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Metamaterials have been previously studied for their ability to tailor the dispersive IR emissivity of a surface. Here, we investigate two metamaterial structures based on an electromagnetic band-gap surface and a dielectric resonator array for use as near-IR emitters with custom angle selectivity. A genetic algorithm is successfully employed to optimize the metamaterial structures to have minimum emissivity in the normal direction and high emissivity at custom off-normal angles specified by the designer. Two symmetry conditions are utilized to achieve emissivity patterns that are azimuthally stable or distinct in the two orthogonal plane cuts.
©2013 Optical Society of America
1. Introduction
In recent years, it has been recognized 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 crucial for applications requiring thermal management, because 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 by engineered metamaterial coatings [1–3]. However, not much attention has been given to the important problem of controlling the angular distribution of the emissivity.
Emissivity, which is both dispersive as well as angle and polarization dependent, is the fraction of blackbody radiation that is emitted from a surface. It can be shown from Kirchoff’s law that the emissivity ε is related to the absorptivity A of a metamaterial, as given by
where R and T are the reflection and transmission magnitudes, respectively [1]. Furthermore, the radiance for a uniform, planar surface is maximum in the normal direction as described by Lambert’s cosine law. Here, we investigate the use of metamaterial coatings for redirecting the peak emissivity from the direction of the surface normal to other, custom angular distributions at the near-IR wavelength of 1.55 µm.In this paper, two structures are considered for use as directional emitters. The first structure is based upon an electromagnetic band-gap (EBG) metamaterial and is composed of a periodic metal screen on top of a thin dielectric layer with a metal backing. EBG absorbers [4] and emitters [1] have been widely explored in the infrared for controlling the spectral dependence of the absorption or emissivity but have yet to be investigated for achieving custom angular absorption or emission control. The second metamaterial structure is based on dielectric resonator building blocks and is similar to all-dielectric frequency selective surfaces (DFSS) and photonic crystal slabs, which have been studied as spatial filters at infrared wavelengths [5]. This structure consists of a pixellated unit cell of dielectric and air that is periodic in two dimensions and backed by a metal ground. This paper also examines two symmetry conditions, 4-fold and 8-fold, which can be enforced on the unit cell geometry to achieve distinct or identical emissivity patterns in the two orthogonal principal planes (φ = 0° and φ = 90°). A robust genetic algorithm (GA) optimization technique [6] is employed to synthesize EBG and dielectric resonator metamaterial coatings that redirect the peak emissivity away from normal incidence to custom off-normal angles.
2. Metamaterial structure
The first 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 as shown in Fig. 1(a) . The pattern for the top Au layer is periodic in two dimensions and defined by a unit cell consisting of 11x11 pixels that specify 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. A finer unit cell pixelization would provide more design flexibility for optimization, but it also increases the number of parameters to optimize and reduces feature size in the design, making fabrication more difficult. By adding loss into the dielectric or metallic layers, EBGs have been exploited for use as absorbers at both RF [7] and optical [4] wavelengths. Here, the intrinsic Au loss at 1.55µm is exploited for emissivity control.
Fig. 1 Metasurface structures used to achieve custom angular emissivity. (a) Electromagnetic bandgap (EBG) structure with a patterned lossy metal screen backed with dielectric and metallic ground plane layers and (b) a dielectric resonator array backed by a metallic ground plane.
The second metamaterial structure consists of a periodic array of dielectric resonators placed on top of a Au ground plane as shown in Fig. 1(b). This type of periodic dielectric resonator array has been previously exploited without a ground plane to make infrared filters and mirrors [5]. Periodic dielectric gratings with optimized nanoscale topology have also been explored for controlling the reflection and transmission at specific angles for optical wavelengths [8]. At resonance, these dielectric structures can support high fields which translate into large absorption losses if there is intrinsic loss in any of the constituent materials. In this work, the Au ground plane is included to both prevent transmission through the metamaterial and to provide intrinsic loss for emission. a-Ge is chosen as the patterned dielectric material because of its high permittivity at 1.55 µm in the near-IR.
