1. Introduction

Negative index metamaterials (NIMs) have attracted a significant amount of research attention since 2000, following the prediction of superlensing using NIMs by John Pendry [1]. Recently, NIMs research has been further inspired by the possibility of optical cloaking [2–4]. These applications have the most to offer in the visible spectrum, necessitating the design of NIMs that operate at less than 700 nm. Progress in visible NIMs, which began at 1.5 um, has recently been pushed to the yellow-light wavelength of 580 nm [5]. Unfortunately, wavelengths for three-dimensional metamaterials are still too long for superlens biological sensing and imaging applications. Therefore, pushing the operational wavelength of negative index metamaterials further into the blue-light and violet-light range will be a vital step in order to realize more exciting metamaterials applications.

The first artificial negative index metamaterial was fabricated by Smith et al., through combining SRRs and continuous wires [6–8]. After that, all kinds of microstructure configurations of NIMs with different shapes were found, such as Omega-shaped [9], S-shaped [10], Double S-shaped [11], H-shaped [12], and so on. In recent years, the fishnet structure was proposed because it has generally been shown to be a robust and scalable design at visible wavelengths [13], including the demonstration of a negative refractive index at 715 nm [14]. Using this fishnet structure, significant progress has been made towards the realization of negative index of refraction at visible wavelengths [15, 16].

However, most of those microstructure were obtained by changing the shape and (or) the size of the components based on human intuition, experience or large numbers of simulation experiments which were time-consuming, ineffective or expensive. Thus, an effective and systematic methodology for guiding and designing the microstructure of negative index metamaterials at visible wavelengths is quite desirable and vital.

Heuristic search techniques have been successfully applied to many electromagnetic problems to pursue novel solutions which are difficult to obtain using the conventional design approaches. For instance, genetic algorithms (GAs) have been used to optimize wide frequency band of NIMs [17], scannable circular antenna arrays [18], fractal antenna-array [19], and so on.

Differential Evolution (DE) algorithm [20, 21] is a new heuristic approach which mainly has three advantages: finding the true global minimum regardless of the initial parameter values, fast convergence, and using few control parameters. Therefore, differential evolution algorithm was introduced to design and optimize microstructure of NIMs for the visible spectrum with low losses in this paper.

The paper was organized as follows. Firstly, a novel negative index metamaterial design methodology for the visible spectrum with low losses was presented in this paper. The robust differential evolution was employed to optimize the metamaterial design to achieve a desired set of values for the index of refraction. Then the effectiveness of the new technique was demonstrated by a fishnet structure design example that was optimized by DE. By using numerical simulation of a wedge-shaped model and S-parameter retrieval method, we found that the DE-designed optimal solution can exhibit a low loss LH frequency band with simultaneously negative values of effective permittivity and permeability at the violet-light wavelength of 408 nm, and the figure of merit is 15.2, that means it may have practical applications because of its low loss and high transmission.

2. DE-based design and optimization model

2.1 Fishnet structure design example

In this paper, a double-layer fishnet structure was used as design example because it has generally been shown to be a robust and scalable design at visible wavelengths. The fishnet structure has five geometrical parameters (a, $r_x$, $r_y$, $s_{{\text{MgF}}_{\text{2}}}$ and $t_{\text{Silver}}$) which are described in Fig. 1 . In Fig. 1, thea is the lattice constant, the $a{r}_x$ and $a{r}_y$ denote the length and the width of rectangular-hole respectively. The $r_x$ and$r_y$ are the ratio of lattice constant between 0 and 1. The $s_{{\text{MgF}}_{\text{2}}}$and $t_{\text{Silver}}$ represent the thicknesses of MgF₂ and Ag respectively.

 

Fig. 1 The five geometrical parameters for fishnet structure design and optimization.

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The optical parameters of silver and the MgF₂ dielectric spacer are chosen as previously [22]. In brief, we use the free-electron Drude model with plasma frequency ${\omega }_{pl}=1.37\cdot {10}^{16}{s}^{-1}$ and collision frequency ${\omega }_{col}=8.5\cdot {10}^{13}{s}^{-1}$ for silver. The MgF₂ refractive index is taken as $n=1.38$. The boundary conditions of fishnet structure are presented in Fig. 2 .

 

Fig. 2 The boundary conditions of fishnet structure.

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2.2 Differential evolution algorithm

The DE algorithm is a population based algorithm like genetic algorithms using the similar operators: crossover, mutation and selection [20]. The main difference in constructing better solutions is that genetic algorithms rely on crossover while DE relies on mutation operation. This main operation is based on the differences of randomly sampled pairs of solutions in the population.

The DE algorithm uses mutation operation as a search mechanism and selection operation to direct the search toward the prospective regions in the search space. The DE algorithm also uses a non-uniform crossover that can take child vector parameters from one parent more often than it does from others. By using the components of the existing population members to construct trial vectors, the recombination (crossover) operator efficiently uses information about successful combinations, enabling the search for a better solution space.

