## Abstract

Optical metamaterials with a negative value of the refractive index can be fabricated by means of patterning techniques developed for microelectronics. One of those is a layered metamaterial, where the electric and magnetic response comes from coupled parallel subwavelength size wires. We simulate propagation of EM waves through such a metamaterial. Its properties depend on the density of pairs of nanowires oriented in parallel in one layer. There is a tradeoff between high transmittance and large negative refractive index value *n*. The smaller is the density of nanowires; 1° – the narrower the range of frequencies, where *n* is negative; 2° – the less negative is *n*; 3° – the higher is the transmission.

©2006 Optical Society of America

## 1. Introduction

Veselago predicted the existence of materials where the vectors of electric **D** and magnetic **B** induction are anti-parallel to the electric **E** and magnetic **H** fields [1]. The idea of metamaterials with a negative value of the refractive index has been brought to reality in several ways. A few years ago artificial materials of various structures with electromagnetically active cells in the forms of thin wire 3D lattices, Swiss rolls as well as split ring resonators combined in one unit with a dipole were proposed [2–3] and since then are under study. They show negative refraction properties in the spectral range from microwaves up to a frequency of hundreds of THz [4, 5]. Feasible metamaterial structures active in the optical range might be fabricated by means of patterning techniques developed for microelectronics [6].

The idea of a scalable and potentially isotropic in 3D metamaterial originates from an early paper of Lagarkov and Sarychev [7]. They considered properties of composites containing elongated metal inclusions embedded in a host dielectric using the effective medium approximation [8]. A metal-in-dielectric metamaterial structure composed of randomly distributed electromagnetically active units in the form of parallel pairs of subwavelength size wires was developed by Podolskiy *et al*. [9,10] and Shalaev *et al*. [4]. Independently, the role of coupled rotated metal nanostripes in an optically active layered chiral medium was considered by Svirko *et al*. [11].

Recently, there has been a growing interest in the fabrication of a multilayer metamaterial in the form of periodic arrays of parallel nanowires [4, 5, 12–14]. A single layer, while interesting from a scientific point of view, will probably have fewer applications when compared to a layered structure. We simulate the metamaterial properties that depend on the density of pairs of nanowires oriented in parallel in all layers. A compromise is necessary between high transmittance and large negative refractive index *n*. The smaller is the density of nanowires; 1° – the narrower the range of frequencies, where *n* is negative; 2° – the less negative is *n*; 3° – the higher is the transmission. Both the first and the second points imply that the density should be high. The third and the most important from a practical point of view implication demands that the contradictory requirements must find a middle ground.

## 2. Simulation details

A single silver wire of a square cross-section is 2*b* = 60 nm thick and 2*l* = 420 nm long. The separation of two coupled wires is *a* = 60 nm. The accepted dimensions are consistent with the theoretical model [9,10] and our recalculations [12]. Production of arrays of wires of that size is possible with nowadays nanoimprint and soft lithography [6]. The wires are embedded in a medium of refractive index *n* = 1.51 (dielectric permeability *ε*_{d}
= 2.28 + 0i). In an experiment it is advisable to differentiate the dielectric: between wires to choose one with high *ε*_{d}
and fill the remainder of the cell with another one of *ε*_{f}
< *ε*_{d}
. In a single layer of the metamaterial pairs of wires are arranged in a rectangular grid of lattice constant ratio of 1:7 that repeats the aspect ratio of a single wire. The values of these lattice constants are chosen to achieve fill factors from 8% to 20%. Simulations are performed for one and three layers of the metamaterial. There is no relative shift of the second and third layers with respect to the first one. Four interlayer separations are considered from 400 nm to 550 nm every 50 nm.

According to Drude’s model of dispersion the dielectric function depends on the frequency *ω* as follows

Simulations are made with our own implementation of the FDTD method [15, 16]. We use the following parameters: *ε*
_{∞}, = 3.70, *ω*
_{p} = 13673 THz and Γ = 27.35 THz calculated by Sönnichsen [17] from experimental data on reflection and transmission of silver films obtained by Johnson and Christy [18].

