Photoelastic coupling in gallium arsenide optomechanical disk resonators
Christopher Baker, William Hease, Dac-Trung Nguyen, Alessio Andronico, Sara Ducci, Giuseppe Leo, Ivan Favero
PPhotoelastic coupling in gallium arsenideoptomechanical disk resonators
Christopher Baker, William Hease, Dac-Trung Nguyen, AlessioAndronico, Sara Ducci, Giuseppe Leo, and Ivan Favero ∗ Universit´e Paris Diderot, Sorbonne Paris Cit´e, Laboratoire Mat´eriaux et Ph´enom`enesQuantiques, CNRS-UMR 7162, 10 rue Alice Domon et L´eonie Duquet, 75013 Paris, France ∗ [email protected] Abstract:
We analyze the magnitude of the radiation pressure and elec-trostrictive stresses exerted by light confined inside GaAs semiconductorWGM optomechanical disk resonators, through analytical and numericalmeans, and find the electrostrictive force to be of prime importance.We investigate the geometric and photoelastic optomechanical couplingresulting respectively from the deformation of the disk boundary and fromthe strain-induced refractive index changes in the material, for variousmechanical modes of the disks. Photoelastic optomechanical coupling isshown to be a predominant coupling mechanism for certain disk dimensionsand mechanical modes, leading to total coupling g om and g reachingrespectively 3 THz/nm and 4 MHz. Finally, we point towards ways tomaximize the photoelastic coupling in GaAs disk resonators, and weprovide some upper bounds for its value in various geometries. © 2018 Optical Society of America OCIS codes: (120.4880) Optomechanics; (230.5750) Resonators; (130.5990) Semiconductors;(130.3120) Integrated optics devices; (999.9999) Optical forces; (999.9999) Radiation pres-sure; (999.9999) Photoelasticity
References and links
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The field of optomechanics [1, 2, 3] offers a rich array of applications spanning mechanicalground-state optical cooling [4, 5], force and acceleration sensing [6, 7], wavelength conver-sion [8, 9] and all-optical tuning of photonic circuits [10, 11, 12]. In this context, semiconductoroptomechanical disk resonators [13, 14, 15, 16] are of particular interest due to their high op-tical quality factors (Q) and ability to confine both optical and mechanical energy in a reduced( ∼ λ ) interaction volume, thus providing very strong optomechanical coupling. AlongsideSilicon (Si), GaAs is a platform of great potential for integrated photonics, as it allows for theintegration of high optical Q and GHz high mechanical Q resonators [17, 18] with strong op-tomechanical coupling [19] directly on-chip [20]. The GaAs platform furthermore enables theaddition of electrically driven optically active elements, as well as the inclusion of quantumdots or quantum wells [21] offering novel hybrid optomechanical coupling schemes [22].The optomechanical resonators described in this work are composed of a micrometer-sizedGaAs disk, isolated from the sample substrate atop an Aluminum Gallium Arsenide (AlGaAs)pedestal (Fig.1(a)). The GaAs disk supports high Q optical WGMs located on the peripheryof the disk, which are identified by their radial order p and azimuthal number m [13, 23]. Thedisk also supports a variety of in- and out-of-plane mechanical modes [13]. A radial contourmechanical mode is schematically depicted in Fig. 1(b).The photons confined inside the semiconductor disk exert two different stresses which will bedetailed in the following: a radiation pressure “pushing the walls of the optical cavity apart” andan electrostrictive stress linked to the material’s photoelasticity. Recently, Rakich et al. showedthat for certain geometries of straight silicon photonic waveguides the electrostrictive stresscould 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 optomechanicaldisk resonators. We investigate the associated geometric and photoelastic optomechanical cou-pling strengths, resulting respectively from the deformation of the disk boundary and from thestrain-induced refractive index changes in the material, for various mechanical modes of thedisk. We propose different computational methods, from