Abstract

Band inversion in one-dimensional superlattices is a strategy to generate topological interface modes in electronics, optics, acoustics, and nanophononics. Despite their potential for the control of topologically robust interactions, most realizations of these states have so far explored only a single kind of excitation. In this work, we design and fabricate GaAs/AlAs devices with simultaneously inverted band structures for light and phonons. We experimentally observe colocalized interface modes for 1.34 eV photons by optical reflectivity and 18 GHz phonons by coherent phonon generation and detection. Through numerical simulations, we demonstrate the ensuing robustness of the Brillouin interaction between them with respect to a specific type of disorder. Furthermore, we theoretically analyze the efficiency of time-domain Brillouin scattering in different topological designs presenting colocalized states and deduce a set of engineering rules. Potential future applications include the engineering of robust optomechanical resonators in a material system compatible with active media such as quantum wells and quantum dots.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

The concept of inverted spatial mode symmetries in periodic lattices, often referred to as band inversion, is one of the cornerstones in the field of topological matter. Historically, this concept was first formalized with a comprehensive explanation for the electrical conduction properties of specific polymers using the Su–Schrieffer–Heeger (SSH) model [13]. Band inversion leads to the generation of an interface mode inside a bandgap by concatenating two periodic lattices with inverted bands. The existence of this interface state is then protected against any perturbation that does not change the underlying topological invariants of the structures. In other words, the mode is pinned at the bandgap center due to the chirality of the system. The robustness of these topologically induced interface modes has been exploited for a wide range of excitations (photons [48], plasmons [9], phonons [1012], vibrations [13,14], polaritons [15,16], and classical sound [14,1722]). Most of these realizations explored a single kind of excitation. Despite its potential in the manipulation and control of interactions, the simultaneous topological confinement of multiple excitations remains an open challenge.

In this work, we construct an optical and phononic interface mode by simultaneous band inversion for the two excitations. We rely on GaAs/AlAs heterostructures that exhibit a naturally occurring colocalization of the optical and acoustic fields, resulting in enhanced optomechanical interactions. We designed, fabricated, and experimentally studied a GaAs/AlAs heterostructure presenting a topological interface mode for photons at 1.34 eV and phonons at 18 GHz. We observe colocalization of light and sound on the nanoscale and theoretically predict a robust photoelastic interaction between them.

2. CONCEPT AND SPECTRAL CHARACTERIZATION

An infinite, periodic superlattice is characterized by a folded dispersion relation with gaps at the center and the edge of the Brillouin zone. For light, the indices of refraction govern the dispersion relation. For acoustic phonons, the relevant quantities are the elastic constants of the materials.

A finite-sized superlattice relates to the infinite system through the correspondence between stop bands in the reflectivity spectrum and bandgaps, and the correspondence between reflection phases and Bloch mode symmetries. The finite system is usually termed a distributed Bragg reflector (DBR). For the sake of simplicity, we will use the terminology for the finite and infinite system interchangeably.

Band inversion is a strategy to systematically generate an interface mode. To this end, we concatenate two DBRs fulfilling two conditions: (1) they share a common bandgap [same central frequency as a necessary condition for topological robustness (see Section 3) and same bandwidth, resulting in similar evanescent decay lengths into both DBRs] and (2) the order of Bloch mode symmetries at the band edges enclosing this bandgap is inverted. Under these conditions, the reflection phases of the two DBRs present opposite signs, one evolving from ${-}\pi$ to zero, and the other from zero to $\pi$ across the gap. At the bandgap center, the phases are equal in magnitude, resulting in a zero roundtrip phase thus giving rise to a resonance. This effect is directly rooted in the topological properties of the individual bands and can be expressed in terms of a sum of Zak phases (the Berry phase in one-dimensional periodic systems) for the bands below the considered bandgap [11,14,21] (see Supplement 1, Section S1).

We control the order of Bloch mode symmetries through the internal structure of the DBR unit cells. In the left panel of Fig. 1(a), we show the dispersion relation of acoustic phonons in a GaAs/AlAs DBR. We have defined an inversion-symmetric three-layer unit cell and introduced the parameter $\delta$, which describes the relative thicknesses of the layers (in units of the acoustic wavelength). The layers of a unit cell with a full thickness of one wavelength are therefore described by $({1} + \delta){\lambda _{{\rm GaAs}}}/{8}$, $({1} -\delta){\lambda _{{\rm AlAs}}}/{4}$, $({1} + \delta){\lambda _{{\rm GaAs}}}/{8}$. Here, $\lambda$ is the acoustic wavelength in the corresponding material at a chosen design frequency. For a full thickness of one wavelength, this frequency is the center of the second minigap. The parameter $\delta$ thus allows us to construct unit cells with different thickness ratios while keeping the overall thickness in terms of acoustic wavelength constant. In the particular case of Fig. 1(a), left panel, $\delta = - {0.5}$, and the chosen frequency is 18.12 GHz. At the second minigap, the Bloch mode at the lower (upper) edge is symmetric (antisymmetric). We achieve a band inversion in the second minigap by choosing an opposite sign of $\delta$. For $\delta = + {0.5}$ [right panel of Fig. 1(a)], the dispersion relation is the same except for the mode symmetries. For other values of $\delta$, the width of the bandgap changes, and for the particular case of $\delta = {0}$, the second bandgap is closed (not shown here).

 figure: Fig. 1.

Fig. 1. Construction principle of an interface state in a photonic–phononic nanocavity by band inversion. (a) Schematic of the unit cells of each DBR parametrized by $\delta$, which describes the relative thickness of the materials in units of wavelength while keeping the overall thickness constant (also in units of wavelength). For the particular case of GaAlAs, the values of $\delta$ are the same for both the optical and acoustic cases. Dark (light) colors represent GaAs (AlAs); blue and green are used to distinguish between left and right superlattices, respectively. The dispersion relations of acoustic (a) phonons and (b) photons present inverted Bloch mode symmetries around the first bandgap at the Brillouin zone center. The dashed red box indicates the bandgap over which the band inversion takes place; mode symmetries are indicated with violet (antisymmetric) and green (symmetric) dots. (c) The spatial displacement pattern $|{ u}({z})|^{2}$ (black) and optical profile $|{ E}({z})|^{2}$ (red) of the topological interface state at 18.12 GHz/1.34 eV are colocalized and present a maximum at the interface between the two DBRs. Both fields decay evanescently away from the interface. The complete heterostructure is represented at the top.

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For the case of light, the corresponding band structures are shown in Fig. 1(b). There, the second bandgap appears around 1.34 eV. Up to a scaling factor, the overall band structures look the same as for the case of acoustic phonons. Usually, for a given unit cell structure made from two arbitrary materials, the resulting values of $\delta$ in the optical and acoustic domain are not the same. For acoustics, $\delta$ depends on the ratio of the speeds of sound in the two materials, while for optics, it is the ratio of the speeds of light. This ratio is usually different, but GaAs and AlAs show a unique parameter coincidence, which consists of equal ratios for the speeds of sound ${v_i}$ and speeds of light ${c_{\!i}}$ [23,24], i.e., $\frac{{{c_{{\rm GaAs}}}}}{{{c_{{\rm AlAs}}}}} \approx \frac{{{v_{{\rm GaAs}}}}}{{{v_{{\rm AlAs}}}}} \approx 0.84$. For any optical frequency in the NIR, an acoustic frequency exists, resulting in equal optical and acoustic wavelengths within each of the two materials. In a superlattice, the parameter $\delta$ then equally describes optical and acoustic structures. Under these conditions, a band inversion characterized by a sign change of $\delta$ happens simultaneously in optical and acoustic domains. In GaAs/AlAs, furthermore, the ratio of optical and acoustic impedances is equal, resulting in equal relative widths of the bandgaps for light and sound, as observed in Figs. 1(a) and 1(b). Together, these parameter coincidences allow the conception of simultaneous optical and acoustic colocalized modes in multilayered structures.

We can extend this design principle to other technologically relevant material combinations such as ${\rm InP}/{{\rm Ga}_{0.53}}{{\rm In}_{0.47}}$, as [25] for which a similar coincidence in material parameters occurs. For some dielectrics such as ${{\rm SiO}_2}/{{\rm TiO}_2}$, we can achieve a simultaneous band inversion with opposite mode symmetries for light and sound [26].

We concatenate two DBRs with an inverted second bandgap. This results in colocalized interface modes at the center of the optical and acoustic bandgaps, at 1.34 eV and 18.12 GHz, respectively. In Fig. 1(c), we show the acoustic displacement $|{u}({ z})|{\rm ^2}$ (black) and optical field $|{E}({z})|^{2}$ (red) of the topological interface modes with a perfect mode overlap. The blue and green colors are a guide to the eye to distinguish DBRs on the left and right, respectively. The optical and mechanical quality factors of the confined mode are typically $Q\sim{2000}$ given by the acoustic and optical reflectivities of the two DBRs. That is, the number of periods in the DBRs and the value of $\delta$ determine $Q$.

We have fabricated a GaAlAs heterostructure consisting of alternating ${{\rm Al}_{0.95}}{{\rm Ga}_{0.05}}{\rm As}$ and GaAs layers by molecular beam epitaxy (MBE) on a (001) GaAs substrate. The structure is composed of two DBRs formed by 14 (16) periods of 65.0 nm/230.7 nm (195.1 nm/76.9 nm) ${\rm GaAs}/{{\rm Al}_{0.95}}{{\rm Ga}_{0.05}}{\rm As}$ layers for the left (right). The different numbers of pairs in the two DBRs account for the different refractive indices of air and the substrate on both sides of the sample. This heterostructure simultaneously confines an optical interface mode at an energy of 1.34 eV and an acoustic phonon interface mode at a frequency of 18.12 GHz. We grew all layers of the sample with a linear thickness gradient of 30% across the 51 mm wafer resulting in a 30% change in optical and acoustic resonance wavelength. We consider this change small enough to be irrelevant for the outcome of our experiments.

To evidence the optical interface mode, we performed optical reflectivity measurements. Figure 2(a) shows the reflectivity of the structure measured by Fourier-transform infrared spectroscopy (FTIR) (Bruker Equinox 55 with Hyperion Microscope, halogen lamp illumination). The two measured stop bands around 0.71 eV and 1.34 eV correspond to the first and second bandgaps illustrated in Fig. 1(a). The dip in the middle of the second bandgap is associated to an optical cavity mode. Figure 2(b) provides a zoomed-in view of the interface state with a full width at half maximum (FWHM) of ${\sim}{0.4}\;{\rm nm}$. The detail of this dip was measured using a home-built reflectometer based on a CW laser (M2 SolsTiS). We have simulated the optical reflectivity using a transfer matrix formalism. The results (gray lines) agree with the experimental spectra.

 figure: Fig. 2.

Fig. 2. Continuous wave optical characterization of the topological interface modes. (a) The measured optical reflectivity reveals an interface state in the middle of the first optical stopband at the Brillouin zone center. (b) The zoomed-in optical reflectivity spectrum shows the interface state to be centered at 922.5 nm (1.34 eV).

