Simulation results of near infrared (100- to 200-THz) fishnet-structure negative-index metamaterials (NIMs) with single and multiple functional layers exhibit bi-anisotropy - inhomogeneous asymmetry - due to the presence of a sidewall-angle. The influence of sidewall-angle resulting from realistic fabrication processes is investigated through the retrieved effective parameters by both a three-dimensional finite-difference time-domain (FDTD) method and a rigorous coupled wave analysis (RCWA).
© 2009 Optical Society of America
Artificially structured composite metamaterials consist of sub-wavelength sized structures that exhibit unusual electromagnetic properties, not found in nature. Research on metamaterials that display the properties of negative refraction, artificial magnetism and negative permeability (permittivity) has seen impressive growth since the first experimental verification [1,2] along with theoretical predictions of potential applications [3,4]. One of the most attractive features of metamaterials is to obtain negative refraction, termed left-handed (LH) materials or negative-index metamaterials (NIMs), over a frequency band. In artificially fabricated materials with permittivity (ε= ε′ + iε′′) and permeability (μ= μ′ + iμ′′), the necessary condition for achieving a negative index is ε′·μ′′ + μ′·ε′′ < 0 and lower losses are achieved if both the real parts are less than zero ε′ < 0, μ′ < 0 . For useful application, interest has been focused on reduction of loss in NIMs. A first candidate to reduce the loss is to design new structured NIMs (optimize the structure) to have negative real parts of ε and μ simultaneously so that losses become small. Another is to approach three dimensional (bulk) rather than planar structures in NIMs to improve the figure of merit (FOM = −Re(n)/Im(n)), that is, to lower Im(n), by stacking multiple functional layers .
The fishnet structured NIM is composed of a 2-dimensional square periodic array of rectangular holes penetrating through a metal-dielectric-metal film stack. The dimension of NIMs with N functional layers based on the fishnet structure  for the optical frequency regime should be at least below ~1 μm (pitch) and on the order of 100×N nm (thickness of NIM). The present fabrication techniques for making one- to multiple-functional layer NIMs, include interferometric lithography (IL) [8,9], E-beam lithography (EBL) [10,11], focused ion beam (FIB) , nanoimprint (NIL) , and so on. In general, the processing procedure to fabricate the fishnet NIMs is to define the hole-size using a polymetric material, usually by lithographically defining polymer posts, followed by deposition of the constitutive materials and dissolution of the polymer (liftoff processing). This processing (fabrication of posts: multi-layer deposition: liftoff) often gives rise to significant sidewall-angles (SWAs), resulting from continuous growth of the tops of the posts during the deposition. NIMs with SWAs are not symmetric structures, but rather are bi-anisotropic, exhibiting different reflectivities at normal incidence from the two sides of the film stack. This fabrication-induced SWA effect requires a modified method to retrieve the effective parameters [14,15] in place of the conventional approach that is appropriate only to symmetric, vertical sidewall structures . A three dimensional finite-difference time-domain (FDTD) approach is used in this work. An Au (gold) - SiO2 (silicon dioxide) - Au sandwich fishnet structure with 1- to 5-functional layers (FLs) is modeled. No substrate is present in the simulation to allow clearer demonstration of the effects of the bi-anisotropy.
The geometrical parameters of the multi-functional layered fishnet NIMs are indicated in tilted view in Fig. 1(a) and listed in Table 1. The orthogonal pitches of the 2D gratings px (pitch along x̂-axis) and py (pitch along ŷ -axis) are both fixed at 800 nm. The thicknesses of Au/SiO2/Au are fixed at 20/20/20 nm, respectively. The rectangular hole size (2a, 2b) is fixed at 65% (520 nm) / 40% (320 nm) of the pitch for the long / short sides, that is, the linewidths of the gratings along the x̂, ŷ axes are 280 / 480 nm, respectively. As discussed above, a free floating sample with no substrate is evaluated.
We performed simulations of an ideal multi-functional layered fishnet NIM with SWAs using both the commercial three dimensional finite-difference time-domain (FDTD) solver (CST Microwave Studio, Computer Simulation Technology GmbH, Darmstadt, Germany)  and a rigorous coupled wave analysis (RCWA) [18,19]. Comparable results were obtained for both techniques; the results shown in this paper were obtained by CST. In the CST simulator, a single unit cell was simulated as shown in Fig. 1(a), with appropriate boundary conditions including both the transverse magnetic field (Ht) equal to zero (perfect magnetic conductor: PMC) in the ŷ, ẑ plane and transverse electric field (Et) equal to zero (perfect electric conductor: PEC) in the x̂, ẑ plane, stimulating a TEM plane wave propagating in the ẑ direction. We find the complex frequency dependent S parameters S 11, S 12, S 21, and S 22, where the subscript 1(2) represents the waveguide port at larger (smaller) aperture. The simulated transmission (∣S 21∣2 = ∣S 12∣2), reflectances (∣S 11∣2, ∣S 22∣2) and reflection anisotropy (∣S 22∣2 − ∣S 11∣2) are plotted in Fig. 2. The direction of polarized incoming light (incident electric field) is parallel to the narrower stripe width (wx) between apertures (ŷ axis) as shown in Fig. 1(a).
The refractive index of the dielectric material (SiO2) used as a spacer between Au (gold) layers in NIMs taken as n SiO2 = 1.5 along with a simple Drude model for the gold dielectric function given by ε = 1−ω 2 p/ω(ω + iωc), with plasma frequency ωp = 9.02 eV and collision frequency ωc = 0.0269 eV .