analytical models leading to usefulscaling formula, to full numerical approaches providing precise values of the coupling strengthas 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 cou-pling strengths g om and g that reach respectively 3 THz/nm and 4 MHz. Finally we proposesome simple rules to maximize the value of this coupling. The disk resonator is described by the standard optomechanical Hamiltonian ˆ H describing anoptical field coupled to a mechanical resonator [3]:ˆ H = ¯ h ω ˆ a † ˆ a + ¯ h Ω M ˆ b † ˆ b − ¯ hg ˆ a † ˆ a (cid:0) ˆ b † + ˆ b (cid:1) (1)with ω and Ω M respectively the optical and mechanical angular resonance frequency and ¯ h the reduced Planck constant. ˆ a † (ˆ b † ) and ˆ a (ˆ b ) are respectively the photon (phonon) creationand annihilation operators. The optomechanical interaction can be defined in terms of the op-tomechanical coupling strength g om = − d ω d x , representing the shift in the optical resonance fre-quency for a given mechanical displacement dx or, in a complementary way, by g = g om x ZPF ,which represents the optical frequency shift for a mechanical displacement equal to the zeropoint fluctuations x ZPF . For completeness, we will quote both g om and g in this work, fixing x to be the maximum amplitude of displacement of the resonator [13]. e r e θ a) b) c) e r e θ e z n=2 ! n=4 ! n=6 ! π /2 ! π /4 ! π /6 ! h Fig. 1. (a) Schematic side-view of a GaAs disk of thickness h (blue), positioned atop anAlGaAs pedestal (grey), along with the cylindrical coordinates used throughout this work.(b) Top view of a GaAs disk of radius R (blue). The dashed green and red lines representthe radial deformation of the disk by a mechanical mode. (c) Schematic view of an opticalcavity composed of n=2, 4 or 6 mirrors, and the associated grazing angles. A confined optical wave in the disk is only resonant provided it closes upon itself in phase aftera round-trip, respecting the condition: 2 π n eff R (cid:39) m λ , with n eff the WGM effective index, λ the optical freespace wavelength, R the disk radius and m ∈ N . From this it appears that theresonance wavelength is modified by a small mechanical displacement dx that changes the cav-ity radius R. But this small displacement, by modifying the whole crystal lattice, also changesthe refractive index via the photoelastic effect and, through this, the resonance wavelength ofthe WGM. The total g om can be split into two independent contributions depending on each ofthese two mechanisms: g om = − d ω ( R , ε ) dx = − ∂ ω ∂ R d R d x (cid:124) (cid:123)(cid:122) (cid:125) geometricg geoom − ∂ ω ∂ ε d ε d x (cid:124) (cid:123)(cid:122) (cid:125) photoelasticg peom (2)where ε is the material’s permittivity, which is no longer necessarily isotropic nor homogeneousinside the disk under stress. The photoelastic contribution g peom is obviously unique to resonatorswhere light is confined inside matter, like semiconductor disks, silica toroids or spheres andphotonic crystal slabs, and would not appear in an empty Fabry-Perot optomechanical cavity.For this reason, it has been little considered in the early optomechanics literature [26, 27, 28,29].Note that assuming a purely radial mechanical displacement with maximal amplitude exactlyat the periphery of the disk, and assuming the separability of the in-plane and out-of-planecomponents of the electric field, the geometric optomechanical coupling in a disk resonator ofradius R takes the exact simple form g geoom = ω / R [13].
2. Radiation pressure in an optomechanical disk resonator
We first calculate the radiation pressure exerted by photons confined by total internal reflectioninside a circular disk resonator in two different ways: 1) through simple analytical energy andomentum conservation arguments; 2) by 3D Finite Element Method (FEM) computations ofthe Maxwell Stress Tensor (MST). Our analytical approach provides original helpful formulafor whispering gallery optomechanics. While both approaches fittingly yield consistent results,each provides specific insights into the radiation pressure mechanism.
