1. Introduction

Metamaterial absorbers (MMAs) have been experimentally demonstrated across much of the electromagnetic spectrum from radio frequencies [1] to the optical [2], and at nearly every band in-between [3–7]. Indeed the nascent field of MMAs is aided by the fact that unit cell geometries demonstrated in one band of the electromagnetic spectrum are easily scaled to other bands - a general feature of metamaterials. Further, unit cell and sub-unit cell symmetries provides great flexibility to absorb various polarized and wavelength sensitive portions of incident light. Thus in only a handful of years MMAs have additionally been shown to be broadband, polarization sensitive, dynamic, thin and efficient [8, 9]. Metamaterial absorbers are commonly fashioned from a metal-insulator-metal geometry, (see inset to Fig. 1) where a dielectric (or other insulting layer) is sandwiched on one side by a ground plane, and on the other by an electrically resonant metamaterial - ELC for example [10,11]. Light impinges on the ELC side of the structure and can be efficiently absorbed over some relatively narrow range of frequencies.

 

Fig. 1 Top panel shows the experimental (open symbols) and simulated absorbance (solid curves) as a function of incident angle for both transverse electric (blue) and transverse magnetic (red) polarization. Inset shows a schematic of a top view of the metamaterial absorber and a SEM picture of the fabricated sample. Experimental (left panels) and simulated (right panels) frequency dependent absorption for various incident angles (θ) for both TM (grey solid curves) and TE (black dash curves) polarizations. The labels P, A, and B indicate the principal absorbing mode, the angle independent mode (at the blue dashed line), and the angle dependent mode (tracked by the red arrow) of the absorber respectively.

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The MMA operates by achieving an impedance match to free space, while at the same time providing a large loss component - both of which can be described as arising due to a frequency dependent effective permittivity (ε_eff(ω)) and effective permeability (μ_eff (ω)) provided by the metamaterial unit cell. The ε_eff(ω) response arises from a combination of a Drude-like term - due to the ground plane - and a Lorentz resonant term that comes from the geometry of the front metamaterial structure. The μ_eff(ω) term is due to regions of parallel metallic segments of the unit cell, both in the ELC and the ground plane, that are perpendicular to a normally incident k-vector. An incident time-varying magnetic field may thus couple to these portions of the unit cell and drive anti-parallel currents which may be described by μ_eff(ω). There is an implicit assumption of homogenization in the application of an effective medium approach. This assumption is not always valid, and criteria for the homogenization assumption are outlined in the literature [12]. Notably, the resonant nature of metamaterials creates regions where the wavelength can be shortened or lengthened compared to the free space wavelength λ₀. Our analysis present here demonstrates that the resonant wavelength λ_r is nearly 20 times larger than the lattice parameter (a) of our metamaterial where we are describing the TM surface electromagnetic wave. We are thus in a regime where it is possible to calculate the local effective material parameters ε_eff(ω) and μ_eff(ω).

Although metamaterial absorbers are a burgeoning research area, with great progress having been made over the last several years, a mechanism has yet to be identified responsible for the drastic difference between the angular dependence of the polarization sensitive absorption. For example, it has been demonstrated that MMAs maintain a high absorption for transverse magnetic (TM) polarized waves, even for relatively large angles of incidence. In-fact, in some designs the absorption only begins to fall-off when incident angles are about 80°, or greater, from the surface normal. On the other hand, for transverse electric (TE) polarized waves the absorption starts to diminish around 45°. A common explanation for the difference in polarization dependent absorption is rotation out of plane of the magnetic field for TE polarized waves and thus the effective magnetic response of the metamaterial unit cell is reduced leading to a lowering of the absorption [9]. In-contrast, the incident magnetic field is always in-plane for the TM polarized case. Another description explains the discrepancy as a mismatch of impedance as TE incidence reaches broad angles, leading to poor coupling with the absorbing surface [13]. Although these descriptions seem to resolve the discrepancy, the physics underlying such a mechanism has yet to be put forth.

Some insight into the fundamental difference between the polarization dependent absorption may be gained by noting that the surface of MMAs may support surface electromagnetic waves (SEWs) which may be coupled to with TM polarized electromagnetic waves [14]. The existence of surface waves for TM incident light may then explain the difference in angular dependent absorption, i.e. a SEW may dissipate energy through propagation loss or re-radiation thus leading to an increase in absorption. Here we theoretically, computationally and experimentally investigate the role surface electromagnetic waves and their connection to the absorptive process in MMAs. We unambiguously demonstrate that the excellent broad-angle TM performance of MMAs is due to coupling to a surface mode that is both supported by, and well described via, the effective optical constants of the metamaterial.