In the direction of propagation *z* the simulation volume is 5,000 nm long with transversal dimensions, width *x* and height *y*, varying to achieve the desired fill factor. To model an infinite layer of nanowires Uniaxial Perfectly Matched Layers (UPML) are used as boundary conditions in the direction of propagation *z* and periodic boundary conditions along the *x* and *y* axes. The space discretization step (spatial resolution) equals *Δr* = 5 nm and the time step *Δt* = *Δr*/*2c* = 8.34 × 10^{-18} s, where *c* is the speed of light. We simulate the propagation of a plane wave that is linearly polarized in the *z* direction (along the wires) for 10,000 simulation steps and then record the field intensity, that is Poynting vector length and the discrete Fourier transform of the electric field.

## 3. Theoretical assessment of refractive index

We recalculate analytically the induction of dipole moments of nanowires with rectangular instead of elliptical cross-sections. The resulting equations for the permittivity and permeability of a single layer of the metamaterial differ only slightly from those obtained by Lagarkov and Sarychev [7]

where 2*l* is wire length, 2*b* is its width, *a* is separation of coupled wires, *p* is fill factor of a layer, *k* is wave vector, and function *f*(Δ) depends on frequency and takes into account the skin effect of conducting wires

*γ* is dimensionless relaxation parameter,

*G*
^{2}=Ω^{2}+2*i*/*γ* where g and *Ω* are frequencies, the latter one dimensionless

with permittivity *ε*_{d}
of the dielectric between wires; permittivity *ε*_{f}
of the dielectric filling the rest of the cell volume; *ε*_{m}
and *μ*_{m}
, permittivity and permeability of the metal; *σ*_{m}
conductivity of the metal; and *J*_{i}
Bessel functions of first kind and zero and first order.

Figure 1 shows the dispersion curves for permittivity *ε* = *ε*_{r}
+ *iε*_{i}
[Fig. 1(a)], permeability *μ* = *μ*_{r}
+*iμ*_{i}
[Fig. 1(b)] and the effective refractive index *n* of a single layer of the metamaterial calculated for the above geometry assuming a fill factor *p* = 12%. When the condition *ε*_{r}
|*μ*|+*μ*_{r}
|*ε*| < 0 is satisfied [19] the effective index of refraction is negative. For *p*= 12% this takes place for a range of wavelengths from 1.13-1.28 μm and is illustrated in Fig. 1(c). Increase of the fill factor value brings three consequences: 1° the spectral range of negative refractive indices widens, 2° the left zero-valued point of *n*(*ω*) plot shifts towards smaller wavelengths, and 3° the absolute value of the negative refraction index becomes larger.

## 4. Transmittance of a single layer vs. fill factor

Figure 2 shows the attenuation of light *a* = 10ln(${\mathit{\text{II}}}_{0}^{-1}$) by a single layer of the metamaterial, where *I* is the intensity of transmitted light integrated over the cross-section in xy plane and *I*_{0}
is the intensity of the illuminating wave. It is calculated 270 nm behind the second wire. Lines correspond to layers with different fill factors from *p* = 6% to 20%. For each curve, two attenuation minima are observed: the first at wavelengths λ = 2.1-2.3 μm and the second in the range λ = 1-1.75 μm.

The first one, with almost the same spectral position for all fill factors is attributed to absorption by wire pairs at the resonance frequency. The resonance frequency observed in FDTD simulations is different than that theoretically assessed in the previous section. The difference results from rough assessment of electric and magnetic moments of coupled wires, where dipole approximation is used and higher electric and magnetic moments are neglected. Moreover, in wires with square cross-section the surface current distribution is not uniform in contrast to those with circular cross-section. Also, the Telegrapher’s equation used to calculate currents in wire pairs holds for infinite lines, which is not exactly the case in our metamaterial.

The attenuation minima from the second set shift their positions with the changing wavelength-to-layer lattice period ratio. Clearly, the metamaterial behaves as a diffraction grating that reflects light.

