1 Introduction

The use of metamaterials to reshape absorption and emission spectra has gained wide interest in recent years [1,2]. Metamaterial absorbers have been demonstrated across the electromagnetic spectrum, from microwave to optical wavelengths [3–6]. In the infrared range in particular [7], one typical design uses a three-layer, metal-insulator-metal structure, where the top metal layer is patterned into a sub-wavelength resonator element, e.g. a cross. Using appropriate designs, both perfect absorption in single resonators [8] and spectral synthesis using multiple resonators [9], have been demonstrated. These advances are impacting a wide range of applications, including thermophotovoltaics [10,11], microbolometers [12] and refractive index sensors [13].

Bright modes are modes which couple to normally-incident light, allowing absorption [5]. Dark modes prohibit this coupling. Coupling between bright- and dark-modes has previously been explored in various photonic systems, uncovering such fascinating phenomena as electromagnetically induced transparency [14–16], plasmon lifetime reduction [17,18], high quality factor Fano resonances [19], and hybrid quantum optomechanical systems [20–22]. Coupling between two dark modes has been less studied, particularly for application as infrared metamaterial absorbers.

Here, we study the impact of dark modes on the infrared absorptivity of dual-resonator systems. We compare the effect of coupling in a system with one dark and one bright resonator (dark-bright) to a system with two dark resonators (dark-dark). In each case, we investigate the dependence of the absorption spectrum on the resonator separation. For the dark-bright system, we observe that coupling results in modal splitting. We use coupled mode theory to establish that increasing resonator separation corresponds to decreasing coupling constant. For the dark-dark system, we identify an emergent bright mode that vanishes in the limit of zero coupling constants (large separations). For this mode, we find that the broadband absorptivity, integrated over a 1 µm bandwidth, decreases exponentially with resonator separation. This result illustrates the ability to achieve large absorptivity (or emissivity) contrasts by varying the resonator separation in the dark-dark system. We thus expect these results to provide new tools for the design of infrared metamaterials, with applications including infrared sensing and emissive pattern generation [23].

2 Bright and dark modes in MIM resonators

Figure 1(a) shows a schematic of a metamaterial thermal emitter in the IR range that supports both bright and dark modes. Following [5], the design consists of a 100 nm thick, 400 nm wide gold cross on top of a 185 nm Al₂O₃ layer and a 100 nm gold back reflector and is polarization independent due to symmetry. By definition, a bright mode can be excited by a normally-incident plane wave, while a dark mode cannot. To identify the bright modes in the structure, we perform a normal-incidence plane wave simulation of reflection using Lumerical FDTD. Since there is no transmission through the structure, the absorptivity can be found as A = 1 - R. To simulate the dark modes in this structure, we use dipole cloud excitation. A cloud of randomly positioned dipoles with random phases is positioned within the micro resonator. Another randomly positioned cloud of time-domain monitors is placed to measure the field amplitude. We then perform a fast Fourier transform (FFT) on the monitor readings to obtain the spectrum.

Figure 1(b) shows the simulated absorptivity spectra resulting from normal-incidence, x-polarized plane-wave simulations (red curve). The simulation shows a single resonance peak around 6.2µm, which corresponds to a bright mode. For the dipole cloud simulation, we plot the FFT intensity in arbitrary units (blue line). Two modes are apparent: the bright mode and an additional mode at lower wavelength. The inset to Fig. 1(b) show the electric field (E_x) profiles of each mode for a horizontal plane in the middle of the Al₂O₃ layer. The symmetries in these profiles show that only the higher wavelength mode (at 6.2µm) can couple to a plane wave, while the lower peak is a dark mode (integrates to zero by symmetry when multiplied by a constant-E_x plane wave).

 

Fig. 1. FDTD simulations of bright- and dark-mode metamaterial resonance (a) Schematic of the simulated design. L, W, and t are the cross length, width, and thickness, respectively, and a is the period. (b) Absorptivity spectrum calculated from x- polarized plane-wave simulation (left axis) and FFT intensity from dipole-cloud simulation (right axis). Insets are electric field profiles for the two resonance modes, and xy coordinate axes are marked on the upper inset.

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3 Dark-bright mode coupling

To achieve coupling between a dark mode and a bright mode, we design a structure where both modes coincide in wavelength. Tuning the length of the top metal-layer cross (L), while keeping all other parameters constant, shifts the wavelengths of the modes as shown in Fig. 2(a). From the figure, crosses with lengths of L₁=1.7 µm and L₂=1.27 µm support dark and bright resonances, respectively, at a wavelength of 4.7 µm. The two resonators are placed at a distance d=0.14 µm as shown in the lower inset to Fig. 2(b) and simulated using Lumerical FDTD using periodic boundary conditions. We use an x-polarized plane wave to calculate the absorptivity spectrum; this procedure yields the bright modes of the coupled system. Two bright modes are apparent. The lower wavelength mode has strong intensity in both resonators, whereas the higher wavelength mode has stronger intensity in the bright resonator.

