Brillouin optomechanics in nanophotonic structures

The interaction between light and mesoscopic mechanical degrees of freedom has been investigated under various perspectives, from spec-troscopy in condensed matter, optical tweezer particle trapping, and long-haul optical fiber communication system penalties to gravitational-wave detector noise. In the context of integrated photonics, two topics with dissimilar origins—cavity optomechanics and guided wave Brillouin scattering—are rooted in the manipulation and control of the energy exchange between trapped light and mechanical modes. In this tutorial, we explore the impact of optical and mechanical subwavelength confinement on the interaction among these waves, coined as Brillouin optomechanics. At this spatial scale, optical and mechanical fields are fully vectorial and the common intuition that more intense fields lead to stronger interaction may fail. Here, we provide a thorough discussion on how the two major physical effects responsible for the Brillouin interaction—photoelastic and moving-boundary effects—interplay to foster exciting possibilities in this field. In order to stim-ulate beginners into this growing research field, this tutorial is accompanied by all the discussed simulation material based on a widespread commercial finite-element solver.


A. Context and brief history
When the phase of a propagating light field is disturbed by a spatially periodic modulation, the direction of light propagation is shifted as determined by momentum conservation. 1 If such a spatial modulation is also periodic in time, the Doppler effect takes place and energy conservation implies that the scattered light is frequencyshifted. As initially pictured by Léon Brillouin in his famous treatise La diffraction de la lumière par des ultra-sons, 2 mechanical waves can precisely induce such spatiotemporal gratings leading to the homonym scattering process: Brillouin scattering (BS). Motivated by Townes' pioneer experiments in nonlinear light-scattering, Brillouin scattering gained importance in early experiments on optical fibers [3][4][5] and planar waveguides. 6 The enhanced interaction length brought by these waveguides fostered the investigation of Brillouin scattering, 6 although it is fair to say that it was within the optical fiber community that Brillouin scattering had the largest impact, either as a detrimental or gainful effect.
At high optical intensities, the interference pattern between the strong pump and the Brillouin back-scattered light-initially induced by thermal phonons-causes a back-action drive of the mechanical waves through electrostriction and radiation pressure forces. 7,8 Such feedback leads to the so-called backward Stimulated Brillouin Scattering (SBS) effect. In such a stimulated regime, Brillouin scattering can ultimately prevent high power signal APL Photonics TUTORIAL scitation.org/journal/app propagation through an optical fiber. 9 Despite the wider literature on the backward Brillouin scattering, a counterpart of it can also occur in the forward direction, first identified as GAWBS (Guided Acoustic Wave Brillouin Scattering) by Shelby et al. 10 in optical fibers experiments. The mechanical modes participating in those initial GAWBS observations were transverse standing waves with rather low resonant frequencies due to the large optical fiber cladding diameter (≈125 μm). Despite its weak thermal-noise origin, the phase noise induced by GAWBS was identified as a major source of noise in fiber-based quantum noise squeezing experiments. 10 These guided mechanical waves can also be stimulated by light through electrostriction forces and may lead to cross talk 11 and long range interaction 12 among short-pulses propagating in optical fibers.
The fundamentals and applications of SBS derived from these pioneer experiments impacted various research fields. For instance, the large and narrow band optical gain available through the Brillouin interaction has enabled efficient fiber-optic based slow and fast light, 13,14 nanosecond-long storage of optical pulses, 15 optical and radio-frequency (RF) tunable filters with unprecedented narrow band, extinction, and tunability. [16][17][18][19] Other technological breakthroughs enabled by the recent advances on Brillouin optomechanics are the generation of high-spectral purity RF sources, 20 distributed strain and temperature sensors, [21][22][23][24][25] high resolution spectrometers, 26,27 liquid-compatible sensors, 28,29 gyroscopes, 30 and narrow linewidth Brillouin lasers. [31][32][33][34][35][36][37] For some of us, the love story with Brillouin started back in 2003 when selecting small-core photonic crystal fiber (PCF) samples for fiber-optical parametric amplification experiments at Bath University (UK). In the following years, in a joint effort between University of Campinas, FEMTO-ST in Besançon, and the Max Planck Research Group at Erlangen (now the Max Planck Institute for the Science of Light), we verified that the famous formula by Smith 38 -used for predicting the onset of SBS-consistently overestimated it for small-core PCFs. 39 Further investigation revealed that the strong optical and mechanical confinement in these fibers was responsible for the nonintuitive reduced overlap between optical and mechanical modes and, therefore, the overshoot of Smith's formula. At the time, similar predictions were made by Kobyakov and collaborators in weakly guiding optical fibers designed to suppress SBS. 40 PCFs not only triggered the possibility of controlling backward Brillouin scattering but also enabled the first observation of forward Brillouin scattering at gigahertz frequencies; 41 the mechanical waves were trapped at the 1.2 μm glass-air core interface, which drastically increased the GAWBS frequency and led to a broader wavevector range with near-zero mechanical group velocity, a property shared by the optical phonon branch in solids and inspired referring to GAWBS as Raman-like scattering. 41 Further work carried out by several groups using optical fibers [42][43][44][45][46][47][48][49] unraveled many remaining mysteries and potential applications of guided Brillouin scattering.
Parallel to these developments, the role of optical forces in photonic integrated cavities and waveguides was going through a revolution after a series of papers demonstrating the strong optomechanical interaction enabled by radiation pressure forces in highindex contrast devices. [50][51][52][53][54][55][56][57][58] In the last ten years, following the growth and availability of advanced top-down nanofabrication tools, a series of research breakthroughs occurred in this field within the classical [59][60][61] and quantum realm. [62][63][64][65][66][67] The advancement of these integrated photonic structures also impacted backward Brillouin scattering, especially after the landmark demonstration of on-chip SBS by Pant et al., 68 which sparked a convergence of the traditional waveguide-based Brillouin concepts with the cavity optomechanics ones. One clear outcome of this convergence was the prediction and subsequent demonstration of giant gain Brillouin scattering in a CMOS-compatible platform, [69][70][71] which culminated in the first generation of on-chip Brillouin lasers 20,35,36,72 and on-chip signal processing devices. 72,73 In the context of these recent demonstrations, it is harder to draw a line dividing the fields of Brillouin scattering in wave-guiding photonic structures and the field of cavity optomechanics. In light of this convergence, in this tutorial, we will use the term Brillouin optomechanics to encompass the whole spectra of experiments involving trapped light and mechanical waves.