The stored electromagnetic energy in the closed resonator is given by: E = N ph ¯ h ω (3)with N ph the number of stored photons in the resonator and ω the photon’s angular frequency.A small mechanical displacement of the disk ∆ x leads to a change in the photon angular fre-quency ∆ ω and stored energy ∆ E = N ph ¯ h ∆ ω . Therefore the force associated to this workreads: F = − ∆ E ∆ x = − N ph ¯ h ∆ ω ∆ x = N ph ¯ hg om (4)Using Eq. 2, we split the total force F into two distinct contributions, linked to radiation pressure F rp and electrostriction F es : F rp = N ph ¯ hg geoom and F es = N ph ¯ hg peom (5) Let us consider the radiation pressure exerted on the outer boundary of a disk resonator by aconfined photon, through momentum conservation arguments. In free space, the momentumassociated with a photon of wavelength λ is ¯ hk , with k = π / λ the free space wavenum-ber. When this photon impinges on a rigid mirror with orthogonal incidence, and is perfectlyreflected, conservation of momentum dictates that the mirror receives 2¯ hk momentum. Wenow wish to describe how much momentum is transferred to a circular resonator by a pho-ton confined by total internal reflection, as this photon performs a round-trip. Using ray opticsconsiderations, 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 momentum2¯ hk sin ( π / n ) (Fig. 1 (c)). The radial momentum transfer as a photon completes one round tripis the limit: 2¯ hk lim n → ∞ n sin ( π / n ) = π ¯ hk (6)where k is used instead of k as we now consider the case of a photon confined inside a dielectricmedium. Note the difference with the often encountered 4¯ hk expression stemming from theFabry-Perot case. The associated radial force per photon is the momentum transfer per round-trip (Eq. 6) divided by the cavity round-trip time τ rt and is written for a disk of radius R : F = dPdt = π ¯ hk π Rn eff / c (cid:124) (cid:123)(cid:122) (cid:125) τ rt = ¯ hkcn eff R (7)with c the speed of light in vacuum. Provided we write the photon momentum ¯ hk in a materialof refractive index n eff as ¯ hk = ¯ hk n eff (Minkowski formulation for the photon momentum in adielectric [30]) and use the geometrical expression g geoom = ω / R for a purely radial displacementof the disk, the radial force F rp exerted by N ph photons takes the simple form: F rp = N ph ¯ h k cR = N ph ¯ h g geoom (8)hich is consistent with what was obtained through energy conservation (Eq. 4). The radiationpressure P rp exerted on the disk resonator’s vertical outer boundary of surface S = π R h is: P rp = F rp / S = N ph × ¯ hk c π R h = N ph × ¯ hc λ R h (9)The R h dependency in Eq. 9 illustrates the benefit of using small-diameter thin disk resonators. Parameter Name Unit Value
Disk weight P disk
N 5.2 · − Radiation pressure force per photon F rp / N ph N 1.5 · − Radiation pressure per photon P rp / N ph Pa 7.5 · − Table 1. Radiation pressure values for a 1 µ m radius, 320 nm thick GaAs disk resonatorand λ =1.32 µ m wavelength light. Since both force and pressure exerted by the stored photons are independent of the refractiveindex of the resonator material, the benefit of using high refractive index materials appears onlythrough the reduced disk radii feasible before incurring significant bending losses. Numericalestimates of F rp and P rp for a 1 µ m radius disk are provided in Table 1. The remarkable op-tomechanical properties of these small resonators are highlighted by the fact that the radiationforce F rp exerted on the disk’s outer boundary by a single photon is larger than the disk’s ownweight. In this section we estimate the magnitude of the radial radiation pressure per confined photonby computing the spatially dependent Maxwell stress tensor (MST) [31]. In a dielectric mediumof relative permittivity ε r ( r , z ) and permeability µ r , the i j components of the MST are givenby: T i j = ε ε r ( r , z ) (cid:20) E i E j − δ i j | E | (cid:21) + µ µ r (cid:20) H i H j − δ i j | H | (cid:21) (10)Here ε = . · − F · m − and µ = π · − H · m − are the vacuum permittivity and per-meability, δ i j is Kronecker’s delta and E i ( H i ) is the i th electric (magnetic) field component. Inthe following we will take ε r ( r , z ) = n ∈ R and µ r =1 inside the GaAs, and ε r = µ r =1 in the sur-rounding air. With the choice of notations of Eq. 10, the radiation pressure induced stress σ rpi j (applied on the face normal to the i direction along the j direction) is expressed as a functionof the MST element T i j as σ rpi j = − T i j . While this approach allows for computing both normal( σ ii ) and shear ( σ i j with i (cid:54) = j ) stresses, in the following we focus only on normal stresses, asthese are the ones producing work when coupled to the radial displacement of a mechanicalRadial Breathing Mode (RBM). Since the disk cannot respond mechanically to rapidly varyingforces at optical frequencies (10 Hz range), we compute the time averaged value of the radialstress over an optical cycle.To calculate the radial radiation pressure due to a photon confined in the resonator in aspecific WGM, we first perform a FEM simulation of the desired WGM. (Throughout thispaper -unless mentioned otherwise- we will be considering Transverse Electric (TE) WGMs,with radial order p=1 and a resonance wavelength λ (cid:39) . µ m). This simulation provides theelectric and magnetic field components needed to compute Eq. 10. The main field componentsare plotted in Figure 2 a, b, c, d, and e. Next, the value of the time-averaged normal radial stress σ rprr = − T rr is calculated at every point in space along the rz cross-section (see Fig. 2, f). The ) b) c) d) e) f) GaAs