 

Fig. 2. (a) Bright and dark resonance wavelengths versus the cross length (L). (b) FDTD simulation of emissivity spectrum of a bright-dark coupled resonator system. Lower inset shows geometry; upper insets show the electric field magnitude at the two resonance peaks.

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The splitting between the two modes is highly dependent on the coupling between the two resonators, which can be varied by tuning the distance (d) between the two crosses, as depicted schematically in Fig. 3(a). Figure 3(b) shows the simulated spectra for values of the separation (d) equal to 1.00, 0.27, 0.20, and 0.14 µm. For d > = 1 µm, the coupling coefficient goes to zero, and a single peak corresponding to the bright mode appears around 4.8 µm. As the distance between crosses is reduced, the coupling increases and a second peak appears in the spectrum. The separation between peaks, or splitting, increases with increasing coupling.

 

Fig. 3. (a) Schematic of coupled bright-dark resonators. (b) Simulated FDTD absorptivity spectrum for varying resonator separations. (c) Schematic of coupled mode theory model. (d) Emissivity calculated from coupled mode theory model.

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Varying the distance between resonators changes both the narrowband and broadband absorptivity. For example, if the wavelength is fixed at 4.74 µm (at the peak of the spectrum labeled d>1 µm), we observe a decrease in absorptivity from 1.0 to 0.36 as the distance is decreased to 0.14 µm. This is due to the shifting and splitting of the absorptivity peaks. The broadband absorptivity also changes. Integrating over the entire wavelength range shown in the figure, we found that the integrated absorptivity for d=0.14 µm is approximately 1.47 times larger than for d>1 µm.

4 Coupled-mode theory model for dark-bright coupling

The FDTD results can be modeled using a coupled mode theory of emissivity, by adapting the approach described in [24]. Figure 3(c) shows a schematic of two resonators with a coupling coefficient β. Both resonators act as thermal emitters with temporal thermal noise magnitude functions sources n₁(t) and n₂(t).

To study the emission of the coupled system, we start by writing the temporal coupled mode theory equations describing the field intensities as follows,

The absorptivity can be calculated by normalizing Eq. (4) to $\mathrm{\Theta }({\omega ,T} )/2\pi $.

To plot the CMT result, we extract the resonance frequencies and decay constants of the bright and dark resonators from FDTD simulations of single, uncoupled resonators. From the simulations, we find ${\tau _{0\textrm{b}}}$, ${\tau _{e\textrm{b}}}$, and ${\tau _{0\textrm{d}}}$ to be 2.6e–13 s, 2.6e–13 s, and 3.4e–13 s, respectively. The resonance wavelengths of the uncoupled modes were found to be ${\lambda _1} = 2\pi /{\omega _1} = 4.81$µm and ${\lambda _0} = 2\pi /{\omega _0}$ = 4.74 µm. We then fit the coupling constant $\beta $ at each distance d to obtain the closest match to the FDTD results.

A comparison of Fig. 3(b) and Fig. 3(d) shows that CMT can qualitatively reproduce the mode splitting observed in the FDTD simulations. Figure 3 clearly shows that the decrease in separation (d) between resonators increases the coupling constant ($\beta $). We will use this result in our analysis of dark-dark mode coupling, below.

5 Dark-dark mode coupling

We next examine coupled resonators that both support dark modes. Figure 4 shows two identical resonators of length 1.7 µm separated by a distance d = 0.05 µm. From Fig. 1(b), a single resonator supports a dark mode at 4.7 µm and a bright mode at 6.2 µm. When the two resonators couple, both the dark mode and the bright mode split, generating four peaks. The dashed lines in Fig. 4 show the dark modes, while the solid lines show the bright modes. As in Section 2, above, we note that bright modes absorb normally incident light, with an absorptivity between 0 and 1 (left axis). Dark modes do not absorb normally incident light; their locations and linewidths are instead identified from dipole simulations. The field intensity in the dipole simulation is plotted in arbitrary units on the right axis. Note that in Fig. 4, we have scaled the dark mode peak heights to match the bright mode peak heights for convenience. The higher wavelength bright mode originates from coupling between two bright modes (Bright-Bright (Bright)). The lower wavelength bright mode originates from coupling between two dark modes (Dark-Dark (Bright)). Interestingly, we note that the linewidth of the Dark-Dark (Bright) peak is smaller than for the Bright-Bright (Dark) peak. A fit to the numerical data found that the extrinsic decay rate of the Dark-Dark (Bright) mode is approximately 5x smaller than for the Bright-Bright (Bright) mode, while their intrinsic decay rates are similar.