2. Design, fabrication and experimental results

An infrared metamaterial absorber was fabricated using thin-film processing combined with UV photolithographic patterning. We use a 190 nm thick layer of benzocyclobutene (BCB) as the dielectric spacer and 80nm of gold for the metamaterial elements and ground plane. A scanning electron microscope (SEM) image of a unit cell of the fabricated MMA is shown as the inset to Fig. 1. The cross shaped metamaterial has a linewidth of w = 500 nm, side length of L = 1.95 μm, and a unit cell size of a = 3.2 μm. In order to study the broad angular dependence of the MMA, we fabricated relatively large samples with a total lateral size of 2.54 × 2.54 cm². In Fig. 1 we show the experimental (left panels) and simulated (right panels) absorbance, (defined as A(ω) = 1−R(ω) where R(ω) is the reflectance), for both TE and TM polarization, at four incident angles of θ =15°, 30°, 45°, and 60°.

There are several noteworthy features observable in Fig. 1, and we begin our discussion with consideration of the experimental data at 15° (the top left panel of Fig. 1). In Fig. 1 we note the presence of three well defined peaks, which we term here Mode P at 6.14μm, Mode A at 4.83μm, and Mode B at 4.12μm. We find that for both TE (dashed black curve) and TM (solid gray curve) polarization, the MMA obtains a peak value of A= 99.8% at 6.14 μm and, outside of this range, the absorbance is relatively low with the exception of several small features observed at 6.67, 6.96, 7.10, 7.99, and 9.57 μm. These features are due to natural absorptions in the BCB dielectric spacer layer [15].

We next draw attention to the first of three significant features, Mode B. The absorbance presented in Fig. 1 shows that Mode B (tracked by the red arrow) is angle dependent in nature, and no such mode can be observed in the TE polarized absorbance. The second significant feature, Mode A, occurs at 4.83μm (dashed vertical blue line) and does not shift with incident angle. We note that Mode A - like Mode B - only appears in the TM polarized absorbance. Third we consider Mode P, the principal absorbing peak. At both 30 and 45 degrees incidence the main absorptive feature at 6.14 μm is largely unchanged for both TE and TM curves. At an incident angle of 60° the primary absorbance peak for TE polarization has begun to fall off and we find a value of 86%.

In order to gain insight into the various absorptive features observed in experiment, we perform 3D full wave electromagnetic field simulations. The MMA was simulated with dimensions identical to that of the fabricated sample. We use a Drude model for all metallic components [16], and a complex frequency independent dielectric constant of ε = 2.06+i0.12 for BCB. Unit cell boundary conditions are assigned which enable us to computationally investigate the angular and polarization dependent absorption. Simulations provide the complex scattering parameters and we calculate the absorption as A = 1 −R = 1 −|S₁₁|². The right panels of Fig. 1 show A(ω) for incident angles consistent with the experimental measurements. We find that in simulation the the features of interest are in good agreement with the experimental results. The first observation is that the modes termed A and B are present in only the TM polarization. Furthermore Mode A remains fixed, while Mode B shows a redshift as a function of incident angle, consistent with the experimental results. It is noted that there is a slight discrepancy in the character of Mode B in the simulation in comparison to the experimental results. We attribute this primarily to the contrast between the perfect lattice parameter attainable in simulation compared to a spread in lattice parameter (random errors) for the experimental sample, which leads to broadening of the width of Mode B in experiment.

Having described the characteristics of the experimental and computational results we find them to be in good agreement, and shift our focus to their implications, beginning with a discussion of Mode B. It is well-known that periodic structures can support spoof surface-plasmon-polaritons for TM polarized incident waves, and though early works considered perfect conductors, more recent works have shown their existence in real metals [14, 17, 18]. Excitation of SPPs requires the momentum of incident light, k_i, to match that of the SPP [19,20], i.e. k_spp = k_i + k_G, where k_spp and k_G are k-vectors of SPP and Bloch waves (reciprocal lattice vector), respectively. By re-writing k_i in terms of free space light k₀ as k_i = k₀sinθ, a strong angular dependence is expected. It is important to again stress that Mode B only appears for TM polarized waves and its peak redshifts as a function of incident angle. If Mode B is indeed a spoof SPP, then its peak value should be a function of sin(θ). To examine this possibility we plotted the wavelength of peak absorption for Mode B against the sine of the incident angle (not shown). We find that with a linear fit our y-intercept, k_G = 2π/a, equals a value of 1.72 × 10⁶(m⁻¹) giving a = 3.65 μm, which agrees qualitatively with the lattice spacing (3.2μm) of the fabricated samples. We thus conclude that the angular dependent feature Mode B is a spoof SPP, which is only related to the periodicity of the metamaterial, and we do not discuss this further [21]. In the following section we provide an additional computational study of Modes A and P in an effort to investigate the existence of a SEW contributing to the absorbance of these modes.

