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Negative index metamaterial designs for the mid-infrared with low absorption and impedance mismatch losses are presented. A robust genetic algorithm is employed to optimize the flexible metamaterial structure for targeted refractive index and impedance values. A new figure of merit is introduced to evaluate the impedance match of the metamaterial to free space. Two designs are presented demonstrating low-loss characteristics for a thin metamaterial with two metal screens and a thick metamaterial stack with five screens. The device performance is analyzed when adding more screens to the structure, revealing that optimizing a thick stack produces a metamaterial with properties approaching those of a bulk material.
©2009 Optical Society of America
1. Introduction
In recent years, there has been a substantial research interest in demonstrating metamaterials with a negative refractive index from the RF [1–3] through the optical [4–7] regions of the electromagnetic spectrum. This research interest was sparked in 2000 when Pendry first proposed a ‘perfect’ flat lens with a refractive index of −1 that could in principle overcome the diffraction limit of conventional optics [8]. Experiments have proven that a negative index metamaterial (NIM) can be realized when the permeability and permittivity are simultaneously negative over the same wavelength range and that a flat lens constructed from such a metamaterial exhibits super-resolution [2, 3]. However, for infrared and optical wavelengths, there are problems that need to be addressed in order to realize a practical flat lens. First, the material losses due to absorption and impedance mismatches to the surrounding media for optical NIMs are quite high. Secondly, most optical NIM demonstrations have been thin structures, which are unsuitable for use in a flat lens. Recently, an optical NIM stack was fabricated that consists of many alternating layers of metal and dielectric insulator [7], representing a step towards realizing a practical, optical NIM.
This paper presents an optimization strategy for realizing low-loss NIMs for the mid-IR spectrum targeting the atmospheric window from 3 μm to 5 μm with very small absorption losses and a good impedance match to free space. The primary goal of this research is to introduce a design tool and methodology for minimizing the losses in mid-IR NIMs. The proposed metamaterial structure consists of a stack of alternating layers of silver (Ag) metal and polyimide dielectric perforated by a two dimensional periodic pattern of air holes. The parallel metallic layers provide control over the permittivity and permeability, which can be adjusted by using a genetic algorithm (GA) optimization technique (see Fig. 1 ) [9] to achieve both a negative refractive index and low loss, as quantified by the following figures of merit (FOM):
where n is the effective refractive index and Z is the effective impedance of the metamaterial normalized to the impedance of free space Z 0 . FOMn is the ratio of the real part of the index to the imaginary part and provides a measure of the absorption loss in the metamaterial. FOMZ describes the impedance match of the metamaterial to the surrounding medium (i.e. free space) and is often neglected in the NIM literature. By optimizing for large values of FOMn and FOMZ, a NIM design can be achieved with the smallest possible absorption and reflection.
Fig. 1 Flowchart showing the genetic algorithm synthesis procedure used to generate low-loss NIM stack designs.
This paper also examines the bulk properties (i.e., approaching those of an infinite stack) of the metamaterial by including an additional polyimide and Ag layer to designs optimized with two and five Ag screens. This analysis demonstrates that it is more effective to optimize a metamaterial stack with many layers in order to achieve a practical low-loss NIM. Optimizing a large stack also results in a metamaterial with properties approaching those of a bulk material.
2. Metamaterial structure
The metamaterial structure proposed here consists of stacked metallic screens sandwiching layers of dielectric. The metal-dielectric stack is perforated by air holes in a periodic pattern defined by a unit cell such as the one shown in Fig. 2 . The unit cell geometry representing the stack and air hole locations is constrained to possess eight-fold symmetry, so that the metamaterial appears the same to normally incident waves polarized with the electric field directed vertically or horizontally with respect to the unit cell geometry. Thus, metamaterials with the proposed structure will have a polarization independent response at normal incidence, unlike the optical NIMs found in the literature, which are designed for linear polarization [4–7]. The materials in the structure were chosen a priori to be Ag and polyimide because they both have low intrinsic losses in the mid-IR wavelength range from 2 μm to 5 μm. We have also previously studied these materials for use in planar metallo-dielectric filters for the far- and mid-IR spectra [10, 11].
Fig. 2 13 x 13 pixel geometry for a negative index metamaterial with two metal screens separated by an insulator. The pixel size, Ag thickness, and polyimide thickness for this design are 107 nm, 75 nm, and 115 nm, respectively. (a) The top and side views of the unit cell, which is periodic in two dimensions. (b) 3D isometric view of the metamaterial.
