## Abstract

Chip-based cavity optomechanical systems are being considered for applications in sensing, metrology, and quantum information science. Critical to their development is an understanding of how the optical and mechanical modes interact, quantified by the coupling rate ${g}_{0}$. Here, we develop GaAs optomechanical resonators and investigate the moving dielectric boundary and photoelastic contributions to ${g}_{0}$. First, we consider coupling between the fundamental radial breathing mechanical mode and a 1550 nm band optical whispering gallery mode in microdisks. For decreasing disk radius from $R=5$ to 1 μm, simulations and measurements show that ${g}_{0}$ changes from being dominated by the moving boundary contribution to having an equal photoelastic contribution. Next, we design and demonstrate nanobeam optomechanical crystals, in which a 2.5 GHz mechanical breathing mode couples to a 1550 nm optical mode, predominantly through the photoelastic effect. We show a significant (30%) dependence of ${g}_{0}$ on the device’s in-plane orientation, resulting from the difference in GaAs photoelastic coefficients along different crystalline axes, with fabricated devices exhibiting ${g}_{0}/2\mathrm{\pi}$ as high as 1.1 MHz, for orientation along the 110 axis. GaAs nanobeam optomechanical crystals are a promising system, which can combine the demonstrated large optomechanical coupling strength with additional functionality, such as piezoelectric actuation and incorporation of optical gain media.

© 2014 Optical Society of America

## 1. INTRODUCTION

Mechanical motion and optical fields are coupled by a number of different mechanisms in cavity optomechanical systems [1–4]. Within microscale and nanoscale geometries, perhaps the most commonly considered one is the change in effective optical path length resulting from moving dielectric boundaries, analogous to a movable mirror in a Fabry–Perot cavity. However, the optical path length also depends on the refractive index of the medium filling the cavity, and in solids this can change due to mechanical motion, because of the photoelastic (PE) effect (electrostriction) [5]. This has been observed in stimulated Brillouin scattering, in suspended silicon waveguides [6], and cooling and excitation of traveling wave acoustic modes, in silica whispering gallery mode resonators [7,8]. More recently, it has been considered in silicon optomechanical crystals, where optimized geometries that exclusively rely on the PE effect have been developed [9].

In this paper, we investigate the moving dielectric boundary and PE contributions to the optomechanical coupling in GaAs devices. GaAs has many desirable properties for cavity optomechanics: relatively large PE coefficients [5], which can produce devices with high optomechanical coupling [9]; piezoelectric properties [10], which can be exploited for driving or readout of mechanical motion; and potential for integration with InAs/GaAs quantum dots that offer nonclassical light emission and a strong resonant nonlinearity [11], and which can be used to probe and control mechanical motion [12,13].

We first present a combined theoretical and experimental analysis of GaAs microdisks of varying radius, where in large radius devices, the moving boundary (MB) effect dominates, while in small radius devices the photoelastic effect is the leading contribution. While these trends have recently been theoretically predicted [14], here we experimentally demonstrate the importance of considering both effects in the overall optomechanical coupling rate ${g}_{0}$. We then present two designs of GaAs nanobeam optomechanical crystals that rely predominantly on the PE effect. We show a significant (30%) dependence of ${g}_{0}$ on the in-plane device angle, in contrast with similar Si devices [9], for which the dependence is much weaker (3%). This dependence originates from the much larger magnitude and opposite sign of the PE coefficient ${p}_{12}$ in GaAs. We experimentally demonstrate this effect, by measuring ${g}_{0}$ in devices fabricated with differing in-plane angles, and measure ${g}_{0}/2\pi $ as high as 1.1 MHz, for devices oriented along the [110] axis of GaAs. Mechanical modes at 2.5 GHz (with a quality factor ${Q}_{\mathrm{m}}\approx 2000$ at room temperature and atmosphere) are observed, as is self-oscillation of the mechanical modes, through radiation-pressure driven dynamical back-action [15].