Air
Fig. 2. 2D axi-symmetric FEM modeling of the normal radial ‘radiation pressure’ stress σ rprr in a 320 nm thick and 1 µ m radius GaAs WGM disk resonator. The considered WGMis a (p=1, m=10). The solid lines show the boundary of the two computational domains:the GaAs disk and the surrounding air. 2D axi-symmetric cross sections are shown here,the whole disk is obtained by revolving around the z axial symmetry axis (dashed red line).The AlGaAs pedestal, being sufficiently remote from the optical field, is not included inthe simulation. Images (a) through (e) show the computed electric and magnetic field cross-sections, normalized such that the total electromagnetic energy in the resonator is equal tothe energy of one photon. (a) E r (b) E θ (c) E z (d) H z (e) H r . Since the simulated WGMis TE, the in-plane electric field and out-of-plane magnetic field components E r , E θ and H z are dominant. (f) Normal radial stress exerted by a confined photon σ rprr . The opticallyinduced stress is largest near the outer boundary of the disk resonator, where most of theelectromagnetic energy is located. normal radial stress is largest near the outer edge of the disk, where the light is confined. Fromthe local stress we can infer a local volume force (force per unit volume) F via the relation: F rpj = − ∂ i σ rpi j = ∂ i T i j (11)The spatial distribution of the radiation volume force F rpr is maximal right at the discontinuousdielectric interface (at r=1 µ m), lending some degree of support to the previously used image ofthe photon as a particle exerting a force as it bounces off the resonator sidewalls. In this image,the photon is “pushing on the boundary”. In order to quantitatively compare the results of theanalytical approach, which considers a radial force applied to the disk boundary, with the MSTapproach, which provides radial, azimuthal and axial stresses distributed throughout the diskresonator, the associated g geoom must be computed. This will be done in section 4.
3. Electrostriction in an optomechanical disk resonator
Electrostriction is a mechanism whereby electric fields induce strain within a material. It differsfrom piezoelectricity in that the induced strain is proportional to the square of the electricfield, and not to the electric field. Since the electric fields we consider are rapidly oscillating atoptical frequencies, the time averaged piezoelectric strain shall be zero, while the time averagedelectrostrictive strain contribution remains. Electrostrictive stresses scale with the fourth powerof the dielectric refractive index, making them of significant importance for high refractiveindex materials such as silicon and GaAs (for which n ≥ train Δ ε r Δ E e r e θ e z a) b) Fig. 3. (a) Illustration of the link between photoelasticity and electrostriction. A strain leadsto change in refractive index (photoelasticity) which itself leads to a change in the storedelectric energy. Electrostriction is the converse mechanism (red arrow), whereby electricfields (stored energy) induce strain in the material. (b) Schematic illustrating the directionof electrostrictive and radiation pressure forces acting on a GaAs disk resonator due tophotons confined in a WGM (black arrows), and represented in the cross section plane overwhich the stress and volume force are plotted in Figs. 2 and 4. p i jkl [32], which links a material strain S i j to a change in the material’s inverse dielectric tensor ε − i j : ε − i j ( S kl ) = ε − i j + ∆ (cid:16) ε − i j (cid:17) = ε − i j + p i jkl S kl (12)The photoelastic tensor has 3 =
81 elements, that reduce to only 3 independent coefficientsfor cubic crystals such as GaAs [33]. These three parameters are p , p and p , written herein contracted notation, where 11 →
1; 22 →
2; 33 →
3; 23, 32 →
4; 31, 13 →
5; 12, 21 → σ es to the electric fieldcomponents in the following way [32]: σ esrr σ es θθ σ eszz σ es θ z = σ esz θ σ esrz = σ eszr σ esr θ = σ es θ r = − ε n p p p p p p p p p p p
00 0 0 0 0 p (cid:124) (cid:123)(cid:122) (cid:125) photoelastictensor E r E θ E z E θ E z E r E z E r E θ (13)The value of the three photoelastic coefficients for GaAs are provided in Table 2. The rela-tion between electrostriction and photoelasticity is seen by considering a disk resonator sud-denly subject to strain. The strain leads to a change in the material’s permittivity ∆ ε , via thephotoelastic properties. Provided some electric energy was stored in the disk at the time, thisstored energy (proportional to ε E ) changes due to the change in permittivity ∆ ε . This changein energy can be seen as the work of the electrostrictive force during the displacement. (A morecomplete version of this argument is developed in [32], see Fig. 3 a).Material Wavelength ( µ m) p p p ReferenceGaAs 1.15 -0.165 -0.140 -0.072 [34]Si 3.39 -0.09 +0.017 -0.051 [35]
Table 2. Photoelastic material parameters for GaAs, and silicon (Si) for comparison. Thephotoelastic coefficients vary little for wavelengths with energies well below the materialbandgap [36].