B. Scope of this tutorial
There have been a series of reviews and tutorials published about the individual topics of Brillouin scattering and optomechanics in the recent years, with focus on optical fibers, 81 experimental progress, 82 integrated photonic applications, 83,84 phononics, 85 historical evolution, 86 surface modes, 87 electrostriction excitation, 88 and cavity optomechanics. 89 More recently, an extensive and detailed review article on current developments and future perspectives of light and sound interaction has been published by Safavi-Naeini et al., 90 which brings together several aspects of electromechanical devices that share many similarities with optomechanical systems. [91][92][93][94][95] In this tutorial, we offer a complementary perspective aimed at providing a thorough discussion of the interaction between optical and mechanical modes in both waveguides and cavities as described by the coupled-mode theory. Since there is an extensive literature covering the derivations of the coupled-mode equations for both waveguides 79,96,97 and cavities, 79,98 and the fundamental 90 and technological 84 outcomes of this interaction, our focus will be on discussing the interplay between the two major mechanisms responsible for the Brillouin and optomechanical effects: the refractive index perturbations through the photoelastic effect 99 (PE) induced by the elastic strain and moving boundary (MB) effect caused by the mechanical mode distortion of the optical boundaries, 100 paying particular attention to highly confined optical and mechanical modes in nanophotonic structures. Doing so, we also cover the two major optical forces that enable driving mechanical modes with light, the electrostriction and radiation pressure forces, which are directly connected to the PE and MB refractive index perturbations. 96 To give an overview of the tutorial's scope, we show in Fig. 1 several structures where forward and backward Brillouin optomechanics have already been demonstrated, divided into fiber-based and integrated waveguides and axisymmetric cavities.
We organized this tutorial as follows: in Sec. II, we discuss the role of optical dispersion relation and momentum conservation (phase-matching) in both cavities and waveguides; general properties of the Brillouin optomechanical interaction are also discussed, including coupled-mode equations, definition of coupling rate and gain coefficient, and the structure of boundary and volume optomechanical overlap integrals. In Sec. III, we explore in detail Brillouin scattering in circular dielectric waveguides -specifically a silica glass rod suspended in air. In Sec. IV, we review in detail the APL Photonics  74,75 (b) integrated As 2 S 3 waveguide buried on SiO 2 , 68 (c) simple Si disk, 76 (d) silica photonic crystal fiber, 39 (e) suspended silicon waveguide, 71 (f) toroidal silica cavity, 77 (g) dual-nanoweb silica fiber, 47 (h) suspended silicon waveguide membrane, 35 and (i) silicon bullseye optomechanical cavity. 78 For the typical dimensions, (a)-(f) present examples of mechanical modes for backward scattering, while (g)-(i) present examples of mechanical modes for forward scattering. In each panel, the top right inset shows the normalized total mechanical displacement (|u|) and the bottom right inset shows the normalized absolute value of the electric field (|E|). In order to compare the coupling strength between the optical and mechanical modes for each device, we extract g 0 [see Eq. (11)] from the simulations of the cavities and follow Ref. 79 to calculateg0 [see Eq. (12)] using the same length for all circular cavities, equivalent to their perimeter (Lwg = 2π × 10 μm). The same approach is used to evaluate the gain coefficient (G B /Qm). The resulting mechanical frequency (Ω/2π), effective coupling rate (g 0 /2π), and the ratio between the gain and mechanical quality factors (G B /Qm) are (a) 6 In (e), forward and backward cases are presented as well as an example on the use of mechanical anisotropy for calculating modes and couplings in this geometry. A simulation file for each one of these devices is hosted on the data repository. 80 optomechanical interaction in axisymmetric dielectric cavities, where the example of a silicon microdisk is explored and contrasted with other devices. Finally, general remarks and perspectives are outlined in the conclusions. In order to fulfill the purpose of this tutorial, we also created an accompanying data repository 80 that includes most of the simulation files and processing scripts used in each of the figures. With the exception of some analytical tools, all the simulations were performed using the commercial finite-element solver COMSOL Multiphysics.
In waveguides and cavities, mechanical and optical fields propagate in the eigenmodes of the structure obeying specific dispersion relations. If a pump laser at frequency ωp is launched in an optical mode with the wavevector βp, it propagates along the waveguide (z-direction) with a phase given by exp(−iβpz + iωpt) and may be scattered by a mechanical mode with frequency and wavevector (Ω, βm) into a distinct optical mode (ωs, βs), provided the following energy and momentum relations are fulfilled: The ± signs in these relations are due to Doppler effect and indicate that the scattered light may be blue-shifted (+ sign, anti-Stokes) or red-shifted (− sign, Stokes) from the pump; the wavenumbers βp,s,m can be positive-or negative-valued depending on the propagation direction. We illustrate such phase-matching conditions in Figs. 2(a) and 2(b). For example, in backward Brillouin scattering [βs(ωs) = −|βs(ωs)|], the phase-matched mechanical wavevector predicted by Eq. (1b) for the Stokes wave is βm(Ω) = βp(ωp) − βs(ωp − Ω) = |βp(ωp)| + |βs(ωp − Ω)|, which corresponds to a short-wavelength forward propagating (βm > 0) mechanical mode, as shown in Fig. 2(a). In the case of forward Stokes scattering, the phase-matched mechanical mode is a long-wavelength forward propagating one. Therefore, in order to infer the phase-matched βm, one needs to decide not only in which direction (forward or backward) the scattered light is being investigated but also whether the pump and scattered waves belong to the same transverse optical mode. Before we discuss some general aspects of the phase-matching in confined photonic structures, we explore how Brillouin scattering in axisymmetric optical cavities share similarities with their waveguide counterparts.
Within an axisymmetric optical cavity, the propagation phases for both optical and mechanical modes are in the form exp(−imϕ + iωt), where m is an integer and ϕ is the azimuthal angle. This phase dependence leads to a modified form of momentum conservation, where m(ω) and M(Ω) represent the optical and mechanical dispersion relations for the cavity, respectively. Equation (2) is analogous to Eq. (1b), but the dispersion relation β(ω) is now attached to the azimuthal number frequency dependence m(ω); the azimuthal numbers mp,s, M can also be positive-valued or negative-valued depending on the propagation direction, clockwise or counterclockwise. In addition, since the cavity resonance condition requires the azimuthal numbers m and M to be integers, the phase-matching points are discrete. An explicit connection between Eqs. (2) and (1b) can be drawn by mapping the propagation phase along the z-direction with the azimuthal coordinate, z → Rϕ, where R is the cavity radius; this implies that exp(−iβRϕ) → exp(−imϕ) and, therefore, β = m/R. This relation is quite useful and predicts the approximate values of the azimuthal numbers for an optical cavity mode with a given effective refractive index (n eff ) as m ≈ k 0 n eff R, where k 0 = 2π/λ is the vacuum wavenumber at an optical wavelength λ.
In Fig. 2(c), we illustrate the most common forward and backward Brillouin scattering configurations studied in photonic structures, restricted to the Stokes scattering for simplicity. We generalize The solid curves represent dispersion branches for different optical modes, and the circles sampling the solid curves represent longitudinal resonances that are formed if the waveguide was wrapped into a cavity geometry; the shaded horizontal regions represent the linewidth of each optical mode resonance (κ). The arrows in the lower (upper) region correspond to the intramode (intermode) scattering. For the intramode set, an optical pump at (ωp, βp) is scattered to the same dispersion branch in the FW (BW) direction by forward propagating mechanical modes represented by the red arrow (green arrow). For the intermode set, an optical pump at (ωp, βp) is scattered to another dispersion branch in the FW (BW) direction by forward propagating mechanical modes represented by the red (green) dotteddashed arrow. (d) Cavity resonance schematic representing the photonic density of states (PDOS) of the cavity modes. Notice that in the intramode FW case, the Stokes is detuned from the cavity mode (open symbol) effectively reducing the Stokes photons at the cavity mode.
this dispersion diagram schematic to represent the optical dispersion relation of both waveguides (solid lines) and cavities (discrete dots). We now distinguish between two scattering configurations: (1) intramodal and (2) intermodal scattering.

