We analyze the magnitude of the radiation pressure and electrostrictive stresses exerted by light confined inside GaAs semiconductor WGM optomechanical disk resonators, through analytical and numerical means, and find the electrostrictive stress to be of prime importance. We investigate the geometric and photoelastic optomechanical coupling resulting respectively from the deformation of the disk boundary and from the strain-induced refractive index changes in the material, for various mechanical modes of the disks. Photoelastic optomechanical coupling is shown to be a predominant coupling mechanism for certain disk dimensions and mechanical modes, leading to total coupling gom and g0 reaching respectively 3 THz/nm and 4 MHz. Finally, we point towards ways to maximize the photoelastic coupling in GaAs disk resonators, and we provide some upper bounds for its value in various geometries.
© 2014 Optical Society of America
The field of optomechanics [1–3] offers a rich array of applications spanning mechanical ground-state optical cooling [4, 5], force and acceleration sensing [6, 7], wavelength conversion [8, 9] and all-optical tuning of photonic circuits [10–12]. In this context, semiconductor optomechanical disk resonators [13–16] are of particular interest due to their high optical quality factors (Q) and ability to confine both optical and mechanical energy in a reduced (∼ λ3) interaction volume, thus providing very strong optomechanical coupling. Alongside Silicon (Si), GaAs is a platform of great potential for integrated photonics, as it allows for the integration of high optical Q and GHz high mechanical Q resonators [17, 18] with strong optomechanical coupling  directly on-chip . The GaAs platform furthermore enables the addition of electrically driven optically active elements, as well as the inclusion of quantum dots or quantum wells  offering novel hybrid optomechanical coupling schemes .
The optomechanical resonators described in this work are composed of a micrometer-sized GaAs disk, isolated from the sample substrate atop an Aluminum Gallium Arsenide (AlGaAs) pedestal, as shown in Fig. 1(a). The GaAs disk supports high Q optical WGMs located on the periphery of the disk, which are identified by their radial order p and azimuthal number m [13, 23]. The disk also supports a variety of in- and out-of-plane mechanical modes . A radial contour mechanical mode is schematically depicted in Fig. 1(b).
The photons confined inside the semiconductor disk exert two different stresses which will be detailed in the following: a radiation pressure “pushing the walls of the optical cavity apart” and an electrostrictive stress linked to the material’s photoelasticity. Recently, Rakich et al. showed that for certain geometries of straight silicon photonic waveguides the electrostrictive stress could be commensurate with the radiation pressure stress commonly studied in optomechanics [24,25]. In this paper we study the magnitude of these optical stresses in GaAs optomechanical disk resonators. We investigate the associated geometric and photoelastic optomechanical coupling strengths, resulting respectively from the deformation of the disk boundary and from the strain-induced refractive index changes in the material, for various mechanical modes of the disk. We propose different computational methods, from analytical models leading to useful scaling formula, to full numerical approaches providing precise values of the coupling strength as a function of the mechanical mode and of the disk radius. For certain mechanical modes, photoelasticity is a predominant optomechanical coupling mechanism, resulting in total coupling strengths gom and g0 that reach respectively 3 THz/nm and 4 MHz, for experimentally achievable disk resonator dimensions . Finally we propose some simple rules to maximize the value of this coupling.
1.1. Optomechanical coupling
The disk resonator is described by the standard optomechanical Hamiltonian Ĥ describing an optical field coupled to a mechanical resonator :13].
1.2. Mechanisms of optomechanical coupling
A confined optical wave in the disk is only resonant provided it closes upon itself in phase after a round-trip, respecting the condition: 2πneffR ≃ mλ, with neff the WGM effective index, λ the optical freespace wavelength, R the disk radius and m ∈ ℕ. From this it appears that the resonance wavelength is modified by a small mechanical displacement dx that changes the cavity radius R. But this small displacement, by modifying the whole crystal lattice, also changes the refractive index via the photoelastic effect and, through this, the resonance wavelength of the WGM. The total gom can be split into two independent contributions depending on each of these two mechanisms:26–29].