Looking at Eq. 13 and the values of the photoelastic coefficients in Table 2, it appears in thecase of a WGM that the electrostrictively induced normal stresses are significantly larger than ) b) c) d) Fig. 4. 2D axi-symmetric FEM modeling of the electrostrictive stress and volume force. (a)and (b) show respectively the rz cross-section of the radial σ esrr and axial σ eszz electrostrictivestress distributions. The azimuthal normal stress σ es θθ (not shown here) is of comparablemagnitude. (c) and (d) plot the associated radial and axial volume force distributions. Blackarrows indicate the overall direction these forces point in. the shear stresses. We will focus here for brevity just on the radial σ esrr and axial σ eszz normalstress components: σ esrr = − ε n (cid:2) p | E r | + p (cid:0) | E θ | + | E z | (cid:1)(cid:3) σ eszz = − ε n (cid:2) p | E z | + p (cid:0) | E r | + | E θ | (cid:1)(cid:3) (14)Figure 4 (a) and (b) show the value of σ esrr and σ eszz due to a single photon confined in the p=1,m=10 WGM of a 1 µ m radius GaAs disk resonator of thickness 320 nm, already consideredin section 2. Figure 4 (c) and (d) represent the associated volume force for both these stresses,where F esr = − ∂ r σ esrr and F esz = − ∂ z σ eszz . The black arrows show the net direction these forcesare pointing in. We see here that the electrostrictive force pushes outwards in both the radialand vertical z directions, adding constructively to the radiation pressure force. The fact thatelectrostriction and radiation pressure add up constructively as they do here is not true for allmaterials and geometries. As we can see from Eq. 14, since the p i j are negative for GaAs, allelectrostrictive stresses are positive, and confined photons tend to expand the material in alldirections. However this would not be the case for silicon disk resonators or waveguides, as thecoefficients p and p are of different sign and significantly different magnitude.