3. Computational results

In order to reveal the nature of Mode A, and the origin of the difference between TM and TE polarized absorption, we proceed with two further sets of analyses. First, to understand the properties of the MMA and whether it could support a SEW, we investigate the dispersion relation of the fabricated metamaterial in conjunction with a more subwavelength structure. Second, we computationally investigate the existence of surface waves directly, through full wave simulation. In the following section we elucidate the results of this twofold analysis and verify that SEWs play a key role in MMA behavior.

We can gain insight into the nature of the MMA optical response by performing eigenmode simulations which permit calculation of the dispersion relation. In Fig. 2(b) we plot the dispersion relation for k-vectors parallel (k) to the MMA surface. Beginning from zero frequency, the dispersion curve reveals light-like behavior over the range plotted with two flat-band, SEW like modes occurring near 6μm and 4.7μm. An important detail of the simulation is the inability of the eigenmode solver to incorporate loss. Although loss is neglected we find that Fig. 2(b) predicts the occurrence of both Mode A and Mode P. Comparison to Fig. 2(a) shows that the predicted SEW modes align well with the experimental data at 15°. We conclude from Figs. 2(a) and (b) that there exist SEW modes near 6μm and 4.7μm for the TM polarized case.

 

Fig. 2 (a) Experimental absorbtion at 15° for transverse-magnetic (TM) polarization. (b) Lossless dispersion relation simulated for the k-vector parallel (k_||) to the metamaterial absorber surface for TM polarized light. (c) Dispersion relation calculated from Eq. (1) for TM polarized light is shown for the real (black curve) and imaginary (green curve) parallel k-vector. Note the blue and red horizontal lines indicate the peak value of Modes A and P respectively. (d) Simulated absorbance and (e) dispersion relation for a more subwavelength MMA for TM polarization, where ω_a = c(2π/a). Inset: subwavelength design investigated. (f) A zoomed portion of (c) showing detail. Vertical purple line shows where k_1|| intersects the dashed horizontal red line, i.e the peak value of Mode P.

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We next seek to deepen our understanding of the principal absorbing peak, Mode P, following the theoretical work of N. V. Ilin et al. [25]. Utilizing effective optical constants of the MMA – calculated via an effective medium approach – we obtain an analytical form for the dispersion relation. The metamaterial cross structure of the absorber possesses π/2 rotational symmetry, (see Fig. 1), and thus we describe the optical properties of the MMAs by a set of effective parameters: $\tilde{\epsilon }={\epsilon }_{xx}\widehat{x}+{\epsilon }_{xx}\widehat{y}+{\epsilon }_{zz}\widehat{z}$ and $\tilde{\mu }={\mu }_{yy}\widehat{x}+{\mu }_{yy}\widehat{y}+{\mu }_{zz}\widehat{z}$, where the anisotropy of the structure is explicitly taken into account. The dielectric function ε_zz is approximated as ε₀ since the metamaterial is sub-wavelength in this direction, i.e. the height of the metamaterials is ~ λ_P/20, where λ_P is the free space wavelength of Mode P, i.e. 6.14μm [26]. The analytical form of the dispersion relation for TM waves traveling along the air/MM interface can be written as [25],

In Fig. 2(c) we plot both the real (solid black curve) and imaginary (solid green curve) portion of the k-vector k_|| for the TM polarized dispersion relation calculated from Eq. 1. We find that our analytical solution shows that k_1|| for TM waves disperse linearly for low frequencies but realizes a SEW-like flat band mode around 6.2μm. Our analytical calculations include loss and thus we notice a small discrepancy between the location of this band compared to the loss-less simulation shown in Fig. 2(b). Notably the dispersion shown in Fig. 2(c) permits calculation of λ_r as λ_r = 2π/k_1|| - denoted by the purple vertical line in 2(f). The method we utilize, based on based on Maxwell’s equations [25], gives λ_r/a = 19.1 which indeed meets the criteria necessary to establish local effective medium parameters ε_eff(ω) and μ_eff(ω) [12]. Thus an effective medium description of the metamaterial absorber predicts a TM surface mode which occurs precisely at the same frequency as the absorption peak for normal incidence.