Note that assuming a purely radial mechanical displacement with maximal amplitude exactly at the periphery of the disk, and assuming the separability of the in-plane and out-of-plane components of the electric field, the geometric optomechanical coupling in a disk resonator of radius R takes the exact simple form .
2. Radiation pressure in an optomechanical disk resonator
We first calculate the radiation pressure exerted by photons confined by total internal reflection inside a circular disk resonator in two different ways: 1) through simple analytical energy and momentum conservation arguments; 2) by 3D Finite Element Method (FEM) computations of the Maxwell Stress Tensor (MST). Our analytical approach provides original helpful formula for whispering gallery optomechanics. While both approaches fittingly yield consistent results, each provides specific insights into the radiation pressure mechanism.
2.1. Analytical approach
2.1.1. Energy conservation argument
The stored electromagnetic energy in the closed resonator is given by:Eq. (2), we split the total force F into two distinct contributions, linked to radiation pressure Frp and electrostriction Fes:
2.1.2. Momentum conservation argument
Let us consider the radiation pressure exerted on the outer boundary of a disk resonator by a confined photon, through momentum conservation arguments. In free space, the momentum associated with a photon of wavelength λ0 is ħk0, with k0 = 2π/λ0 the free space wavenumber. When this photon impinges on a rigid mirror with orthogonal incidence, and is perfectly reflected, conservation of momentum dictates that the mirror receives 2ħk0 momentum. We now wish to describe how much momentum is transferred to a circular resonator by a photon confined by total internal reflection, as this photon performs a round-trip. Using ray optics considerations, a photon confined inside a regular cavity with n sidewalls will strike the side-walls n times per round-trip at an angle of π/n, each time transferring radially a momentum 2ħk sin(π/n), see Fig. 1(c). The radial momentum transfer as a photon completes one round trip is the limit:Eq. (6)) divided by the cavity round-trip time τrt and is written for a disk of radius R: 30]) and use the geometrical expression for a purely radial displacement of the disk, the radial force Frp exerted by Nph photons takes the simple form: Eq. (5)). Conversely, if one considers that the force expression obtained from energy conservation arguments is general (Eq. (5)), our present derivation, based on the Minkowski assumption, actually indicates that is a valid expression if a purely radial displacement is considered. The radiation pressure Prp exerted on the disk resonator’s vertical outer boundary of surface S = 2πRh is: Eq. (9) illustrates the benefit of using small-diameter thin disk resonators. Since both force and pressure exerted by the stored photons are independent of the refractive index of the resonator material, the benefit of using high refractive index materials appears only through the reduced disk radii feasible before incurring significant bending losses. Numerical estimates of Frp and Prp for a 1 μm radius disk are provided in Table 1. The remarkable optomechanical properties of these small resonators are highlighted by the fact that the radiation force Frp exerted on the disk’s outer boundary by a single photon is larger than the disk’s own weight.
2.2. Numerical approach
In this section we estimate the magnitude of the radial radiation pressure per confined photon by computing the spatially dependent Maxwell stress tensor (MST) . In a dielectric medium of relative permittivity εr (r, z) and permeability μr, the ij components of the MST are given by:Eq. (10), the radiation pressure induced stress (applied on the face normal to the i direction along the j direction) is expressed as a function of the MST element Tij as . While this approach allows for computing both normal (σii) and shear (σij with i ≠ j) stresses, in the following we focus only on normal stresses, as these are the ones producing work when coupled to the radial displacement of a mechanical Radial Breathing Mode (RBM). Since the disk cannot respond mechanically to rapidly varying stresses at optical frequencies (1014 Hz range), we compute the time averaged value of the radial stress over an optical cycle.