4. Optomechanical coupling in GaAs disks geoom
Reference [37] provides a perturbation theory for Maxwell’s equations in the case of shiftingmaterial boundaries. This theory can be applied to determine the frequency shift of an opticalWGM to an arbitrary mechanical deformation of the confining dielectric disk. Following thismethod, the term g geoom is calculated as a surface integral of the unperturbed optical fields over ) b) c) d) “Bowl” mode st RBM nd RBM “Pinching” mode
Fig. 5. (a) through (d): Displacement profile for the four mechanical eigenmodes listed inTable 3, with exaggerated deformation. The surface color code illustrates the total displace-ment, with red as maximum and blue as minimum. the perturbed dielectric interface: g geoom = ω (cid:90)(cid:90) disk ( (cid:126) q · (cid:126) n ) (cid:104) ∆ ε | (cid:126) e (cid:107) | − ∆ (cid:0) ε − (cid:1) | (cid:126) d ⊥ | (cid:105) d A (15)Here (cid:126) q and (cid:126) n are respectively the normalized mechanical displacement vector and surface nor-mal vector. (cid:126) e (cid:107) (resp. (cid:126) d ⊥ ) is the parallel (orthogonal) component to the surface of the electric field(electric displacement field). (cid:126) q and (cid:126) e are normalized such that max | (cid:126) q | =1 and (cid:82) ε | e | d V = ∆ ε = ε − ε is the difference in permittivity between the materials on either side of the bound-ary and ∆ (cid:0) ε − (cid:1) = ε − − ε − . Here we are only interested in the geometric contribution to theg om , so ε is simply n over the entire disk, while ε =1. g geoom is computed from Eq. 15 using aFEM simulation software (COMSOL Multiphysics). The results for the four mechanical modesshown in Fig. 5 are summarized in Table 3. For a given mechanical mode, the displacement ofevery point of the disk is spatially non uniform and g geoom = − d ω geo d x is therefore dependent on thesomewhat arbitrary choice of the reduction point which experiences the displacement dx . Thenormalization choice max | (cid:126) q | =1 in Eq. 15 means that the point of maximal displacement is usedas reduction point. As evidenced in Table 3, different mechanical modes have vastly differentvalues of g geoom . Note for instance how the g geoom value for the 1 st RBM is roughly 10 000 timeslarger than for the out of plane ‘bowl’ mode. This difference illustrates how efficiently eachmechanical mode modulates the total cavity length (the d R / d x term in Eq. 2) and confirms thatthe first RBM is the mechanical mode with the highest g geoom .Mechanical mode ‘bowl’ 1 st RBM 2 nd RBM ‘pinching’Frequency 494 MHz 1.375 GHz 3.5 GHz 5.72 GHzg geoom (GHz/nm) 0.11 1080 412 82g peom (GHz/nm) 0 984 1720 231g totalom (GHz/nm) 0.11 2064 2132 313x
ZPF (m) 2.95 · − · − · − · − g geo (MHz) 3.2 · − pe (MHz) 0 1.21 1.98 0.50g total (MHz) 3.2 · − Table 3. Comparison between the geometric and photoelastic optomechanical couplingstrengths g geoom and g peom , for four mechanical modes of a 320 nm thick, 1 µ m radius GaAsdisk, and a p=1 m=10, λ (cid:39) . µ m WGM, obtained through FEM simulations. The me-chanical deformation profiles are shown in Fig 5. Due to their extremely miniaturized dimensions, 1 µ m GaAs disks exhibit remarkably largeoptomechanical coupling, with g geoom reaching over 1 THz/nm in the case of the first RBM.(Here we see that the ∼ .0 0.2 0.4 0.6 0.8 1.005. ¥ - ¥ - ¥ - Radial coordinate H – m L R a d i a l d i s p l ace m e n t H m L - ¥ - ¥ - ¥ - ¥ - ¥ - ¥ - ¥ - Radial coordinate H – m L R a d i a l s t r a i n - ¥ - - ¥ - - ¥ - - ¥ - ¥ - ¥ - ¥ - Radial coordinate H – m L R a d i a l d i s p l ace m e n t H m L - ¥ - - ¥ - ¥ - Radial coordinate H – m L R a d i a l s t r a i n a) b) c) d) Fig. 6. Radial displacement and normal radial strain S =S rr as a function of the radialcoordinate, for the first RBM ((a) and (b)) and for the second RBM ((c) and (d)) of a1 µ m radius and 320 nm thick GaAs disk resonator. The values are obtained through FEMmodeling, using the approximation of an isotropic Young’s modulus for GaAs. The orangehighlighted zone between r=0.6 µ m and r=1 µ m and the black ring in the inset pictures markthe region of highest electromagnetic energy density for a p=1 WGM. (The displacementsare normalized such that for both modes the mechanical energy is equal to k B T at 300 K). value provided by the simplified expression g om = ω / R (cid:39) . ZPF are obtained by equaling the mechanical energy in the resonator to ¯ h Ω M /
2, yieldingx
ZPF =1.2 · − m for the first RBM, using the same reduction point as above. The single pho-ton optomechanical coupling strength for this mechanical mode is g = g om x ZPF (cid:39) . ≥