Intramodal scattering
In intramodal scattering, both pump and the scattered light propagate in the same transverse mode, in either of two directions, and therefore fall in the same dispersion relation branch; as illustrated in the lower part of Fig. 2(c). The backward case is the most common form of Brillouin scattering and can be easily observed in optical fibers. 3,5 In this configuration, the forward propagating pump at mode (ωp, βp) is back-scattered to (ω BW s , β BW s ). Assuming

APL Photonics
where we approximated βs(ωp ± Ω) ≈ βs(ωp) since the mechanical wave frequency is orders of magnitude lower than the optical frequency. This backward intramode process is represented by the solid green arrow in the lower part of Fig. 2(c). Albeit the waveguide phase-matching (β) can always be fulfilled, the cavity case (M) is far more restrictive: the frequency difference between the pump and scattered modes must match the mechanical frequency within the linewidth of the optical modes. In Fig. 3(a), we illustrate the mechanical dispersion relation of a waveguide (Ω, βm) or axisymmetric cavity (Ω, M/R)-the straight line represents the bulkwave dispersion relation, and the hyperbolic curve with nonzero cutoff frequency represents the confined mechanical mode. The phase-matching condition for backward scattering given in Eq. (3) is represented by a dashed-gray line identified as BW. In either the waveguide or cavity geometry, the mechanical mode is a shortwavelength mode (large βm), as indicated in Fig. 3(b). The symmetry of Eq. (3) with respect to Stokes and anti-Stokes scattering suggest that both are equally phase-matched, giving rise to two scattering peaks symmetrically positioned around the central pump frequency. For intramodal backward scattering, a reasonably good estimate of the phase-matched mechanical frequency can be obtained by assuming a linear dispersion relation for both optical and mechanical modes, βp = k 0 n eff and βm = Ω/V l , with V l being the longitudinal speed of sound. Inserting into Eq. (3), we obtain Ω ≈ (4π/λ)n eff V l . For a pump at λ = 1550 nm and a silica glass waveguide (n eff = 1.45 and V l = 5968 m/s), the phase-matched frequency is predicted to be Ω/2π = 11.16 GHz, which agrees pretty well with the experiments in pure-silica core fibers. 41 Similar predictions 76 can be made for axisymmetric optical cavities by employing the approximation mp ≈ βpR. Tight confinement structures have multiple mechanical modes with dispersion relations that substantially deviate from the linear dispersion relation of bulk longitudinal waves and, as a result, exhibit multiple phase-matching frequencies that differ from the above estimate. Further in this tutorial, several examples are explored to illustrate this point.
The case of forward intramodal scattering represents the usual interaction observed in the forward Brillouin scattering experiments 10,41,102 and many of the cavity optomechanics ones. 51,53,56 In this configuration, the forward propagating pump at mode (ωp, βp) is forward-scattered to (ω FW s , β FW s ) and the phase-matching conditions [Eqs. (1b) and (2)] result in The phase-matching condition for forward intramodal scattering is represented in Fig. 3(a) as a vertical dashed line identified with FW. Notice that in the limit of (βm, M) ≈ 0, forward intramodal scattering involves a mechanical mode at the cut-off frequency (i.e., at the waveguide transverse mechanical resonances). In either waveguide or cavity structures, these transverse resonances correspond to a uniform spatial phase along the propagation direction (z for waveguides or ϕ for axisymmetric cavities), as illustrated in Fig. 3(c). From the mechanical dispersion relation of Fig. 3(a), we note that the forward scattering phase-matching intercepts the mechanical modes very close to their cut-off frequencies. In contrast to the bulk mechanical dispersion [the straight solid line in Fig. 3(a)], these transverse resonances dispersion curves flatten out in the limit of null wavevector, resulting in very large mechanical phase velocities Ω/βm. Such near-cutoff dispersion relation characteristics render forward intramodal Brillouin scattering analogous to Raman scattering in solids due to the flattened-out dispersion of the optical phonon branch. 41 Rigorously, the mechanical wavevector is not exactly null and can be estimated by exploring the fact that both scattered and pump waves have nearly the same frequency. Taylor expansion of the scattered optical wavevector around the pump frequency, i.e., βs ≈ βp + (ωs − ωp)∂β/∂ω, results in βm ≈ ±Ω/υg (υg is the group velocity of the optical dispersion branch), which