To calculate the radial radiation pressure due to a photon confined in the resonator in a specific WGM, we first perform a FEM simulation of the desired WGM. (Throughout this paper -unless mentioned otherwise- we will be considering Transverse Electric (TE) WGMs, with radial order p=1 and a resonance wavelength λ0 ≃ 1.3μm). This simulation provides the electric and magnetic field components needed to compute Eq. (10). The main field components are plotted in Figs. 2(a)–2(e). Next, the value of the time-averaged normal radial stress is calculated at every point in space along the rz cross-section, see Fig. 2(f). The normal radial stress is largest near the outer edge of the disk, where the light is confined. From the local stress we can infer a local volume force (force per unit volume) ℱ via the relation:
3. Electrostriction in an optomechanical disk resonator
Electrostriction is a mechanism whereby electric fields induce strain within a material. It differs from piezoelectricity in that the induced strain is proportional to the square of the electric field, and not to the electric field. Since the electric fields we consider are rapidly oscillating at optical frequencies, the time averaged piezoelectric strain shall be zero, while the time averaged electrostrictive strain contribution remains. Electrostrictive stresses scale with the fourth power of the dielectric refractive index, making them of significant importance for high refractive index materials such as silicon and GaAs (for which n ≥ 3.3 at telecom wavelengths) . The electrostrictively induced stress can be expressed in terms of the material photoelastic tensor pijkl , which links a material strain Sij to a change in the material’s inverse dielectric tensor :33]. These three parameters are p11, p12 and p44, written here in contracted notation, where 11→ 1; 22 → 2; 33 → 3; 23, 32 → 4; 31, 13 → 5; 12, 21 → 6. Using this definition, Eq. (13) links the electrostrictively induced stresses σes to the electric field components in the following way : Table 2. The relation between electrostriction and photoelasticity is seen by considering a disk resonator suddenly subject to strain. The strain leads to a change in the material’s permittivity Δε, via the photoelastic properties. Provided some electric energy was stored in the disk at the time, this stored energy (proportional to ε E2) changes due to the change in permittivity Δε. This change in energy can be seen as the work of the electrostrictive force during the displacement. (A more complete version of this argument is developed in , see Fig. 3(a).
Looking at Eq. (13) and the values of the photoelastic coefficients in Table 2, it appears in the case of a WGM that the electrostrictively induced normal stresses are significantly larger than the shear stresses. We will focus here for brevity just on the radial and axial normal stress components:Figures 4(a) and 4(b) show the value of and due to a single photon confined in the p=1, m=10 WGM of a 1 μm radius GaAs disk resonator of thickness 320 nm, already considered in section 2. Figures 4(c) and 4(d) represent the associated volume force for both these stresses, where and . The black arrows show the net direction these forces are pointing in. We see here that the electrostrictive force pushes outwards in both the radial and vertical z directions, adding constructively to the radiation pressure force.
The fact that electrostriction and radiation pressure add up constructively as they do here is not true for all materials and geometries. As we can see from Eq. (14), since the pij are negative for GaAs, all electrostrictive stresses are positive, and confined photons tend to expand the material in all directions. However this would not be the case for silicon disk resonators or waveguides, as the coefficients p11 and p12 are of different sign and significantly different magnitude.
4. Optomechanical coupling in GaAs disks
4.1. Geometric contribution
In , Johnson et al. provide a perturbation theory for Maxwell’s equations in the case of shifting material boundaries. This theory can be applied to determine the frequency shift of an optical WGM to an arbitrary mechanical deformation of the confining dielectric disk. Following this method, the term is calculated as a surface integral of the unperturbed optical fields over the perturbed dielectric interface:Eq. (15) using a FEM simulation software (COMSOL Multiphysics). The results for the four mechanical modes shown in Fig. 5 are summarized in Table 3. For a given mechanical mode, the displacement of every point of the disk is spatially non uniform and is therefore dependent on the somewhat arbitrary choice of the reduction point which experiences the displacement dx. The normalization choice max|q⃗|=1 in Eq. (15) means that the point of maximal displacement is used as reduction point. As evidenced in Table 3, different mechanical modes have vastly different values of . Note for instance how the value for the 1st RBM is roughly 10 000 times larger than for the out of plane ‘bowl’ mode. This difference illustrates how efficiently each mechanical mode modulates the total cavity length (the dR/dx term in Eq. (2)) and confirms that the first RBM is the mechanical mode with the highest . Due to their extremely miniaturized dimensions, 1 μm GaAs disks exhibit remarkably large optomechanical coupling, with reaching over 1 THz/nm in the case of the first RBM. (Here we see that the ∼1.1 THz/nm numerically computed value is roughly 20% below the value provided by the simplified expression gom = ω0/R ≃ 1.4 THz/nm). The zero point fluctuations xZPF are obtained by equaling the mechanical energy in the resonator to ħΩM/2, yielding xZPF=1.2 · 10−15 m for the first RBM, using the same reduction point as above. The single photon optomechanical coupling strength for this mechanical mode is g0=gomxZPF ≃ 1.3 MHz. Note that these calculations are carried out without any AlGaAs pedestal under the disk, and are therefore only valid for small pedestal radii (≥ 90 % undercut). For larger radii the stated mechanical frequencies and may differ significantly.