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To acoustically characterize the structure and evidence the existence of a topological phononic interface mode, we performed time-domain Brillouin scattering experiments [27,28] using an optical pump–probe technique.

We use a titanium:sapphire laser (Spectra Physics Tsunami) producing 3.4 ps long pulses (0.4 nm linewidth) at an 80 MHz repetition rate. As shown in Fig. 3(a), we split the laser beam into pump and probe with powers of 50 and 10 mW, respectively. The probe beam passes through a delay line to control the relative delay between pump and probe pulses. We send the pump beam through an acousto-optical modulator for synchronous detection. Both beams are focused onto the sample into a spot of 5 µm in a collinear geometry. The pump pulse impulsively generates coherent phonons with periods much longer than the pulse duration. These phonons modulate the optical properties of the sample with gigahertz frequencies. Using a cross-polarization scheme, a photodetector measures the resulting changes in the delay-dependent instantaneous reflectivity experienced by the probe pulses [29,30]. A lock-in amplifier extracts differential reflectivity time traces, as the ones we show in Fig. 3(b).

 figure: Fig. 3.

Fig. 3. Time-domain opto-phononic characterization of the topological interface mode. (a) Experimental coherent phonon generation setup. A mechanical delay line is used to control the delay between pump and probe pulses arriving on the sample. A cross-polarization scheme is used to probe the sample’s instantaneous reflectivity while suppressing the reflected pump pulses. (b) Differential reflectivity time trace (0–11.000 ps) measured with laser-cavity detuning of ${-}{0.786}\;{\rm nm}$, 50 mW pump power, and 10 mW probe power. (c) Photoelastic model calculation of the pump–probe spectrum considering a resolution of 90 MHz. (d) Fast Fourier transform (FFT) of the experimental time trace featuring the topological resonator mode at 18.12 GHz. The inset shows a detail of the resonance with a full width at half maximum (FWHM) of 158 MHz. The experimental result agrees well with the simulation.

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The sharp change in transient reflectivity at ${t} = {0}$ is induced by the direct electronic excitation of the system. The signal then slowly increases up to a maximum at ${\sim}{2}\;{\rm ns}$, and descends until close to its initial value at ${\sim}{11}\;{\rm ns}$. These slow changes result from the electronic and thermal evolution of the sample [30,31], both inducing a time dependence of the optical cavity mode. Essentially, these slow dynamics result in a time-dependent sensitivity to the presence of coherent phonons. The phonons manifest as oscillations of small amplitude on a much shorter time scale. The main experimental parameters influencing the slow dynamics of the cavity are the pump power and the initial detuning of the laser wavelength with respect to the unperturbed optical cavity mode. The maximum generation of coherent acoustic phonons is reached when the pump laser is initially on resonance with the cavity mode. A maximum detection sensitivity affords tuning the probe laser to the highest slope in reflectivity.

We scan a total of ${\sim}{11}\;{\rm ns}$ delays between pump and probe, covering almost the full laser repetition period. We then Fourier transform (FT) the derivative of the reflectivity trace in Fig. 3(b) to obtain the acoustic spectrum of the topological resonator. The resulting spectrum is shown in Fig. 3(d) and presents a clear peak at 18.12 GHz, corresponding to the acoustic mode illustrated in Fig. 1(b).

The inset in Fig. 3(d) is a zoom-in of the resonance at 18.12 GHz, with a FWHM of 158 MHz. We attribute the FWHM mainly to the dynamics of the optical cavity mode (for details, see Fig. 4) and the finite length of the time trace (11 ns corresponds to a Fourier limited linewidth of 90 MHz). Other contributions are the interference between phonons generated by two consecutive pump pulses, and lower signal detection efficiency for large delays. By substantially reducing the pump power, we can avoid the slow optical cavity dynamics and measure a linewidth of 97 MHz at the expense of signal-to-noise ratio.

 figure: Fig. 4.

Fig. 4. (a) Differential reflectivity time trace (0–11.000 ps) taken at a spectral detuning of ${-}{0.18}\;{\rm nm}$ of the laser from the cavity mode, 50 mW pump power, and 8 mW probe power. (b) Fast Fourier transform (FFT) of the experimental time trace featuring the topological resonator mode at 18.12 GHz. The inset shows a detail of the resonance that appears split with a separation of 140 MHz. (c) Time trace from (a) after passing a bandpass filter (17.97 GHz to 18.25 GHz). Coherent phonon oscillations are clearly visible with a modulated envelope. (d) The envelope in (c) is determined by the sensitivity of the structure to measure the modulation induced by the phonons. The red line indicates the position of the cavity as a function of delay, the dashed line the laser wavelength (fixed), and the gray line the unperturbed cavity resonance wavelength. The arrow shows the moment when the laser matches the cavity mode and the system is completely insensitive to phonons.

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To account for our observations, we performed photoelastic model calculations based on a transfer matrix formalism. In one-dimensional geometries, the frequency-dependent coherent phonon generation cross-section $\sigma (\omega)$ can be approximated by the overlap integral

$${\sigma}(\omega ) = \int {\left| {E(z )} \right|^2}p(z )\frac{{\partial u\!\left({\omega ,z} \right)}}{{\partial z}}{\rm d}z.$$
Here, $\partial {u}(\omega ,{ z})/\partial {z}$ is the structure’s frequency-dependent stationary phononic strain field (the boundary condition on the substrate side is a monochromatic input field of unit amplitude), $|{E}({ z}){|^2}$ is the electric intensity of the optical laser field at the wavelength of optical resonance, and ${p}({z})$ is the material-dependent photoelastic constant [32]. We approximate ${{p}_{{\rm GaAs}}} = {1}$ for GaAs layers and ${{p}_{{\rm AlAs}}} = {0}$ for AlAs [29,33]. We then approximate the frequency-dependent detection efficiency $D(\omega)$ as the overlap integral
$$D(\omega ) = \int E{(z )^2}p(z )\sigma (\omega )\frac{{\partial u\!\left({\omega ,z} \right)}}{{\partial z}}{\rm d}z.$$

Equations (1) and (2) indicate that the stronger electric field of a confined optical mode leads to enhanced photoelastic generation and detection processes. For extremely high acoustic frequencies (with acoustic wavelengths much shorter than the optical wavelength, i.e., no colocalization) this enhancement could be approximated in Eqs. (1) and (2) by a constant electric field of increased amplitude [3437]. For a colocalized opto-phononic mode, the overlap between the two fields in Eqs. (1) and (2) automatically dictates the selective generation and detection of the confined acoustic mode. This implication has been theoretically and experimentally validated in Fabry–Perot resonators [23,38].

Figure 3(c) presents the simulated power spectrum ${| {D(\omega)} |^2}$ for the sample structure shown in Fig. 1. The only free parameters in this simulation are a global scaling factor of the overall structure thickness accounting for the spatial gradient of the sample and a global normalization factor of the intensity spectrum. Furthermore, we convoluted the simulated spectrum with a Gaussian of 90 MHz FWHM to incorporate the finite spectral resolution of the experimental setup, which is ultimately dictated by the laser repetition rate. We observe good agreement between the measured and simulated spectra.

In the measured time trace in Fig. 3(b), we observed that two contributions modulate the optical cavity: acoustic phonons on the picosecond time scale and electronic excitations as well as thermal effects on the nanosecond time scale [27,28,31]. Typically, the modulation induced by acoustic phonons is much smaller than the linewidth of the optical cavity, while the slow modulation is usually on the order of the linewidth. Their coherent interplay could lead to interesting signatures in the pump–probe spectra: if over the course of the pump–probe trace the laser was to sense phonons through both slopes of the optical cavity, the phonon-induced transient reflectivity would change sign. This configuration would lead to more complex line profiles of the acoustic resonances. [30] Experimentally, we demonstrate this effect by tuning the laser power and sweeping the wavelength across the cavity mode.

Figure 4(a) is the time trace obtained for a laser-cavity detuning of ${\lambda _{{\rm Laser}}} - {\lambda _{{\rm cav}}} = - {0.18}\;{\rm nm}$ (compare with the linewidth of the laser and the optical cavity mode of 0.4 nm), a pump power of 50 mW, and probe power of 8 mW. In Fig. 4(a), there is a sharp change of reflectivity at ${t} = {0}$ followed by a slow variation where the signal reaches a maximum ($\sim 1\,\,{\rm ns}$), then passes through a minimum ($\sim 6\,\,{\rm ns}$) and finally reaches a value close to the starting point of the transient reflectivity curve ($\sim 12\,\,{\rm ns}$). Figure 4(b) presents the FT of the time trace with the inset presenting a zoomed-in version of the resonance. Note that in contrast to Fig. 3(c), there are two peaks instead of one. Figure 4(c) presents the same time trace as in (a) after a bandpass filter between 17.97 GHz and 18.25 GHz. The fast oscillations correspond to coherent acoustic phonons at 18.12 GHz. The envelope of the time trace is modulated with a node at 7 ns. The splitting of the peak in Fig. 4(b) is a proof of the mutual coherence between the time dynamics of the optical mode due to coherent phonons and due to electronic effects. The latter lead to modulation of the envelope in Fig. 4(c).The evolution of the differential reflectivity can be directly associated with evolution of the optical cavity mode over time, as shown by time-resolved spectroscopy in Fig. 4(d). We observe that the laser is tuned to both flanks of the cavity mode. During measurement, the optical resonance crosses the central laser wavelength, causing the minimum in the time trace near 6000 ps.

In the absence of the slow optical cavity dynamics observed in Fig. 4(d), the envelope of the fast oscillations would be a simple exponential decay dictated by the acoustic cavity lifetime. The additional slow modulation of the optical cavity mode results in the splitting of the measured peak in the acoustic spectrum.

We heuristically model the resulting split spectrum shown in Fig. 4(b) by considering the FT of a periodic signal (${s}$) as a function of time delay (${t}$) with an amplitude modulation such as $s({t}) = {\cos}({\omega _{{\rm amp}}}{\rm t}){\cdot \exp}[\; - {i}\;{\omega _{{\rm phonon}}}\;{t}]$. The resulting function ${\rm FT}({s}({t}))= {S}(\omega) = {\rm sqrt}(\pi /{2}){\cdot}(\delta (\omega - ({\omega _{{\rm phonon}}} - {\omega _{{\rm amp}}})) + \delta (\omega - ({\omega _{{\rm phonon}}} + {\omega _{{\rm amp}}})))$ shows a splitting of ${2}\;{\omega _{{\rm amp}}}$ around the resonant frequency ${\omega _{{\rm phonon}}}$. The measured splitting in Fig. 4(b) is $\Delta \omega \;\sim{140}\;{\rm MHz}$; therefore, we deduce ${\omega _{{\rm amp}}} = {70}\;{\rm MHz}$. This simple argument supports our image of the dynamic interaction between the photonic topological mode, determining the sensitivity to the present phonons, and the topological phononic mode, which is at the origin of the modulated signal in the gigahertz regime.

