## Abstract

We demonstrate that rapidly switched high-$Q$ metasurfaces enable spectral regions of negative optical extinction.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

The extinction coefficient of light passing through any photonic structure is defined as the deficit of optical power in transmission. Because optical extinction is caused by absorption, reflectance, and scattering, it must be positive as long as the structure is composed of gain-free time-invariant material. Moreover, for dispersive materials and photonic structures, the frequency-dependent extinction coefficient $E(\lambda )\equiv 1-T(\lambda )>0$ must be positive for *every* wavelength $\lambda $ of the incident light, where $T(\lambda )$ is the transmission coefficient.

This situation can be dramatically altered by time-variant media whose optical properties (e.g., their refractive index) are an explicit function of time. To illustrate this effect, we calculate the extinction of an incident optical pulse with complex-valued amplitude ${s}^{+}(t)$ by a single-mode time-varying metasurface (TVM) characterized by its amplitude $a(t)$, natural frequency ${\omega}_{0}$, time-dependent damping rate $\gamma (t)$, and radiative coupling rate ${\gamma}_{r}$. Within the framework of coupled-mode theory (CMT) [1], the transmitted wave ${s}^{-}(t)$ is calculated according to the equations below:

where ${\gamma}_{nr}(t)=\gamma (t)-{\gamma}_{r}>0$ is a non-radiative damping rate of the mode. When the quality factor $Q(t)\equiv {\omega}_{0}/2[{\gamma}_{r}+{\gamma}_{\mathrm{nr}}(t)]$ rapidly decreases from its high initial value of ${Q}_{i}\gg 1$ to its final value of ${Q}_{f}<{Q}_{i}$ due to rapid increase of non-radiative losses, the TVM is assumed to be $Q$-switched.For a Gaussian input signal ${s}^{+}(t)={s}_{0}\text{\hspace{0.17em}}\mathrm{exp}(-i{\omega}_{0}t-{t}^{2}/{\tau}_{\mathrm{probe}}^{2})$ incident on an instantaneously $Q$-switched TVM (from ${Q}_{i}=100$ to ${Q}_{f}=5$ at $t=\tau $), the mode evolution $a(t)$ is plotted in Fig. 1(a). We assume the following pulse and TVM parameters: ${\omega}_{0}=2\pi c/{\lambda}_{0}$ (where ${\lambda}_{0}=3.3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ and $c=3\times {10}^{10}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{cm}/\mathrm{s}$ is the speed of light) and $\tau =200\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{fs}$. The transmission spectrum $T(\lambda ,\tau )\equiv {|{s}^{-}(\lambda ,\tau )|}^{2}/{|{s}^{+}(\lambda )|}^{2}$, which depends on the $Q$-switching time $\tau $ and extinction $E(\lambda ,\tau )\equiv 1-T(\lambda ,\tau )$, are calculated. The latter is plotted in Fig. 1(b). While $E>0$ for a metasurface that does not vary during the trapping time of the pulse (dashed line), negative extinction (NE) spectral regions corresponding to $E<0$ emerge for a TVM with $\tau <{Q}_{0}/{\omega}_{0}$. Intuitively, NE originates from (i) spectral broadening of the captured (resonant) photons caused by the dynamic $Q$-switching and (ii) their subsequent constructive interference with the non-resonant photons present in the broadband incident pulse. The spectral spacing $\mathrm{\Delta}\lambda \approx {\lambda}_{0}^{2}/2\pi c\tau $ between the NE regions is determined by the delay $\tau $ between TVM excitation at $t=0$ and switching at $t=\tau $.

In a more realistic CMT calculation, we have used finite switching times ${\tau}_{s}<\tau $ to establish the conditions for the emergence of the NE spectral region. According to Fig. 1(c), $E(\lambda ,\tau )<0$ is achieved for at least one value of $\tau $, as long as the switching is fast (${\tau}_{\mathrm{s}}\ll {\gamma}_{r}^{-1}$) and deep (small $\eta \equiv {Q}_{f}/{Q}_{i}$).