APL Photonics
TUTORIAL scitation.org/journal/app compared to the pump wavevector is βm/βp ≈ Ω/ωp ≈ 10 −5 (for ng = c/υg ≈ n eff ). One can see that the Stokes and anti-Stokes scattered waves interact with very long wavelength mechanical modes (λm ≈ 20 cm for a 1 GHz mode). Since either the mechanical attenuation length is shorter than such a wavelength or than the waveguide itself, the optical waves essentially interact with the same mechanical mode. For cavities, the existence of a finite lifetime (1/κ) allows for the pump and scattered light to couple to the same optical mode, although the number of intracavity photons will depend on the cavity photonics density of states (PDOS) at a given detuning [see Eq. (10)]. 89

Intermodal scattering
In intermodal scattering, the scattered light couples to a transverse optical mode distinct from the pump-wave mode, either in the forward or backward direction. In the backward configuration, represented by the dashed-dotted green arrow in the upper part of On the other hand, the forward intermodal scattering, represented by the dashed-dotted red arrow in Fig. 2(c), contrasts with the forward intramodal case as the mechanical wavevector may be considerably larger, as given by the phase-matching condition, Note that the intermode phase-matching given by Eqs. (5) and (6) may also be applied between distinct nondegenerate polarizations of an given mode family, such as in between polarizations of fundamental modes in PCFs. 103 The mechanical wavevector imposed by the phase-matching is an important aspect of the mechanical mode engineering, and it is crucial in determining the mechanical group velocity and the attenuation length for a mode with a given dispersion relation βm(Ω); it may also define whether mechanical bandgap guidance is feasible. 104 Therefore, although we cannot make any further assumption about the relation between βs and βp in Eqs. (5) and (6), it is instructive to analyze and compare the limiting cases of BW and FW scattering on a dielectric waveguide based on the optical total internal reflection. In this case, the guided optical wavevector is limited to the range β clad < β < βcore, where β clad = k 0 n clad (βcore = k 0 ncore) is the cladding (core) material wavenumber; nj is their respective refractive indices. For BW intermodal scattering, the shortest wavevector mismatch is given by β inter,BW m ≈ ∓(βcore + β clad ). In a low contrast silica-on-air structure, such an intermodal wavevector differs slightly from the intramodal case [Eq.  105 For comparison, the largest wavevector mismatch for FW intermodal scattering can be estimated as β inter,FW m ≈ ∓(βcore − β clad ). In a silicon-on-air structure, it could result in a larger forward intermodal wavevector, (β inter,FW m /β inter,BW m ) = (3.45 − 1)/(3.45 + 1) = 0.55. In practice, this ratio will be smaller due to modal guidance constraints, but it shows that intermodal forward scattering can also involve short-wavelength (large wavevector) propagating mechanical modes. At such large wavevectors, the mechanical wave bounces from the guiding structure boundaries at an angle, which may drastically change the mode behavior; for instance, mechanical total internal reflection and phononic bandgaps may be completely modified at blazing incidence.
As a historical note of forward intermodal scattering, it was first reported by Engan et al. in a two-mode optical fiber with the mechanical wave driven by a piezoactuator, 106 and a couple of years later, it was observed as stimulated scattering by Russell et al. 107 Recently, it has been the configuration of choice in important Brillouin experiments in microphotonic structures. 35,108,109 Within optical microcavities, such short-wavelength mechanical modes can also be explored to ensure that the scattered light is resonant with one of the cavity modes, a technique that has been used 35,72,108 to demonstrate Brillouin lasing in integrated photonic cavities.

B. Photoelasticity and moving boundary perturbations: Electrostriction and radiation pressure forces
Although the phase-matching conditions discussed so far are necessary to ensure efficient optomechanical energy transfer, they are not sufficient. The optical and mechanical modes must have a reasonable spatial overlap and effectively perceive each other through some underlying physical effect. In the PE effect, the dielectric permittivity is disturbed by the elastic strain, 1,8 while in the MB, geometrical distortions of the optical boundary induced by the mechanical wave perturb the guided mode effective refractive index. 100 The latter was first noticed as a secondary contribution in thin-film scattering experiments 6 but has been demonstrated to be critical in subwavelength scale photonic structures. 62,69,74,88,96 These two effects together are responsible for light scattering by mechanical waves in the various configurations described in Sec. II A. It is worth noting that although photoelasticity and moving boundary perturbation are the main forward effects responsible for the Brillouin optomechanical interaction considered here, other coupling mechanisms exist, such as thermomechanical 8 and magnetoelastic 110 coupling.
Within optical waveguides, the Brillouin optomechanical interaction is described through the spatiotemporal evolution of slowly varying optical and mechanical field amplitudes along the propagation direction, while in optical cavities, the time evolution of the slowly varying fields describes the interaction. One may follow a few distinct approaches to derive such coupled equations. A classical one starts from Maxwell's and elastodynamics wave equations with a forward effect, describing how optical fields are perturbed by the mechanical waves (such as photoelastic and moving boundary effects), and a backward effect representing the electromagnetic forces driving the mechanical modes (such as electrostriction and radiation pressure). This path has been explored for both waveguides 96,111 and cavities. 64 Another common approach to derive the APL Photonics TUTORIAL scitation.org/journal/app coupled-mode equations, which only relies on energy conservation, starts directly from the classical or quantum mechanical Hamiltonian for waveguides 71,90,[112][113][114][115][116] or cavities 89,90,117 where the photoelastic and moving boundary effects are directly incorporated in the Hamiltonian. The cavity and waveguide descriptions may also be directly linked to each other by employing a mean-field approach that averages out the cavity field dependence along the round-trip path-similar to the derivation of the Lugiatto-Lefever equation for optical frequency combs. 79,118,119 Regardless of choosing the force density approach or the energy-based (Hamiltonian) one, the derived coupled mode equations and their describing physics should be the same. This will be the case provided that the corresponding forward and backward effects have been properly accounted for in the Hamiltonian. It is fair to say that up to date Brillouin optomechanics experiments are quite accurately described by the electrostrictive and moving boundary effects, and these terms can be derived by exploring only coupledmode theory and thermodynamics arguments 96 -thus showing that forward and backward effects are directly related to each other in the absence of optical and mechanical dissipation. In other words, photoelasticity is directly related to electrostriction, while moving boundary effect is directly related to radiation pressure. In fact, we discuss later in this tutorial that the overlap integrals describing the forward effects are identical to the those describing the backward effects. 96