90 % undercut). For larger radii the statedmechanical frequencies and g geoom may differ significantly. peom
To compute the photoelastic coupling contribution, first the unperturbed resonance frequency ofthe 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 thecomplete deformation profile and strain distributions inside the resonator (we will focus in thefollowing discussion on the first RBM). While the radial displacement is zero at the center andmaximum near the periphery, the behavior for the normal radial strain is reversed. The normalradial strain S rr =S (in contracted notation) is maximal at the center of the disk and changessign right by the edge of the disk (this is a normal consequence of the circular geometry), seeFig. 6 (a) and (b). The behavior is similar for the normal azimuthal and axial strains S andS , which are of similar magnitude and largest near the center of the disk. The S and S straincomponents are zero over the whole disk, while the S strain component is roughly three ordersof magnitude smaller than S , , . We now use Eq. 12 to relate the strain distribution inside thedisk to changes in the dielectric tensor. Since S , S and S are negligible, the off-diagonalerms in the dielectric tensor can be neglected. The dielectric tensor modified by the RBMdisplacement therefore takes the form: ε ε ε with ε = (cid:0) / n + p S + p S + p S (cid:1) − ε = (cid:0) / n + p S + p S + p S (cid:1) − ε = (cid:0) / n + p S + p S + p S (cid:1) − (16)Note that it is now both anisotropic and dependent upon the position inside the disk resonator.The problem of finding the new WGM resonance frequency under these conditions is solvedthrough another FEM simulation (with unperturbed geometric boundaries). This provides thephotoelastic frequency shift due to the mechanical displacement dx . In the linear limit of small dx , the procedure leads to the photoelastic optomechanical coupling g peom , which is found toamount to 0.98 THz/nm for the first RBM of the above considered disk and WGM. This valueis remarkably high, considering how inefficient the refractive index modulation is through thefirst RBM. Indeed in order to maximize the photoelastic frequency shift the optical mode shouldbe localized in the region of highest strain . In the case of the first RBM the radial strain is notonly weak but also changes sign right around the area of highest optical energy density (see thehighlighted area of Fig. 6 b). In contrast, this condition is much better fulfilled for the secondorder RBM (Fig. 6 (c) and (d)). Accordingly, this translates into a remarkably high g peom of nearly2 THz/nm for this mechanical mode, see Table 3. Because of the reduced geometric couplingfor the second RBM, its total optomechanical coupling g total is comparable to that of the firstRBM, around 2.5 MHz, albeit at a much higher mechanical frequency of 3.5 GHz. Table 3summarizes the g values for the four considered mechanical modes of Fig. 5.Figure 7 plots the dependency of g pe and g geo with disk radius for the first and second RBM.Each purple (blue) dot corresponds to a distinct FEM simulation of g pe (g geo ), while the solidblue line for g geo corresponds to the value given by the following analytical formula: g geo = g geoom x ZPF (cid:39) ω R (cid:115) ¯ h m eff Ω PM with Ω PM = λ P R (cid:115) E ρ ( − ν ) (17)Here m eff , λ P , E , ρ and ν are respectively the effective mass of the mode, calculated with a re-duction point sitting on the disk boundary, a frequency parameter, the Young Modulus, densityand Poisson ratio of GaAs, the values of which are provided in Table 4. Ω PM is the mechanicalfrequency of the RBM of order P [38, 39]. We have obtained Eq. 17 through the analyticaltreatment of a free elastic circular plate. For the first RBM, Eq. 17 accurately reproduces thetrend provided by FEM simulations, but overestimates the coupling by 20% because it neglectsthe out-of-plane component of the mechanical motion. For the second RBM the overestimationis more pronounced, reflecting a larger out-of-plane component of the mechanical mode.Since the effective mass scales with R , g geo scales as (cid:0) R (cid:1) / . Interestingly, g pe rises fasterthan g geo with decreasing disk radius. For instance for the 1st RBM (Fig 7 a), g pe goes frombeing two times smaller than g geo for disks of radius R=10 µ m, before reaching comparablevalues for 1 µ m radius disks. For the 2nd RBM (Fig 7 b), the photoelastic coupling is alwaysthe dominant coupling mechanism. Note that the maximal photoelastic coupling is reached for R (cid:39) µ m. Further reducing the disk dimensions reduces the coupling as the optical mode is nolonger well localized on the region of highest strain. The link between radiation pressure and boundary deformation and electrostriction and pho-toelasticity can be understood by looking at the work done by the optical forces during a me-chanical displacement. Incidently, these energy considerations provide an additional way of H – m L g H M H z L Photoelastic