The Ag screens in the structure provide control over both the effective permittivity ε and permeability μ of the metamaterial. Operating under the same principals as the paired metallic nanowire arrays studied theoretically [12] and experimentally [4] for use in optical NIMs, the neighboring layers of Ag in the proposed structure can form parallel plate magnetic resonators for H-field components transverse to the surface of the metamaterial. At resonance the transverse magnetic field excites a circulating current on the neighboring Ag patches, which, in turn, induces a magnetic field opposing the external magnetic field [4]. Similarly, isolated metallic patches can provide electric resonances for transverse E-field components [4, 12]. A fully connected geometry results in a Drude behavior for ε, similar to a metallic mesh that exploits the inherent Drude properties of a metal film while lowering the plasma frequency ω p of the effective permittivity by introducing air holes into the film [6]. The unit cell geometry is pixilated, so that complex structures can be considered in the design process. This provides great flexibility in controlling Lorentz and Drude characteristic resonances in the ε and μ profiles to simultaneously achieve a negative index with a small imaginary component (i.e., low absorption loss) and an impedance match to free space.
3. Design methodology
The design parameters that need to be optimized to achieve a desired n and Z include the unit cell dimension, the thicknesses of the Ag and polyimide layers, and the pixilated geometry differentiating between Ag/polyimide pixels and air holes. We employ a robust GA to optimize the metamaterial design parameters for a given n and Z [9]. GAs are proven to be an effective design tool for electromagnetics as they have been applied to a variety of metamaterials applications such as artificial magnetic conductors (AMCs) [13–15], electromagnetic bandgap absorbers [16], magnetic metamaterials [17], metaferrites [18], and metallo-dielectric zero index metamaterials (ZIMs) [19] at RF, and a rotationally stacked Drude checkerboard ZIM at IR wavelengths [20]. GAs are also known to be useful in optimizing problems with many parameters such as the flexible metamaterial structure described in Section 2 and for simultaneously optimizing multiple, weighted goals, such as described in this section. A flowchart showing the operation of the GA used in this work is presented in Fig. 1. Operating based on the principals of natural selection, GA’s evolve optimum designs from a pool of randomized candidate designs. The design parameters, including the unit cell dimension, thickness, and screen geometry, are encoded into a binary string called a chromosome. The unit cell dimension and thickness are 8-bit binary numbers, whereas the screen geometry is divided into eight mirrored triangular folds, one of which is encoded into the chromosome with each pixel being either “0” (air) or “1” (metal/dielectric). The initial population contains randomized chromosomes, spread across the design parameter space. After evaluating the fitness of each member and ranking the population according to performance, mates are paired using tournament selection, and their genetic data is combined using single-point crossover to generate two new designs per set of parents. Random single bit mutations are made to a small percentage of the new members in order to constantly explore new areas of the parameter space. Finally, the best member from the previous generation is copied into the new generation so that the fitness is always maintained or improved across subsequent generations. Convergence is determined when the fitness has not improved in several generations.
In order to evaluate the performance of each population member, the design parameters are used to build a model of the metamaterial for a finite element boundary integral (FE-BI) full-wave electromagnetic simulator [21]. This code calculates the scattering from the metamaterial structure for a normally incident wave. Fabrication constraints are enforced on the pixelized geometry during the fitness evaluation. The geometry is simplified to remove single isolated metal/dielectric pixels. Diagonal connections between pixels are eliminated to ease fabrication by removing one of the metal/dielectric pixels. Measured dispersive metal and dielectric properties for Ag and polyimide are also incorporated into the simulation [22]. The scattering parameters calculated by the full-wave simulation are then inverted using the Nicolson, Ross, and Weir (NRW) method [23] to obtain the effective medium parameters n and Z of the metamaterial. The first goal for the design is to achieve a refractive index of −1 with a large FOMn to minimize absorption loss in the metamaterial. The second goal is to obtain an impedance match to free space as indicated by a large FOMZ in order to minimize the reflection from the metamaterial. These goals are combined to obtain a cost function given by
where n target = −1 + 0i is the desired refractive index, Z target = 1 + 0i is the desired normalized impedance for the metamaterial, and freqs is a range of frequency points over which the GA will search for the best performance. While the weighting for each of these design goals is equal, one goal can be emphasized over the other using weighting coefficients. Also, a fitness function that integrates across frequency such as described in [24] could be used to produce a more broadband response, but such a broadband response would reduce the peak performance of the device.. The cost function in (2) highlights the usefulness and importance of employing the GA to design low loss NIMs. While the physical mechanisms that give rise to a negative refractive index are understood, a delicate balance between ε and μ must be obtained in order to realize a design with high FOMn and FOMZ. The GA is an excellent tool for evolving a complex geometry with balanced absorption and impedance losses, where the loss mechanisms are difficult to interpret and fine-tune using simple physical models.4. Numerical results
In order to demonstrate the effectiveness of the proposed design approach for synthesizing low loss NIMs, two examples will be presented. The first design consists of two Ag screens separated by a polyimide film, and the second design is a stack of five Ag screens separated by polyimide layers. Further analysis is presented to show the effect of adding another polyimide/Ag layer to the structure and to evaluate how well the effective n and Z values match the bulk properties of a cascaded metamaterial.