## 2. MICRODISKS

Figure 1(a) shows a scanning electron microscope image of a microdisk cavity, fabricated in a 220 nm thick GaAs layer, using typical lithography and dry etching processes (Supplement 1, Section S1). Finite-element method simulations are used to calculate the whispering gallery optical modes and radial breathing mechanical modes of such devices [Fig. 1(a)], with disk radius $R$ varying between 1 and 5 μm. For each value of $R$, we calculate optical modes of transverse electric (TE) polarization (dominant electric field components are in the plane of the disk), and determine the azimuthal mode number $m$, that places a first-order radial mode in the 1550 nm band. We focus on the ${\mathrm{TE}}_{(1,m)}$ mode, because of its comparatively high radiation-limited optical quality factor (${Q}_{o}$) for small disks ($R\gtrsim 1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$). Similarly, we focus on the fundamental radial breathing mechanical mode, as it is expected to have a higher mechanical quality factor (${Q}_{m}$) than higher-order modes, for a given supporting pedestal size.

The optomechanical coupling rate ${g}_{0}$, defined as the optical mode frequency shift due to the mechanical mode’s zero-point motion [16], has MB (${g}_{0,\mathrm{MB}}$) and PE (${g}_{0,\mathrm{PE}}$) contributions obtained from the calculated modes as in [9]

Figure 2 shows the calculated contribution to ${g}_{0}$, due to the MB (blue) and PE (green) effects, as a function of $R$. For $R\gtrsim 1.25\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, the MB effect dominates, whereas for $R\lesssim 1.25\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, the PE effect is comparable, or even slightly larger. This is consistent with recent simulation results for similar GaAs microdisks [14]. To verify this scaling behavior experimentally [Fig. 1(b)], we measure fabricated GaAs microdisks of varying radius, following an approach similar to [16], where a calibration signal of known modulation index ${\beta}_{\mathrm{pm}}$, from a phase modulator driven close to the mechanical resonance frequency, is used to determine the magnitude of ${g}_{0}$ (Supplement 1, Section S3). Figures 1(c)–1(d) show representative optical and mechanical modes for a device, where the mechanical mode spectrum also includes the phase modulator calibration tone. Compared to microdisks fabricated previously, using an essentially identical process [18], ${Q}_{o}$ in these devices ($<5\times {10}^{4}$) is an order of magnitude lower, likely as a result of 100 nm length scale roughness present on the underside of the GaAs layer (Supplement 1, Section S2). Such lower ${Q}_{o}$ values do not influence the estimate of ${g}_{0}$, but do prevent operation in the sideband-resolved regime, needed for a number of applications.

The optomechanical coupling rate ${g}_{0}$ can be estimated from a mechanical mode spectrum as (Supplement 1, Section S3)

The experimentally measured ${g}_{0}$ values, for disks of varying $R$, are plotted in Fig. 2 (black circles), where the uncertainty in each measurement is dominated by the uncertainty in the phase modulator ${V}_{\pi}$. The data shows good agreement with the red curve, which plots the sum of the MB and the PE contributions to ${g}_{0}$. Especially for small disk radii, the data shows significant deviation from the MB contribution alone, which might help explain some discrepancies observed in previous measurements of GaAs disk optomechanical resonators, where only MB effects were considered in simulation comparisons [19]. We note that for nominally identical disks, a spread in ${g}_{0}$ of $\approx 10\%$ is observed. We attribute this to the specifics of the fiber taper coupling for each device which, we have observed, can perturb the optical and mechanical modes, and the resultant optomechanical coupling (Supplement 1, Section S5).

## 3. NANOBEAM OPTOMECHANICAL CRYSTALS

The measurements of ${g}_{0}$ in GaAs microdisks demonstrated that the contribution due to the PE effect (${g}_{0,\mathrm{PE}}$) can become comparable to (and even exceed that due to) the MB effect (${g}_{0,\mathrm{MB}}$), as the disk radius $R$ becomes comparable to the wavelength. Intuitively, as $R$ is reduced, the volume of both the optical and mechanical modes decrease, and ${g}_{0}$ increases, due to increased spatial overlap. Given that bending loss starts to dominate ${Q}_{o}$, for $R<0.7\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu m}$, one can estimate that ${g}_{0}/2\pi \lesssim 450\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kHz}$, based on the data shown in Fig. 2. For higher ${g}_{0}$, one needs to consider geometries that support more tightly confined optical and mechanical modes.