We next simulate a more deeply sub-wavelength MMA (λ_P/a=3.7, see inset to Fig. 2(e)) in order verify that our surface wave at Mode P is not related to any underlying periodicity of the unit cell. In Fig. 2(d) we show the simulated absorbance at an incident angle of 60 degrees and find that there exist analogous modes to Mode P and Mode A from our primary MMA under study. These modes lie at 0.275ω/ω_a and 0.313ω/ω_a, respectively, where ω_a is the frequency of the light-line at the first Brillouin zone, i.e. ω_a = (2π/a)c where a is the periodicity and c is the speed of light. We again perform an eigenmode simulation (see Fig. 2(e)) and find two flat bands, one at 0.275ω/ω_a and the other at 0.312ω/ω_a, which matches quite well to values obtained from our free-space simulated results shown in Fig. 2(d). From the dispersion analysis of the two structures, we conclude that the SEW Mode P is well described by the optical constants of the MMA and not related to the periodicity of the unit cell.

Our experimental measurements, eigenmode simulations, and analytical calculations verify the existence of SEWs at Modes A and P; we seek to further the description with a final computational analysis. We simulate a one dimensional array consisting of 32 unit cells with a total length of 102.4μm, see Fig. 3. The orientation of the simulation domain is as follows: the surface normal of the MMA lies in the $\widehat{z}$-direction, the one dimensional array is oriented in the $\widehat{x}$-direction, and the ŷ-direction serves as the boundary by which the incident mode is enforced. Along the $\widehat{z}$-direction, an open boundary is assigned to the top of the bounding box while a perfect electric boundary is assigned to the bottom and used in place of the metallic ground plane. By setting either a perfect electric or perfect magnetic boundary condition in the ŷ-direction, light incident along the $\widehat{x}$-direction can be approximated as TE or TM polarized. In Figs. 3(a) and 3(b) we plot the electric field over the simulation volume for TE and TM polarized waves for Mode A and Mode P respectively. We note two prominent absorptive features. These absorptive features appear only for the TM radiation; one at a frequency coincident with Mode A and the other near the frequency of the main metamaterial absorbance peak at λ_P = 6.17μm, Mode P. It can be observed that, for both modes, the TE polarized wave does not couple with the surface and stays relatively plane-wave like throughout the region. In contrast, TM polarized waves realize drastically different behavior and are strongly confined to the surface of the MMA and the electric field strength falls off as a function of distance along the metamaterial. It can be observed that the phase advance of the TM polarized wave along the surface, compared to that of the free space radiation, is different (for both Modes A and P) which causes interference – a signature of light coupling to the surface [22].

 

Fig. 3 Simulated electric field along the surface of the metamaterial absorber at (a) 4.83 μm and (b) 6.14 μm for both TE and TM polarizations. (c) |E_z| as a function of position in the x-direction at 4.83μm (open triangles) and 6.14 μm (open circles). The horizontal axis is in units of the free space wavelengths of λ_P and λ_A, where λ_A denotes free space wavelength of Mode A.

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Results presented in Fig. 3 demonstrate that TE and TM polarized waves show qualitatively disparate behavior when incident upon a metamaterial absorber. However, our simulations allow us to further obtain quantitative information by approximating loss, wavelength, and propagation distance of the SEWs [23, 24]. From simulations we obtain the absolute value of the electric field strength in the z-direction |E_z| along the surface of the MMA for both Mode A (4.80μm) and Mode P (6.17μm) - see Fig. 3(c). From a decaying exponential fit to max E_z we are able to calculate the propagation length L_SEW by determining where max |E_z| has fallen to 1/e of its initial value. We find L_SEW =64.5μm (open triangles) and L_SEW =102 μm |(open circles) for Mode P and Mode A, respectively, plotted in Fig. 3(c). These distances corresponds to about 10 and 21 free space wavelengths for Mode P and Mode A, respectively. One noticeable difference is that Mode P realizes a propagation length of 64.5 μm, which is only about half of that found for Mode A, shown in Fig. 3(c). The results indicate that there are different loss mechanisms at play for the individual modes and suggests an interesting avenue for future study.

4. Conclusion

We have demonstrated that a surface electromagnetic wave is responsible for the high angular performance of the metamaterial absorber for TM polarized waves. Results from eigenmode, free-space normal incidence, and free-space parallel incidence simulations match very well to experiments. Further simulation of more subwavelenght metamaterials verify that the SEW cannot be associated with the periodicity of the unit cell. Rather the SEW is plasmon-like and well described by the effective optical constants of the metamaterial. Thus we note that it may be possible to tailor the angular absorbance of the MMA by modification of the effective optical constants of the unit-cell. The symmetry of Maxwell’s equations then further suggests that one may also achieve good angular absorbance in TE polarization by using a unit-cell geometry which supports magneto-plasmons [30].