Waveguides
Since we aim at a unified discussion of the waveguide and cavity Brillouin optomechanics, we chose to write the coupled mode equations in the form derived in Refs. 90,113,114, and 116 for the Stokes wave-interaction. Note that the anti-Stokes also plays an important role in waveguides, especially in the forward intramodal scattering where the almost null wavevectors lead a strong symmetry between the two scattered waves. 114 Nevertheless, considering only the Stokes wave suffices our purpose of discussing the fundamentals of the Brillouin optomechanical coupling. Assuming slowly varying envelopes for the pump (ãp), Stokes (ãs), and mechanical waves (b) (see Appendix A), the following coupled equations are obtained: where υp,s,m represent the moduli of pump, Stokes, and mechanical group velocities, the upper and lower signs in the ± symbols in Eq. (7b) account explicitly for either copropogating (upper sign) or counterpropagating (lower sign) optical modes, 120  The important aspects of the Brillouin interaction follow from Eq. (7). 79,97 The mechanical wave envelope is driven by the beating between the pump and Stokes envelopes, which are slowly varying functions of z under typical values of power and coupling strength. 96,116 In steady-state (∂t → 0) and for a small υm/γm ratio (large mechanical damping), a common experimental condition, 71,103 the mechanical wave adiabatically follows the optical drive,b = −ig * 0ã * sãp /(iΔ + γm/2). Coupled equations involving only the optical waves can be derived revealing the power-exchange (Pp,s = ̵ hωp,sυp,s|ap,s| 2 ) along the waveguide length, 79,90,96,116,120 where we introduced the Brillouin gain coefficient GB = 4|g 0 | 2 L(Ω)/(υsυpγm ̵ hω), with ω = ωp ≈ ωs and L(Ω) = (γm/2) 2 / [(γm/2) 2 + (Ω − Ωm) 2 ] being the Lorentzian gain profile for a mechanical mode. The gain Lorentzian expresses the role of phasematching in the Brillouin interaction; if the pump and Stokes frequency are offset from the mechanical mode by more than one mechanical linewidth (γm), the Brillouin gain is largely suppressed. In the undepleted pump approximation (∂zPp ≈ 0), Eq. (8) ensures that the Stokes optical power grows exponentially with a power gain coefficient GBPp per unit length, leading to GBPp/αs > 1 as the net Brillouin Stokes gain condition. The power exchange interaction described by Eq. (8) is an important property explored in Brillouin amplifiers, filters, and lasers, 121 and other interesting regimes of Brillouin scattering can also be derived from Eq. (7) and have been discussed in detail elsewhere. 84,90,114,116,[122][123][124] The discussion above shows that within waveguides, the Brillouin gain coefficient GB is often more relevant than the fundamental optomechanical coupling constantg 0 . Thus, GB will be our metric of choice when comparing different waveguides throughout this tutorial. The relation between GB and the optical and mechanical mode fields can be expressed as 96,125 where Qm is the mechanical energy quality factor,m eff = ∫ ρ|um| 2 / max |um| 2 dA is the effective linear mass density of a mechanical mode with displacement profile um, ρ is the mass density, and f wg mb and f wg pe are the line and area overlap integrals representing the MB and PE effects, respectively. We discuss their properties after introducing the optical cavity figure-of-merit expressions. It is important to note that the choice of optical and mechanical mode normalization in the coupled mode-theory affects the expression for the gain, and in Appendix A, we discuss our choices leading to Eq. (9).

Cavities
In order to keep the cavity discussion and notation as close as possible to waveguides, we write the cavity coupled mode equations for the slowly varying envelopes for the pump (ap), Stokes (as), and mechanical waves (b) as 79,90 (∂t

TUTORIAL scitation.org/journal/app
where κp,s are the pump and Stokes optical energy damping rates, Δp,s = ωp,s − ωcp,cs are the frequency detunings between the driving fields (ωp,s) and the cavity modes (ωcp,cs); the optomechanical coupling rate is g 0 . The driving field amplitudes are represented by sp,s and coupled to the cavities at a rate κep,es. In writing Eq. (10), we ignored the weak thermal-noise drive for the mechanical modes. 116 The normalization for the optical and mechanical slowly varying amplitudes in Eq. (10) are such that they represent the average number of photons or phonons stored in the cavity, 79 and the coupling constant has SI units of [g 0 ] = (rad/s). A plethora of Brillouin optomechanical effects are predicted by Eq. (10), 90 among which lasing is one important example. In this case, it is important to distinguish between optical and mechanical lasing, both of which have been experimentally observed. 33,34,36,72,77,105,[126][127][128] In the mechanical lasing regime, 51,56,127,129 the mechanical mode has a lower dissipation rate (Γm ≪ κs) and experiences the Schawlow-Townes linenarrowing often associated with lasing behavior, 130,131 whereas in the optical lasing regime, 34,36,72,77,105,126 which is often associated with the label Brillouin laser due to historical hierarchy, the Stokes mode dissipation is the smallest (κs ≪ Γm) and the linewidth-narrowing is experienced by the Stokes field. 32 One way to see why such dissipation hierarchy defines the lasing field (Stokes or mechanical) is by inspecting Eq. (10) in the undepleted pump approximation: the field experiencing the largest dissipation responds quickly to the interaction terms [proportional to g 0 in right-hand side of Eq. (10)] and can be adiabatically eliminated (∂t ≈ 0). Such adiabatic elimination leaves only one dynamical variable, which is the slowly responding (small dissipation) field, and the lasing threshold occurs when a Hopf bifurcation of the slow field occurs. 132,133 When only a single optical mode is involved, a common situation in many cavity optomechanics experiments, 89 the back-action to the mechanical mode is driven by the sidebands of the pump wave. 89 Two important effects in this single-mode regime are the optical spring, 57,134 amplification 51,135 and cooling, 62,136,137 which have been extensively discussed in other reviews. 89,90 A thorough discussion of these effects has been recently published by Safavi-Naeini et al. 90 It is also worth mentioning that the pump and Stokes laser frequency detunings in Eqs. (10a) and (10b) impact the cavity dynamics more strongly than those in waveguides (which are nonresonant structures): the steady-state amplitudes for the pump and Stokes fields are attenuated as |ap,s| ∝ |sp,s| 2 /Δ 2 p,s for large detunings (Δp,s ≫ κp,s), which may cause a substantial increase in the threshold power for Brillouin lasing. 76,138 They also play a very important role in phase-sensitive coherent effects such as electromagnetic induced transparency (EIT), which can be explored to create Brillouinenabled optomechanical switches 139,140 and nonreciprocal optical devices. 141,142 Within optical cavities, the most common figure-of-merit is the single-photon optomechanical coupling rate g 0 , representing the frequency shift of the optical mode induced by a single quantum of mechanical excitation. 89 Its relation with the optical and mechanical modes can be derived from perturbation theory 100 and is given by where x zpf = √ ̵ h/(2m eff Ω) is the zero-point fluctuation of the mechanical mode with effective mass m eff = ∫ρ|um| 2 /max |um| 2 dV. f cav mb and f cav pe are the area and volume overlap integrands representing the MB and PE effects, respectively.