Optomechanical crystals [20], structures that spatially co-localize optical modes within a photonic bandgap and mechanical modes within a phononic bandgap, have been demonstrated in a number of materials [9,21–25], and in both one- and two-dimensional geometries. Here, we focus on one-dimensional nanobeam geometries, where the nanobeams are patterned with a series of holes, whose dimensions are graded quadratically from the center (cavity section) to the edge (mirror section). The general design principle for the optical cavity [26] relies on choosing the center hole dimension to support a guided mode, and then quadratically tapering the hole dimension down to the mirror section, where the mode lies in the forbidden band, and hence is reflected. The cavity is constructed by putting two such tapers back-to-back. The quadratic taper ensures that the electric field amplitude of the mode has a Gaussian profile, and retains high ${Q}_{o}$. Similar design concepts have been used in the development of GaAs nanobeam photonic crystal cavities, for applications in lasing [27] and cavity quantum electrodynamics [28]. Here, we demonstrate these GaAs cavities in the context of cavity optomechanics, where we note that the tailoring of the hole dimensions also enables localization of mechanical modes [20].

Figure 3 shows two different nanobeam optomechanical crystal cavity designs, in 220 nm thick GaAs, using circular holes [Figs. 3(a)–3(d)], as described previously, and elliptical holes [Figs. 3(e)–3(h)], in an approach similar to [9]. For each design, we show a schematic of the geometry, the variation in device parameters (as a function of hole number), and the optical and mechanical modes. We designed the devices to have a nominal operating wavelength of 1550 nm, and ran a parameter sweep to find designs with the highest ${g}_{0}{Q}_{o}$, where both MB and PE contributions to ${g}_{0}$ are calculated, as described previously for microdisks.

The circular hole design is one realization of the more general elliptical hole design, and is thus comparatively simple, in terms of the number of design parameters. In particular, we fix the lattice constant $a$, and adjust only the hole radius $r$ at the different lattice sites. More specifically, we vary the hole radius in the center of the cavity and the mirror sections, as well as the steepness of the quadratic grading profile (i.e., the number of holes over which the radius is tapered). In comparison, the elliptical hole designs have quadratic grades for the lattice constant ($a$) and lengths of the principal axes of the ellipse (${h}_{x}$ and ${h}_{y}$). Thus, for each dimension $a$, ${h}_{x}$, and ${h}_{y}$, we vary the value in the center of the cavity and the mirror sections, as well as the steepness of the quadratic grade.

While for both the circular and elliptical hole designs, we find parameters for which ${Q}_{o}>{10}^{6}$, ${g}_{0}$ is higher for the elliptical hole designs. In particular, the optimized elliptical hole design has ${g}_{0,\mathrm{PE}}/2\pi =860\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kHz}$ and ${g}_{0,\mathrm{MB}}/2\pi =-94\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kHz}$, for coupling between the $\lambda \approx 1535\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ optical mode and ${\mathrm{\Omega}}_{m}/2\pi \approx 2.14\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{GHz}$ mechanical mode, compared to the optimized circular hole design, with ${g}_{0,\mathrm{PE}}/2\pi =563\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kHz}$ and ${g}_{0,\mathrm{MB}}/2\pi =-43\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kHz}$, for coupling between the $\lambda \approx 1545\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{nm}$ optical mode and ${\mathrm{\Omega}}_{m}/2\pi \approx 2.31\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{GHz}$ mechanical mode. The elliptical design ensures a higher ${g}_{0}$ by having a higher GaAs volume fraction in the center of the beam. We note that while in microdisks, the MB and PE contributions to ${g}_{0}$ are comparable in magnitude, and of the same sign, for the optimized nanobeams, ${g}_{0}$ is dominated by the PE effect, and the MB contribution is opposite in sign, and thus reduces the net optomechanical coupling rate. This behavior is consistent with [9,24], where an optimized nanobeam geometry, dominated by the PE effect, was developed for Si [9] and diamond [24].