3. ROBUSTNESS OF THE BRILLOUIN CROSS-SECTION

The construction principle of the topological resonator studied in this work dictates that both optical and acoustic resonance frequencies are pinned at the bandgap center for any perturbation that preserves chirality. In terms of the SSH model, these are perturbations that act only on the hopping terms and do not introduce on-site potentials. Here, we numerically show that this robustness of resonance frequencies implies robustness of the interaction between the two fields. That is, the Brillouin scattering cross-section $\sigma$ shows signatures of robustness.

To this end, we compare the topological resonator presented in Fig. 1 to a Fabry–Perot cavity formed by non-centrosymmetric bilayer unit cells of GaAs/AlAs. Using the definitions introduced in Fig. 1(a), fluctuations preserving chirality are those that lead to local changes in the parameter $\delta$. We numerically implement this type of noise using a flat distribution of random numbers with an amplitude $\Delta \delta /\delta$ ranging from zero (unperturbed system) to 0.999 (local bandgap can almost completely close due to fluctuations). The results are presented in Fig. 5. Figure 5(a) shows a comparison between the unit cells on the left and right superlattices of the topological resonator and the Fabry–Perot resonator. Both are constructed from an equal number of unit cells, and we consider GaAs as the background medium on both sides. Figure 5(b) compares the acoustic resonance frequencies of the two systems. When introducing local fluctuations in $\delta$, the resonance of the topological resonator stays pinned at the bandgap center, whereas the mode of the Fabry–Perot resonator fluctuates. Note that due to the perfect colocalization of light and sound in GaAlAs based systems, the optical resonances behave in precisely the same way. Figure 5(c) compares the corresponding acoustic quality factors. Here, both systems behave very similarly. We observe a fluctuation-induced drop in the quality factor of almost one order of magnitude. This effect is expected since fluctuations in the parameter $\delta$ always reduce the width of the local bandgap, i.e., the evanescent decay length of the confined mode grows, enhancing leakage radiation through the DBRs into the background medium. Again, the colocalization of light and sound dictates the same behavior in the optical domain.

 figure: Fig. 5.

Fig. 5. Robust Brillouin scattering in a topological opto-phononic resonator under random perturbations. (a) Unit cells at the center of a topological resonator and a Fabry–Perot resonator. (b) Upon perturbing the structures by random variations of the parameter $\delta$ or each unit cell, the acoustic resonance frequency stays pinned at the bandgap center for the topological mode (black) but undergoes substantial variations for the Fabry–Perot case (gray). We chose random variations with a uniform distribution of width $\Delta \delta/\delta$. (c) For both types of resonators, the acoustic quality factor drops for fluctuations $\Delta \delta/\delta \gt {0.2}$. (d) This drop in quality factor fully accounts for a moderate drop in Brillouin scattering cross-section for the topological resonator. In contrast, the cross-section fluctuates over four orders of magnitude for the Fabry–Perot resonator.

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Interestingly, despite this similar behavior of quality factors, the resulting Brillouin cross-sections of the two systems plotted in Fig. 5(d) show marked differences. For all the simulations, we consider a fixed laser wavelength of 920 nm. For the topological system, we find a drop in cross-section of one order of magnitude, which is fully accounted for by the loss in quality factor. However, for the Fabry–Perot resonator, we find huge fluctuations of four orders of magnitude even for small perturbation amplitudes of $\Delta \delta /\delta \sim 0.2$. This finding demonstrates that indeed a gain in the robustness of an interaction between two different excitations can be attained when both benefit from topological protection such as through preserved chirality. In other words, even though the perfect colocalization of light and sound is guaranteed by the choice of materials, only the topological construction principle allows for a robust interaction between them. Alternatively, we could follow the position of the fluctuating cavity mode with the laser wavelength. In this case, the measured Brillouin spectrum from the Fabry–Perot resonator would be subject to stronger wavelength instabilities than the Brillouin spectrum of the topological resonator. Intensity fluctuations, however, would be similar for both types of resonators in this case.

4. BRILLOUIN CROSS-SECTION EFFICIENCY

We showed a possible configuration in which two concatenated superlattices with inverted bands give rise to an opto-phononic interface mode. By following the same principle, we can define a full class of resonators, which we explore in this section. We consider structures of two DBRs presenting inverted bands and centrosymmetric unit cells. Two different operations lead to a band inversion: (i) changing the sign of delta or (ii) changing the superlattice inversion centers that coincide with the origin of the unit cells. That is, the materials of the layers forming a centrosymmetric unit cell are swapped. For all the discussion that follows, we consider resonators formed by two superlattices of 16 periods each with GaAs half spaces surrounding the structure.

In Figs. 6(a)–6(d), we present four particular topological structures: (a) the same structure as presented in Fig. 1; (b) $\delta \lt {0}$ for the left DBR and $\delta \gt {0}$ for the right DBR, all unit cells centered on GaAs; (c) $\delta \gt {0}$; and (d) $\delta \lt {0}$ with the left DBR centered on GaAs and the right DBR centered on AlAs. In Fig. 6(e), we present the particular case of non-centrosymmetric unit cells formed by bilayers of GaAs/AlAs. This case results in a conventional Fabry–Perot resonator formed with two identical DBRs composed of ${3}\lambda{\rm /4}$ (AlAs) and $\lambda/{4}$ (GaAs) layers enclosing a $\lambda/{2}$ spacer. Figures 6(a)–6(e) display the normalized acoustic displacement (dashed black) and optical intensity (red) patterns of the interface modes. [32] For each structure, we plot the value of the integrand of the overlap integral [Eq. (1)] for the photoelastic interaction [Figs. 6(f)–6(j)]. The cross-section is determined by two conditions: (i) the overlap between the antinodes of the opto-phononic field $|{E}({z}){|^2}(\partial { u}/\partial { z})$ and the maxima of the photoelastic constant distribution ${p}({ z})$; (ii) the relative sign of the opto-phononic antinodes in the regions where the photoelastic constant is non-zero. In addition, it is also important to consider that the antinodes with maximum amplitude at the interface between the two superlattices contribute the most to the overall cross-sections.

 figure: Fig. 6.

Fig. 6. Comparison between different topological structures and a Fabry–Perot resonator. The structures are designed to confine an acoustic mode at 18.12 GHz and an optical mode at 1.34 eV. (a)–(e) For each structure, we display the spatial displacement pattern $|{u}({z})|^{2}$ (dashed black) and the optical profile $|{E}({z})|^{2}$ (solid red). (a)–(d) Topological structures composed of varying thicknesses and material composition; (e) conventional Fabry–Perot resonator (see text for details). The topmost panel represents the unit cell of the DBRs at the interface of each structure. (f)–(j) Brillouin cross-section integrand of structures in (a)–(e).

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The integrands in Figs. 6(f)–6(j) can be analyzed by quadrants (left/right superlattice, positive/negative contributions). The signal in each quadrant is formed by either single or double peaks (thin/thick lines in the plot). The four cases confining phonons and light through band inversion [Figs. 6(a)–6(d)] fall into two categories: In cases (c) and (d), the positive signal formed by double peaks in one superlattice compensates mostly for the negative signals formed by single peaks in both superlattices. This results in an overall small Brillouin cross-section $\sigma$. Cases (a) and (b) feature positive signals composed of double peaks in one superlattice and single peaks in the other one, while only one superlattice contributes negative single peaks. Effectively, the negative peaks compensate for half of the positive double peaks, leaving an overall signal with positive single peaked contributions from both superlattices. The main difference between cases (a) and (b) is the material of the layers forming the interface. The GaAs in case (a) contributes to the cross-section, while the AlAs in case (b) does not.

The maximum cross-section is reached for the case of the Fabry–Perot resonator in Fig. 6(e) ($\sigma = {2939}$). Note that in this case, the integrand is definite-positive all over the structure, i.e., two quadrants with positive single peaks. Note that the particular case of a Fabry–Perot resonator formed by two DBRs separated by a $\lambda$ spacer (not shown here) would result in a vanishing cross-section, where the integrand would present positive peaks for one superlattice and negative peaks for the other.

5. DISCUSSION AND CONCLUSION

Based on the band inversion concept applied to an opto-phononic GaAs/AlAs heterostructure, we conceived an interface mode where both NIR photons and gigahertz phonons are simultaneously confined. We reached $Q$-factors on the order of 2000 for both fields, which are phonons at 18.12 GHz and photons at 1.34 eV. The full structure thickness was on the order of 10 µm, reachable with standard MBE techniques. The GaAs/AlAs platform dictates the perfect overlap between electric and atomic displacement fields.

We performed coherent phonon generation experiments based on pump–probe differential reflectivity measurements. We observed acoustic-phonon lifetimes longer than 10 ns. The experimental signatures show the convoluted responses of the optical and acoustic cavities, unveiling the interplay between the two confined fields. The frequency of the acoustic phonons does not change with the optical interactions; however, the measured spectrum can show altered frequencies due to the modulation of the optical cavity mode. In our case, the split peak in Fig. 4(b) is a direct proof of mutual coherence between the coherent phonons and modulation of the optical cavity mode due to electronic and thermal effects. The use of an optical cavity mode allows us to flip the phase of the detection in the middle of the measurement by choosing just the appropriate initial excitation conditions, pump power, and detuning.

Our optical experiments evidenced the existence of the optical and phononic topological interface modes. The chosen structure is not the only topological resonator that can be constructed in GaAs/AlAs heterostructures. By comparing different combinations of concatenated superlattices, we numerically showed that despite similar $Q$-factors, the Brillouin cross-section of the experimentally chosen structure is the highest, almost matching the performance of a topologically trivial Fabry–Perot resonator.

A significant advantage of the studied topological opto-phononic resonator is the robustness of its modes when subject to chirality-preserving thickness fluctuations. Note that these specific perturbations are different from random thickness variations of each individual layer. To preserve chirality, we specifically consider fluctuations in the detuning parameter $\delta$. Under these properly chosen perturbations, the reported resonator not only presents topological robustness of its optical and acoustic modes [11,14], but also a robust Brillouin interaction between them. We compared this robustness to a standard Fabry–Perot resonator of similar characteristics. Although colocalization is preserved in both cases, the Brillouin interaction presents a different dependence on fluctuations in the parameter $\delta$. For a Brillouin interaction with fixed excitation laser wavelength, we numerically showed that the stronger fluctuations of the cavity resonance for the Fabry–Perot resonator manifest in pronounced intensity fluctuations of the Brillouin signal. If, alternatively, the laser wavelength follows the position of the cavity mode, the measured Brillouin spectrum from the Fabry–Perot resonator would instead be subject to stronger wavelength instabilities than the Brillouin spectrum of the topological opto-phononic resonator.

