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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 17 — Aug. 18, 2008
  • pp: 12806–12818

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Synthesis design of artificial magnetic metamaterials using a genetic algorithm

P. Y. Chen, C. H. Chen, H. Wang, J. H. Tsai, and W. X. Ni  »View Author Affiliations


Optics Express, Vol. 16, Issue 17, pp. 12806-12818 (2008)
http://dx.doi.org/10.1364/OE.16.012806


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Abstract

In this article, we present a genetic algorithm (GA) as one branch of artificial intelligence (AI) for the optimization-design of the artificial magnetic metamaterial whose structure is automatically generated by computer through the filling element methodology. A representative design example, metamaterials with permeability of negative unity, is investigated and the optimized structures found by the GA are presented. It is also demonstrated that our approach is effective for the synthesis of functional magnetic and electric metamaterials with optimal structures. This GA-based optimization-design technique shows great versatility and applicability in the design of functional metamaterials.

© 2008 Optical Society of America

1. Introduction

Recently, there has been growing interest in the study of artificial electromagnetic materials, metamaterials [1–15

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996). [CrossRef] [PubMed]

]. Metamaterials are artificially synthesized periodic structures with lattice constant that is much smaller than the wavelength of the incident electromagnetic wave (EM wave), thus considered as effectively homogeneous media [3

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [CrossRef] [PubMed]

]. In general, these metamaterials are deliberately designed to manipulate the effective permittivity (εeff ) and permeability (µeff ) of the effectively homogenous media

Of particular interest is the magnetic metamaterial (metaferrites). The first magnetic metamaterial is the split-ring resonator (SRR) proposed by Pendry [2

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999). [CrossRef]

]. The SRR is expected to exhibit resonant magnetic response to the external EM wave, polarizing with the magnetic field parallel to the axis of the SRR (P-polarization). Therefore, the permeability can be negative over a finite frequency range around the resonance frequency. The effective permeability of the periodic arrays of SRR can be characterized by,

μeff (ω)= 1− ωmp2− ω m02 ω2− ω m02+i Γmω= μeff′ (ω)+ μeff″ (ω),
(1)

where ω0m is the magnetic resonance frequency, ωpm is the magnetic plasma frequency, and Γ m is the damping factor.

Furthermore, if both εeff and µeff are simultaneously negative, a negative-index metamaterial (NIM), also called left-handed metamaterial (LHM), can be realized [4

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000). [CrossRef] [PubMed]

]. The negative εeff can be produced by an array of thin metallic wires [1

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996). [CrossRef] [PubMed]

], which behave as the electric resonators. For the electric field of the external EM wave polarizing parallel to the wire, εeff can be characterized by:

εeff (ω)= 1− ωp2 ω2,
(2)

where ωp is the electric plasma frequency; the εeff is predicted to be negative when ω<ωp .

A composite material composed of the SRRs and wires can therefore possess negative effective refractive index ( n= μeff εeff) over a finite frequency range, thereby resulting in a NIM.

As a key component of the NIM, the magnetic metamaterials have been extensively studied. Besides the SRR, a number of candidate magnetic resonant structures have been proposed; these include hexagonal split-ring resonators [10

W. Zhu, X. Zhao, and N. Ji, “Double bands of negative refractive index in the left-handed metamaterials with asymmetric defects,” Appl. Phys. Lett. 90, 011911-1–3 (2007). [CrossRef]

], S-shaped resonators [11

H. Chen, L. Ran, J Huangfu, X. Zhang, and K. Chen, “Left-handed materials composed of only S-shaped resonators,” Phys. Rev. E 70, 057605-1–4 (2004). [CrossRef]

], double S-shaped resonators [12

H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Negative refraction of a combined double S-shaped metamaterial,” Appl. Phys. Lett. 86, 151909-1–3 (2005). [CrossRef]

], Ω-shaped resonators [13

K. Aydin, Z. Li, M. Hudlička, S. A. Tretyakov, and E. Ozbay, “Transmission characteristics of bianisotropic metamaterials based on omega shaped metallic inclusions,” New J. Phys. 9, 326–336 (2007). [CrossRef]

], quasi-periodic dendritic-cell [14

X. Zhou, Q. H. Fu, J. Zhao, Y. Yang, and X. P. Zhao, “Negative permeability and subwavelength focusing of quasi-periodic dendritic cell metamaterials,” Opt. Express 14, 7188–7197 (2006). [CrossRef] [PubMed]

], and labyrinth resonators [15

M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A: Pure Appl. Opt. 7, S12–S22 (2005). [CrossRef]

]. The traditional magnetic metamaterial design is based on the analysis of the equivalent LC-circuit, consisting of the plate capacitors and magnetic coils on a scale much smaller than the wavelength of incident wave [2

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999). [CrossRef]