Given that the PE effect is represented by a tensor, one would expect the contribution to ${g}_{0}$ to depend on the in-plane orientation of the nanobeam. Moreover, the fact that ${p}_{12}$ has a much larger magnitude in GaAs than in Si (${p}_{12,\mathrm{GaAs}}=-0.14$, ${p}_{12,\mathrm{Si}}=0.017$) suggests that the dependence of ${g}_{0}$ on in-plane orientation will be much more significant in GaAs. Figure 4(b) shows ${g}_{0,\mathrm{PE}}$ for the elliptical hole nanobeam, shown in Figs. 3(e)–3(h), as a function of the angle the long axis of the nanobeam makes with the [100] direction. Here, the coupling rate is calculated using the rotated PE tensor (Supplement 1, Section S8), and an isotropic elastic tensor is used for both GaAs and Si. As we discuss subsequently in the text, using the full anisotropic elastic tensor [29,30] leads to corrections in both the MB and PE coupling rates, although the total ${g}_{0}$ values are within 10% of the values calculated with the isotropic elastic tensor.

We can see from Fig. 4(b) that the effect of in-plane orientation on ${g}_{0}$ is quite significant, with a variation of almost 35%, and a peak value ${g}_{0,\mathrm{PE}}/2\pi \approx 1.2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$ at 45° (device orientation along [110]). We also plot the orientation dependence of ${g}_{0,\mathrm{PE}}$ for a silicon nanobeam optomechanical crystal, similar to that of [9] (Supplement 1, Section S7). We see that again there is a dependence of ${g}_{0,\mathrm{PE}}$ on in-plane orientation, although in this case, the optomechanical coupling rate is minimized at 45°, and the variation between 0° and 45° is less than 5%. To understand this, we plot in Fig. 4(c) the contributions due to the (${p}_{11}+{p}_{44}$) and ${p}_{12}$ terms of Eq. (2), as a function of in-plane angle. The in-plane anisotropy of ${g}_{0,\mathrm{PE}}$ is seen to arise primarily from the contribution due to the ${p}_{12}$ term, and the previously mentioned difference in the ${p}_{12}$ values for GaAs and Si helps us understand why this rotational dependence is so weak in Si. The MB contribution ${g}_{0,\mathrm{MB}}$ is insensitive to in-plane orientation, under the assumption of GaAs being an isotropic elastic material [as can be seen from the form of Eq. (1)]. The total optomechanical coupling rate for the GaAs elliptical hole design, including both PE and MB contributions, is therefore as high as ${g}_{0}/2\pi \approx 1.1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$.

We study this orientation-dependent optomechanical coupling by fabricating a series of GaAs nanobeam devices, according to the elliptical hole geometry described previously, while varying the orientation of the long axis of the nanobeam, between [110] and [100], as shown in Fig. 5(a). Devices are tested in the same measurement setup used to test the microdisk samples [Fig. 1(b)]. In this case, the sample is kept at atmospheric pressure, as gas damping is expected to have limited influence on the mechanical modes, due to their high frequencies [31], an effect that has also been observed recently in ${\mathrm{Si}}_{3}{\mathrm{N}}_{4}$ nanobeam optomechanical crystals [23]. Optical modes in the 1550 nm band are observed with typical quality factors ${Q}_{o}\approx 4.0\times {10}^{4}$ [Figs. 5(b) and 5(c)]. Mechanical quality factors are typically around ${Q}_{m}\approx 2\times {10}^{3}$.

Using the phase modulator calibration approach described in Section 2, we extract the optomechanical coupling rate for the different devices, with ${g}_{0}/2\pi =870\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kHz}\pm 45\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kHz}$ for the device aligned along [100] [Fig. 5(c)], and ${g}_{0}/2\pi =1.12\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}\pm 0.06\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$ for the device aligned along [110] [Fig. 5(b)]. These values correspond reasonably well with simulations, which when taking into account both the PE [Fig. 4(b)] and MB [Fig. 4(a)] contributions, predict ${g}_{0}/2\pi =770\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kHz}$ and ${g}_{0}/2\pi =1.09\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$, respectively. Measurement of devices fabricated at an intermediate angle of 15° with respect to [100] (not shown) yield ${g}_{0}/2\pi =920\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kHz}\pm 50\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kHz}$, which also matches reasonably well with the simulation result of ${g}_{0}/2\pi =850\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kHz}$. We note that while the uncertainty values we have quoted for the measured ${g}_{0}$ are the 1 standard deviation value due to the uncertainty in the phase modulator ${V}_{\pi}$, another source of uncertainty is in the precise angle with which the cavity was fabricated relative to the GaAs crystal planes (i.e., the alignment of the GaAs chip within the electron-beam lithography system). In particular, for intermediate device orientation angles, between [100] and [110], Fig. 4(b) predicts a variation in ${g}_{0}/2\pi $ of $\approx 50\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kHz}$ for a 5° offset in orientation.