4.2. Photoelastic contribution
To compute the photoelastic coupling contribution, first the unperturbed resonance frequency of the desired WGM is obtained through a FEM simulation with uniform and isotropic ε. Second, the desired mechanical eigenmode is solved for in another FEM simulation, which provides the complete deformation profile and strain distributions inside the resonator. In the isotropic elasticity limit, the mechanical modes exhibit rotation invariance. Since the same holds true for WGMs and for the photoelastic tensor, the complete coupling problem can be described using axial symmetry around the disk rotation axis. We will focus in the following discussion on the first RBM. While the radial displacement is zero at the center and maximum near the periphery, the behavior for the normal radial strain is reversed. The normal radial strain Srr=S1 (in contracted notation) is maximal at the center of the disk and changes sign right by the edge of the disk (this is a normal consequence of the circular geometry), see Figs. 6(a) and 6(b). The behavior is similar for the normal azimuthal and axial strains S2 and S3, which are of similar magnitude and largest near the center of the disk. The S4 and S6 strain components are zero over the whole disk, while the S5 strain component is roughly three orders of magnitude smaller than S1,2,3. We now use Eq. (12) to relate the strain distribution inside the disk to changes in the dielectric tensor. Since S4, S5 and S6 are negligible, the off-diagonal terms in the dielectric tensor can be neglected. The dielectric tensor modified by the RBM displacement therefore takes the form:Fig. 6(b)). In contrast, this condition is much better fulfilled for the second order RBM, as shown in Figs. 6(c) and 6(d). Accordingly, this translates into a remarkably high of nearly 2 THz/nm for this mechanical mode, see Table 3. Because of the reduced geometric coupling for the second RBM, its total optomechanical coupling is comparable to that of the first RBM, around 2.5 MHz, albeit at a much higher mechanical frequency of 3.5 GHz. Table 3 summarizes the g0 values for the four considered mechanical modes of Fig. 5.
Figure 7 plots the dependency of and with disk radius for the first and second RBM. Each purple (blue) dot corresponds to a distinct FEM simulation of ( ), while the solid blue line for corresponds to the value given by the following analytical formula:Table 4. is the mechanical frequency of the RBM of order P [38, 39]. We have obtained Eq. (17) through the analytical treatment of a free elastic circular plate. For the first RBM, Eq. (17) accurately reproduces the trend provided by FEM simulations, but overestimates the coupling by 20% because it neglects the out-of-plane component of the mechanical motion, which affects the effective refractive index of the disk slab and hence the value of . For the second RBM the overestimation is more pronounced (close to 40%), reflecting a larger out-of-plane component of the mechanical mode, as visible in Fig. 5(c).
Since the effective mass scales with R2, scales as . Interestingly, rises faster than with decreasing disk radius. For instance for the 1st RBM (see Fig. 7(a)), goes from being two times smaller than for disks of radius R=10μm, before reaching comparable values for 1μm radius disks. For the 2nd RBM, the photoelastic coupling is always the dominant coupling mechanism (see Fig. 7(b)). Note that the maximal photoelastic coupling is reached for R ≃ 1μm. Further reducing the disk dimensions reduces the coupling as the optical mode is no longer well localized on the region of highest strain.