Two symmetry conditions are considered for the screen geometries in both the EBG and dielectric resonator type of metamaterials as shown in Fig. 2 . These symmetry conditions aid in controlling the emissivity patterns during optimization. For the case of the 8-fold mirrored screen geometry, the emissivity pattern is expected to vary in azimuth only over the range φ = [0°,45°]. Thus, this symmetry condition will be used for designs in which azimuthal stability is desired. The second symmetry condition illustrated in Fig. 2(b) is 4-fold mirrored and results in emissivity patterns that vary over the range φ = [0°,90°]. Thus, this 4-fold symmetry condition is chosen when an emissivity pattern is desired that changes for the two orthogonal plane cuts. Both of these symmetry conditions are implemented in the GA optimization procedure as described in detail in the next section.
Fig. 2 Symmetry conditions utilized for azimuthal emissivity control. (a) 8-fold symmetry is used for azimuth stability and (b) 4-fold symmetry is used for achieving distinct responses in the two orthogonal plane cuts.
3. Design methodology
The design parameters that need to be optimized for both metamaterial types 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 given angular distribution of the emissivity at a 1.55 µm wavelength in the near-IR. The flowchart in Fig. 3 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 bottom Au layer thickness is fixed at 60 nm for all designs, so that the wave is attenuated completely before passing through the metamaterial structure. The screen geometry, on the other hand, is divided up into either eight mirrored triangular folds or four mirrored quadrants depending on the problem at hand as illustrated in Fig. 2. A single triangle or quadrant is encoded into the chromosome with each pixel being either “0” (air) or “1” (filled in metal or dielectric). 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. Tournament selection is then 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.
Fig. 3 Flowchart showing the genetic algorithm synthesis procedure used to evolve angle-selective emitter designs.
The performance of each population member is evaluated by first building the metamaterial geometry from the design parameters encoded in the chromosome. Fabrication constraints are enforced for both the 4-fold or 8-fold symmetric pixellized screen geometries, which eliminate diagonal connections between pixels that are difficult to accurately replicate in experiment. The unit cell dimension is also constrained to be less than a half-wavelength and large enough that the pixel size is larger than the 30 nm resolution for e-beam lithography. The metamaterial unit cell is then 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 [9]. Because the unit cell size is constrained to be smaller than a half-wavelength, the higher order diffraction modes are suppressed, and only the specular diffraction needs to be calculated in order to determine the emissivity for all oblique incidence angles. Measured dispersive properties for Au, polyimide, and/or a-Ge at 1.55 µm are incorporated into the simulation [10,11]. Once the scattering parameters are obtained, the emissivity is calculated from Eq. (1) and used to determine the cost according to
where (θ,φ)l are angles where low emissivity is desired, (θ,φ)h are angles where high emissivity is desired, and Cl is a weighting coefficient used for balancing the contributions from the low and high emissivity angles in the cost function.4. Numerical results
In order to demonstrate the effectiveness of the proposed design approach for synthesizing angle-selective metamaterial emitters, two examples will be presented for each of the EBG and dielectric resonator structures with 8-fold and 4-fold symmetry. In all cases, the optimization goal is to re-direct the peak emissivity away from the direction normal to the surface into either an azimuthally stable or distinct pattern.