By etching micropillars out of the studied heterostructures, we could achieve 3D control of the optical and acoustic densities of states. Potential future applications of the reported results thus include the engineering of topologically robust optomechanical resonators, for which the overlap of the optical and acoustic fields is a fundamental requirement. Since we use standard III-V semiconductor materials, the system is also a potential testbed of novel concepts involving active media such as quantum wells and quantum dots, for which the control of light–matter interactions might be a key asset.

Funding

H2020 European Research Council (715939); H2020 Future and Emerging Technologies (824140); French RENATECH Network; Agence Nationale de la Recherche (ANR-10-LABX-0035); Deutsche Forschungsgemeinschaft (401390650).

Acknowledgment

The authors acknowledge funding by the European Research Council Starting Grant No. 715939, Nanophennec. This work was supported by the European Commission in the form of the H2020 FET Proactive project TOCHA (No. 824140). The authors acknowledge funding by the French RENATECH network and through a public grant overseen by the ANR as part of the “Investissements d'Avenir” program (Labex NanoSaclay Grant No. ANR-10-LABX-0035). M.E. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) Project 401390650. NDLK and ME proposed the concept and designed the device. AL fabricated the sample. All the authors performed the experiments and simulations, discussed and analyzed the results and wrote the paper. NDLK guided the research.

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

Supplemental document

See Supplement 1 for supporting content.

REFERENCES AND NOTES

1. A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60, 781–850 (1988). [CrossRef]  

2. W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42, 1698–1701 (1979). [CrossRef]  

3. W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22, 2099–2111 (1980). [CrossRef]  

4. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7, 1001–1005 (2013). [CrossRef]  

5. M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011). [CrossRef]  

6. T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019). [CrossRef]  

7. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821–829 (2014). [CrossRef]  

8. P. A. Kalozoumis, G. Theocharis, V. Achilleos, S. Félix, O. Richoux, and V. Pagneux, “Finite-size effects on topological interface states in one-dimensional scattering systems,” Phys. Rev. A 98, 023838 (2018). [CrossRef]  

9. J. C. G. Henriques, T. G. Rappoport, Y. V. Bludov, M. I. Vasilevskiy, and N. M. R. Peres, “Topological photonic Tamm-states and the Su-Schrieffer-Heeger model,” Phys. Rev. A 101, 043811 (2020). [CrossRef]  

10. G. Arregui, O. Ortíz, M. Esmann, C. M. Sotomayor-Torres, C. Gomez-Carbonell, O. Mauguin, B. Perrin, A. Lemaître, P. D. García, and N. D. Lanzillotti-Kimura, “Coherent generation and detection of acoustic phonons in topological nanocavities,” APL Photon. 4, 030805 (2019). [CrossRef]  

11. M. Esmann, F. R. Lamberti, P. Senellart, I. Favero, O. Krebs, L. Lanco, C. Gomez Carbonell, A. Lemaître, and N. D. Lanzillotti-Kimura, “Topological nanophononic states by band inversion,” Phys. Rev. B 97, 155422 (2018). [CrossRef]  

12. M. Esmann, F. R. Lamberti, A. Lemaître, and N. D. Lanzillotti-Kimura, “Topological acoustics in coupled nanocavity arrays,” Phys. Rev. B 98, 161109 (2018). [CrossRef]  

13. L. M. Nash, D. Kleckner, A. Read, V. Vitelli, A. M. Turner, and W. T. M. Irvine, “Topological mechanics of gyroscopic metamaterials,” Proc. Natl. Acad. Sci. USA 112, 14495–14500 (2015). [CrossRef]  

14. M. Xiao, G. Ma, Z. Yang, P. Sheng, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 11, 240–244 (2015). [CrossRef]  

15. S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature 562, 552–556 (2018). [CrossRef]  

16. M. Milicevic, G. Montambaux, T. Ozawa, O. Jamadi, B. Real, I. Sagnes, A. Lemaître, L. Le Gratiet, A. Harouri, J. Bloch, and A. Amo, “Type-III and tilted Dirac cones emerging from flat bands in photonic orbital graphene,” Phys. Rev. X 9, 031010 (2019). [CrossRef]  

17. Y. Meng, X. Wu, R.-Y. Zhang, X. Li, P. Hu, L. Ge, Y. Huang, H. Xiang, D. Han, S. Wang, and W. Wen, “Designing topological interface states in phononic crystals based on the full phase diagrams,” New J. Phys. 20, 073032 (2018). [CrossRef]  

18. J. Yin, M. Ruzzene, J. Wen, D. Yu, L. Cai, and L. Yue, “Band transition and topological interface modes in 1D elastic phononic crystals,” Sci. Rep. 8, 6809 (2018). [CrossRef]  

19. X. Li, Y. Meng, X. Wu, S. Yan, Y. Huang, S. Wang, and W. Wen, “Su-Schrieffer-Heeger model inspired acoustic interface states and edge states,” Appl. Phys. Lett. 113, 203501 (2018). [CrossRef]  

20. Z. Wang, D. Zhao, J. Luo, R. Wang, and H. Yang, “Broadband modulation of subwavelength topological interface states in a one-dimensional acoustic system,” Appl. Phys. Lett. 116, 013102 (2020). [CrossRef]  

21. M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014). [CrossRef]  

22. L.-Y. Zheng, V. Achilleos, O. Richoux, G. Theocharis, and V. Pagneux, “Observation of edge waves in a two-dimensional Su-Schrieffer-Heeger acoustic network,” Phys. Rev. Appl. 12, 034014 (2019). [CrossRef]  

23. A. Fainstein, N. D. Lanzillotti-Kimura, B. Jusserand, and B. Perrin, “Strong optical-mechanical coupling in a vertical GaAs/AlAs microcavity for subterahertz phonons and near-infrared light,” Phys. Rev. Lett. 110, 037403 (2013). [CrossRef]  

24. G. Arregui, N. D. Lanzillotti-Kimura, C. M. Sotomayor-Torres, and P. D. García, “Anderson photon-phonon colocalization in certain random superlattices,” Phys. Rev. Lett. 122, 043903 (2019). [CrossRef]  

25. Ioffe Institute, “NSM archive—physical properties of semiconductors,” http://matprop.ru/semicond.

26. In materials such as ${{\rm Si}_x}{{\rm Ti}_{1 - x}}{{\rm O}_2}$, the speed of longitudinal acoustic phonons and the speed of light change in opposite directions as a function of composition. When tuning from ${{\rm TiO}_2}\;({ x} = {0})$ to ${{\rm SiO}_2}\;({x} = {1})$, the speed of longitudinal acoustic phonons drops from ${{v}_{\rm TiO2}} = {10}{,}{300\,\,\rm m/s}$ to ${{v}_{\rm SiO2}} = {5}{,}{970\,\,\rm m/s}$, while the speed of light increases from ${{ c}_{\rm TiO2}} = {0.{43c}_0}$ to ${{c}_{\rm SiO2}} = {0.{69c}_0}$. Under these conditions, it is possible to select pairs of materials for which the ratios of speed of light and speed of sound are each other’s inverse. That is, ${{c}_{x1}}/{{v}_{x1}}\; = {{v}_{x2}}/{{c}_{x2}}$, with the index xi referring to the proportion of silicon in material ${i} = {1},{2}$. Following the construction principle shown in Fig. 1, superlattices from a pair of materials like these show simultaneous band inversion for light and sound. However, the order of spatial mode symmetries at the band edges [violet and green dots in Figs. 1(a) and 1(b)] is opposite for the two fields (if for one of the two lattices the order from bottom to top is symmetric–antisymmetric for light, it is antisymmetric–symmetric for sound).

27. C. Thomsen, J. Strait, Z. Vardeny, H. J. Maris, J. Tauc, and J. J. Hauser, “Coherent phonon generation and detection by picosecond light pulses,” Phys. Rev. Lett. 53, 989–992 (1984). [CrossRef]  

28. C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B 34, 4129–4138 (1986). [CrossRef]  

29. N. D. Lanzillotti-Kimura, A. Fainstein, B. Jusserand, and A. Lemaître, “Resonant Raman scattering of nanocavity-confined acoustic phonons,” Phys. Rev. B 79, 035404 (2009). [CrossRef]  

30. P. Sesin, P. Soubelet, V. Villafañe, A. E. Bruchhausen, B. Jusserand, A. Lemaître, N. D. Lanzillotti-Kimura, and A. Fainstein, “Dynamical optical tuning of the coherent phonon detection sensitivity in DBR-based GaAs optomechanical resonators,” Phys. Rev. B 92, 075307 (2015). [CrossRef]  

31. S. Anguiano, A. E. Bruchhausen, I. Favero, I. Sagnes, A. Lemaître, N. D. Lanzillotti-Kimura, and A. Fainstein, “Optical cavity mode dynamics and coherent phonon generation in high-Q micropillar resonators,” Phys. Rev. A 98, 013816 (2018). [CrossRef]  

32. M. Trigo, A. Bruchhausen, A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Confinement of acoustical vibrations in a semiconductor planar phonon cavity,” Phys. Rev. Lett. 89, 227402 (2002). [CrossRef]  

33. M. F. Pascaul Winter, G. Rozas, A. Fainstein, B. Jusserand, B. Perrin, A. Huynh, P. O. Vaccaro, and S. Saravanan, “Selective optical generation of coherent acoustic nanocavity modes,” Phys. Rev. Lett. 98, 265501 (2007). [CrossRef]  

34. N. D. Lanzillotti-Kimura, A. Fainstein, A. Huynh, B. Perrin, B. Jusserand, A. Miard, and A. Lemaître, “Coherent generation of acoustic phonons in an optical microcavity,” Phys. Rev. Lett. 99, 217405 (2007). [CrossRef]  

35. N. D. Lanzillotti-Kimura, A. Fainstein, B. Perrin, B. Jusserand, L. Largeau, O. Mauguin, and A. Lemaitre, “Enhanced optical generation and detection of acoustic nanowaves in microcavities,” Phys. Rev. B 83, 201103 (2011). [CrossRef]  

36. Y. Li, Q. Miao, A. V. Nurmikko, and H. J. Maris, “Picosecond ultrasonic measurements using an optical cavity,” J. Appl. Phys. 105, 083516 (2009). [CrossRef]  

37. N. D. Lanzillotti-Kimura, A. Fainstein, B. Perrin, and B. Jusserand, “Theory of coherent generation and detection of THz acoustic phonons using optical microcavities,” Phys. Rev. B 84, 064307 (2011). [CrossRef]  

38. N. D. Lanzillotti-Kimura, A. Fainstein, and B. Jusserand, “Towards GHz–THz cavity optomechanics in DBR-based semiconductor resonators,” Ultrasonics 56, 80–89 (2015). [CrossRef]  

References

  • View by:

  1. A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60, 781–850 (1988).
    [Crossref]
  2. W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42, 1698–1701 (1979).
    [Crossref]
  3. W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22, 2099–2111 (1980).
    [Crossref]
  4. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7, 1001–1005 (2013).
    [Crossref]
  5. M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
    [Crossref]
  6. T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
    [Crossref]
  7. L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821–829 (2014).
    [Crossref]
  8. P. A. Kalozoumis, G. Theocharis, V. Achilleos, S. Félix, O. Richoux, and V. Pagneux, “Finite-size effects on topological interface states in one-dimensional scattering systems,” Phys. Rev. A 98, 023838 (2018).
    [Crossref]
  9. J. C. G. Henriques, T. G. Rappoport, Y. V. Bludov, M. I. Vasilevskiy, and N. M. R. Peres, “Topological photonic Tamm-states and the Su-Schrieffer-Heeger model,” Phys. Rev. A 101, 043811 (2020).
    [Crossref]
  10. G. Arregui, O. Ortíz, M. Esmann, C. M. Sotomayor-Torres, C. Gomez-Carbonell, O. Mauguin, B. Perrin, A. Lemaître, P. D. García, and N. D. Lanzillotti-Kimura, “Coherent generation and detection of acoustic phonons in topological nanocavities,” APL Photon. 4, 030805 (2019).
    [Crossref]
  11. M. Esmann, F. R. Lamberti, P. Senellart, I. Favero, O. Krebs, L. Lanco, C. Gomez Carbonell, A. Lemaître, and N. D. Lanzillotti-Kimura, “Topological nanophononic states by band inversion,” Phys. Rev. B 97, 155422 (2018).
    [Crossref]
  12. M. Esmann, F. R. Lamberti, A. Lemaître, and N. D. Lanzillotti-Kimura, “Topological acoustics in coupled nanocavity arrays,” Phys. Rev. B 98, 161109 (2018).
    [Crossref]
  13. L. M. Nash, D. Kleckner, A. Read, V. Vitelli, A. M. Turner, and W. T. M. Irvine, “Topological mechanics of gyroscopic metamaterials,” Proc. Natl. Acad. Sci. USA 112, 14495–14500 (2015).
    [Crossref]
  14. M. Xiao, G. Ma, Z. Yang, P. Sheng, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 11, 240–244 (2015).
    [Crossref]
  15. S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature 562, 552–556 (2018).
    [Crossref]
  16. M. Milicevic, G. Montambaux, T. Ozawa, O. Jamadi, B. Real, I. Sagnes, A. Lemaître, L. Le Gratiet, A. Harouri, J. Bloch, and A. Amo, “Type-III and tilted Dirac cones emerging from flat bands in photonic orbital graphene,” Phys. Rev. X 9, 031010 (2019).
    [Crossref]
  17. Y. Meng, X. Wu, R.-Y. Zhang, X. Li, P. Hu, L. Ge, Y. Huang, H. Xiang, D. Han, S. Wang, and W. Wen, “Designing topological interface states in phononic crystals based on the full phase diagrams,” New J. Phys. 20, 073032 (2018).
    [Crossref]
  18. J. Yin, M. Ruzzene, J. Wen, D. Yu, L. Cai, and L. Yue, “Band transition and topological interface modes in 1D elastic phononic crystals,” Sci. Rep. 8, 6809 (2018).
    [Crossref]
  19. X. Li, Y. Meng, X. Wu, S. Yan, Y. Huang, S. Wang, and W. Wen, “Su-Schrieffer-Heeger model inspired acoustic interface states and edge states,” Appl. Phys. Lett. 113, 203501 (2018).
    [Crossref]
  20. Z. Wang, D. Zhao, J. Luo, R. Wang, and H. Yang, “Broadband modulation of subwavelength topological interface states in a one-dimensional acoustic system,” Appl. Phys. Lett. 116, 013102 (2020).
    [Crossref]
  21. M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).
    [Crossref]
  22. L.-Y. Zheng, V. Achilleos, O. Richoux, G. Theocharis, and V. Pagneux, “Observation of edge waves in a two-dimensional Su-Schrieffer-Heeger acoustic network,” Phys. Rev. Appl. 12, 034014 (2019).
    [Crossref]
  23. A. Fainstein, N. D. Lanzillotti-Kimura, B. Jusserand, and B. Perrin, “Strong optical-mechanical coupling in a vertical GaAs/AlAs microcavity for subterahertz phonons and near-infrared light,” Phys. Rev. Lett. 110, 037403 (2013).
    [Crossref]
  24. G. Arregui, N. D. Lanzillotti-Kimura, C. M. Sotomayor-Torres, and P. D. García, “Anderson photon-phonon colocalization in certain random superlattices,” Phys. Rev. Lett. 122, 043903 (2019).
    [Crossref]
  25. Ioffe Institute, “NSM archive—physical properties of semiconductors,” http://matprop.ru/semicond .
  26. In materials such as ${{\rm Si}_x}{{\rm Ti}_{1 - x}}{{\rm O}_2}$, the speed of longitudinal acoustic phonons and the speed of light change in opposite directions as a function of composition. When tuning from ${{\rm TiO}_2}\;({ x} = {0})$ to ${{\rm SiO}_2}\;({x} = {1})$, the speed of longitudinal acoustic phonons drops from ${{v}_{\rm TiO2}} = {10}{,}{300\,\,\rm m/s}$ to ${{v}_{\rm SiO2}} = {5}{,}{970\,\,\rm m/s}$, while the speed of light increases from ${{ c}_{\rm TiO2}} = {0.{43c}_0}$ to ${{c}_{\rm SiO2}} = {0.{69c}_0}$. Under these conditions, it is possible to select pairs of materials for which the ratios of speed of light and speed of sound are each other’s inverse. That is, ${{c}_{x1}}/{{v}_{x1}}\; = {{v}_{x2}}/{{c}_{x2}}$, with the index xi referring to the proportion of silicon in material ${i} = {1},{2}$. Following the construction principle shown in Fig. 1, superlattices from a pair of materials like these show simultaneous band inversion for light and sound. However, the order of spatial mode symmetries at the band edges [violet and green dots in Figs. 1(a) and 1(b)] is opposite for the two fields (if for one of the two lattices the order from bottom to top is symmetric–antisymmetric for light, it is antisymmetric–symmetric for sound).
  27. C. Thomsen, J. Strait, Z. Vardeny, H. J. Maris, J. Tauc, and J. J. Hauser, “Coherent phonon generation and detection by picosecond light pulses,” Phys. Rev. Lett. 53, 989–992 (1984).
    [Crossref]
  28. C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B 34, 4129–4138 (1986).
    [Crossref]
  29. N. D. Lanzillotti-Kimura, A. Fainstein, B. Jusserand, and A. Lemaître, “Resonant Raman scattering of nanocavity-confined acoustic phonons,” Phys. Rev. B 79, 035404 (2009).
    [Crossref]
  30. P. Sesin, P. Soubelet, V. Villafañe, A. E. Bruchhausen, B. Jusserand, A. Lemaître, N. D. Lanzillotti-Kimura, and A. Fainstein, “Dynamical optical tuning of the coherent phonon detection sensitivity in DBR-based GaAs optomechanical resonators,” Phys. Rev. B 92, 075307 (2015).
    [Crossref]
  31. S. Anguiano, A. E. Bruchhausen, I. Favero, I. Sagnes, A. Lemaître, N. D. Lanzillotti-Kimura, and A. Fainstein, “Optical cavity mode dynamics and coherent phonon generation in high-Q micropillar resonators,” Phys. Rev. A 98, 013816 (2018).
    [Crossref]
  32. M. Trigo, A. Bruchhausen, A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Confinement of acoustical vibrations in a semiconductor planar phonon cavity,” Phys. Rev. Lett. 89, 227402 (2002).
    [Crossref]
  33. M. F. Pascaul Winter, G. Rozas, A. Fainstein, B. Jusserand, B. Perrin, A. Huynh, P. O. Vaccaro, and S. Saravanan, “Selective optical generation of coherent acoustic nanocavity modes,” Phys. Rev. Lett. 98, 265501 (2007).
    [Crossref]
  34. N. D. Lanzillotti-Kimura, A. Fainstein, A. Huynh, B. Perrin, B. Jusserand, A. Miard, and A. Lemaître, “Coherent generation of acoustic phonons in an optical microcavity,” Phys. Rev. Lett. 99, 217405 (2007).
    [Crossref]
  35. N. D. Lanzillotti-Kimura, A. Fainstein, B. Perrin, B. Jusserand, L. Largeau, O. Mauguin, and A. Lemaitre, “Enhanced optical generation and detection of acoustic nanowaves in microcavities,” Phys. Rev. B 83, 201103 (2011).
    [Crossref]
  36. Y. Li, Q. Miao, A. V. Nurmikko, and H. J. Maris, “Picosecond ultrasonic measurements using an optical cavity,” J. Appl. Phys. 105, 083516 (2009).
    [Crossref]
  37. N. D. Lanzillotti-Kimura, A. Fainstein, B. Perrin, and B. Jusserand, “Theory of coherent generation and detection of THz acoustic phonons using optical microcavities,” Phys. Rev. B 84, 064307 (2011).
    [Crossref]
  38. N. D. Lanzillotti-Kimura, A. Fainstein, and B. Jusserand, “Towards GHz–THz cavity optomechanics in DBR-based semiconductor resonators,” Ultrasonics 56, 80–89 (2015).
    [Crossref]

2020 (2)

J. C. G. Henriques, T. G. Rappoport, Y. V. Bludov, M. I. Vasilevskiy, and N. M. R. Peres, “Topological photonic Tamm-states and the Su-Schrieffer-Heeger model,” Phys. Rev. A 101, 043811 (2020).
[Crossref]

Z. Wang, D. Zhao, J. Luo, R. Wang, and H. Yang, “Broadband modulation of subwavelength topological interface states in a one-dimensional acoustic system,” Appl. Phys. Lett. 116, 013102 (2020).
[Crossref]

2019 (5)

L.-Y. Zheng, V. Achilleos, O. Richoux, G. Theocharis, and V. Pagneux, “Observation of edge waves in a two-dimensional Su-Schrieffer-Heeger acoustic network,” Phys. Rev. Appl. 12, 034014 (2019).
[Crossref]

G. Arregui, N. D. Lanzillotti-Kimura, C. M. Sotomayor-Torres, and P. D. García, “Anderson photon-phonon colocalization in certain random superlattices,” Phys. Rev. Lett. 122, 043903 (2019).
[Crossref]

G. Arregui, O. Ortíz, M. Esmann, C. M. Sotomayor-Torres, C. Gomez-Carbonell, O. Mauguin, B. Perrin, A. Lemaître, P. D. García, and N. D. Lanzillotti-Kimura, “Coherent generation and detection of acoustic phonons in topological nanocavities,” APL Photon. 4, 030805 (2019).
[Crossref]

M. Milicevic, G. Montambaux, T. Ozawa, O. Jamadi, B. Real, I. Sagnes, A. Lemaître, L. Le Gratiet, A. Harouri, J. Bloch, and A. Amo, “Type-III and tilted Dirac cones emerging from flat bands in photonic orbital graphene,” Phys. Rev. X 9, 031010 (2019).
[Crossref]

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

2018 (8)