A common method to retrieve the effective constitutive parameters is based on the complex transmission and reflection coefficients (S-parameters). The standard retrieval method assumes isotropic constitutive parameters (ε, μ) for a symmetric structure . However, this fails if the unit cell of the NIM is not symmetric in the propagation direction as a result of the sidewall-angles induced during fabrication. In case of a NIM with SWAs, the S-parameters, especially S 11 and S 22 are different depending on the propagation direction of incident light with respect to the unit cell (S 11 ≠ S 22, S 21 = S 12), e.g. the structure exhibits inhomogeneous asymmetry or bi-anisotropy . The constitutive relations for fishnet-structured NIMs with SWA-induced bi-anisotropy in the x̂, ŷ plane is
This 3D constitutive relation can be simplified to a 1D case for the restricted polarization relevant to the experimental measurements as shown in Fig. 1(a) because there is only a y (x) component in electric (magnetic) field. Therefore, 1D constitutive equation is derived from Eq. (1):
Where c, ε 0, μ 0 and ξ, are the speed of light, permittivity and permeability of free space and the material bi-anisotropic parameter, which are related by n 2 = εy μx-ξ 2. This result was used previously to interpret experiments on deliberately anisotropic metamaterial structures .
Structures with different SWAs were modeled to investigate the effect of the SWA on the strength and position of the effective parameters – effective refractive index and permeability. From 1 to 5 functional layers, the wavelength at the minimum value of Re[μeff] decreases as the SWA increases as shown in Figs. 3(b) and 3(d). This tendency can be qualitatively interpreted by the magnetic resonance of an equivalent LC circuit: a larger SWA corresponds to smaller capacitance and inductance , which in turn leads to a shorter resonance wavelength. As shown in Fig. 3(d), the wavelength of the minimum value of Re[μeff] is linearly dependent on the SWA. The trend of a decreasing minimum value of Re[μeff] can be explained as a more complicated coupling between magnetic and electric dipoles. In an ideal fishnet structured NIM with vertical sidewalls (the bi-anisotropic parameter ξ for fishnet NIMs with zero sidewall-angle is identically zero, independent of number of functional layers). The structure is composed of independent electric atoms and magnetic atoms: the electric response results from the array of thin metal wires parallel to the electric field direction and the magnetic response from pairs of metal stripes separated by a dielectric spacer along the direction of magnetic field that support an anti-symmetric current as shown in Fig. 4(a). However, in realistic fishnet structured NIMs with a fabrication-induced sidewall-angle, the NIMs are bi-anisotropic In the thin metal wires (electric response), a magnetic dipole in the x̂ -direction is induced by an electric field Ē in the ŷ -direction, resulting from the disparity of the current distribution in the two metal films, so the magnetic and electric dipoles induced by Ē are coupled to each other. Figure 5 shows that the bi-anisotropic parameter ξ for multiple functional layered fishnet NIMs as a function of the sidewall-angle. Figure 6 shows the current density at a specific frequency (at the minimum value of Re [μeff]. From Figs. 6(a) and 6(c), two and three fishnet structured NIMs with vertical sidewalls have a symmetric current distribution in the upper and lower Au plates, but NIM with SWA (20°) show an unbalanced current distribution as shown in Figs. 6(b) and 6(d). Similarly, the electric dipoles (PH) induced by H̄ (magnetic loop response) now include a net ŷ -component, so the electric and magnetic dipoles induced by H̄ are also mutually coupled as shown in Fig. 4(b). An equal current distribution is still obtained in ideal NIMs (vertical sidewalls) with 2, 3 FLs as shown in Figs. 6(a) and 6(c). For this case there are no electric dipoles induced by H̄ in the ŷ -direction. However, the sidewall-angle effect shown in Figs. 6(b) and 6(d) results in the asymmetric current distribution resulting from unequal Au plate size. It also induces coupling between electric and magnetic dipoles as shown in Fig. 5(b).
Next, it is of interest to study the extracted real part of the effective refractive index depending on the SWA and number of functional layers. In Figs. 3(a) and 3(b), the Re[μeff] of a single functional layered NIM is positive, but Re[neff] is negative because the real part of the effective permittivity has a large negative value and overall the structure satisfies the necessary condition for negative refraction (ε′·μ′′ + μ′·ε′′ < 0). The magnitude of negative refractive index in NIMs with 1 FL depends on the sidewall-angle, that is, the minimum value of Re[neff] increases as the SWA increases as shown in Fig. 3(a). The reason for the increase is clear: as the sidewall-angle increases, the absolute values of Re[εeff] decrease (not shown) due to the decrease of the Au linewidth along the direction of electric field. The wavelength at the minimum of Re[neff] decreases with increase of sidewall-angle, which is a result of the blue-shifted resonance peaks of both Re[μeff] and Re[εeff]. Two- to five-functional layer fishnet structured NIMs with SWA exhibit negative Re[μeff] and Re[εeff], i.e. are double negative materials. Also, the blue-shifted resonance frequency of effective permittivity and permeability as mentioned above accounts for decrease of the wavelength at minimum effective refractive index in multi-functional layered NIMs with the increase of the sidewall-angle.
In conclusion, we have numerically demonstrated the influence of a fabrication-induced sidewall-angle on the effective parameters for one- to five-functional-layered negative-index metamaterials based on a fishnet structure. For the application of negative-index metamaterials, the bulk structure (one of candidates) for NIMs is attractive because of a lower loss (Im[n]). However, the simulation results show that as the number of functional layers in NIMs is increased, the impact of any sidewall-angle should be understood, especially at optical frequencies due to the high aspect ratios (height/width) required. This is shown in Fig. 7, which shows the FOM, quality indicator for multiple layer NIMs as a function of the sidewall-angle. As the number of layers is increased the fabrication tolerance should be improved to minimize the sidewall-angle and retain the improvement associated with the multiple layers.
This work was supported by DARPA under the University Photonics Research Center program. We are grateful to Dr. Shuang Zhang for useful technical discussions.
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