]. However, this may be very complicated for irregular shapes and therefore restrict the magnetic metamaterial to regular shape. It is also difficult to find the optimal structures for the magnetic or other functional metamaterials using the intuition and empirical testing. To solve this problem, we, in this paper, intend to propose an intelligent computer-aided design (CAD) approach employing the genetic algorithm (GA) and the filling element methodology to synthesize the magnetic metamaterials with whatever regular or irregular structures. A representative design example, magnetic metamaterial with a desired permeability value of µeff =-1, is detailed. For the filling element structural design methodology, we use the typical filling square-pixel (FSP) method and a novel filling beehive-cell (FBC) method, which aims at improving the patch contact problem in the FSP method. Furthermore, different types of the magnetic and electric metamaterials with different desired material properties are studied to illustrate the versatility and applicability of our approach for the metamaterial synthesis designs. A more detailed implantation of the GA-optimization for the structural design of metamaterials will be described in the following section.

2. Genetic algorithm and structural design

The principle of how to automatically generate structures by computer is illustrated in Fig. 1; Fig. 1(a) shows the typical FSP method, which has widely been used for structural design problems [16–17

F. J. Ville, T. Cwik, Y. Rahmat-Samii, and M. Manteghi, “A parallel electromagnetic genetic-algorithm optimization (EGO) application for patch antenna design,” IEEE Trans. Antenna Propag. 52, 2424–2435 (2004). [CrossRef]

]; Fig. 1(b) shows the novel FBC method. For comparison, both methods will be used to synthesize a unit cell of the metamaterial. The incident field propagates along the x-direction, with E and H along the x- and y-directions, respectively. The metamaterial is formed by the elementary metallic patches printed on one side (x-z plane) of the dielectric substrate, which is a 1.5-mm-thick FR4 with relative permittivity εr =4.4. Since our interest frequency is in the range of 4~10 GHz, the artificial lattice constant is ax =ay =az =7.5 mm, which assures to be 1/4~1/10 the wavelength of the EM wave. We then discretize the unit cell into many elements (square-pixels or beehive-cells); each element can be filled by metal or in free space (ε0 =1). Here we define Ns and Nc as the number of elementary patches at each side for the FSP and FBC method, respectively. The structure is forced to be mirrored to ensure the 1D-symmetry of the metamaterial (see Fig. 1). Although this CAD approach seems to be ideal and a wide variety of structures can be discovered, the most critical problem is that the total number of possible structures is extremely huge. Take the FSP method for example, the total number of possible structures is 2 Ns(Ns+1)/2 (i.e. for Ns =7, there are more than 2.6×108 different structures) and it will increase exponentially with the number of pixels. Therefore, the full search method is not practical, especially when the number of the elements is very large. In this work, the simulation experiments of Ns =Nc =7 and 11 are performed. The search space would be 49-pixel and 21-pixel square patch for Ns =7 and 11, respectively, and that would be 37-cell and 91-cell beehive for Nc =7 and 11, respectively. For problems involving such a large search space, it is also impossible to land the result in the global optimum by the gradient-based local search method. As a result, the GA, as a direct and adaptive search technique, should be introduced to automatically generate the optimal structure. The GA is a very powerful and attractive optimization tool, and has been applied to several electromagnetic problems, such as the microwave antennas [16

F. J. Ville, T. Cwik, Y. Rahmat-Samii, and M. Manteghi, “A parallel electromagnetic genetic-algorithm optimization (EGO) application for patch antenna design,” IEEE Trans. Antenna Propag. 52, 2424–2435 (2004). [CrossRef]

], photonic crystals [18–19

J. W. Rinne and P. Wiltzis, “Design of holographic structures using Genetic Algorithm,” Opt. Express 14, 9909–9916 (2006). [CrossRef] [PubMed]

], ion-optics [20

P. Y. Chen, C. H. Chen, J. S. Wu, H. C. Wen, and W. P. Wang, “Optimal design of integrally gated CNT field-emission devices using a genetic algorithm,” Nanotechnology 18, 395203-1–10 (2007).

], and some metamaterial designs even in the synthesis design of metamaterials by FSP method [17

D. J. Kern, D. H. Werner, and M. Lisovich, “Metaferrites using electromagnetic bandgap structures to synthesis metamaterial ferrites,” IEEE Trans. Antenna Propag. 53, 1382–1389 (2005). [CrossRef]

] or automatic parameter extraction [21–22

D. H. Kwon and D. H. Werner, “Low-index metamaterial designs in visible spectrum,” Opt. Express 14, 9267–9272 (2007). [CrossRef]

].

Fig. 1. Illustration of (a) filling square-pixel (FSP) and (b) filling beehive-cell (FBC) method for the structural design of a metamaterial unit cell. The lattice constant ax =ay =az =a is set to 7.5mm. The dimensions of the metallic patches are w1 =0.8 mm, w2 =0.55 mm for Ns =N c=7, and w1 =0.5 mm, w2 =0.35 mm for Ns =Nc =11. The thickness of the substrate (FR4 with εr = 4.4) and the metallic patches are 1.5 mm and 0.035 mm, respectively.