P. A. Kalozoumis, G. Theocharis, V. Achilleos, S. Félix, O. Richoux, and V. Pagneux, “Finite-size effects on topological interface states in one-dimensional scattering systems,” Phys. Rev. A 98, 023838 (2018).
[Crossref]

Y. Meng, X. Wu, R.-Y. Zhang, X. Li, P. Hu, L. Ge, Y. Huang, H. Xiang, D. Han, S. Wang, and W. Wen, “Designing topological interface states in phononic crystals based on the full phase diagrams,” New J. Phys. 20, 073032 (2018).
[Crossref]

J. Yin, M. Ruzzene, J. Wen, D. Yu, L. Cai, and L. Yue, “Band transition and topological interface modes in 1D elastic phononic crystals,” Sci. Rep. 8, 6809 (2018).
[Crossref]

X. Li, Y. Meng, X. Wu, S. Yan, Y. Huang, S. Wang, and W. Wen, “Su-Schrieffer-Heeger model inspired acoustic interface states and edge states,” Appl. Phys. Lett. 113, 203501 (2018).
[Crossref]

M. Esmann, F. R. Lamberti, P. Senellart, I. Favero, O. Krebs, L. Lanco, C. Gomez Carbonell, A. Lemaître, and N. D. Lanzillotti-Kimura, “Topological nanophononic states by band inversion,” Phys. Rev. B 97, 155422 (2018).
[Crossref]

M. Esmann, F. R. Lamberti, A. Lemaître, and N. D. Lanzillotti-Kimura, “Topological acoustics in coupled nanocavity arrays,” Phys. Rev. B 98, 161109 (2018).
[Crossref]

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature 562, 552–556 (2018).
[Crossref]

S. Anguiano, A. E. Bruchhausen, I. Favero, I. Sagnes, A. Lemaître, N. D. Lanzillotti-Kimura, and A. Fainstein, “Optical cavity mode dynamics and coherent phonon generation in high-Q micropillar resonators,” Phys. Rev. A 98, 013816 (2018).
[Crossref]

2015 (4)

P. Sesin, P. Soubelet, V. Villafañe, A. E. Bruchhausen, B. Jusserand, A. Lemaître, N. D. Lanzillotti-Kimura, and A. Fainstein, “Dynamical optical tuning of the coherent phonon detection sensitivity in DBR-based GaAs optomechanical resonators,” Phys. Rev. B 92, 075307 (2015).
[Crossref]

L. M. Nash, D. Kleckner, A. Read, V. Vitelli, A. M. Turner, and W. T. M. Irvine, “Topological mechanics of gyroscopic metamaterials,” Proc. Natl. Acad. Sci. USA 112, 14495–14500 (2015).
[Crossref]

M. Xiao, G. Ma, Z. Yang, P. Sheng, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 11, 240–244 (2015).
[Crossref]

N. D. Lanzillotti-Kimura, A. Fainstein, and B. Jusserand, “Towards GHz–THz cavity optomechanics in DBR-based semiconductor resonators,” Ultrasonics 56, 80–89 (2015).
[Crossref]

2014 (2)

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821–829 (2014).
[Crossref]

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).
[Crossref]

2013 (2)

A. Fainstein, N. D. Lanzillotti-Kimura, B. Jusserand, and B. Perrin, “Strong optical-mechanical coupling in a vertical GaAs/AlAs microcavity for subterahertz phonons and near-infrared light,” Phys. Rev. Lett. 110, 037403 (2013).
[Crossref]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7, 1001–1005 (2013).
[Crossref]

2011 (3)

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

N. D. Lanzillotti-Kimura, A. Fainstein, B. Perrin, B. Jusserand, L. Largeau, O. Mauguin, and A. Lemaitre, “Enhanced optical generation and detection of acoustic nanowaves in microcavities,” Phys. Rev. B 83, 201103 (2011).
[Crossref]

N. D. Lanzillotti-Kimura, A. Fainstein, B. Perrin, and B. Jusserand, “Theory of coherent generation and detection of THz acoustic phonons using optical microcavities,” Phys. Rev. B 84, 064307 (2011).
[Crossref]

2009 (2)

Y. Li, Q. Miao, A. V. Nurmikko, and H. J. Maris, “Picosecond ultrasonic measurements using an optical cavity,” J. Appl. Phys. 105, 083516 (2009).
[Crossref]

N. D. Lanzillotti-Kimura, A. Fainstein, B. Jusserand, and A. Lemaître, “Resonant Raman scattering of nanocavity-confined acoustic phonons,” Phys. Rev. B 79, 035404 (2009).
[Crossref]

2007 (2)

M. F. Pascaul Winter, G. Rozas, A. Fainstein, B. Jusserand, B. Perrin, A. Huynh, P. O. Vaccaro, and S. Saravanan, “Selective optical generation of coherent acoustic nanocavity modes,” Phys. Rev. Lett. 98, 265501 (2007).
[Crossref]

N. D. Lanzillotti-Kimura, A. Fainstein, A. Huynh, B. Perrin, B. Jusserand, A. Miard, and A. Lemaître, “Coherent generation of acoustic phonons in an optical microcavity,” Phys. Rev. Lett. 99, 217405 (2007).
[Crossref]

2002 (1)

M. Trigo, A. Bruchhausen, A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Confinement of acoustical vibrations in a semiconductor planar phonon cavity,” Phys. Rev. Lett. 89, 227402 (2002).
[Crossref]

1988 (1)

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60, 781–850 (1988).
[Crossref]

1986 (1)

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B 34, 4129–4138 (1986).
[Crossref]

1984 (1)

C. Thomsen, J. Strait, Z. Vardeny, H. J. Maris, J. Tauc, and J. J. Hauser, “Coherent phonon generation and detection by picosecond light pulses,” Phys. Rev. Lett. 53, 989–992 (1984).
[Crossref]

1980 (1)

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22, 2099–2111 (1980).
[Crossref]

1979 (1)

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42, 1698–1701 (1979).
[Crossref]

Achilleos, V.

L.-Y. Zheng, V. Achilleos, O. Richoux, G. Theocharis, and V. Pagneux, “Observation of edge waves in a two-dimensional Su-Schrieffer-Heeger acoustic network,” Phys. Rev. Appl. 12, 034014 (2019).
[Crossref]

P. A. Kalozoumis, G. Theocharis, V. Achilleos, S. Félix, O. Richoux, and V. Pagneux, “Finite-size effects on topological interface states in one-dimensional scattering systems,” Phys. Rev. A 98, 023838 (2018).
[Crossref]

Amo, A.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

M. Milicevic, G. Montambaux, T. Ozawa, O. Jamadi, B. Real, I. Sagnes, A. Lemaître, L. Le Gratiet, A. Harouri, J. Bloch, and A. Amo, “Type-III and tilted Dirac cones emerging from flat bands in photonic orbital graphene,” Phys. Rev. X 9, 031010 (2019).
[Crossref]

Anguiano, S.

S. Anguiano, A. E. Bruchhausen, I. Favero, I. Sagnes, A. Lemaître, N. D. Lanzillotti-Kimura, and A. Fainstein, “Optical cavity mode dynamics and coherent phonon generation in high-Q micropillar resonators,” Phys. Rev. A 98, 013816 (2018).
[Crossref]

Arregui, G.

G. Arregui, N. D. Lanzillotti-Kimura, C. M. Sotomayor-Torres, and P. D. García, “Anderson photon-phonon colocalization in certain random superlattices,” Phys. Rev. Lett. 122, 043903 (2019).
[Crossref]

G. Arregui, O. Ortíz, M. Esmann, C. M. Sotomayor-Torres, C. Gomez-Carbonell, O. Mauguin, B. Perrin, A. Lemaître, P. D. García, and N. D. Lanzillotti-Kimura, “Coherent generation and detection of acoustic phonons in topological nanocavities,” APL Photon. 4, 030805 (2019).
[Crossref]

Bandres, M. A.

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature 562, 552–556 (2018).
[Crossref]

Bloch, J.

M. Milicevic, G. Montambaux, T. Ozawa, O. Jamadi, B. Real, I. Sagnes, A. Lemaître, L. Le Gratiet, A. Harouri, J. Bloch, and A. Amo, “Type-III and tilted Dirac cones emerging from flat bands in photonic orbital graphene,” Phys. Rev. X 9, 031010 (2019).
[Crossref]

Bludov, Y. V.

J. C. G. Henriques, T. G. Rappoport, Y. V. Bludov, M. I. Vasilevskiy, and N. M. R. Peres, “Topological photonic Tamm-states and the Su-Schrieffer-Heeger model,” Phys. Rev. A 101, 043811 (2020).
[Crossref]

Bruchhausen, A.

M. Trigo, A. Bruchhausen, A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Confinement of acoustical vibrations in a semiconductor planar phonon cavity,” Phys. Rev. Lett. 89, 227402 (2002).
[Crossref]

Bruchhausen, A. E.

S. Anguiano, A. E. Bruchhausen, I. Favero, I. Sagnes, A. Lemaître, N. D. Lanzillotti-Kimura, and A. Fainstein, “Optical cavity mode dynamics and coherent phonon generation in high-Q micropillar resonators,” Phys. Rev. A 98, 013816 (2018).
[Crossref]

P. Sesin, P. Soubelet, V. Villafañe, A. E. Bruchhausen, B. Jusserand, A. Lemaître, N. D. Lanzillotti-Kimura, and A. Fainstein, “Dynamical optical tuning of the coherent phonon detection sensitivity in DBR-based GaAs optomechanical resonators,” Phys. Rev. B 92, 075307 (2015).
[Crossref]

Cai, L.

J. Yin, M. Ruzzene, J. Wen, D. Yu, L. Cai, and L. Yue, “Band transition and topological interface modes in 1D elastic phononic crystals,” Sci. Rep. 8, 6809 (2018).
[Crossref]

Carusotto, I.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

Chan, C. T.

M. Xiao, G. Ma, Z. Yang, P. Sheng, Z. Q. Zhang, and C. T. Chan, “Geometric phase and band inversion in periodic acoustic systems,” Nat. Phys. 11, 240–244 (2015).
[Crossref]

M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface impedance and bulk band geometric phases in one-dimensional systems,” Phys. Rev. X 4, 021017 (2014).
[Crossref]

Demler, E. A.

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

Egorov, O. A.

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature 562, 552–556 (2018).
[Crossref]

Emmerling, M.

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature 562, 552–556 (2018).
[Crossref]

Esmann, M.

G. Arregui, O. Ortíz, M. Esmann, C. M. Sotomayor-Torres, C. Gomez-Carbonell, O. Mauguin, B. Perrin, A. Lemaître, P. D. García, and N. D. Lanzillotti-Kimura, “Coherent generation and detection of acoustic phonons in topological nanocavities,” APL Photon. 4, 030805 (2019).
[Crossref]

M. Esmann, F. R. Lamberti, P. Senellart, I. Favero, O. Krebs, L. Lanco, C. Gomez Carbonell, A. Lemaître, and N. D. Lanzillotti-Kimura, “Topological nanophononic states by band inversion,” Phys. Rev. B 97, 155422 (2018).
[Crossref]

M. Esmann, F. R. Lamberti, A. Lemaître, and N. D. Lanzillotti-Kimura, “Topological acoustics in coupled nanocavity arrays,” Phys. Rev. B 98, 161109 (2018).
[Crossref]

Fainstein, A.