4.1 Angle-selective emitters with EBG metamaterial structure
The first design example based on an EBG structure with 8-fold symmetry targeted a conical emissivity pattern. The angles specified in the cost function targeting low and high emissivity were (θ,φ)l = {(0°,0°)} and (θ,φ)h = {(40°,0°), (40°,15°), (40°,30°), (40°,45°)}, respectively. Only a single normal incidence angle is required as the total emissivity (including TE and TM polarizations) is equivalent for all azimuth angles, but a weighting coefficient Cl = 4 is specified to balance the contributions from the four oblique angles. Additional oblique angles could be specified (at an increased computational burden) to further smooth or customize the emissivity pattern. The GA evolved a population of 24 over 105 generations for a total of 2520 cost evaluations to arrive at the design shown in Fig. 4 with a unit cell dimension of 730 nm on a side and thicknesses of 15 nm, 391 nm, and 60 nm for the top Au, polyimide, and bottom Au layers, respectively. This design has a suppressed total emissivity of less than 0.083 in the normal direction and a high total emissivity greater than 0.923 at the targeted off-normal angles as shown by the patterns in Fig. 5(a) . The 3D emissivity pattern in Figs. 5(b) and 5(c) show that this EBG metamaterial effectively redirects the peak emissivity away from the normal direction into a conical pattern. Because four oblique angles were included in the cost evaluation, the pattern contains minimal azimuth variation over φ = 0° to φ = 45°. The wavelength variation at each of the optimized angles is shown in Fig. 6 . While the metamaterial performance over wavelength was not included in the optimization, this EBG metamaterial maintains oblique emissivity above 0.80 and normal incidence emissivity under 0.10 at the design angles over an approximately 30 nm line width. As a 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. Ansoft HFSS was also used to validate the response of this metamaterial as shown in Fig. 6(b). The emissivity profile predicted by HFSS corresponds well with FE-BI with a slight shift in peak performance from 1.55 µm to 1.575 µm, confirming the accuracy of the FE-BI code for EBG metamaterials.
Fig. 4 GA optimized Au and polyimide EBG emitter with azimuthal stability. (a) 3D unit cell and (b) 3x3 tiling.
Fig. 5 Near-IR emissivity patterns at λ = 1.55µm for the EBG emitter design in Fig. 4 showing low emissivity in the normal direction and high off-normal emissivity at θ = 40°. (a) Total emissivity for angle cuts at φ = 0°, 15°, 30°, and 45° are shown as well as 3D patterns for (b) TE and (c) TM waves.
Fig. 6 Wavelength dependence of the emissivity for the EBG emitter design in Fig. 4 at each of the optimized angles predicted by (a) FE-BI and (b) Ansoft HFSS.
The second design example is based on an EBG structure with 4-fold symmetry and targets an emissivity pattern that is different in the two plane cuts. The specified angles for low and high emissivity were (θ,φ)l = {(0°,0°), (40°,90°)} and (θ,φ)h = {(40°,0°)}, respectively, and a weighting coefficient of Cl = 2 was chosen. Only one normal incidence angle is required in the optimization because (0°,0°) and (0°,90°) have swapped TE and TM responses. A population of 24 was optimized by the GA over 253 generations for a total of 5072 cost evaluations to produce the final design geometry shown in Fig. 7 with a unit cell dimension of 628 nm on a side and thicknesses of 39 nm, 188 nm, and 60 nm for the top Au, polyimide, and bottom Au layers, respectively. This metamaterial design suppresses the total emissivity below 0.084 over all (θ,φ)l angles and enhances the total emissivity to over 0.951 at the targeted oblique angle as shown by the plots in Fig. 8(a) . The 3D emissivity patterns in Figs. 8(b) and 8(c) resemble a “butterfly wing”, where the emissivity is suppressed in the φ = 90° plane and enhanced away from the surface normal in the φ = 0° plane.
Fig. 7 GA optimized Au and polyimide EBG emitter with 4-fold symmetry. (a) 3D unit cell and (b) 3x3 tiling.
Fig. 8 Near-IR emissivity patterns at λ = 1.55µm for the EBG emitter design in Fig. 7 showing low emissivity in the normal direction and high off-normal emissivity at θ = 40° in the φ = 0° plane cut. (a) Angle cuts at φ = 0°,90° are shown as well as 3D patterns for (b) TE and (c) TM waves.