The idea of the GA came from Charles Darwin’s theory of evolution, natural selection or survival of the fittest [23

J. Holland, Adaptation in Nature and Artificial System (Ann Arbor: The University of Michigan Press, 1975). CST Microwave Studio 2006.b http://www.CST.com.

]. In the GA, the structure of the metamaterial is encoded using a binary string filled with “1’s” and “0’s” representing metal and free space, respectively (see Fig. 1). This binary string is called “chromosome” and each 1’s and 0’s on the chromosome can be seen as a “gene.” At the start of GA, Nchromosome (here Nchromosome =80) chromosomes are randomly generated (each gene on the chromosome is randomly set to 1’s and 0’s with the uniform probability) to form the initial population, G0 . Each chromosome is then assigned a fitness value, F, providing an indication of quality of chromosome to determine whether this chromosome is the most probable to evolve towards a better next generation. Under the pressure of the fitness, GA will evolve, over time, toward the optimal solution. The fitness is defined as:

fitness=min(∣ μeff′+ Ra∣+∣ μeff″+ Rb∣)| 4GHz<f<10GHz
(3)

where µ′ eff and µ″ eff represents the real part and the imaginary part of µeff . Since our goal aims at producing negative magnetic metamaterial with µeff =-1, we must choose Ra =1 and Rb =0. We note that the imaginary part determines the absorption of the EM wave. Hence, for low loss, the real part should be negative unity while at the same time the image part should be zero. The commercial available software, CST Microwave Studio [24

X. Chen, Tomasz, M. Grzegorczyk, B. Wu, J. Pacheco, Jr., and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608-1–7 (2004). [CrossRef]

], is employed as the Maxwell’s equation solver, which is a finite-different time-domain (FDTD) simulation based on perfect boundary approximation and finite integration technique. The metallic patches with the thickness of 0.035 mm are modeled as PEC because in our interest frequency range the effect of the skin depth (δ~0.7 µm) can be neglected. Due to the periodicity of the metamaterials, we limit our simulation to a single unit cell with periodic boundary condition. After the transmission and reflection coefficients of each chromosome are calculated, the robust retrieval method [25

http://www.python.org.

] is applied to compute the effective impedance z, the effective refractive index n, and effective permittivity εeff , and the effective permeability µeff of the metamaterial.

Based on the fitness value given in Eq. (3), the next generation is produced by the “reproduction” process utilizing the biologically analogous operators of selection (S), crossover (C), and mutation (M). In these steps, we define Gk and Gk+1 as the current and the next generations, respectively. The operation steps of GA are described as follows:

  1. Selection: in this step, a pair of chromosomes is selected from the current generation as parents for mating. Here the tournament selection strategy is used, four chromosomes are randomly chosen and the best two individuals are selected as the parents. Therefore, chromosomes with higher fitness values are more likely to be selected for generating new chromosomes or children.
  2. Crossover: once the parents are selected, two chromosomes will interchange their gene material, with a crossover probability Pc =0.8, to create a pair of children. Here, we use the two-point crossover strategy, in which the parents will exchange the segments cut by two randomly selected points to create two children. The purpose of crossover is to rearrange the genes, thereby creating better combinations of genes to result in “fitter” chromosome.
  3. Mutation: mutation is also necessary to maintain the diversity in the population and explore the solutions which are not yet present, thus preventing the results to be trapped in the local minimum. Here, we use the uniform mutation, in which each gene could be mutated under a mutation probability Pm =0.08. In the case of binary coding “1s” will be inverted to “0” and vice versa.
  4. Elitist strategy: the top 5% chromosomes, as the “elite chromosomes,” from the current generation are preserved and directly inserted into a new generation. This procedure ensures the elite of each population survive to be used as the parents in the next generation.

These processes, including selection, crossover, and mutation are repeated until the size of the new generation is the same as the current generation. The relationship of the next generation and the current generation can be described as Gk+1 =M{C{S{Gk }}}. The GA processes are iterated until Ng (here Ng =50) generations are calculated. Since the GA’s result cannot guarantee the global optimum, three runs with different randomly generated initial population are performed and the best chromosome is saved. A computer intelligence technique framework, CITO lab, included the GA and middleware between the GA and the CST Microwave Studio, has been developed at National Nano Device Laboratories (NDL). This framework is completely written in the open-source programming language, Python [26

F. Bilotti, L. Nucci, and L. Vegni, “An SRR based microwave absorber,” Microwave Opt. Technol. Lett. 48, 2171–2175 (2006). [CrossRef]

].

