1. Introduction

Recently, an exact theoretical recipe for complete electromagnetic cloaking was unveiled in [1]. Furthermore, the feasibility of electromagnetic cloaks has been demonstrated in the microwave regime [2]. Achieving perfect electromagnetic invisibility will require the use of metamaterials to form an anisotropic volumetric cloak with continuously varying (inhomogeneous) permittivity and permeability together with a perfect impedance match to free space. The prescribed permittivity and/or permeability encompass a range of non-negative real values including zero. In particular, low-index metamaterials (LIMs), i.e. metamaterials with a refractive index anywhere between zero and unity (0≤n≤1), play a critical role in the ability to realize an electromagnetic cloak. The LIMs as defined here are bounded by two important special cases; namely the zero-index metamaterials (ZIMs) on one hand and on the other hand metamaterials which exhibit the “transparency” or “invisibility” condition when n=1.

In contrast to negative-index optical metamaterials [3, 4, 5], ZIMs and LIMs have received much less attention in the literature. Wave interactions of ZIMs having the impedance matched to free space were theoretically investigated in [6]. Reflection and refraction properties of LIM slabs consisting of an array of metallic wires have been analyzed at optical wavelengths [7]. At lower frequencies, use of a ZIM was suggested to enhance the gain of an antenna embedded inside the metamaterial [8]. In [9], the design parameters of a frequency selective surface were optimized using a genetic algorithm (GA) to exhibit LIM behavior in the GHz regime. A GA optimization technique was also successfully applied in [10], where a non-dispersive dielectric material was combined with a Drude-type material in a checkerboard fashion to achieve a ZIM/LIM characteristic.

This paper presents low-index metamaterial (LIM) designs in the visible spectrum that could be used as building blocks for a variety of devices ranging from perfect electric or magnetic mirrors to an optical invisibility cloak. The proposed periodic metamaterial structure incorporates magnetic resonators and silver meshes for nearly independent control of the effective permeability and the permittivity respectively. This type of design allows a joint optimization of independent values for the index of refraction and the intrinsic impedance, unlike in [7] where the LIM designs inevitably lead to high impedance characteristics.

2. Metamaterial architecture

Let the effective index of refraction n and the intrinsic impedance z of a metamaterial be defined as n=n′+in″ and z=z′+iz″, respectively, normalized by their corresponding values in free space. These quantities may be expressed in terms of the relative effective permittivity ε=ε′+iε″ and the relative effective permeability μ=μ′+iμ″ such that

where proper branches of the square roots should be chosen for a passive medium according to Im{n}≥0 and Re{z}≥0. Equation (1) suggests that it is possible to realize a desired set of values for n and z by independently adjusting the values of ε and µ. This can be achieved by incorporating sub-structures into the unit cell of a metamaterial that have dominant effects on either ε or µ.

 

Fig. 1. Unit cell geometry of a doubly-periodic metamaterial slab on a thick glass substrate: (a) A view from the +ŷ direction. (b) A perspective view.

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The unit cell of a doubly periodic metamaterial slab having the period p both in the x̂ and the ŷ directions is illustrated in Fig. 1. A pair of square silver plates separated by an alumina (Al₂O₃) layer form a magnetic resonator. These periodic magnetic resonators are bounded on the top and bottom by a silver mesh. Each mesh is formed by silver strips running both in the ±x̂ and the ±ŷ directions with grid points located at the center of each magnetic resonator. The regions which are not occupied by either silver or alumina are filled with silica (SiO₂). Finally, the entire metamaterial slab is placed on a thick glass substrate. The metamaterial slab is illuminated by a normally incident plane wave propagating in the $+\widehat{z}$ direction with the incident electric field vector polarized in the x̂ direction.

Although cross-coupling effects do exist between the electric and the magnetic structures inside a unit cell, the magnetic resonator and the silver mesh have a dominant influence on µ and ε, respectively. At a given wavelength, the value of µ will be strongly dependent upon the geometrical parameters of the magnetic resonator — w, t and d. Similarly, the dimensional parameters of the silver mesh s and t_m will strongly affect the value of ε. It is expected that the period p will contribute to both µ and ε.

