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We use spectroscopic ellipsometry to investigate the angular-dependent optical modes of fishnet metamaterials fabricated by nanoimprint lithography. Spectroscopic ellipsometry is demonstrated as a fast and efficient method for metamaterial characterization and the measured polarization ratios significantly simplify the calibration procedures compared to reflectance and transmittance measurements. We show that the modes can be well identified by a combination of comparing different substrates and considering the angular dependence of the Wood’s anomalies. The lack of angular dispersion of the anti-symmetric gap-modes does not agree with the model and requires further theoretical investigation.
©2012 Optical Society of America
1. Introduction
The rapid expansion of the field of metamaterials has been facilitated by the development of new techniques to fabricate nanostructured metallic arrays. Nanoimprint lithography (NIL) is one such technique, allowing the relatively fast and cost effective production of arrays of split ring resonators and fishnet metamaterials [1]. These are designed to exhibit effective magnetic resonances and a negative effective refractive index at optical frequencies. Simultaneous with fabrication advances, optical characterization of metamaterials has also developed, although progress in this endeavor has undoubtedly lagged, partly due to the dimensions of the samples fabricated thus far. The availability of relatively large area metamaterials (cm2) produced by NIL [1,2] now allows accurate optical measurements by well-established plane-wave reflection and transmission techniques.
The standard optical characterization of metamaterials uses combined normal incidence reflection and transmission measurements of both amplitude and phase (the S-parameter method) [3]. At optical frequencies an experimental determination of the absolute phase is extremely demanding [4]. In many cases the optical properties are determined by parameter retrieval from calculated data using full wave techniques such as Rigorous Coupled Wave Analysis (RCWA) [5]. Despite the potential advantages of obtaining additional phase information, the use of ellipsometry to characterize metamaterials is not widespread [6]. The development of this application is therefore of importance for the metamaterials community. There are also to date very few investigations of the angular dependence of fishnet metamaterials [7].
In this work we characterize a fishnet metamaterial fabricated by NIL using spectroscopic ellipsometry (SE) and transmission spectroscopy. To our knowledge, these are the first large area samples with effective magnetic-type resonances below wavelengths of 800 nm. A complete optical characterization should consider the angular-dependent optical response, which is particularly important for ellipsometry measurements. This complicates any attempts at parameter retrieval if the material is anisotropic [8]. However, we will show here that oblique incidence aids in the identification of the physical origin of the optical modes. We additionally show that a comparative study of the same metamaterials on different substrates (silicon and glass) assists in mode identification.
2. Metamaterial design, fabrication and spectroscopic characterization
The metamaterial design is based on that by Garcia-Meca et al. [5]. It is a three layer stack (metal, insulator, metal) periodic square mesh array, shown schematically in Fig. 1(a) . Three samples were fabricated by nanoimprint lithography, similar to the process described in [1] consisting of either a single silver layer (sample A) or a functional fishnet layer consisting of three layers of Ag/SiO2/Ag (samples B and C). Sample A and B are deposited on a silicon substrate with native oxide, and sample C is deposited on a borofloat glass substrate. All layers of all samples are patterned with the grid structure of Fig. 1(a). The nominal dimensions of the structures are period d = 365 nm, hole side length a = ax = ay = 205 nm, film thickness h = 40 nm (Ag) and s = 20 nm (SiO2), in all structures. From the scanning electron microscope image of sample C (Fig. 1(b)) it can be observed that the side lengths of the holes are larger than the nominal values due to the fabrication process, in particular those of the upper layers. The actual values measured from the images are a = 240 ± 5 nm for the bottom layer and a = 310 ± 5 nm for the upper layer.
Fig. 1 (a) Schematic of the fishnet. (b) Scanning electron micrograph of the three layer sample on glass (sample C). The scale bar is 500 nm.