3. Results and Discussion

Figure 2 shows the best fitness as a function of the number of generations. For all cases, it is seen that the best fitness eventually converges to a fixed value. The corresponding optimized structures for the FBC design at different generations during the evolution process of the GA are shown in the bottom of Fig. 2. For FBC method with Nc =7, the structure is eventually evaluated to a structure that is analogous to the typical double SRR, having one outer ring and one inner ring with oppositely-oriented splits. The insert of inner ring is to generate a large capacitance in the small gap between two rings, thus lowering the resonance frequency and concentrate the electric field to enhance the response [3

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [CrossRef] [PubMed]

]. It is surprising that the GA has such an ability to find out an optimized structure, nearly consisting with that designed by people. This also implies that double SRR might be the best suitable for FBC method with small number of elements (spatial resolution). It is also found from Fig. 2 that the value of the best fitness decreases with increasing the number of elements (search space is enlarged). This is valid for both the FSP and FBC methods.

Fig. 2. The best fitness plotted as a function of generation (top). The unit cells show the best structures (elite) at different generations during the evolution process of the GA (bottom).

Applying the GA with different structural design methodology, different GA-optimized structures are shown in Fig. 3. Figures 3(a) and (b) show the resulting structures using the FSP method with Ns =7 and Ns =11, respectively. The corresponding µeff and εeff are also shown in the right-hand side. In both cases, a symmetrical ring structure with a large capacitance at the split is presented. It is not hard to image that a symmetric ring with a distinct capacitance and an inductance will form a LC-circuit to response to the external magnetic field of the EM wave, thus producing the magnetic resonance. A typical resonant behavior in µeff (ω) can be clearly seen in both Fig. 3(a) and (b). At resonance, a considerably large positive µeff in the low frequency side of the resonance and negative µeff in the high frequency side of the resonance are obtained. In addition, an antiresonant behavior [7

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602-1–4 (2003). [CrossRef]

] of εeff accompanied with the resonant behavior of µeff is shown, in which the imaginary parts of µeff and εeff are opposite in sign. For Ns =7 the optimal values of µeff is given by µeff =-0.965+i0.469 (fitness=0.504) at 4.25 GHz, while for Ns =11 an improved optimized value of µeff is given by µeff =-0.992+i0.208 (fitness=0.216) at 4.05 GHz.

Fig. 3. GA-optimized unit cell and its corresponding µeff and εeff for the FSP method with (a) Ns =7 and (b) Ns =11, and the FBC method with (c) Nc =7and (d)Nc =11 (desired permeability: µeff =-1 over the range of 4~10 GHz).

Figures 3(c) and (d) show the resulting structures and the corresponding µeff and εeff using the FBH method with Nc =7 and Nc =11, respectively. For Nc =7, the double SRR-like structure is presented. The optimized values of µeff is given by µeff =-0.668+i0.319 (fitness=0.651) at 5.2 GHz. However, for Nc =11, a more complex resonant structure, as shown in Fig. 3(d), is presented, in which several capacitors and inductors are shunted together to form a LC-circuit, thus producing the magnetic resonance. As compared to the case of Nc =7, this complex structure has a stronger resonant behavior as well as an enhanced magnitude of µeff . The optimized value of µeff for Nc =11 has been improved to µeff =-1.011+i0.237 (fitness=0.248) at 6.8 GHz. The resonance frequency increases because more capacitive elements are shunted to build up the magnetic resonator. This phenomenon is analogous to that in multi-cut SRR: the resonance frequency increases with increasing the number of cuts of SRR.

Comparing two structural design methods, there are advantages for the FBC method. Firstly, the electric connection problem for two diagonally-neighbor elements in the FSP method can be improved. Moreover, for both small and large number of elements, the FBC design is found to result in a faster convergence speed than that of the FSP design. This is because the computer-generated pattern by the FBC method is much easier to evaluate toward a closed and conductive circuit due to the nature of beehive. As a result, the FBC method can improve electric connection, while at the same time, yield to a faster convergence speed. The single-processor execution time for the fitness evaluation (CST simulation) of a chromosome takes approximately 1.5~2 minutes (depends on the complexity of the structures) on a HP Pentium IV-3.06 GHz PC with 3.37GB RAM.

Based on the same concept in the prior example, we further use the GA to synthesize the magnetic metamaterials with the desired permeability values of µeff =-0.5 and µeff =-2 over the same frequency range. Here the value of Ra in fitness function (Eq. 3) is replaced by Ra =0.5 and Ra =2 to achieve the design goals of µeff =-0.5 and µef =-2, respectively. In the following, the FBC method will be used throughout this study.

Figure 4 shows the resulting structure and the corresponding µeff and εeff of the magnetic metamaterial with a desired permeability µeff =-0.5. The optimized values of µeff is given by µeff =-0.470+i0.055 (fitness=0.085) at 7.02 GHz.