 

Fig. 4. Bright and dark modes of two coupled, identical resonators at d = 0.05 µm. Insets are electric field profiles in the x-direction for the bright modes formed by coupling between two dark resonators (Dark-Dark (Bright)) and two bright modes (Bright-Bright (Bright)). Note that the dark peak amplitudes (arbitrary units) are scaled in height to match the bright peaks.

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We focus on the Dark-Dark (Bright) mode. Figure 5(a) shows the emissivity for various separations. As separation increases, the peak amplitude decreases. The peak wavelength decreases with separation, approaching the isolated resonator value of 4.7 µm. In the limit of infinitely large separation (zero coupling), the mode becomes dark. The quality factor and peak absorptivity for the Dark-Dark (Bright) mode are plotted in Fig. 5(b). The peak value can be tuned between 1.0 and <0.001 by tuning the distance between the two resonators from 0.05 to 1.00µm. The quality factor of the Dark-Dark (Bright) mode takes values between ∼40 and ∼130, which is much higher than the quality factor of the original single resonator bright peak (Q ∼10). Figure 5(c) shows that integrated emissivity over the shown spectral range in Fig. 5(a) decreases exponentially as coupling decreases. Here, the integrated emissivity is normalized by setting the value to 1 at a reference separation of d=0.05 µm. The exponential fit yields a 1/e decay length of 0.1 µm.

 

Fig. 5. Dark-dark mode coupling: (a) Absorptivity spectrum of two coupled identical resonators for separation distance d=0.05, 0.15, 0.25, 0.35, and 1 µm. (b) Peak absorptivity (left axis) and quality factor (right axis) of the dark-dark bright mode as a function of separation d. (c) Integrated absorptivity of the spectrum in (a) as a function of separation, fitted to an exponential function.

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The dependence of broadband absorptivity on resonator separation for the dark-dark system (Fig. 5) can be compared to the bright-dark system studied above (Fig. 3). For the dark-dark system, the integrated broadband absorptivity decays to zero exponentially with resonator separation. In the limit of large separations, the peak is extinguished. This is contrast to the dark-bright system, where increasing resonator separation merges the two absorptivity peaks into a single feature. In the latter case, the absorptivity is finite and nonzero even for large resonator separations. For a given pair of resonator separations, the contrast is integrated absorptivity is larger for the dark-dark case than the dark-bright one. For example, analysis of our calculations showed that for d_s = 0.15 µm and d_l = 1.0 µm, E_s/E_l = 2.8 for the dark-dark system and only 1.5 for the dark-bright system. Moreover, for the dark-dark system, both the peak absorptivity and the integrated absorptivity can be varied by as much as an order of magnitude by increasing the resonator separation from 0.05 µm to 0.5 µm, which is not possible in the dark-bright system.

6 Conclusion

In this paper, we studied metamaterial coupled-resonator systems for infrared spectral shaping. We compared structures supporting one dark and one bright mode to those supporting two dark modes. In dark-bright systems, we observe mode splitting that increases with decreasing resonator separation. We find that the integrated absorptivity of the coupled system is larger than the sum of the individual resonators. We use coupled-mode theory to establish definitively that decreasing separation corresponds to increasing coupling constant. For dark-dark systems, we observe the appearance of a bright mode that vanishes at large resonator separation (small coupling constant). The integrated absorptivity decays exponentially with resonator separation, going to zero in the infinite separation limit. The dark-dark system can thus yield much stronger contrast in the integrated absorptivity between different values of resonator separation than the dark-bright system.

This work varied the coupling coefficient by simulating systems with different separation distances between the resonators. In future work, it will be interesting to explore schemes for in situ tuning of the coupling constant. For example, recent work has demonstrated stretchable metamaterial absorbers using flexible PDMS substrates and metals such as silver conductive ink [25], and liquid metal [26]. This technology could provide a practical route to tunable inter-resonator coupling, allowing spectral modulation schemes similar to those studied here. Moreover, the high sensitivity of the spectral response to inter-resonator separation may provide new modalities for displacement and/or temperature sensing. While the designs presented here are polarization dependent, polarization-independent devices can also be explored by using 2 × 2 unit cells with 90-degree rotational symmetry. We thus envision that the results here will contribute to advances in mid-infrared sensing and imaging.