The block diagram of the DE design for fishnet structure optimization is illustrated in Fig. 3 . Firstly, an initial population is generated randomly. Then the five parameters (a, $r_x$, $r_y$, $s_{{\text{MgF}}_{\text{2}}}$ and $t_{\text{Silver}}$) of each individual is used to generate the geometrical microstructure of fishnet structure metamaterial for the finite-difference time-domain (FDTD) method [23, 24] to call. The fitness function is then calculated based on the S parameters of fishnet structure metamaterial. The DE judges if the termination criteria is met. If not, the population is regenerated through a DE process including selection, crossover and mutation. In this paper, the exponential crossover strategy of DE/rand-to-best/1/exp is applied in the DE. The DE process is repeated until the maximum number of the generation is reached.

 

Fig. 3 The block diagram of the DE design for fishnet structure optimization.

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In this paper, the goal of DE is to pursue an optimal microstructure of fishnet structure NIMs for the visible spectrum with low losses. Therefore, the fitness function of DE can be defined as follows:

In the formula (1), the${\lambda }_{\min }$,${\lambda }_{\max }$ denote the lower and the upper limits of the wavelength range in visible spectrum respectively over which the DE will search for the best performance. In this paper, our goal is to pursue a low loss negative index metamaterial for the violet-light range because the NIMs that operate at violet-light spectrum have rarely been reported by far. Therefore, the ${\lambda }_{\min }$ is set to 380 nm and the ${\lambda }_{\max }$ is set to 470 nm in our optimization experiment, in other words, the lower limit $f_{\min }$ and the upper limit $f_{\max }$ of the frequency range are equal to 638 THz and 789 THz respectively.

Furthermore, $n_{\text{target}}=-1+0i$ is the desired refractive index for the fishnet structure in order to achieve a negative index metamaterial microstructure with lower loss.

The ranges of the five geometrical parameters that define the DE optimization search space were constrained as follows: $200nm\le a\le 300nm$,$0\le r_x\le 1$,$0\le r_y\le 1$, $0nm\le s_{{\text{MgF}}_{\text{2}}}\le 100nm$ and $0nm\le t_{\text{Silver}}\le 100nm$.

Finally, when the S parameters of fishnet structure are obtained by a FDTD solver, we can determine the wave impedance Z, refractive index n, permittivity ε and permeability μ of this NIMs microstructure by the S-parameter retrieval methods [25–27] which can be described as following:

3. Design results and discussion

3.1 Parameters for the DE optimization

In this paper, the fishnet structure design was optimized to operate at 638 - 789 THz, and the parameters for the DE optimization are listed in Table 1 .

Table 1. Parameters for the DE Optimization

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The DE optimization experiments were executed under Microsoft Windows Server 2003 with 3.00 GHz of Intel(R) Pentium(R) 4 CPU and 2 GB of RAM. In our simulation experiments, the CPU calculation time for fitness evaluation of one individual is about 2.5 minutes or so. Therefore, the whole CPU calculation time for 300 generations with 25 individuals is about two weeks.

The convergence curves of DE for fishnet structure optimization are presented in Fig. 4 . From Fig. 4 it can be seen that the differential evolution algorithm almost converges before the maximum number of the generation is reached.

 

Fig. 4 The convergence curves of DE for fishnet structure optimization.

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3.2 Experimental results and discussion

The optimal solutions of fishnet structure design at 50, 100, 150, 200 and 300 generations are listed in Table 2 . From the optimization results of Table 2, it can be seen that the objective function Fof DE is getting smaller and smaller which decreases from 0.0348 to 0.0117, and the figure of merit FOM (-$n\text{'}$/$n\text{'}\text{'}$) of fishnet structure metamaterial is getting higher and higher which increases from 8.7 to 15.2, that means the loss of fishnet structure NIMs becomes lower and lower.

Table 2. The Optimal Solutions of Fishnet Structure Design at Different Generations

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For example, when the generations of DE is 50, the optimal geometrical parameters (a,$r_x$,$r_y$,$s_{{\text{MgF}}_{\text{2}}}$and $t_{\text{Silver}}$) for fishnet structure design are 234.68 nm, 0.694, 0.274, 47.68 nm and 46.03 nm respectively. The optimal index of refraction is n = −0.84 + 0.10i, and the FOM is 8.7 correspondingly at the frequency of 733 THz or at the wavelength of 409 nm. However, when the generations of DE is 300, the optimal geometrical parameters for fishnet structure design are 220.00 nm, 0.741, 0.170, 52.98 nm and 27.64 nm respectively. The optimal index of refraction becomes n = −0.91 + 0.06i, and the FOM increases to 15.2 correspondingly at the frequency of 736 THz or at the wavelength of 408 nm. Therefore, the design methodology presented in this paper that uses a FDTD solver optimized by differential evolution technique is a more convenient means to pursue a novel microstructure of NIMs with low loss for the visible spectrum.