Fig. 4. The GA-optimized unit cell for the metamaterial with permeability µeff =-0.5 and its corresponding µeff and εeff .

Figure 5 shows the resulting structure and the corresponding µeff and εeff of the magnetic metamaterial with a desired permeability µeff =-2; the optimized values of µeff is given by µeff =-1.955+i0.448 (fitness=0.493) at 5.53 GHz. To obtain a larger negative value of permeability, the magnitude enhancement of µeff is required. As can be seen in Fig. 5, the magnetic resonance becomes stronger as compared to the prior case.

Fig. 5. The GA-optimized unit cell for the metamaterial with permeability µeff =-2 and its corresponding µeff and εeff .

In contrast to the negative and low-loss magnetic metamaterials, a metamaterial with positive and large µ′ eff and µ″ eff can also be obtained in the low frequency side of the magnetic resonance. Therefore, magnetic metamaterial can also act as an effective electromagnetic absorber at the desired operating frequency [27

S. Enoch, G. Tayeb, F. Sabouroux, N. Guerin, and P. Vincent, “A Metamaterial for Directive Emission,” Phys. Rev. Lett. 89, 213902-1–4 (2002). [CrossRef]

]. This encourage us to use the GA to synthesize a lossy magnetic metamaterial with permeability µeff =2+i4. In this case, Ra =-2 and Ra =-4 is applied. Fig. 6 shows the resulting structure and the corresponding µeff and εeff of such a metamateria is given by µeff =2.067+i4.025 (fitness=0.092) at 5.08 GHz.

Fig. 6. The GA-optimized unit cell for the metamaterial with permeability µeff =2+i4 and its corresponding µeff and εeff .

So far, the object in all cases aims to search the optimized structure in terms of the desired permeability for all frequencies between 4 GHz and 10 GHz. Next, we intend to use the GA to synthesize a particular magnetic metamaterial at a fixed frequency under the same unit cell size. Here we take the design of a magnetic metamaterial with µeff =-1 at 5 GHz for example. In this case, the fitness is defined as:

fitness=min(∣ μeff′+1∣+∣ μeff"∣)∣ f=5GHz
(5)

Figure 7 shows the resulting structure and the corresponding µeff and εeff . From Fig. 7, we can find that, at 5.0 GHz, µeff =-0.970+i0.506 (fitness=0.536) is obtained.

Fig. 7. The GA-optimized unit cell for the metamaterial with permeability µeff =-1 at 5 GHz and its corresponding µeff and εeff .

In addition to the NIMs, there is also increasing interest in the zero-index and low-index metamaterials (ZIMs, LIMs) due to their fascinating physical properties and potential applications, such as the high directivity antenna according to Snell’s refraction law [28

D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, 041109-1–3 (2006). [CrossRef]

] and metamaterial invisibility cloak [9

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]

, 21

D. H. Kwon and D. H. Werner, “Low-index metamaterial designs in visible spectrum,” Opt. Express 14, 9267–9272 (2007). [CrossRef]

]. In these applications, like the operation of NIMs, zero or low refractive index are tuned through the manipulation of the µeff and εeff . Here we take µeff =0 as a design example to testify GA’s capability of synthesis functional metamaterial, but alter the fitness function to:

fitness= ( ∣ μeff′∣+ ∣ μeff"∣)∣ μeff′=min ( μeff′)
(6)

Our purpose is to design a zero or low permeability metamaterial. The resulting metamaterial and its µeff and εeff are shown in Fig. 8, where the minimum µeff saturates to zero at the high frequency side of the resonance and its value is 0.086+i0.595 (fitness=0.681) at 6.52 GHz.

Fig. 8. The GA-optimized unit cell for the metamaterial with permeability of zero and its corresponding µeff and εeff .

Since metamaterials are usually composed of strong magnetic or electric resonant structure, in the last part, we would like use the GA to synthesize an electric-LC (ELC) resonator with εeff =-1and a NIM or LHM with n=-1. We note that the electric metamaterial, as a discontinuous metallic pattern, could also possess a cut-wires-like electric resonant behavior [8

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402-1–4 (2004). [CrossRef] [PubMed]

]. The effective permittivity can characterized as:

εeff (ω)= 1− ωep2− ω e02 ω2− ω e02+i Γeω= εeff′ (ω)+ εeff" (ω)
(4)

where ωe0 is the electric resonance frequency, ωep is the electric plasma frequency, and Γe is the damping factor. When an electric metamaterial with a desired permittivity εeff =-1 is considered, we should define the fitness as:

fitness=min(∣ εeff′+1∣+∣ εeff"∣)∣ 4GHz<f<10GHz
(7)

The resulting metamaterial structure and its εeff and µeff are shown in Fig. 9. During the electric resonance, the optimized values of εeff is found to be -1.011+i0.001 (fitness=0.012) at 7.63 GHz. From Fig. 9, a resonant behavior similar to the magnetic resonance but in εeff (ω) is clearly seen. The antiresonant behavior of µeff accompanied by the resonant behavior of εeff is also observed.