The normal incidence reflection and transmission properties of the structure were simulated assuming the nominal dimensions (with a sidewall slope of 10°) using RCWA. Figure 2(a) shows the simulated values in the wavelength range near the “magnetic” resonance (the origin of which is discussed further below), compared with measured transmission data (also discussed further below). Using the standard retrieval method [3] we calculated the effective refractive index for normal incidence, showing a wavelength region of negative real part of the effective refractive index between 750 and 790 nm. Having obtained the standardized criteria for the fishnet metamaterials under investigation by this well established method, it is the key contribution of this work to complement them by the use of large-scale ellipsometry.
Fig. 2 (a) Simulated reflection (red solid line), transmission (blue solid line) compared to the measured transmission (black dotted line). (b) Effective real (black line) and imaginary (red line) refractive index of the fishnet design on a glass substrate. Input dimensions are equal to sample C, with a simulated sidewall angle of 10°.
3. Optical characterization
Ellipsometry is the method of choice for the experimental determination of the permittivity of bulk and thin film materials. Ellipsometry measures obliquely incident reflected and transmitted polarization ratios. Oblique incidence lifts the polarization degeneracy and expands the number of measurable parameters to four complex values (for both reflection and transmission). Thus even without phase measurements, four real values are available by which the complex permittivity ε, and permeability µ, may be retrieved. However to simplify the normalization procedures, which can be particularly demanding for small samples, ellipsometry uses the ratio, ρ, of the polarized reflection coefficients. In the Jones formalism one defines the polarization states as orthogonal electric field components Ep and Es. Reflection of an incident light ray from a surface is expressed by the Jones matrix
where matrix elements are the reflection coefficients and the subscripts r and i denote the reflected and incident rays. In isotropic materials the off-diagonal elements, rps and rsp, are zero. The diagonal elements rpp and rss may then be simply written rp and rs, the ratio of which defines the ellipsometric angles, Ψ and Δ, asThe angles Ψ and Δ correspond to the amplitude ratio and the phase difference of the reflection coefficients, respectively. If the film and/or substrate are optically anisotropic, and the measurement is not performed along an optical axis, a proportion of incident p-polarized light is converted to s-polarized light (and vice versa) and the off-diagonal elements of the Jones matrix are non-zero. They may be determined using generalized ellipsometry or Mueller matrix ellipsometry (MME). In this work we will confine our measurements to the high symmetry axes.
The samples were measured by normal-incidence unpolarized transmission spectroscopy and variable-angle spectroscopic ellipsometry (VASE) with an angle of incidence (AOI) from 45° to 75° at 2° intervals. The transmission spectra of sample C is shown in Fig. 3 . The maximum transmission peak is observed at a wavelength near 650 nm and the plasma frequency transmission peak at a wavelength of 320 nm. There are 5 resonant absorption bands at wavelengths of 785, 580, 525, 420 and 365 nm (and a weaker shoulder at 480 nm). The expected magnetic resonance is the 785 nm band. Based on the array period we can attribute the 365 nm and 525 nm bands to the first order Rayleigh wavelengths at the air superstrate and glass substrate, respectively [9]. The second order Rayleigh wavelength at the glass interface is expected near 375 nm and thus is merged with the first order air wavelength. The origin of the bands at 420 and 580 nm will be discussed below.
Representative VASE data (Ψ and Δ) for the three samples at an AOI of 55° are shown in Fig. 4 . The Ψ data may be interpreted as showing dips and peaks for p-polarized and s-polarized modes, respectively. If one considers that a perfect conductor would generate a constant Ψ value of 45° across the whole spectrum, then s-modes appear as peaks in Ψ with values greater than 45° (in the case that p- and s-modes do not significantly overlap).
Fig. 4 Ellipsometric spectra for the three samples (a) A, (b) B and (c) C, at an incident angle of 55°.