The GA optimization for both designs is conducted on a computer cluster with a parallelized fitness evaluation to enhance the convergence speed. In both cases, a population of 32 members spread across 8 computational nodes with 3.0 GHz Intel Xeon Quad-Core processors is evolved over approximately 100 to 200 generations. The Ag film thicknesses are fixed at 75nm, so that they are larger than the skin depth, but the separation distance between Ag films is optimized. In the FEBI model, the Ag film is represented by an impedance sheet, which has provided excellent accuracy when comparing with previous experiments [10, 11]. Fabrication constraints in the GA restrict diagonal connections between pixels of Ag/polyimide.
4.1 Low loss NIM with 2 metallic screens
The first design example with two Ag screens is shown in Fig. 2. The distance between Ag screens was permitted to vary between 150 nm and 300 nm, and the unit cell dimension ranged between 0.8 μm and 2.0 μm. The GA searched for an optimum NIM band around the shorter end of the 3 μm to 5 μm atmospheric window, where the polyimide loss is very low. A search range from 2.85 μm to 3.15 μm in wavelength was explored by the GA in order to speed up convergence. Convergence was achieved in approximately 5 hours with a cost value of 0.0415 after 100 generations. The optimized design geometry shown in Fig. 2(a) resembles the fishnet structure, which has frequently been used to achieve optical NIMs with low absorption (high FOMn) [5–7]. However, the shape of the air holes are more complex than traditional fishnet structures, having been tuned by the GA for both high FOMn and high FOMZ. They also possess eight-fold symmetry for a non-polarized response at normal incidence.
The scattering parameters and effective n and Z for the design are shown in Fig. 3 . It is notable that this design possesses broadband high transmission extending from the negative index region to shorter wavelengths where the index crosses zero and becomes positive (i.e., ZIM and PIM regions). Part of the reason for high transmission over a large wavelength range is that the normalized impedance Z has a broad band that is slowly varying around the free space impedance, giving a reflection coefficient lower than −10 dB from 2.53 μm to 3.04 μm in wavelength. Also, n” is small, resulting in low absorption from shorter wavelengths up to the optimum wavelength. The index approaches −1 at λ = 2.93 μm wavelength, where n = −1.04 + 0.21i and Z = 1.01 – 0.03i. The figures of merit at λ = 2.93 μm for this design are calculated to be FOMn = 5.0 and FOMZ = 31.6. The scattering parameter magnitudes at λ = 2.93 μm are transmission |T| = −1.1 dB, reflection |R| = −34.7 dB, and absorption |A| = −6.7 dB. The reflection null is indicative of an excellent impedance match to free space, and the absorption loss is minimized along with n”.
Fig. 3 Effective and scattering parameters for the NIM design shown in Fig. 2: (a) Refractive index n and impedance normalized to free space Z. The optimization range (gray box) and optimum wavelength (vertical dashed line) are highlighted. (b) Reflection R, transmission T, and absorption A.
4.2 Low loss NIM stack with 5 metallic screens
The second design example contains five Ag screens sandwiching four layers of polyimide. For this optimization, the distance between Ag layers could vary between 130 nm and 500 nm, and the unit cell dimension ranged between 0.8 μm and 2.0 μm. The same wavelength search range was explored by the GA for the optimum NIM band as in the previous example. After 220 generations and 24 hours of computation time, the GA converged on the design shown in Fig. 4 , which has a cost value of 0.0249. The unit cell geometry for this design in Fig. 4(a) also resembles a fishnet structure, modified with notches in the corners. These notches serve to improve the impedance match (FOMZ) of the metamaterial, which is one of the goals sought by the GA. Also, this geometry possesses eight-fold symmetry and polarization insensitivity for normal incidence, an advantage over the fishnet structures previously considered in the literature.
Fig. 4 13 x 13 pixel geometry for a NIM stack with five metal layers. The pixel size, Ag thickness, and polyimide thickness for this design are 138 nm, 75 nm, and 36 nm, respectively. (a) Top and cross-section views of the structure. (b) 3D isometric view of the metamaterial.