3. Optimization methodology

For a given pair of desired values for n and z, six geometrical parameters (p, w, s, t, d, and t_m) in the LIM design need to be properly chosen to achieve the goal. In this study, a GA [11] is employed for the optimization task. The GA is a powerful optimization methodology based on the principle of natural selection. An initial population is formed to comprise a generation, where ‘individuals’ of the population correspond to random realizations of the given designs. A performance metric, usually referred to as a ‘fitness’ in GA terminology, quantifies the appropriateness of an individual in light of the desired properties. Based on the fitness values for each member of the population, poor-performing individuals are discarded and only the best-performing individuals are allowed to ‘mate’ to produce the next generation. This evolution process is repeated until convergence is achieved to a desired fitness.

A specific implementation of the GA, called the micro-GA [12], is employed with a population size of six. For each individual, the corresponding electromagnetic scattering problem of an infinitely-periodic metamaterial slab is rigorously solved using the finite element boundary integral (FE-BI) method [13] with periodic boundary conditions.

We have chosen to employ the retrieval procedure for the equivalent material parameters given in [14, 15] from the reflection and the transmission coefficients. This inversion method achieves accurate reconstruction of the scattering parameters by choosing proper numerical values for n and z of a homogeneous isotropic material slab of the same thickness. However, although widely used, there is a controversy as to whether the retrieved material parameters correspond to homogeneous, isotropic, and passive materials [16, 17, 18, 19]. The inversion method is known to produce negative values for either ε″ or µ″ at times, which appears to contradict the passivity requirement [19, 20], even though the corresponding refractive index is indicative of a passive medium.

In this publication, we concentrate on the metamaterial design and optimization of the structure to achieve desirable bulk electrical properties. It is stressed that they can be coupled with any other parameter inversion procedures equally well.

4. Numerical results

To demonstrate the flexibility of the proposed design approach in realizing optical LIMs, two example designs are presented. These examples consider the two boundary cases for a LIM, i.e. a zero-index design (n→0) and a unity-index design (n→1). In both of these cases, an additional condition was imposed that the metamaterial be impedance matched to free space (z→1) over the range 400 nm≤λ≤800 nm. The ranges of the six geometrical parameters that define the GA optimization search space were constrained as follows: 133 nm≤p≤400 nm, 0≤w≤p, 0≤s≤p, 20 nm≤t≤60 nm, 20 nm≤d≤100 nm, and 20 nm≤t_m≤100 nm. The values of 1.445, 1.62, and 1.5 were used in the simulations to represent the refractive index of silica, alumina, and glass, respectively. In addition, the refractive index of silver was based on the measured values reported in [21].

4.1. Impedance-matched zero-index metamaterial

In the first design, a metamaterial with a zero index of refraction (n→0) and the intrinsic impedance matched to free space (z→1) is desired. An appropriate fitness function is chosen as

Convergence was achieved to a maximum fitness value of f=21.5 at λ=0.71 µm (edge of red light) with the effective material parameters given by n=0.159+i0.094 and z=0.973−i0.108. The optimized geometrical parameter values were found to be p=345 nm, w=153 nm, s=307 nm, t=39.1 nm, d=94.9 nm, and t_m=20.0 nm. The total thickness was found to be 213 nm. Figure 2 shows plots of the effective material parameters for the optimized geometry as well as the reflectance (R), transmittance (T), and absorbance (A) spectra with respect to the free space wavelength. Figure 2(a) shows that a very low index of refraction is achieved during the transition from a positive-index material to a negative-index material as the wavelength is increased. Figure 2(b) shows that the transmittance T reaches the maximum value of 68 % at the optimal wavelength of λ=0.71 µm. The reduced T from the ideal 96 % (corresponding to that of a free space-glass interface) is attributed to an imperfect impedance match together with a non-zero value of n″. A magnetic resonance typically accompanies enhanced absorption [16]. The relatively high value of A=29 % is attributed to the magnetic resonance around λ=0.72 µm that is associated with the immediately-following negative index band.

 

Fig. 2. Effective parameters of the LIM design optimized for a zero index of refraction and a free-space matched impedance: (a) n and z. (b) The reflectance (R), transmittance (T), and absorbance (A) spectra.