4. Discussion
For a physical interpretation of the optical modes of a two-dimensional periodic arrays of holes under oblique incidence, we follow the discussion by Garcia-Vidal et al. [10]. The main features in such arrays arise from Wood's anomalies. As shown by Hessel and Oliner [11], these may be divided into two subsets; (1) surface modes at the interface between a conductor (permittivity εm) and an insulator (permittivity εa), which are also known as surface electromagnetic waves (SEW), and (2) diffraction orders which become parallel to the surface and are “passed off” into the other orders, corresponding to the Rayleigh condition. At long wavelengths metals behave as perfect electrical conductors (PEC) with, while in the visible region εm is finite and SEWs are described as surface plasmon polaritons (SPP). The dispersion relation for a SPP propagating at the interface of a dielectric and a metal is
where is the vacuum wavevector. For a PEC εm>>εa and , such that the SPP and Rayleigh wavelengths coincide. Equation (3) may thus be generalized to k = k0X, where for the Rayleigh condition and for the SPP. An SPP- Bloch wave (BW) may be excited on a grating if matches an in-plane lattice vector, , with additional momentum from the in-plane component of the plane wave , such thatwhere i and j are integers and θ is the angle makes with the normal. At non-normal incidence there will thus be a minimum in the reflection at λ = P(X ± sinθ) for the ( ± 1,0) modes and λ = P(X2−sin2θ)1/2 for the (0, ± 1) modes [10].The above theory has been used in numerous works to characterize metallic gratings [12], periodic hole arrays [13] and fishnet metamaterials [14]. In our metamaterials we must consider the four different interfaces; the metal-superstrate (Ag-air), the metal-substrate (Ag-Si or Ag-glass), and the two internal interfaces (Ag-SiO2). If both top and bottom internal interfaces are similar, their modes couple due to the fact that the separation between the layers is small enough to allow overlap of the evanescent fields [14], forming a so-called gap SPP. The modes of the gap SPP hybridize and split into low frequency (odd) and high frequency (even) modes. In the low-frequency limit the odd mode may be approximated by the linear relation [15]
where in our structure s is the thickness of the SiO2, h is the thickness of the silver layers and λp is the plasma wavelength of silver. Note that the term in the square brackets is a constant for a given geometry and does not show angular dispersion. The odd mode is considered to have a magnetic resonance character, and may be described by an effective µ. Near to the resonance frequency µ may be negative which, when combined with a negative ε (e.g. a metal below the plasma frequency) results in a negative refractive index metamaterial (NIM).The SPP theory assumes that the films are flat and continuous, which in our case is violated. It is also well documented that the SPP dispersion is significantly modified if the depth of the grooves is of the order of 20 nm [16]. In our samples the hole size is large in relation to the period of the array, and the depth of the holes is around 40 nm. In theory, as the size of the holes increases, the fraction of metal in the film reduces and the BW gradually transits from the SPP condition for a solid metal film to that of the light line in the medium, i.e. the Rayleigh condition. The application of the SPP theory to fishnet structures with large holes is therefore questionable and the Rayleigh formula may be more appropriate.
Figure 5 shows a comparison of the Ψ data for the three samples plotted as a function of AOI. The plot allows us to assess the angular dispersion of the modes observed in the ellipsometry measurements. Often such data is plotted as an E-kx dispersion diagram, which contains exactly the same information as shown here, however our kx range is not extensive enough to warrant a full dispersion plot. The three plots have the same grey-scale which shows light (dark) areas above (below) 45°; therefore the dark and light bands may be attributed to absorption and scattering of p- and s-polarized light, respectively.
Fig. 5 Grey scale plots of the Ψ spectra as a function of the incident angle for the three samples (a) A (b) B and (c) C.