Fig. 9. The GA-optimized unit cell for the metamaterial with permittivity of εeff =-1 and its corresponding εeff and µeff .

We further construct a composite material consists both the magnetic and electric resonator on the opposite sites of the dielectric substrate for making a NIM. With the magnetic resonator from Fig. 3(d) on the back site of the substrate, we use the GA to synthesize an electric resonator on the top site. Hence, the refractive index is expected to be negative through the combination of the electric resonator with εeff <0 and the magnetic resonator with µeff <0. To this end, the fitness is defined as:

fitness=min(∣ neff′+1∣+∣ μeff"∣)∣ 4GHz<f<10GHz
(8)

Figure 10 shows the optimized NIM as well as the retrieval µeff , εeff , n and z. At 6.90 GHz, n=-1.109+i0.326 (fitness=0.435) is obtained. We note that a dual-band is found in the curve of µeff . The first magnetic response, which is caused by the electric resonator on the top site, is weak (the peak is slightly below zero). This has been verified by removing the magnetic resonator on the back site. The second response at the negative-index frequency region is actually produced by the magnetic resonator.

Fig. 10. Left-handed side shows the unit cell of a NIM, which is composite material made of the GA-optimized electric and magnetic resonators on the opposite sides of the substrate; the right-handed side shows its corresponding µeff , εeff , n and z.

So far we have demonstrated the possibility, stability as well as flexibility of using AI, such as the GA in this paper, to design metamaterial with the permeability or permittivity almost fit to the desired values. We note that the difference between the fitness (simulated result) and the design goal (desired material properties) can be further reduced by optimizing the fixed design parameters for each GA-optimized structure, such as the size of the unit cell a and metallic patch w, the thickness t and permittivity of the dielectric substrate εr . The GA-optimized structures can promise to be fabricated by the standard circuit board lithography. Although, in this work, GA is employed to make metamaterial in the microwave region, these results have also implications for other regions of the electromagnetic spectrum, such as infrared or even optical frequencies. Our design approach would also potentially enable us to design versatile functional metamaterials not presented here (i.e. low-loss, high-absorption, dual-band, or wide-band (Δf/f) metamaterial at the desired frequencies).

4. Conclusion

We have illustrated that the GA can work well in the synthesis of the metamaterials with permeability of negative unity. Two structural design methodologies, the typical FSP and the novel FBC methods, are performed and the stability of the GA has been proved in both cases. It is also worth mentioned that the FBC method can improve electric contact problem and give a faster convergence speed, while the performance of the resulting structure is comparable to that by using the FSP method. To further demonstrate the stability and flexibility of our design approach, using the GA and FBC method, various metamaterials are synthesized; these include four low-loss magnetic metamaterials with permeability µeff =-0.5, -2 and 0, a lossy magnetic metamaterials with permeability µeff =2+4i, an electric metamaterial with permittivity εeff =-1, and a NIM with refractive index n=1. All these cases can be synthesized successfully with the material properties approximate to our design goals. With this intelligent CAD technique, we are able to design the functional metamaterials with the desired material properties according to the user-defined object function (fitness function in the GA). Such encouraging results have shed bright light on using artificial intelligence to make artificial metamaterials.

Acknowledgments

The authors acknowledge Prof. C. T. Sun of Department of Computer Science, National Chiao Tung University, Taiwan for discussions in genetic algorithm, evolutionary computation, and system development. We also want to thank Prof. T. J. Yen of Department of Material Science Engineering, National Tsin Hua University, Taiwan for the help in CST simulation. This work is funded by National Nano Device Laboratory, Taiwan.

References and links

1.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996). [CrossRef] [PubMed]

2.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999). [CrossRef]

3.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [CrossRef] [PubMed]

4.

D. R. Smith and N. Kroll, “Negative refractive index in left-handed materials,” Phys. Rev. Lett. 85, 2933–2936 (2000). [CrossRef] [PubMed]

5.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]

6.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a Negative Index of Refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

7.

T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E 68, 065602-1–4 (2003). [CrossRef]

8.

T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, “Effective medium theory of left-handed materials,” Phys. Rev. Lett. 93, 107402-1–4 (2004). [CrossRef] [PubMed]

9.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). [CrossRef] [PubMed]

10.

W. Zhu, X. Zhao, and N. Ji, “Double bands of negative refractive index in the left-handed metamaterials with asymmetric defects,” Appl. Phys. Lett. 90, 011911-1–3 (2007). [CrossRef]

11.

H. Chen, L. Ran, J Huangfu, X. Zhang, and K. Chen, “Left-handed materials composed of only S-shaped resonators,” Phys. Rev. E 70, 057605-1–4 (2004). [CrossRef]

12.