4.2 Angle-selective emitters with dielectric resonator metamaterial structure
The first dielectric resonator metamaterial design example possesses 8-fold symmetry and targets a conical emissivity pattern. The target angles for low and high emissivity were specified to be (θ,φ)l = {(0°,0°)} and (θ,φ)h = {(40°,0°), (40°,15°), (40°,30°), (40°,45°)}, and a weighting coefficient of Cl = 4 was chosen. Evolving a population of 24 members over 68 generations for a total of 1632 cost evaluations, the GA converged to the structure shown in Fig. 9 with a unit cell size of 758 nm on a side and a-Ge and Au layer thicknesses of 164 nm and 60 nm, respectively. The emissivity plane cuts in Fig. 10(a) show that this metamaterial design suppresses the total emissivity in the direction of the surface normal to under 0.037 and enhances the targeted off-normal emissivity to over 0.882. The 3D emissivity patterns in Figs. 10(b) and 10(c) show a conical shape with a minimum in the direction of the surface normal and large off-normal emissivity for all azimuth angles. The wavelength variation at each of the optimized angles is presented in Fig. 11 . While the emissivity in the normal direction is suppressed effectively over a large wavelength range, the oblique emissivity varies more quickly than for the 8-fold EBG design with the oblique emissivity maintained above 0.80 over an approximately 10 nm line width. Ansoft HFSS was used to validate the response of this metamaterial as shown in Fig. 11(b). The emissivity profile predicted by HFSS corresponds well with FE-BI with a slight shift in peak performance from 1.55 µm to 1.565 µm, confirming the accuracy of the FE-BI code for dielectric resonator metamaterials.
Fig. 9 GA optimized a-Ge and Au dielectric resonator emitter with azimuthal stability. (a) 3D unit cell and (b) 3x3 tiling.
Fig. 10 Near-IR emissivity patterns at λ = 1.55µm for dielectric resonator design in Fig. 9 showing low emissivity in the normal direction and high off-normal emissivity at θ = 40°. (a) Total emissivity patterns for angle cuts at φ = 0°, 15°, 30°, and 45° are shown as well as 3D patterns for (b) TE and (c) TM waves.
Fig. 11 Wavelength dependence of the emissivity for the dielectric resonator emitter design in Fig. 9 at each of the optimized angles predicted by (a) FE-BI and (b) Ansoft HFSS.
The second dielectric resonator metamaterial example employed 4-fold symmetry to achieve distinct profiles in the orthogonal plane cuts. For this design the targeted low and high emissivity angles were specified to be (θ,φ)l = {(0°,0°), (40°,90°)} and (θ,φ)h = {(40°,0°)}, respectively, and a weighting coefficient of Cl = 2 was chosen. The GA evolved a population of 24 members over 153 generations for a total of 3672 cost evaluations to produce the design shown in Fig. 12 with a unit cell size of 740 nm on a side and a-Ge and Au layer thicknesses of 382 nm and 60 nm, respectively. The total emissivity for this metamaterial structure was suppressed to under 0.064 for all target angles and enhanced to be greater than 0.946 at the target peak emissivity angle as demonstrated by the plot in Fig. 13(a) . The 3D emissivity patterns in Figs. 13(b) and 13(c) show that this metamaterial surface has a “butterfly wing” shape with high off-normal emissivity in the φ = 0° plane and low emissivity for all elevation angles in the in the φ = 90° plane.
Fig. 12 GA optimized a-Ge and Au dielectric resonator emitter with 4-fold symmetry. (a) 3D unit cell and (b) 3x3 tiling.
Fig. 13 Near-IR emissivity patterns at λ = 1.55µm for dielectric resonator design in Fig. 12 showing low emissivity in the normal direction and high off-normal emissivity at θ = 40° in the φ = 0° plane cut. (a) Angle cuts at φ = 0°,90° are shown as well as 3D patterns for (b) TE and (c) TM waves.
5. Conclusion
Two types of metamaterial coatings based on an EBG structure and a dielectric resonator array were introduced for controlling the angular distribution of the emissivity from a surface at the near-IR wavelength of 1.55 µm. While a uniform surface will have a peak emissivity in the direction of the surface normal, the metamaterial coatings described here can redirect the peak emissivity to off-normal angles. Two symmetry conditions were also studied that can provide further control over the azimuthal variation of the emissivity. An 8-fold mirrored symmetry was employed for designs where a conical, azimuthally stable response was desired, whereas a 4-fold symmetry was used to produce a “butterfly wing” emissivity pattern with high off-normal emissivity in only one plane cut. These examples demonstrate that metamaterial coatings can be effectively synthesized by a GA to achieve custom angle-selective emitters
Acknowledgment
This work was supported by the Office of Naval Research under contract N00014-10-G-0259.
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