In Fig. 5(a) there is a strong p-polarized mode at the screened plasma wavelength (320 nm), as is commonly observed in ellipsometry measurements [17]. The next strongest feature is an angular-dependent mode at a wavelength around 700 nm. We attribute this to the (−1,0) mode at the air-silver interface. This is also prominent in Figs. 5(b) and 5(c). A weaker mode of similar shape around 1600 nm in Figs. 5(a) and (b) is the corresponding (−1,0) mode at the silver-silicon interface. In sample C this mode is absent, replaced by the (−1,0) mode at the glass-silver interface just below 900 nm. The (1,0) and (0, ± 1) modes at the Si interface are also easily observed in (a). Comparing Figs. 5(a) and (b) (both on Si substrate) the addition of the SiO2 and second Ag layer results in significantly more modes, as expected. The intensity of the Ag-Si modes is strongly reduced; only the (−1,0) mode is still visible at a wavelength around 1600 nm. The most prominent new dark band is around 1150 nm and corresponds to the coupled (−1,0) modes at the SiO2-Ag interfaces, which are red-shifted from the case of a semi-infinite dielectric due to the coupling across the gap. This is the gap-mode outlined in Eq. (5). The (1,0) and (0,+/−1) modes of this resonance are also visible at wavelengths slightly shorter than 600 nm and slightly longer than 800 nm, respectively.
In Fig. 6(b) the SPP and Rayleigh wavelengths Eqs. (3) and (4) are shown as a function of the incidence angle, in comparison with the measured data (Fig. 6(a)) for sample A. The dielectric functions of silver and silicon are taken from Refs [18]. and [19], respectively. Only the ( ± 1,0) and (0, ± 1) modes are plotted, although a number of weaker modes are experimentally observed which are ascribed to higher order modes. Note that we observe only uncoupled modes at the separate interfaces due to the high contrast between the permittivity of silicon and air. Although the theory matches almost perfectly the resonance at the air interface, the Si interface modes are slightly mismatched. The first reason for this is the presence of a native oxide layer on the silicon which has the effect of blue shifting the resonances. The second reason is noted above; the SPP theory assumes a continuous film, while we have comparatively large and deep holes in the film. Thus it is not surprising that the modes are observed closer to the Rayleigh wavelength than the SPP wavelength.
Fig. 6 Comparison of the measured Ψ data (a) with theory (b) for the single layer silver hole array on silicon (sample A). The solid lines represent Eqs. (3) and (4), while the dashed lines correspond to the Rayleigh wavelengths. The blue and black modes are at the Si-Ag and Ag-air interfaces, respectively. The green dotted line corresponds to the dipolar Mie resonance for a 130 nm silver sphere.
The effect of the appreciable hole depth results in a Fabry-Perot type resonance in the holes, which is the origin of the large transmission peak observed in Fig. 3. In our films we must also consider the polarizability of the silver strips. Of note is the strong mode at a wavelength of 420 nm. One is tempted to attribute this band to the (0,1) mode at the Air-Ag interface. However this mode is predicted at wavelengths shorter than the period d = 365 nm. It also should appear as an absorption in s-polarized light [20], while we observe a p-polarized absorption here. We attribute this mode to the localized resonance of the metal strips. For the dimensions of our structures, the metal strips are thinner than the skin depth, resulting in a polarization of the metal close to the plasma wavelength (similar to dipolar Mie resonances). For comparison, an isolated single silver sphere of radius 130 nm has a dipolar Mie resonance at 420 nm. We may expect some red-shift due to the presence of the other wires.
Figure 7 shows the angular dependence of Ψ data for sample B (one functional fishnet layer on Si), compared with the predictions of Eqs. (3) and (4). The blue and black curves show the Rayleigh wavelengths at the Ag-Si and Ag-Air interfaces, as before. The red curves show the ( ± 1,0) and 0, ± 1) modes of the internal SPP. To generate these curves we empirically matched the square root condition in Eq. (4) to the (−1,0) curve in the experimental data, and subsequently generated the other curves using Eqs. (3) and (4). We note that the gap SPP condition from Eq. (5) predicted wavelengths much lower than those we observe. We speculate that the equation is not valid for the case presented here where the layers are perforated with holes larger than half the period. The bright lines (s-polarized modes) show little or no angular dispersion. The magnetic resonance at 785 nm wavelength is close to the (−1,1) gap mode and the dark mode at longer wavelengths is ascribed to the (0,1) mode, although the SPP band shows angular dispersion which is not present in the measured data. Note that all the bright bands are in general paired by a dark band (p-polarized mode) at a slightly higher wavelength. In the case of the “magnetic resonance” at 785 nm the bright band fades with increasing AOI and the dark band grows stronger. The band pairs may be due to the asymmetry of the structure attributed to the presence of the substrate.