Interference of light transmitted by a periodic object with wires elongated along the y axis forms self-image-like field distributions at distances *z* = *2mL*^{2}*λ*^{-1}
behind the grating, where *L* denotes the structure period along the *y* axis and *m* is integer. Periodic distribution of light intensity for λ = 0.85μm calculated behind the metamaterial layer with *p* = 16%, that is a lattice period *L* = 1.05μm, is shown in Fig. 3. This quasi-Talbot effect is observed for wide range of wavelengths *Δλ*, where *Δλ*/*L* ≈ 0.35.

A single layer of the metamaterial exhibits a negative index of refraction at wavelengths λ for which the imaginary part of *n* increases up to its maximum value. For the same spectral range the real part of *n* is negative. For the case of considered fill factors we expect *n* to be negative for the range λ= 1.9-2.2 μm. At that spectral range strong resonance interaction of radiation with coupled wires modifies electric D and magnetic B inductions what results in strong attenuation. For a fill factor *p* = 8% we calculate that attenuation is about -10 dB (i.e. transmission of 35%). Layers with fill factors bigger than *p* = 14% transmit less than 25% of incident light, thus stacking them leads to very high absorption.

## 5. Transmittance of three layers vs. fill factor

Figure 4 shows the intensity transmittance of three layers for three fill factor values and four separation distances between the layers. The thickness of each layer is 180 nm and wires are separated with *ε*_{d}
medium. The space between layers of thickness 400, 450, 500, 550 nm is filled with *ε*_{f}
dielectric, in our case *ε*_{f}
= *ε*_{d}
. The transmittance of three layers resembles that of a single one, however, the transmission minima are broader. For high fill factors three layers completely absorb light in the spectral range where the negative index is expected. For *p* = 8% three layers have transmission greater than 10% at the spectral range *λ* = 1.9-2.2 μm.

## 6. Phase shift

In a metamaterial slab with negative *n* the phase of light is delayed in comparison to that of transmitted through a similar dielectric layer with a positive index. The analysis of phase changes indicates whether within a narrow spectral range the metamaterial slab has a negative refractive index or not. This is done for the spectral region where resonant interaction is observed. The procedure is justified because on planes parallel to the direction of propagation for wavelengths smaller than 1.5 μm the dispersion of phase (normalized to 2π and expressed in %) exceeds 1%, which is the accepted cutoff value. Waves from the remaining range (1.5-3.0 μm) have smaller variations across phase planes and after interacting with the metamaterial they remain plane.

Figure 5 shows phase shifts *Δϕ* = *ϕ*
_{∥} − *ϕ*
_{⊥} between **E** field orientations parallel and perpendicular to wires in a single metamaterial layer. At the wavelength range where attenuation considerations predicted negative *n* values we observe negative phase advancement for all fill factors. Although they are small (up to -20 deg), they increase with bigger fill factors due to an increasingly larger negative value of *n*.

## 7. Conclusions

We simulate propagation of EM waves through metamaterial layers where the electric and magnetic response comes from pairs of parallel nanowires distributed regularly with different density. To reach negative refractive index values and high intensity transmittance of such a composite the density of coupled wires and their geometry has to optimized. The wire size and pair separation determine the spectral position of the resonance. The fill factor decides upon the balance between high intensity transmittance and a large negative value of *n*. The smaller is the density of nanowires; 1° - the narrower the range of frequencies, where *n* is negative; 2° - the less negative is *n*; 3° - the higher is the transmission. Additional parameters of a metamaterial are the dielectric permittivities of a material between wires ed and that of a medium filling the remainder of a cell *ε*_{f}
> 1.The relation *ε*_{f}
< *ε*_{d}
assures a large negative refractive index. Moreover, a proper choice of the geometry of a layer is crucial to eliminate diffraction effects for negatively refracted wavelengths.

## Acknowledgments

This research was sponsored by Polish grant 3 T08A 081 27. We also acknowledge the support from the 6FP Network of Excellence Metamorphose, EC Contract #500 252.

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