4.3. Energy considerations
The link between radiation pressure and boundary deformation and electrostriction and photoelasticity can be understood by looking at the work done by the optical forces during a mechanical displacement. Incidently, these energy considerations provide an additional way of calculating both the geometric and photoelastic optomechanical couplings. Following Eq. (4), we can write a generalized expression:40]. The Sij are the mechanical strain components resulting from the displacement Δx, and the σij are the radiation pressure or electrostrictive stress components described respectively in Eqs. 10 and 13. This formulation and the method discussed in 4.2 yield values in very good agreement, within less than 1 % difference. The analytical overestimation of discussed previously can be understood in terms of ‘wasted’ work by the radiation pressure, which pushes outwards in the vertical direction, resisting the disk radial expansion. Note that both for radiation pressure and electrostriction, in the case of the 1st RBM at least, a large part of the work is done by the optically induced azimuthal stress σθθ. Furthermore, for the same mechanical mode, the larger axial stress σzz in the case of electrostriction produces more negative work as the disk expands in the radial direction but contracts in the axial direction. These considerations shed light on two seemingly contradictory observations. On one hand the photoelastic coupling is slightly smaller than the geometric coupling for the first RBM, on the other hand the radial stress per photon is several times larger for electrostriction than radiation pressure. As a consequence, even though the movement of the 1st RBM is predominantly radial, the full picture of optomechanical coupling can not be obtained looking solely at the stresses along the radial direction.
We show that the second order RBM is an interesting mechanical mode thanks to its large total optomechanical coupling and high mechanical frequency. While this type of mode tendentially has a lower mechanical Q due to larger mechanical coupling to the pedestal, its anchoring losses could be overcome with a carefully engineered pedestal geometry .
We verify that both the geometric and photoelastic coupling magnitudes are comparable when considering a transverse magnetic (TM) WGM instead of a TE WGM. We focused here on p=1 WGMs, as these are the modes with the highest radiative optical Qs . When considering different WGMs, the same rule of thumb remains: in order to maximize the photoelastic optomechanical coupling, the regions of high electromagnetic energy should be co-localized with regions of high mechanical strain.
For comparison the photoelastic optomechanical coupling has been computed on Si disks of identical dimensions using the photoelastic parameters of Table 2. The obtained for the 1st RBM is roughly three times lower than for GaAs, notably because of the reduced photoelastic coefficients of Si, but should nevertheless not be neglected.
Finally recent work investigating the optomechanical coupling in distributed Bragg reflector GaAs/AlAs vertical cavities  shows these geometries are also well suited to take advantage of the photoelastic coupling mechanism, thanks to an efficient overlap between the optical field and strain maxima resulting in values of reaching several THz/nm.
We investigated the magnitude of the optical forces due to confined photons in GaAs semiconductor optomechanical disk resonators, successively addressing the case of radiation pressure and electrostriction. We showed these forces add up constructively in the case of GaAs disks. Next, we provided a comparison between the photoelastic and geometric optomechanical coupling for various modes of a GaAs disk, and the scaling of these couplings with disk radius. An interpretation of this coupling in terms of the work done by the optical forces during a mechanical displacement is proposed and numerically verified, leading to an additional estimation of g0. Photoelasticity provides an efficient tool when designing structures for optomechanical applications. The large photoelastic coupling in GaAs underscores the strength of this material for optomechanical applications, in complement with other coupling mechanisms proposed in GaAs membranes and cantilevers [17, 42]. What is more, taking advantage of the increase in the photoelastic coefficients near the GaAs band gap , can boost the photoelastic optomechanical coupling more than one order of magnitude, well into the several tens of MHz range.
This work is supported by the French ANR through the NOMADE project and by the ERC through the GANOMS project. The authors would like to thank Bernard Jusserand for insightful discussions on photoelasticity.
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