H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, “Negative refraction of a combined double S-shaped metamaterial,” Appl. Phys. Lett. 86, 151909-1–3 (2005). [CrossRef]

13.

K. Aydin, Z. Li, M. Hudlička, S. A. Tretyakov, and E. Ozbay, “Transmission characteristics of bianisotropic metamaterials based on omega shaped metallic inclusions,” New J. Phys. 9, 326–336 (2007). [CrossRef]

14.

X. Zhou, Q. H. Fu, J. Zhao, Y. Yang, and X. P. Zhao, “Negative permeability and subwavelength focusing of quasi-periodic dendritic cell metamaterials,” Opt. Express 14, 7188–7197 (2006). [CrossRef] [PubMed]

15.

M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, “Left-handed metamaterials: detailed numerical studies of the transmission properties,” J. Opt. A: Pure Appl. Opt. 7, S12–S22 (2005). [CrossRef]

16.

F. J. Ville, T. Cwik, Y. Rahmat-Samii, and M. Manteghi, “A parallel electromagnetic genetic-algorithm optimization (EGO) application for patch antenna design,” IEEE Trans. Antenna Propag. 52, 2424–2435 (2004). [CrossRef]

17.

D. J. Kern, D. H. Werner, and M. Lisovich, “Metaferrites using electromagnetic bandgap structures to synthesis metamaterial ferrites,” IEEE Trans. Antenna Propag. 53, 1382–1389 (2005). [CrossRef]

18.

J. W. Rinne and P. Wiltzis, “Design of holographic structures using Genetic Algorithm,” Opt. Express 14, 9909–9916 (2006). [CrossRef] [PubMed]

19.

J. Goh, I. Fushman, D. Englund, and J. Vuckovic, “Genetic optimization of photonic band structures,” Opt. Express 15, 8218–8230 (2007). [CrossRef] [PubMed]

20.

P. Y. Chen, C. H. Chen, J. S. Wu, H. C. Wen, and W. P. Wang, “Optimal design of integrally gated CNT field-emission devices using a genetic algorithm,” Nanotechnology 18, 395203-1–10 (2007).

21.

D. H. Kwon and D. H. Werner, “Low-index metamaterial designs in visible spectrum,” Opt. Express 14, 9267–9272 (2007). [CrossRef]

22.

G. Mumcu, M. Valerio, K. Sertel, and J. L. Volakis, “Applications of the finite element method to designing composite metamaterials,” International conference on electromagnetic in advanced applications , 818–821 (2007) [CrossRef]

23.

J. Holland, Adaptation in Nature and Artificial System (Ann Arbor: The University of Michigan Press, 1975). CST Microwave Studio 2006.b http://www.CST.com.

24.

X. Chen, Tomasz, M. Grzegorczyk, B. Wu, J. Pacheco, Jr., and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E 70, 016608-1–7 (2004). [CrossRef]

25.

http://www.python.org.

26.

F. Bilotti, L. Nucci, and L. Vegni, “An SRR based microwave absorber,” Microwave Opt. Technol. Lett. 48, 2171–2175 (2006). [CrossRef]

27.

S. Enoch, G. Tayeb, F. Sabouroux, N. Guerin, and P. Vincent, “A Metamaterial for Directive Emission,” Phys. Rev. Lett. 89, 213902-1–4 (2002). [CrossRef]

28.

D. Schurig, J. J. Mock, and D. R. Smith, “Electric-field-coupled resonators for negative permittivity metamaterials,” Appl. Phys. Lett. 88, 041109-1–3 (2006). [CrossRef]

OCIS Codes
(350.4010) Other areas of optics : Microwaves
(160.3918) Materials : Metamaterials

ToC Category:
Metamaterials

History
Original Manuscript: January 8, 2008
Revised Manuscript: March 15, 2008
Manuscript Accepted: March 17, 2008
Published: August 8, 2008

Citation
P. Y. Chen, C. H. Chen, H. Wang, J. H. Tsai, and W. X. Ni, "Synthesis design of artificial magnetic metamaterials using a genetic algorithm," Opt. Express 16, 12806-12818 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-12806