S. Anguiano, A. E. Bruchhausen, I. Favero, I. Sagnes, A. Lemaître, N. D. Lanzillotti-Kimura, and A. Fainstein, “Optical cavity mode dynamics and coherent phonon generation in high-Q micropillar resonators,” Phys. Rev. A 98, 013816 (2018).
[Crossref]

P. Sesin, P. Soubelet, V. Villafañe, A. E. Bruchhausen, B. Jusserand, A. Lemaître, N. D. Lanzillotti-Kimura, and A. Fainstein, “Dynamical optical tuning of the coherent phonon detection sensitivity in DBR-based GaAs optomechanical resonators,” Phys. Rev. B 92, 075307 (2015).
[Crossref]

N. D. Lanzillotti-Kimura, A. Fainstein, and B. Jusserand, “Towards GHz–THz cavity optomechanics in DBR-based semiconductor resonators,” Ultrasonics 56, 80–89 (2015).
[Crossref]

A. Fainstein, N. D. Lanzillotti-Kimura, B. Jusserand, and B. Perrin, “Strong optical-mechanical coupling in a vertical GaAs/AlAs microcavity for subterahertz phonons and near-infrared light,” Phys. Rev. Lett. 110, 037403 (2013).
[Crossref]

N. D. Lanzillotti-Kimura, A. Fainstein, B. Perrin, and B. Jusserand, “Theory of coherent generation and detection of THz acoustic phonons using optical microcavities,” Phys. Rev. B 84, 064307 (2011).
[Crossref]

N. D. Lanzillotti-Kimura, A. Fainstein, B. Perrin, B. Jusserand, L. Largeau, O. Mauguin, and A. Lemaitre, “Enhanced optical generation and detection of acoustic nanowaves in microcavities,” Phys. Rev. B 83, 201103 (2011).
[Crossref]

N. D. Lanzillotti-Kimura, A. Fainstein, B. Jusserand, and A. Lemaître, “Resonant Raman scattering of nanocavity-confined acoustic phonons,” Phys. Rev. B 79, 035404 (2009).
[Crossref]

N. D. Lanzillotti-Kimura, A. Fainstein, A. Huynh, B. Perrin, B. Jusserand, A. Miard, and A. Lemaître, “Coherent generation of acoustic phonons in an optical microcavity,” Phys. Rev. Lett. 99, 217405 (2007).
[Crossref]

M. F. Pascaul Winter, G. Rozas, A. Fainstein, B. Jusserand, B. Perrin, A. Huynh, P. O. Vaccaro, and S. Saravanan, “Selective optical generation of coherent acoustic nanocavity modes,” Phys. Rev. Lett. 98, 265501 (2007).
[Crossref]

M. Trigo, A. Bruchhausen, A. Fainstein, B. Jusserand, and V. Thierry-Mieg, “Confinement of acoustical vibrations in a semiconductor planar phonon cavity,” Phys. Rev. Lett. 89, 227402 (2002).
[Crossref]

Fan, J.

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7, 1001–1005 (2013).
[Crossref]

Favero, I.

M. Esmann, F. R. Lamberti, P. Senellart, I. Favero, O. Krebs, L. Lanco, C. Gomez Carbonell, A. Lemaître, and N. D. Lanzillotti-Kimura, “Topological nanophononic states by band inversion,” Phys. Rev. B 97, 155422 (2018).
[Crossref]

S. Anguiano, A. E. Bruchhausen, I. Favero, I. Sagnes, A. Lemaître, N. D. Lanzillotti-Kimura, and A. Fainstein, “Optical cavity mode dynamics and coherent phonon generation in high-Q micropillar resonators,” Phys. Rev. A 98, 013816 (2018).
[Crossref]

Félix, S.

P. A. Kalozoumis, G. Theocharis, V. Achilleos, S. Félix, O. Richoux, and V. Pagneux, “Finite-size effects on topological interface states in one-dimensional scattering systems,” Phys. Rev. A 98, 023838 (2018).
[Crossref]

García, P. D.

G. Arregui, O. Ortíz, M. Esmann, C. M. Sotomayor-Torres, C. Gomez-Carbonell, O. Mauguin, B. Perrin, A. Lemaître, P. D. García, and N. D. Lanzillotti-Kimura, “Coherent generation and detection of acoustic phonons in topological nanocavities,” APL Photon. 4, 030805 (2019).
[Crossref]

G. Arregui, N. D. Lanzillotti-Kimura, C. M. Sotomayor-Torres, and P. D. García, “Anderson photon-phonon colocalization in certain random superlattices,” Phys. Rev. Lett. 122, 043903 (2019).
[Crossref]

Ge, L.

Y. Meng, X. Wu, R.-Y. Zhang, X. Li, P. Hu, L. Ge, Y. Huang, H. Xiang, D. Han, S. Wang, and W. Wen, “Designing topological interface states in phononic crystals based on the full phase diagrams,” New J. Phys. 20, 073032 (2018).
[Crossref]

Ge, R.

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature 562, 552–556 (2018).
[Crossref]

Goldman, N.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

Gomez Carbonell, C.

M. Esmann, F. R. Lamberti, P. Senellart, I. Favero, O. Krebs, L. Lanco, C. Gomez Carbonell, A. Lemaître, and N. D. Lanzillotti-Kimura, “Topological nanophononic states by band inversion,” Phys. Rev. B 97, 155422 (2018).
[Crossref]

Gomez-Carbonell, C.

G. Arregui, O. Ortíz, M. Esmann, C. M. Sotomayor-Torres, C. Gomez-Carbonell, O. Mauguin, B. Perrin, A. Lemaître, P. D. García, and N. D. Lanzillotti-Kimura, “Coherent generation and detection of acoustic phonons in topological nanocavities,” APL Photon. 4, 030805 (2019).
[Crossref]

Grahn, H. T.

C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, “Surface generation and detection of phonons by picosecond light pulses,” Phys. Rev. B 34, 4129–4138 (1986).
[Crossref]

Hafezi, M.

T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, “Topological photonics,” Rev. Mod. Phys. 91, 015006 (2019).
[Crossref]

M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Taylor, “Imaging topological edge states in silicon photonics,” Nat. Photonics 7, 1001–1005 (2013).
[Crossref]

M. Hafezi, E. A. Demler, M. D. Lukin, and J. M. Taylor, “Robust optical delay lines with topological protection,” Nat. Phys. 7, 907–912 (2011).
[Crossref]

Han, D.

Y. Meng, X. Wu, R.-Y. Zhang, X. Li, P. Hu, L. Ge, Y. Huang, H. Xiang, D. Han, S. Wang, and W. Wen, “Designing topological interface states in phononic crystals based on the full phase diagrams,” New J. Phys. 20, 073032 (2018).
[Crossref]

Harder, T. H.

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature 562, 552–556 (2018).
[Crossref]

Harouri, A.

M. Milicevic, G. Montambaux, T. Ozawa, O. Jamadi, B. Real, I. Sagnes, A. Lemaître, L. Le Gratiet, A. Harouri, J. Bloch, and A. Amo, “Type-III and tilted Dirac cones emerging from flat bands in photonic orbital graphene,” Phys. Rev. X 9, 031010 (2019).
[Crossref]

Hauser, J. J.

C. Thomsen, J. Strait, Z. Vardeny, H. J. Maris, J. Tauc, and J. J. Hauser, “Coherent phonon generation and detection by picosecond light pulses,” Phys. Rev. Lett. 53, 989–992 (1984).
[Crossref]

Heeger, A. J.

A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W.-P. Su, “Solitons in conducting polymers,” Rev. Mod. Phys. 60, 781–850 (1988).
[Crossref]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Soliton excitations in polyacetylene,” Phys. Rev. B 22, 2099–2111 (1980).
[Crossref]

W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42, 1698–1701 (1979).
[Crossref]

Henriques, J. C. G.

J. C. G. Henriques, T. G. Rappoport, Y. V. Bludov, M. I. Vasilevskiy, and N. M. R. Peres, “Topological photonic Tamm-states and the Su-Schrieffer-Heeger model,” Phys. Rev. A 101, 043811 (2020).
[Crossref]

Höfling, S.

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature 562, 552–556 (2018).
[Crossref]

Hu, P.

Y. Meng, X. Wu, R.-Y. Zhang, X. Li, P. Hu, L. Ge, Y. Huang, H. Xiang, D. Han, S. Wang, and W. Wen, “Designing topological interface states in phononic crystals based on the full phase diagrams,” New J. Phys. 20, 073032 (2018).
[Crossref]

Huang, Y.

Y. Meng, X. Wu, R.-Y. Zhang, X. Li, P. Hu, L. Ge, Y. Huang, H. Xiang, D. Han, S. Wang, and W. Wen, “Designing topological interface states in phononic crystals based on the full phase diagrams,” New J. Phys. 20, 073032 (2018).
[Crossref]

X. Li, Y. Meng, X. Wu, S. Yan, Y. Huang, S. Wang, and W. Wen, “Su-Schrieffer-Heeger model inspired acoustic interface states and edge states,” Appl. Phys. Lett. 113, 203501 (2018).
[Crossref]

Huynh, A.

M. F. Pascaul Winter, G. Rozas, A. Fainstein, B. Jusserand, B. Perrin, A. Huynh, P. O. Vaccaro, and S. Saravanan, “Selective optical generation of coherent acoustic nanocavity modes,” Phys. Rev. Lett. 98, 265501 (2007).
[Crossref]

N. D. Lanzillotti-Kimura, A. Fainstein, A. Huynh, B. Perrin, B. Jusserand, A. Miard, and A. Lemaître, “Coherent generation of acoustic phonons in an optical microcavity,” Phys. Rev. Lett. 99, 217405 (2007).
[Crossref]

Irvine, W. T. M.

L. M. Nash, D. Kleckner, A. Read, V. Vitelli, A. M. Turner, and W. T. M. Irvine, “Topological mechanics of gyroscopic metamaterials,” Proc. Natl. Acad. Sci. USA 112, 14495–14500 (2015).
[Crossref]

Jamadi, O.

M. Milicevic, G. Montambaux, T. Ozawa, O. Jamadi, B. Real, I. Sagnes, A. Lemaître, L. Le Gratiet, A. Harouri, J. Bloch, and A. Amo, “Type-III and tilted Dirac cones emerging from flat bands in photonic orbital graphene,” Phys. Rev. X 9, 031010 (2019).
[Crossref]

Joannopoulos, J. D.

L. Lu, J. D. Joannopoulos, and M. Soljačić, “Topological photonics,” Nat. Photonics 8, 821–829 (2014).
[Crossref]

Jusserand, B.