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4.2. Impedance-matched unity-index metamaterial

The goal of the second metamaterial design is to achieve an effective refractive index equal to one (n→1) with the impedance matched to free space (z→1). This corresponds to the metamaterial “transparency” condition. The GA fitness function in this case was chosen to be

The GA converged to the optimized design with a maximum fitness of f=117 at λ=0.69 µm. The corresponding effective material parameters for this design are given by n=0.959+i0.019 and z=0.974-i0.077. The values of the optimal geometrical parameters were found to be p=345 nm, w=153 nm, s=38.3 nm, t=23.2 nm, d=77.1 nm, and t_m=56.8 nm. The effective material parameters along with the three spectra (i.e., R, T, and A) are plotted with respect to wavelength in Fig. 3. In contrast to the previous design example, no negative index band is observed in Fig. 3(a). No magnetic resonance occurs at or near the optimal wavelength λ=0.69 µm so that the small value of A=9.5 % allows the corresponding value of T to reach 88 %. Although the structure was optimized to maximize the fitness at a single visible wavelength, it is noted from Fig. 3(b) that the metamaterial is highly transmissive with the value of T maintained above 70 % over a wide spectral range from 0.52 (green) to 0.71 µm (red). This may be attributed to the nearly real values which are close to unity for both n and z over the indicated range. Moreover, this is in sharp contrast to the range from 0.74 to 0.89 µm, over which the almost purely imaginary intrinsic impedance dramatically reduces T below 30 % and significantly increases R without seriously affecting A.

During the GA optimizations for the two designs detailed above, the objective in each case was to search the visible spectrum for the best wavelength such that the fitness value is maximized. If we desire to achieve a particular LIMbehavior at a fixed wavelength, the metamaterial design can be optimized for such a goal and the flexibility of the proposed design accommodates a comparable performance. For example, assume that an impedance-matched unity-index metamaterial is desired at the wavelength λ=0.55 µm (green light). A GA optimization for this problem with the fitness function given in (3) leads to a maximum fitness f=34.1 associated with the optimized parameters n=1.127+i0.030 and z=1.023–i0.109.

In the above design examples, metamaterial slabs containing a single layer of unit cells in the ẑ direction (i.e., in the direction of propagation) were considered, in contrast to numerous bulk-type metamaterial realizations reported in the GHz and microwave regimes. This is mainly due to (i) significantly higher losses associated with metals at optical wavelengths and (ii) fabrication difficulty with multi-layer structures including maintaining structural uniformity in the ẑ direction using lithographic processes. The latest fabrication and characterization efforts for optical metamaterials involve designs having 1–3 layers [3, 5, 22]. If thicker optical metamaterials are desired, multi-layer versions of the proposed LIM designs need to be optimized independently. In such cases, it is understood that slightly different equivalent parameter values than those obtained in the single-layer designs may result due to mutual coupling effects between multiple layers of the metamaterial.

 

Fig. 3. Effective parameters of the LIM design optimized for a unity index of refraction and a free-space matched impedance: (a) n and z. (b) The reflectance (R), transmittance (T), and absorbance (A) spectra.

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5. Conclusion

A flexible LIM design has been presented that is well-suited for applications in the visible spectrum. The unit cell of the doubly periodic metamaterial slab incorporates a planar magnetic resonator and silver meshes for effectively controlling the permeability and the permittivity. The LIM architecture can support a wide range of values for the desired refractive index n and the intrinsic impedance z simply by properly adjusting the geometrical parameters of the electric and magnetic sub-structures. This design flexibility allows these LIMs to be employed in a variety of applications such as perfect electric and magnetic mirrors, optically transparent metamaterials, and perhaps even as building blocks for a cloak of invisibility in the visible spectrum.

A GA was used to optimize the values of six geometrical parameters by maximizing properly defined fitness functions. Two example designs were presented to achieve the refractive index value of zero and unity together with the impedance match condition to free space. A very low (near zero) index of refraction with a good impedance match was demonstrated at λ=0.71 µm (red light), where a transmittance value of 68 % was achieved. For the design targeted at achieving a perfect transparency condition, a value of T=88%was realized at λ=0.69 µm.