Geometric FEM H – m L g H M H z L Photoelastic
Geometric FEM st RBM nd RBM a) b) Geometric analytical
Fig. 7. Comparison between the geometric and photoelastic optomechanical couplingstrength g for the TE (p=1) WGM of a GaAs disk of thickness 320 nm, as a functionof radius, for the first RBM (a) and second RBM (b), in log scale. The vertical dashed redline represents the radius r=0.7 µ m, at which bending losses become limiting at the con-sidered WGM wavelength of 1.3 µ m. For the first RBM, the combined optomechanicalcoupling g pe + g geo reaches 4 MHz for the smallest disks. (The dashed blue line representsthe values given by Eq. 17 reduced by 20%). For the second RBM, photoelasticity is thedominant optomechanical coupling mechanism, with g pe reaching 2 MHz for R=1 µ m. Frequency parameter λ P Effective mass ratio GaAs material parameters λ λ λ m eff1 / m m eff2 / m m eff3 / m ρ [kg · m − ] E [GPa] ν Table 4. First three values of the frequency parameter λ P and effective mass ratios for GaAsdisk RBMs, and GaAs material parameters used in the calculations. The effective mass ratiois defined as the effective mass associated to a reduction point on the disk boundary m eff divided by the disk mass m . calculating both the geometric and photoelastic optomechanical couplings. Following Eq. 4,we can write a generalized expression: g om = (cid:82)(cid:82)(cid:82) disk Σ σ i j S i j d V ∆ x ¯ h (18)Here the numerator corresponds to the work produced by the optical stress due to a single con-fined photon, during the displacement ∆ x , in the case of a linear elastic solid starting at rest [40].The S i j are the mechanical strain components resulting from the displacement ∆ x , and the σ i j are the radiation pressure or electrostrictive stress components described respectively in Eqs. 10and 13. This formulation and the method discussed in 4.2 yield values in very good agreement,within less than 1 % difference. Note that both for radiation pressure and electrostriction, in thecase of the 1st RBM at least, a large part of the work is done by the optically induced azimuthalstress σ θθ . Furthermore, for the same mechanical mode, the larger axial stress σ zz in the case ofelectrostriction produces negative work as the disk expands in the radial direction but contractsin the axial direction. These considerations shed light on two seemingly contradictory observa-tions. On one hand the photoelastic coupling g pe is slightly smaller than the geometric coupling g geo for the first RBM, on the other hand the radial stress per photon is several times larger forelectrostriction than radiation pressure. As a consequence, even though the movement of the 1stRBM is predominantly radial, the full picture of optomechanical coupling can not be obtainedlooking solely at the forces exerted in the radial direction. .4. Discussion We show that the second order RBM is an interesting mechanical mode thanks to its large totaloptomechanical coupling and high mechanical frequency. While this type of mode tendentiallyhas a lower mechanical Q due to larger mechanical coupling to the pedestal, its anchoring lossescould be overcome with a carefully engineered pedestal geometry [18].We verify that both the geometric and photoelastic coupling magnitudes are comparablewhen considering a transverse magnetic (TM) WGM instead of a TE WGM, with values vary-ing by less than 20 %. We focused here on p=1 WGM, as these are the modes with the highestradiative optical Qs [13]. When considering different WGMs, the same rule of thumb remains:in order to maximize the photoelastic optomechanical coupling, the regions of high electromag-netic energy should be co-localized with regions of high mechanical strain.For comparison the photoelastic optomechanical coupling has been computed on Si disks ofidentical dimensions using the photoelastic parameters of Table 2. The obtained g pe for the 1stRBM is roughly three times lower than for GaAs, notably because of the reduced photoelasticcoefficients of Si, but should nevertheless not be neglected.Finally recent work investigating the optomechanical coupling in distributed Bragg reflectorGaAs/AlAs vertical cavities [41] shows these geometries are also well suited to take advantageof the photoelastic coupling mechanism, thanks to an efficient overlap between the optical fieldand strain maxima resulting in values of g peom reaching several THz/nm.
5. Conclusion