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References

  1. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996). [CrossRef] [PubMed]
  2. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999). [CrossRef]
  3. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000). [CrossRef] [PubMed]
  4. D. R. Smith and N. Kroll, "Negative refractive index in left-handed materials," Phys. Rev. Lett. 85, 2933-2936 (2000). [CrossRef] [PubMed]
  5. J. B. Pendry, "Negative refraction makes a perfect lens," Phys. Rev. Lett. 85, 3966-3969 (2000). [CrossRef] [PubMed]
  6. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a Negative Index of Refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
  7. T. Koschny, P. Markos, D. R. Smith, and C. M. Soukoulis, "Resonant and antiresonant frequency dependence of the effective parameters of metamaterials," Phys. Rev. E 68, 065602-1-4 (2003). [CrossRef]
  8. T. Koschny, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, "Effective medium theory of left-handed materials," Phys. Rev. Lett. 93, 107402-1-4 (2004). [CrossRef] [PubMed]
  9. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, "Metamaterial electromagnetic cloak at microwave frequencies," Science 314, 977-980 (2006). [CrossRef] [PubMed]
  10. W. Zhu, X. Zhao, and N. Ji, "Double bands of negative refractive index in the left-handed metamaterials with asymmetric defects," Appl. Phys. Lett. 90, 011911-1-3 (2007). [CrossRef]
  11. H. Chen, L. Ran, J Huangfu, X. Zhang, and K. Chen, "Left-handed materials composed of only S-shaped resonators," Phys. Rev. E 70, 057605-1-4 (2004). [CrossRef]
  12. H. Chen, L. Ran, J. Huangfu, X. Zhang, K. Chen, T. M. Grzegorczyk, and J. A. Kong, "Negative refraction of a combined double S-shaped metamaterial," Appl. Phys. Lett. 86, 151909-1-3 (2005). [CrossRef]
  13. K.  Aydin, Z. Li, M.  Hudli�?ka, S. A.  Tretyakov, and E.  Ozbay, "Transmission characteristics of bianisotropic metamaterials based on omega shaped metallic inclusions," New J. Phys. 9, 326-336 (2007). [CrossRef]
  14. X. Zhou, Q. H. Fu, J. Zhao, Y. Yang, and X. P. Zhao, "Negative permeability and subwavelength focusing of quasi-periodic dendritic cell metamaterials," Opt. Express 14, 7188-7197 (2006). [CrossRef] [PubMed]
  15. M. Kafesaki, T. Koschny, R. S. Penciu, T. F. Gundogdu, E. N. Economou, and C. M. Soukoulis, "Left-handed metamaterials: detailed numerical studies of the transmission properties," J. Opt. A: Pure Appl. Opt. 7, S12-S22 (2005). [CrossRef]
  16. F. J. Ville, T. Cwik, Y. Rahmat-Samii, and M. Manteghi, "A parallel electromagnetic genetic-algorithm optimization (EGO) application for patch antenna design," IEEE Trans. Antenna Propag. 52, 2424-2435 (2004). [CrossRef]
  17. D. J. Kern, D. H. Werner, and M. Lisovich, "Metaferrites using electromagnetic bandgap structures to synthesis metamaterial ferrites," IEEE Trans. Antenna Propag. 53, 1382-1389 (2005). [CrossRef]
  18. J. W. Rinne and P. Wiltzis, "Design of holographic structures using Genetic Algorithm," Opt. Express 14, 9909-9916 (2006). [CrossRef] [PubMed]
  19. J. Goh, I. Fushman, D. Englund, and J. Vuckovic, "Genetic optimization of photonic band structures," Opt. Express 15, 8218-8230 (2007). [CrossRef] [PubMed]
  20. P. Y. Chen, C. H. Chen, J. S. Wu, H. C. Wen, and W. P. Wang, "Optimal design of integrally gated CNT field-emission devices using a genetic algorithm," Nanotechnology 18, 395203-1-10 (2007).
  21. D. H. Kwon and D. H. Werner, "Low-index metamaterial designs in visible spectrum," Opt. Express 14, 9267-9272 (2007) [CrossRef]
  22. G. Mumcu, M. Valerio, K. Sertel, and J. L. Volakis, "Applications of the finite element method to designing composite metamaterials," International conference on electromagnetic in advanced applications, 818-821 (2007) [CrossRef]
  23. J. Holland, Adaptation in Nature and Artificial System (Ann Arbor: The University of Michigan Press, 1975). CST Microwave Studio 2006.b http://www.CST.com.
  24. X. Chen, Tomasz, M.  Grzegorczyk, B. Wu, J. Pacheco, Jr., and J. A. Kong, "Robust method to retrieve the constitutive effective parameters of metamaterials," Phys. Rev. E 70, 016608-1-7 (2004). [CrossRef]
  25. http://www.python.org.
  26. F. Bilotti, L. Nucci, and L. Vegni, "An SRR based microwave absorber," Microwave Opt. Technol. Lett. 48, 2171-2175 (2006). [CrossRef]
  27. S. Enoch, G. Tayeb, F. Sabouroux, N. Guerin, and P. Vincent, "A Metamaterial for Directive Emission," Phys. Rev. Lett. 89, 213902-1-4 (2002). [CrossRef]
  28. D. Schurig, J. J. Mock, and D. R. Smith, "Electric-field-coupled resonators for negative permittivity metamaterials," Appl. Phys. Lett. 88, 041109-1-3 (2006). [CrossRef]

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