N. D. Lanzillotti-Kimura, A. Fainstein, and B. Jusserand, “Towards GHz–THz cavity optomechanics in DBR-based semiconductor resonators,” Ultrasonics 56, 80–89 (2015).
[Crossref]

P. Sesin, P. Soubelet, V. Villafañe, A. E. Bruchhausen, B. Jusserand, A. Lemaître, N. D. Lanzillotti-Kimura, and A. Fainstein, “Dynamical optical tuning of the coherent phonon detection sensitivity in DBR-based GaAs optomechanical resonators,” Phys. Rev. B 92, 075307 (2015).
[Crossref]

A. Fainstein, N. D. Lanzillotti-Kimura, B. Jusserand, and B. Perrin, “Strong optical-mechanical coupling in a vertical GaAs/AlAs microcavity for subterahertz phonons and near-infrared light,” Phys. Rev. Lett. 110, 037403 (2013).
[Crossref]

N. D. Lanzillotti-Kimura, A. Fainstein, B. Perrin, and B. Jusserand, “Theory of coherent generation and detection of THz acoustic phonons using optical microcavities,” Phys. Rev. B 84, 064307 (2011).
[Crossref]

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P. Sesin, P. Soubelet, V. Villafañe, A. E. Bruchhausen, B. Jusserand, A. Lemaître, N. D. Lanzillotti-Kimura, and A. Fainstein, “Dynamical optical tuning of the coherent phonon detection sensitivity in DBR-based GaAs optomechanical resonators,” Phys. Rev. B 92, 075307 (2015).
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G. Arregui, O. Ortíz, M. Esmann, C. M. Sotomayor-Torres, C. Gomez-Carbonell, O. Mauguin, B. Perrin, A. Lemaître, P. D. García, and N. D. Lanzillotti-Kimura, “Coherent generation and detection of acoustic phonons in topological nanocavities,” APL Photon. 4, 030805 (2019).
[Crossref]

G. Arregui, N. D. Lanzillotti-Kimura, C. M. Sotomayor-Torres, and P. D. García, “Anderson photon-phonon colocalization in certain random superlattices,” Phys. Rev. Lett. 122, 043903 (2019).
[Crossref]

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APL Photon. (1)

G. Arregui, O. Ortíz, M. Esmann, C. M. Sotomayor-Torres, C. Gomez-Carbonell, O. Mauguin, B. Perrin, A. Lemaître, P. D. García, and N. D. Lanzillotti-Kimura, “Coherent generation and detection of acoustic phonons in topological nanocavities,” APL Photon. 4, 030805 (2019).
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Nature (1)

S. Klembt, T. H. Harder, O. A. Egorov, K. Winkler, R. Ge, M. A. Bandres, M. Emmerling, L. Worschech, T. C. H. Liew, M. Segev, C. Schneider, and S. Höfling, “Exciton-polariton topological insulator,” Nature 562, 552–556 (2018).
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New J. Phys. (1)

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Phys. Rev. A (3)

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Phys. Rev. B (8)

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P. Sesin, P. Soubelet, V. Villafañe, A. E. Bruchhausen, B. Jusserand, A. Lemaître, N. D. Lanzillotti-Kimura, and A. Fainstein, “Dynamical optical tuning of the coherent phonon detection sensitivity in DBR-based GaAs optomechanical resonators,” Phys. Rev. B 92, 075307 (2015).
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W. P. Su, J. R. Schrieffer, and A. J. Heeger, “Solitons in polyacetylene,” Phys. Rev. Lett. 42, 1698–1701 (1979).
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M. F. Pascaul Winter, G. Rozas, A. Fainstein, B. Jusserand, B. Perrin, A. Huynh, P. O. Vaccaro, and S. Saravanan, “Selective optical generation of coherent acoustic nanocavity modes,” Phys. Rev. Lett. 98, 265501 (2007).
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Proc. Natl. Acad. Sci. USA (1)

L. M. Nash, D. Kleckner, A. Read, V. Vitelli, A. M. Turner, and W. T. M. Irvine, “Topological mechanics of gyroscopic metamaterials,” Proc. Natl. Acad. Sci. USA 112, 14495–14500 (2015).
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Sci. Rep. (1)

J. Yin, M. Ruzzene, J. Wen, D. Yu, L. Cai, and L. Yue, “Band transition and topological interface modes in 1D elastic phononic crystals,” Sci. Rep. 8, 6809 (2018).
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Other (2)

Ioffe Institute, “NSM archive—physical properties of semiconductors,” http://matprop.ru/semicond .

In materials such as ${{\rm Si}_x}{{\rm Ti}_{1 - x}}{{\rm O}_2}$, the speed of longitudinal acoustic phonons and the speed of light change in opposite directions as a function of composition. When tuning from ${{\rm TiO}_2}\;({ x} = {0})$ to ${{\rm SiO}_2}\;({x} = {1})$, the speed of longitudinal acoustic phonons drops from ${{v}_{\rm TiO2}} = {10}{,}{300\,\,\rm m/s}$ to ${{v}_{\rm SiO2}} = {5}{,}{970\,\,\rm m/s}$, while the speed of light increases from ${{ c}_{\rm TiO2}} = {0.{43c}_0}$ to ${{c}_{\rm SiO2}} = {0.{69c}_0}$. Under these conditions, it is possible to select pairs of materials for which the ratios of speed of light and speed of sound are each other’s inverse. That is, ${{c}_{x1}}/{{v}_{x1}}\; = {{v}_{x2}}/{{c}_{x2}}$, with the index xi referring to the proportion of silicon in material ${i} = {1},{2}$. Following the construction principle shown in Fig. 1, superlattices from a pair of materials like these show simultaneous band inversion for light and sound. However, the order of spatial mode symmetries at the band edges [violet and green dots in Figs. 1(a) and 1(b)] is opposite for the two fields (if for one of the two lattices the order from bottom to top is symmetric–antisymmetric for light, it is antisymmetric–symmetric for sound).

Supplementary Material (1)

NameDescription
Supplement 1       Band inversion and topological invariants in superlattices.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)

Fig. 1.
Fig. 1. Construction principle of an interface state in a photonic–phononic nanocavity by band inversion. (a) Schematic of the unit cells of each DBR parametrized by $\delta$ , which describes the relative thickness of the materials in units of wavelength while keeping the overall thickness constant (also in units of wavelength). For the particular case of GaAlAs, the values of $\delta$ are the same for both the optical and acoustic cases. Dark (light) colors represent GaAs (AlAs); blue and green are used to distinguish between left and right superlattices, respectively. The dispersion relations of acoustic (a) phonons and (b) photons present inverted Bloch mode symmetries around the first bandgap at the Brillouin zone center. The dashed red box indicates the bandgap over which the band inversion takes place; mode symmetries are indicated with violet (antisymmetric) and green (symmetric) dots. (c) The spatial displacement pattern $|{ u}({z})|^{2}$ (black) and optical profile $|{ E}({z})|^{2}$ (red) of the topological interface state at 18.12 GHz/1.34 eV are colocalized and present a maximum at the interface between the two DBRs. Both fields decay evanescently away from the interface. The complete heterostructure is represented at the top.
Fig. 2.
Fig. 2. Continuous wave optical characterization of the topological interface modes. (a) The measured optical reflectivity reveals an interface state in the middle of the first optical stopband at the Brillouin zone center. (b) The zoomed-in optical reflectivity spectrum shows the interface state to be centered at 922.5 nm (1.34 eV).
Fig. 3.
Fig. 3. Time-domain opto-phononic characterization of the topological interface mode. (a) Experimental coherent phonon generation setup. A mechanical delay line is used to control the delay between pump and probe pulses arriving on the sample. A cross-polarization scheme is used to probe the sample’s instantaneous reflectivity while suppressing the reflected pump pulses. (b) Differential reflectivity time trace (0–11.000 ps) measured with laser-cavity detuning of ${-}{0.786}\;{\rm nm}$ , 50 mW pump power, and 10 mW probe power. (c) Photoelastic model calculation of the pump–probe spectrum considering a resolution of 90 MHz. (d) Fast Fourier transform (FFT) of the experimental time trace featuring the topological resonator mode at 18.12 GHz. The inset shows a detail of the resonance with a full width at half maximum (FWHM) of 158 MHz. The experimental result agrees well with the simulation.
Fig. 4.
Fig. 4. (a) Differential reflectivity time trace (0–11.000 ps) taken at a spectral detuning of ${-}{0.18}\;{\rm nm}$ of the laser from the cavity mode, 50 mW pump power, and 8 mW probe power. (b) Fast Fourier transform (FFT) of the experimental time trace featuring the topological resonator mode at 18.12 GHz. The inset shows a detail of the resonance that appears split with a separation of 140 MHz. (c) Time trace from (a) after passing a bandpass filter (17.97 GHz to 18.25 GHz). Coherent phonon oscillations are clearly visible with a modulated envelope. (d) The envelope in (c) is determined by the sensitivity of the structure to measure the modulation induced by the phonons. The red line indicates the position of the cavity as a function of delay, the dashed line the laser wavelength (fixed), and the gray line the unperturbed cavity resonance wavelength. The arrow shows the moment when the laser matches the cavity mode and the system is completely insensitive to phonons.
Fig. 5.
Fig. 5. Robust Brillouin scattering in a topological opto-phononic resonator under random perturbations. (a) Unit cells at the center of a topological resonator and a Fabry–Perot resonator. (b) Upon perturbing the structures by random variations of the parameter $\delta$ or each unit cell, the acoustic resonance frequency stays pinned at the bandgap center for the topological mode (black) but undergoes substantial variations for the Fabry–Perot case (gray). We chose random variations with a uniform distribution of width $\Delta \delta/\delta$ . (c) For both types of resonators, the acoustic quality factor drops for fluctuations $\Delta \delta/\delta \gt {0.2}$ . (d) This drop in quality factor fully accounts for a moderate drop in Brillouin scattering cross-section for the topological resonator. In contrast, the cross-section fluctuates over four orders of magnitude for the Fabry–Perot resonator.
Fig. 6.
Fig. 6. Comparison between different topological structures and a Fabry–Perot resonator. The structures are designed to confine an acoustic mode at 18.12 GHz and an optical mode at 1.34 eV. (a)–(e) For each structure, we display the spatial displacement pattern $|{u}({z})|^{2}$ (dashed black) and the optical profile $|{E}({z})|^{2}$ (solid red). (a)–(d) Topological structures composed of varying thicknesses and material composition; (e) conventional Fabry–Perot resonator (see text for details). The topmost panel represents the unit cell of the DBRs at the interface of each structure. (f)–(j) Brillouin cross-section integrand of structures in (a)–(e).

Equations (2)

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σ ( ω ) = | E ( z ) | 2 p ( z ) u ( ω , z ) z d z .
D ( ω ) = E ( z ) 2 p ( z ) σ ( ω ) u ( ω , z ) z d z .

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