The effective n and Z as well as the scattering magnitudes are shown plotted in Fig. 5 . At λ = 2.86 μm the optimum effective parameters are n = −0.99 + 0.13i and Z = 1.01 – 0.08i. At this wavelength FOMn = 7.6, which is higher than the figure of merit for the design containing only two Ag screens, meaning that this design will have lower absorption per unit thickness. On the other hand, FOMZ = 12.4 is lower than the first design, resulting in a higher reflection. The scattering parameter magnitudes at the optimum wavelength λ = 2.86 μm are transmission |T| = −1.3 dB, reflection |R| = −23.6 dB, and absorption |A| = −5.9 dB. Despite the better FOMn for this design, the transmission is slightly lower because this design has approximately double the thickness of the two-screen design, resulting in an increased absorption. Nevertheless, the transmission properties are still remarkably good. The excellent performance of this design indicates that for practical, thicker NIMs at mid-IR wavelengths, the losses can be minimized by optimizing a metamaterial stack in its entirety. In moving from 2 metal screens to 5 metal screens, the number of generations required for convergence doubled, and the computation time required on the cluster increased by nearly 5 times. Thus, further increasing the number of screens in the optimization is expected to greatly increase the computational resources required for the GA to converge on a solution.
Fig. 5 Effective and scattering parameters for the NIM in Fig. 4: (a) Refractive index n and normalized impedance Z. The optimization range (gray box) and optimum wavelength (vertical dashed line) are highlighted. (b) Reflection R, transmission T, and absorption A from a normally incident wave.
4.3 Effect of adding a metallic screen
In order to evaluate how closely the effective parameters for the two designs match the desired bulk metamaterial properties, the effect of adding an Ag screen to each design will be analyzed. Figure 6 shows a cross-sectional view of the first design when adding a metal screen and an overlay plot showing the effect of the third screen on n. While the NIM band for the modified metamaterial is still present, the location of the band has shifted and the loss characteristics have changed significantly. The optimum wavelength where n’ = −1 has shifted to λ = 2.86 μm, where n = −1.03 + 0.22i and Z = 1.85 – 0.04i. The scattering magnitudes at this wavelength are |T| = −3.0 dB, |R| = −7.7 dB, and |A| = −4.8 dB. The reduced FOMn = 4.64 as well as the increased metamaterial thickness has contributed to a higher absorption, and the poor impedance match has resulted in a small FOMZ = 1.13 and much higher reflection. The drastic changes seen in n and Z when increasing the number of metal screens in this design indicate that the effective properties are strongly affected by the coupling with the third screen. Hence, this design does not possess bulk-like effective properties.
Fig. 6 Metamaterial cross-section and refractive index when adding a third metallic screen to the design in Fig. 2: (a) Cross-section view of structure. (b) Refractive index n for two (blue curves) and three (green curves) metal screens.
The results of adding a metal screen to the second design are shown in Fig. 7 . As can be seen in the overlay showing the index for the five screen design and the modified design, the NIM band has not changed (moved) significantly and retains its loss characteristics at the optimum wavelength. At λ = 2.85 μm the optimum effective parameters for the modified structure are n = 0.99 + 0.14i and Z = 1.27 – 0.14i, and the scattering magnitudes at this wavelength are |T| = −1.9 dB, |R| = −13.2 dB, and |A| = −5.0 dB. The slightly lower NIM FOMn = 7.07 and the increased metamaterial size contribute to an increased absorption, and the impedance match is not as good as the original design, resulting in a higher reflection and lower FOMZ = 3.3. However, this design still performs very well with the added Ag screen, indicating that by optimizing for a larger metamaterial stack, the resulting design yields effective properties approaching those of a bulk material.
Fig. 7 Metamaterial structure and refractive index when adding an additional metal screen to the design in Fig. 4: (a) Cross-section view showing the device with five and six metal layers. (b) Refractive index n for the device with five (blue curves) and six (green curves) metal screens. The performance of this design does not change significantly when another metal screen is added.
5. Conclusion
A flexible metamaterial platform consisting of stacked metal and dielectric screens perforated with a periodic array of air holes was presented along with a synthesis procedure for realizing low loss NIMs. The metamaterial structure provides control over the electric and magnetic responses of the device, and GA optimization can be used to exploit this flexibility to achieve a negative index along with low absorption and impedance mismatch losses. Designs were presented to demonstrate that these low loss characteristics can be achieved both for thin metamaterials with two metal screens as well as for thicker metamaterial stacks with many alternating metal and dielectric layers. These designs possessed high figures of merit with FOMn = 5.0 and 7.6 and FOMZ = 31.6 and 12.4, respectively, as well as low transmission attenuations of −1.1 dB and −1.3 dB. While the values obtained for FOMn represent the state of the art when compared with experimental IR/optical NIMs found in the literature [6, 25, 26], introducing FOMZ as a second optimization goal advances the field by minimizing reflection losses due to impedance mismatch. Further analysis conducted by adding a metal screen to both designs revealed that by optimizing a thicker metamaterial stack, the recovered effective properties n and Z are approaching those of a bulk metamaterial. In the future metamaterials with larger stacks will be investigated theoretically and experimentally for low loss and bulk effective properties as well as the extension of this design strategy to near-IR and optical wavelengths.
Acknowledgements
This work was supported by the Penn State MRSEC under NSF grant DMR-0820404.
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