To experimentally demonstrate the NE phenomenon, we have chosen a TVM comprised of an array of subwavelength resonators [see Fig. 2(a) for a scanning electron microscope image] made of amorphous germanium (a-Ge). High-Q photonic structures, such as microring and photonic crystal resonators [2–5], as well as fiber-based cavities [6], have been used to redistribute spectral components of light at the nanosecond and picosecond time scales. Recently, time-variant semiconductor metasurfaces have been used to generate new light frequencies over a broad spectral range [7,8], revealing their promise for ultrafast operation and spatiotemporal shaping [9]. To our knowledge, the NE phenomenon has not been reported. Here, we employ a TVM to demonstrate negative optical extinction in the mid-infrared (MIR) spectral range on a femtosecond time scale.

The TVM was designed to exhibit a sharp transmittance dip with a quality factor of ${Q}_{i}\approx 100$ at the resonant MIR wavelength ${\lambda}_{0}=3.32\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ as shown in Fig. 2(b). A spectroscopic pump–probe apparatus [Fig. 2(c)] was used to study the optical extinction of a short MIR pulse by a TVM that was $Q$-switched by a delayed near-infrared (NIR) pulse of the same duration; see Fig. 2 caption for parameters. Single-photon pump absorption $Q$-switches the TVM over the ${\tau}_{s}={\tau}_{\mathrm{pump}}$ time via electron-hole plasma generation. Next, the measured differential transmission spectra $\mathrm{\Delta}T(\lambda ,\tau )/T(\lambda )$, where $\mathrm{\Delta}T(\lambda ,\tau )\equiv T(\lambda ,\tau )-T(\lambda ,\infty )$, were converted into the extinction spectra $E(\lambda ,\tau )=1-[T(\lambda )+\mathrm{\Delta}T(\lambda ,\tau )]/{T}_{0}$ using a Fano-resonance fit [10] of $T(\lambda )$ [Fig. 2(b), solid line]. Note that the coefficient ${T}_{0}=0.89$ was applied to adjust the data for the residual reflectance that is not related to the properties of the metasurface. In contrast with the recently reported blue-shifting of the entire spectrum due to the plasma-driven increase of the refractive index [8], here the spectral reshaping happens on both the red and blue sides of the TVM resonance as shown in Fig. 2(d). In Fig. 2(e), extinction is replotted for four delay times. Indeed, as predicted in Fig. 1, at least one MIR spectral range corresponding to $E<0$ can be identified for all but the largest time delays; the latter corresponds to the effectively stationary metasurface. We have found good qualitative agreement between the experiment and theory for both cases of the static metasurface and the TVM [Figs. 1(b) and 2(e)], suggesting that the NE regime is indeed realized in our TVM. Appreciable NE is found throughout the $3.14\u20133.28\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$ spectral range and is likely to extend further toward shorter wavelengths outside the detection range of our setup. Crucially, the spectral locations of the NE are tunable, controlled by the time delay $\tau $. The minimal measured extinction coefficient ${E}_{\mathrm{min}}^{(\mathrm{exp})}\approx -0.05$ [Fig. 2(e): green curve] is in good agreement with the theoretical value of ${E}_{\mathrm{min}}^{(\mathrm{th})}\approx -0.03$ [Fig. 1(c): black dot] corresponding to the estimated experimental parameters of ${\eta}^{(\mathrm{exp})}\approx 0.28$ and ${\tau}_{s}=35\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{fs}$.

In conclusion, we have predicted and experimentally demonstrated the existence of negative optical extinction of a femtosecond MIR laser pulse in a time-variant semiconductor metasurface. Negative extinction by ultrafast control of the loss in ultrathin resonators opens new opportunities for spectral shaping of short light pulses. The demonstrated approach can be extended to other spectral ranges, intensities (including single photons), and form-factors (e.g., in integrated resonators), and it may find use in novel light sources, pulse shaping schemes, optical amplifiers, and new wavelength-division multiplexing strategies.

## Funding

Office of Naval Research (N00014-17-1-2161); National Science Foundation (CHE-1726536, DMR-1719875, NNCI-1542081); U.S. Department of Energy (DE-FG02-12ER16347, DE-SC0020101).

## Disclosures

The authors declare no conflicts of interest.

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