}} = \bar{\bar{R_{m_{21}}}} = 0

Because cross-polarized reflection is required to achieve a non-zero circularly polarized emissivity component, we conclude that a slab backed by PEC must be bi-anisotropic as a minimum requirement for producing circularly polarized emission at normal incidence.

Acknowledgments

This work was supported by the Office of Naval Research under contract N00014-10-G-0259.

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	± 0.05 change in _$ε$				± 0.10 change in _$ε$
Design	Screen	Polyimide	Ground	Unit Cell	Screen	Polyimide	Ground	Unit Cell
Figure 4(a)	3%	2%	>10%	<1%	8%	3.5%	>10%	1%
Figure 4(b)	4.5%	5.5%	>10%	1%	7.5%	>10%	>10%	2%
Figure 4(c)	>10%	9.5%	>10%	5.5%	>10%	>10%	>10%	>10%
Figure 9	3.5%	5.5%	>10%	1.5%	3.5%	>10%	>10%	2%

Metamaterials with custom emissivity polarization in the near-infrared

Abstract

1. Introduction

2. Stokes-vector representation of the emissivity polarization

3. EBG metamaterial synthesis with arbitrary emissivity polarization

4. Theoretical bi-anisotropic slab with circular emissivity polarization

5. Bi-anisotropic EBG emitters with circular polarization

6. Conclusion

7. Appendix A

Acknowledgments

References and links

Cited By

Figures (13)

Tables (1)

Equations (20)

Optics Express