Cavities vs waveguides
As a final remark on the connection between the distinct figures-of-merit chosen for the cavity and waveguide, it is worth mentioning the so-called optomechanical cooperativity. It is clear that the Brillouin gain GB depends on mechanical dissipation γm, whereas the coupling constantsg 0 and g 0 do not. For cavities, a condition that is analogous to the net Brillouin gain GBPp > αs is expressed as C > 1, where C = 4g 2 0 ncav/(γmκs) is the cavity cooperativity, with ncav = |ap| 2 representing the average number of photons stored in the cavity. Although the dissipation hierarchy is distinct for the mechanical or optical lasing discussed above, the condition C > 1 represents the lasing threshold condition in both cases. 79 Albeit not so popular, an equivalent waveguide cooperativity (C) can also be defined, 79 and similar to cavities, theC = 1 condition also represents net waveguide gain GBPp ≥ αs. Despite the apparent difference between the figures of merit in Eqs. (9) and (11), they can be directly compared to each other if a given waveguide is closed into a path forming an optical cavity -provided that the cavity mode shape resembles the waveguide counterpart as in ring resonators or photonic crystal cavities. For instance, a waveguide with length Lwg has an equivalent cavity single-photon coupling rate given by 79,90 In the waveguide examples shown in Fig. 1, we compare the GB/Qm calculated from Eq. (9) and the equivalent g 0 from Eq. (12), whereas in the cavity examples, we compare g 0 calculated from Eq. (11) with the equivalent GB/Qm obtained from Eq. (12). To avoid specifying the mechanical quality factor for each structure, we quote the Brillouin gain as GB/Qm throughout the text examples. To highlight the material dependence of GB, Appendix B provides a brief discussion on photoelastic Brillouin scaling with material properties.

Boundary and volume overlap integrals
The overlap integrals appearing in Eqs. (9) and (11) depend on the spatial behavior of the two integrands arising from the moving boundary and photoelastic perturbations. These are the quantities that we aim to thoroughly discuss in this tutorial. As such, we choose to write the expressions for the Brillouin gain [Eq. (9)] differently from elsewhere 96,125 (see Appendix A for comparison). The moving boundary overlap integrands in Eqs. (9) and (11) are given by 100 where the permittivity differences are given by δε mb = ε 1 − ε 2 and δε −1 mb = (1/ε 1 − 1/ε 2 ) in which ε 1 = ε 0 n 2 1 and ε 2 = ε 0 n 2 2 are the permittivities of the guiding material and surroundings, respectively. u ⋅n is the surface-normal component of the displacement vector u; the fields Ej, ∥ and Dj, are material interface tangential electric and normal displacement fields for the pump (j = p) or scattered APL Photonics TUTORIAL scitation.org/journal/app (j = s) optical mode. The denominator in Eq. (13) represents the energy or power normalization integrals, N cav It is worth noting that Eq. (13) has dimensions of [f wg mb ] = N/W/m 2 and [f cav mb ] = N/J/m 2 , so they represent force surface densities per unit power or energy; therefore, the MB overlap integrals appearing in Eq. (9) for waveguides have dimensions of linear force densities per unit power [N/W/m]; for cavities, the overlap integrals of Eq. (11) have dimensions of force per unit energy [N/J]. This correspondence is expected based on a virtual work principle 69 or following thermodynamic arguments 143 and suggests that understanding the Brillouin optomechanical coupling based on perturbation of the optical fields due to photoelastic and moving boundary effects is equivalent to follow an optical force description when only conservative forces are considered; another way of justifying the appearance of a single coupling coefficient g 0 in Eq. (7) org 0 in Eq. (10) is through the Manley-Rowe relations for the coupled-mode equations, 8,79 which are based on energy conservation among all the waves involved in a lossless nonlinear interaction. 144,145 The photoelastic overlap integrands in Eqs. (9) and (11) are given by 1 where δεpe = −ε 0 n 4 p:S is the photoelastic tensor perturbation in the permittivity, n is the material refractive index, p is the photoelastic tensor, and S = ∇su is the symmetric strain tensor induced by the mechanical waves, all spatial dependent quantities. Note that by using a symmetric strain tensor, we are assuming a nearly optically isotropic material where the roto-optical effect can be neglected. 146,147 The expression given by Eq. (14) has dimensions of [f wg pe ] = N/W/m 3 and [f cav pe ] = N/J/m 3 and represents force volume densities per unit optical power or energy. It worth noting that even though waveguides and cavities are often discussed through distinct figures of merit, they share essentially the same underlying physical mechanisms and therefore share the same coupling expressions of Eqs. (13) and (14), except for the optical mode normalizations. As pointed out by Van Laer et al., 79 this means that we can anticipate the Brillouin optomechanical coupling in cavities by understanding an equivalent waveguide problem. The material properties used in the simulations discussed here are given in Appendix B.