Fig. 7 Comparison of the measured Ψ data (a) with theory (b) for the fishnet on silicon (sample B). The blue and black modes are at the Si-Ag and Air-Ag interfaces, respectively, while the red modes are at the internal Ag-SiO2 interfaces.
Figures 8(b) and 8(c) show the same analysis for sample C for wavelengths between 300 and 1000 nm, with the transmission data from Fig. 3 additionally included in Fig. 8(a). In Fig. 8(c) the black, blue and red curves correspond to the Rayleigh condition at the air, glass and internal interfaces (generated as described above), respectively. The agreement is good. The green dotted lines also show the normal incidence transmission minima which very closely match the bright, angle-independent modes in (b). We attribute the bright bands at wavelengths of 365 nm and 525 nm to the (0,1) modes of the air and glass interfaces, respectively (note that the 525 nm band was absent in sample B), while the 420 nm band is the localized resonance. We also note that the bright (s-polarized) and dark (p-polarized) bands near 800 nm are more strongly split than in sample B.
Fig. 8 Data for sample C (fishnet on glass). Note the reduced wavelength range. (a) Transmission data from Fig. 3. (b) Measured Ψ data as a function of angle. (c) Simulated modes. The black mode is the Ag-Air interface, while the blue lines are the Ag-Glass modes. The red lines are at the internal Ag-SiO2 interfaces. The green dotted lines corresponds to the transmission minima.
Finally we attribute the 580 nm band to the (1,1) gap mode of the structure (which was identified in [5] as the magnetic resonance), although we would expect this mode to show significant angular dispersion. We note that in [5] the lack of angular dispersion was also observed, and was “ascribed to the large size of the apertures compared to the unit cell size and the hybridization of the SPP mode” with a Fabry-Perot type resonance in the holes. This effect clearly requires further theoretical investigation.
5. Conclusion
This work used the well developed technique of spectroscopic ellipsometry to investigate the angular-dependent optical modes of fishnet metamaterials fabricated by nanoimprint lithography. Spectroscopic ellipsometry is demonstrated as a fast and efficient method for metamaterial characterization and the use of polarization ratios significantly simplifies the calibration procedures compared to reflectance and transmittance measurements. We showed that the modes could be well identified by a combination of comparing different substrates and considering the angular dependence of the Wood’s anomalies. Due to the large hole sizes the modes are closer to the Rayleigh anomaly wavelengths than the SPP wavelengths. The lack of dispersion in the magnetic gap-modes does not agree with the model and requires further consideration. Our work establishes spectroscopic ellipsometry as a valuable tool for the future design and characterization of large-area metamaterials.
Acknowledgments
The authors acknowledge funding by the European Community’s 7th Framework Programme under grant agreement no. 228637 NIM NIL (www.nimnil.org). C.H. gratefully acknowledges a postdoctoral fellowship from the German Academic Exchange Service (DAAD). G.I. is grateful for the support from the Serbian Ministry of Education and Science Project No. OI171005. The financial support by the Senatsverwaltung für Wissenschaft, Forschung und Kultur des Landes Berlin, the Bundesministerium für Bildung, Wissenschaft, Forschung und Technologie is gratefully acknowledged. We thank K.-J. Eichhorn and R. Schultz from the IPF, Dresden, for assistance with the ellipsometry measurements.
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