Furthermore, the electromagnetic characteristic of DE-designed optimal solution at 300 generations is analyzed in detail. The S parameters and the retrieved material parameters of optimal solution are presented in Fig. 5 .

 

Fig. 5 (a) Magnitude and (b) phase of the simulated S parameters for the DE-designed optimal solution. Retrieved index (c), impedance (d), permittivity (e) and permeability (f) are also shown.

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From Fig. 5(a) it can be seen that the transmission coefficient $S_{21}$ is quite small on the band lower than 734.94 THz because the silver metal exhibits a resonant electric response ($\epsilon <0$) which is shown in Fig. 5(e). But the effect permeability μ is still greater than zero which is seen in Fig. 5(f). However, when the frequency increases, both an electric and a magnetic resonance are exhibited, associated with a negative permittivity and negative permeability regime, respectively. Therefore a LH transmission regime can be achieved in the frequency band from 734.94 THz to 761.82 THz which is described in Fig. 5(c). The optimal index of refraction is n = −0.91 + 0.06i with a high FOM of 15.2 at the frequency of 736 THz or at the wavelength of 408 nm which is marked with a blue ellipse in Fig. 5(c). Therefore, the DE-designed optimal solution is indeed a LHM, it can exhibit a low loss LH frequency band with simultaneously negative values of effective permittivity and permeability at the violet-light wavelength of 408 nm, and the figure of merit is 15.2, that means it may have practical applications because of its low loss and high transmission.

Moreover, as shown in Fig. 5(e), the retrieved imaginary part of the effective permittivity is negative in special region. This is easy to understand mathematically [28, 29]. Relation $n^2=\epsilon \mu$ gives the real and the imaginary part of the refractive index:

Finally, in order to further verify the existence of the frequencies range with negative refractive index and give a direct image of the negative refraction of the EM wave for the DE-designed optimal solution, a two-dimensional wedge-shaped model with an inclined angle of $18.7°$ is designed for negative refraction simulation which is shown in Fig. 6 .

 

Fig. 6 The wedge-shaped model for negative refraction simulation.

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Figure 7 displays the magnitude distribution of electric field for the wedge-shaped model at the violet-light wavelength of 408 nm or at the frequencies of 736 THz, at which the negative refractive behavior is demonstrated. The unambiguous negative refraction is observed in the negative index transmission band which has been ascertained from the effective parameter description as discussed above.

 

Fig. 7 The magnitude distribution of electric field for the wedge-shaped model with the negative refraction at the violet-light wavelength of 408 nm. The reference line (black dashed line) is drawn normal to the bevel edge of the model.

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To sum up, the design methodology presented in this paper that uses a FDTD solver optimized by differential evolution technique is a more convenient means to pursue a novel metamaterial microstructure with desired electromagnetic characteristics. Experimental results demonstrate that the DE-designed optimal solution can exhibit a low loss LH frequency band with simultaneously negative values of effective permittivity and permeability at the violet-light wavelength of 408 nm, and the loss of fishnet structure negative index metamerials can be minimized by optimizing using differential evolution technique.

4. Conclusion

In this paper, differential evolution algorithm was introduced to design and optimize microstructure of NIMs for the visible spectrum with low losses. Firstly, a novel negative index metamaterial design methodology for the visible spectrum with low losses was presented in this paper. The robust differential evolution was employed to optimize the metamaterial design to achieve a desired set of values for the index of refraction. Then the effectiveness of the new technique was demonstrated by a fishnet structure design example that was optimized by DE. By using numerical simulation of a wedge-shaped model and S-parameter retrieval method, we found that the DE-designed optimal solution can exhibit a low loss LH frequency band with simultaneously negative values of effective permittivity and permeability at the violet-light wavelength of 408 nm, and the figure of merit is 15.2, that means it may have practical applications because of its low loss and high transmission. Therefore, the design methodology presented in this paper is a very convenient and efficient way to pursue a novel metamaterial with desired electromagnetic characteristics in the visible spectrum.

Future research directions are as follows: 1) Further extend the design methodology presented in this paper to other optical spectrum, including red-light range, yellow-light range, green-light range, and so on. 2) Conduct the DE optimization on computer clusters with parallelized fitness evaluation in order to enhance the convergence speed of algorithm. 3) The current reported negative index metamaterials in visible spectrum often operate at a very short wavelength range, such as in the red-light range, or in the yellow-light range separately. Therefore, how to broaden the LH frequency band of NIMs to cover the whole visible spectrum will be a vital step in order to realize the optical cloaking for the whole visible spectrum.