III. BRILLOUIN COUPLING IN A SILICA GLASS ROD: AN INSIGHTFUL WAVEGUIDE CASE
A silica rod with circular cross section suspended in air represents a simple and instructive platform to understand the fundamentals of Brillouin optomechanical coupling in waveguides. To begin with, analytical solutions are available for both optical 148 and mechanical 149-151 modes, allowing a clear understanding of the modal structure, of its dependence on material properties and waveguide dimensions, and infer symmetry-based selection rules that allow computing coupling coefficients only for relevant modes -more difficult to do in numerical methods. For optical modes, the index contrast between silica and air is high enough to allow strong confinement. Similarly, high-frequency mechanical waves are reflected at the free glass-air interface and form mechanical modes confined to the glass rod. Experimentally, this structure can be easily fabricated in the form of optical fiber tapers, 152 with diameters ranging from tens of micrometers down to a few hundred nanometers while still exhibiting excellent uniformity along its length; 153 such a platform has been explored recently in a number of Brillouin experiments. 49,74,75,[154][155][156] Intuitively, having both modes confined to the same small region leads to stronger interaction. As we discuss in this section, the physics of Brillouin optomechanical coupling in high-confinement structures is quite more involved and richer, 74,75 to a point that it is even possible to have no interaction whatsoever-an effect we reported recently and referred to as Brillouin self-cancellation. 74 A. Confined mechanical modes vs plane mechanical waves We begin discussing the difference in the permittivity perturbation caused by a tightly confined mechanical mode as compared to a simple plane mechanical wave. In bulk, plane longitudinal and transverse (or shear) waves are independent solutions of the mechanical wave equation. A longitudinal mechanical plane wave has only the displacement component uz = u 0 e −ik m z along its propagation direction z, where u 0 is a constant amplitude and km = Ω/V l is the propagation constant at frequency Ω/2π; V l is the longitudinal bulk velocity. This plane wave leads to alternating positive and negative strain layers in the material Szz = ∂uz/∂z = −ikmu 0 e −ik m z , as illustrated in Fig. 4(a). Through the photoelastic effect, these plane strain layers lead to a refractive index grating in the material (homogeneous in the plane transverse to the propagation direction). In waveguides, mechanical modes are generally more complex due to coupling of longitudinal and shear waves at the surface. 149 As a result, the displacement fields in a mode differ significantly from a pure plane longitudinal displacement. In general, a given mode may exhibit all three components ur, u ϕ and uz, and in addition to the dependence on z through the propagating phase e −iβ m z , each component also varies spatially in the transverse plane (r, ϕ). These mechanical modes exhibit a dispersion relation that detours significantly from the bulk, as shown in Fig. 4(b) for the axial-radial (AR 0m ) and torsional-radial (TR 2m ) mode families; the first subscript indicates the azimuthal number of the mode, while the second one indicates the mode index in order of increasing frequency; Figs. 4(c) and 4(d) illustrate the fundamental modes for these two families. At the cut-off frequency (βm = 0), the modes in the axialradial AR 0m family exhibit either radial displacement (ur) or longitudinal displacement (uz), and away from the cutoff, they exhibit both displacement components. In addition to the axisymmetric AR 0m mode family, we note that for azimuthal index 0 a purely torsional mode family T 0m also exists, with only nonzero displacement being u ϕ 149 (this family is not shown in Fig. 4 for simplicity). The torsional-radial TR 2m modes shown in Figs. 4(c) and 4(d) contain all three components ur, u ϕ , and uz. Notice that both of these mode families, T 0m and TR 2m , may induce cross-polarization scattering. 5,10,41,103,154,[157][158][159] The presence of various displacement components and their complex spatial dependence have two immediate consequences to the Brillouin optomechanical interaction. First, various elements of the strain tensor are generated and their complex spatial profiles cause perturbations in the dielectric permittivity that varies spatially not only along the propagation direction z but also in the transverse plane (r, ϕ). For example, in the AR 01 in Fig. 4(c), uz depends on both (r, z) and thus generates two nonzero strain components: Szz, in analogy to the bulk, as well as Szr. Similarly, ur depends on both (r, z) and causes Srr, S ϕϕ , and Srz. These various strain elements must all be taken into account in the permittivity perturbation tensor, and how well they overlap with the optical modes determines the strength of photoelastic effect. A second consequence of the nature of mechanical modes in waveguides is that in addition to a local perturbation in the dielectric permittivity due to various strain fields, the existence of transverse displacement distorts or shifts the boundary shape of the waveguide. In the example of Figs. 4(c) and 4(d), the AR 01 mode leads to an increase/decrease in the rod radius while maintaining a circular cross section, while for the TR 21 modes, the displacement components exhibit a cos(2ϕ) dependence, deforming the cross section from a circular to a somewhat elliptical shape. This shift in the waveguide boundary can also be interpreted as a local and abrupt dielectric permittivity perturbation. For the AR 01 , a shift outwards of the circular rod boundary is equivalent to an increase in the refractive index from n air = 1 (air) to n glass ≈ 1.45 (glass) in the immediate surrounding region. For the TR 21 , the index increases in the regions where the core is stretched out and decreases where it contracts. A detailed discussion is provided further in this section with specific examples illustrating how both manifestations discussed here-presence of spatially varying strain fields and presence of transverse displacement-influence the overall efficiency of the Brillouin optomechanical coupling. One final important comment here is regarding selection rules 160 that determine which families of mechanical modes interact with the optical modes. In a circularly symmetric waveguide, the Brillouin optomechanical coupling coefficients, given by Eqs. (9), (13), and (14), involves azimuthal integrals over [0, 2π] of the product of three fields: (i) the pump electric field, (ii) the scattered wave electric field, and (iii) either directly the mechanical displacement field (for the MB effect) or its spatial derivatives (representing strain in the PE effect). In this case, there is a sinusoidal azimuthal dependence for both optical and mechanical modes 148,149 and we can express the azimuthal dependence of the pump, scattered, and mechanical fields as f (qpϕ), g(qsϕ), and h(qmϕ), respectively, where the functions f, g, and h are either cos or sin functions and qp, qs, and qm are integer azimuthal indices. Performing the azimuthal integral results in a necessary condition for nonzero overlap, qp − qs = qm or qp + qs = qm. For example, this explains why intramode scattering involving the fundamental optical mode HE 11 , with azimuthal index qp = qs = 1, involves only mechanical modes with azimuthal index qm = 0 or qm = 2.