While our experimental results are in reasonable agreement with the simulations presented thus far, one potential source of discrepancy is the use of an isotropic elastic tensor in the calculations. The anisotropy of GaAs leads to an orientation-dependent Young’s modulus, Poisson’s ratio, and shear modulus. One consequence of this is an orientation-dependent mechanical mode frequency, with devices oriented along [110] expected to have a frequency that is $\approx 100\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$ higher than devices oriented along [100], resulting primarily from the anisotropy in the Young’s modulus (Supplement 1, Section S9). This corresponds well with the experiments, where a shift of $\approx 85\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$ was measured for the corresponding devices.

In addition, the precise displacement profile of the mechanical mode will change (Supplement 1, Section S10). This in turn causes a change in both the MB and PE coupling rates, with the former no longer orientation-independent, as was previously assumed. Using the full anisotropic tensor, we calculate ${g}_{0,\mathrm{MB}}/2\pi =-73\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kHz}$ and ${g}_{0,\mathrm{PE}}/2\pi =923\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kHz}$ along [100], and ${g}_{0,\mathrm{MB}}/2\pi =-15\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kHz}$ and ${g}_{0,\mathrm{PE}}/2\pi =1.07\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$, along [110]. The total ${g}_{0}/2\pi $, along [100] (${g}_{0}/2\pi =850\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{kHz}$) and [110] (${g}_{0}/2\pi =1.06\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$), are within 10% of the values reported in Fig. 4, on account of being dominated by the PE effect. We note that the ${g}_{0}$ values calculated using the anisotropic elastic tensor for GaAs show closer agreement with our experimental results than the results shown in Fig. 4, where GaAs was treated as an isotropic elastic material.

Finally, by injecting increasing levels of optical power into the devices, while keeping the laser frequency blue-detuned and on the shoulder of the optical cavity mode, we can drive the system into regenerative mechanical oscillation [15]. Figure 6(a) shows a series of mechanical mode spectra for a device oriented along [100], as a function of increasing optical power, from which a clear linewidth narrowing (and peak amplitude increase) are observed. The peak amplitude is plotted as a function of input optical power in Fig. 6(b). Here, a characteristic threshold behavior at an input power $<35\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{\mu W}$ is seen, indicating that the system is indeed self-oscillating.

## 4. CONCLUSIONS

In summary, we have compared the moving dielectric boundary and photoelastic contributions to the optomechanical coupling rate in GaAs optomechanical resonators. Simulations and experiments in microdisk cavities correspond closely, and show that these two effects have near equal magnitude for devices with a radius near 1 μm. Simulations and experiments on nanobeam optomechanical crystals, optimized for PE coupling (1 order of magnitude larger than the MB effect), show a significant dependence on the in-plane orientation of the nanobeam, with overall coupling rates ${g}_{0}/2\pi =1.1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{MHz}$ achieved for coupling between 1550 nm optical modes and 2.5 GHz mechanical modes. Along with recent works exploiting the PE effect in silicon-based devices [6,9], our results indicate the importance of considering the effect in the development of cavity optomechanical devices. Moreover, the anisotropy of both the PE and elastic tensors indicates that device orientation must be taken into account during layout. Finally, while our experimental results are consistent with simulations based on the values of the PE coefficients found in the literature, in general, there is a limited amount of data available on this topic, particularly as a function of wavelength, and in different material systems. Such measurements would be extremely valuable for the continued development of optomechanical platforms, especially as they move toward higher mechanical frequencies involving acoustic wave resonances.

## FUNDING INFORMATION

DARPA (MESO); KIST (Future Convergence Pioneer Program, Flagship Program, GRL program); Cooperative Research Agreement between University of Maryland and NIST-CNST (70NANB10H193).

## ACKNOWLEDGMENT

We acknowledge J. Kim and S. Krishna, from the Center of High Technology Materials at the University of New Mexico, for providing epitaxial GaAs material used in fabricating microdisk devices. J. Y. L. is now with the Laser Technology Research Center, Korea Photonics Technology Institute.

See Supplement 1 for supporting content.

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