B. Confined optical modes: Boundary field strength and longitudinal component
The moving boundary effect perturbs the dielectric permittivity only near the waveguide glass-air interface. Although the high index difference between glass and air causes large perturbation, its contribution to the Brillouin optomechanical interaction is only relevant if the optical mode extends significantly toward the waveguide boundary. This is the case for a small diameter, as is illustrated in Fig. 5(a) that shows the transverse |Et| = √ |Er| 2 + |E ϕ | 2 and longitudinal |Ez| field profiles (normalized by the maximum of the total field norm), for the y-polarized fundamental mode in a 1 μm diameter rod. In Fig. 5(b), the ratio |Et(a)|/|Et(0)| of the electric field strength at the waveguide boundary r = a relative to its value in the center r = 0 is plotted as a function of the waveguide diameter. Since the surface discontinuity of the normal electric field Er leads to an asymmetry in the transverse field profile, to build Fig. 5(b), we chose to evaluate |Et(a)| along the y-axis outside the surface. As the diameter is reduced, the mode expands out toward the surface and eventually its strength at the surface becomes comparable to and even larger than its value at the center. This is the regime where the moving boundary is expected to give the most significant contribution to overall Brillouin optomechanical coupling.
Another relevant point in high confinement waveguides is the role of the longitudinal electric field component, Ez, in the Brillouin scattering process. The fundamental optical mode HE 11 is not purely transverse-electric and always contains the longitudinal components Ez. Figures 5(a) and 5(b) show that for a diameter near the diffraction limit-where mode confinement is maximum-the longitudinal field Ez is comparable to the transverse field Et. The ratio max(|Ez|)/|Et(0)| reaches a maximum of ≈0.5 near the diameter of 0.7 μm and decreases too far above or below it as the mode size is significantly increased at these limits. The presence of a strong longitudinal field impacts the optomechanical coupling in waveguides in two ways: first, it is no longer enough to consider the overlap between the permittivity perturbation and the transverse electric field only, but one must also consider the contribution from the longitudinal field in the overlap integrands of Eqs. (13) and (14). Second, and particularly in backward scattering, the longitudinal field changes sign for backward-propagating mode vs forwardpropagating mode. 101 This represents a sort of mode mismatch that results in a reduction of the scattering efficiency when compared to the forward case.

C. Brillouin optomechanical coupling
To illustrate how these aspects resulting from the confinement of mechanical and optical modes impact the characteristics of Brillouin optomechanical interaction, we discuss specific examples in detail. We start with the moving boundary effect induced by mechanical modes with different transverse field profiles, covering elements of both forward and backward scattering. We then discuss the more complex photoelastic effect and finally add the contributions from the two effects to obtain the overall Brillouin optomechanical gain. Throughout this discussion, we identify regimes in which one or the other mechanism dominates and regimes in which both contributions are relevant and, combined either constructively or destructively, lead to enhanced or suppressed Brillouin optomechanical interaction, respectively.

Moving boundary effect
The optomechanical coupling in a waveguide due to the moving boundary effect, given by Eqs. (9) and (13) where we have considered both pump and scattered fields propagating in the same transverse optical mode, the ± sign represents forward and backward scattering, and the proportionality symbol is used to avoid writing the optical and mechanical normalization factors of Eq. (13). The change in sign arises due to the opposite direction of Ez for forward and backward propagating fields. The factor n 2 glass /n 2 air multiplying the term containing the normal component of the electric field Er arises due to the field discontinuity, where we chose to evaluate Er inside the boundary. For simplicity, we assume that the optical pump and scattered fields propagate in the fundamental HE 11 mode (y-polarization), for which the azimuthal dependencies are E (r ,z ) (a, ϕ) = E (r ,z ) sin(ϕ) and E ϕ (a, ϕ) = E ϕ cos(ϕ). For the mechanical modes, we consider both families illustrated in Fig. 4, the AR 0m and TR 2m , the former being circularly symmetric ur(a, ϕ) = ur and the latter varying as APL Photonics TUTORIAL scitation.org/journal/app BW AR 0m ε 0 (n 2 glass − n 2 air )πaur(  Table I. A first observation is that if the longitudinal component Ez is negligible compared to the transverse fields, such as in the bulk or far below and far above the diffraction limit in Fig. 5(b), the results are no different for forward and backward cases. For a 1 μm diameter rod, on the other hand, its impact is significant. The term |Ez| 2 /(n 2 glass /n 2 air |Er| 2 + |E ϕ | 2 ) is 0.46, and this factor alone means that the term in parentheses for the AR 0m is 2.7 times larger for forward than for backward. For the TR 2m , |Ez| 2 /(n 2 glass /n 2 air |Er| 2 − |E ϕ | 2 ) = 2, and the term in parentheses is 3 times larger in absolute value for forward than for backward.
In Figs. 6(a) and 6(b), we show the results of the moving boundary overlap integral for different mechanical modes for the case of forward scattering. Broadly speaking, we can see that for all modes the overlap integrals increase (in absolute value) as the diameter is reduced until the diffraction limit is reached, and beyond it, the optical mode expands again, reducing the overall intensity. The overlap integral for the AR 02 and TR 21 modes is represented by the blue solid lines (dashed lines for higher order modes) in Fig. 6; note that the AR 01 mode frequency approaches zero at cutoff (see Fig. 4), and therefore, we chose to plot the AR 02 in Fig. 6. It is not surprising that the moving boundary contribution is larger for the symmetrical AR 02 than for the TR 21 , simply because in the former the entire waveguide boundary shifts outwards (taking positive ur) causing a positive permittivity perturbation along the complete rod circumference. In the case of TR 21 , on the other hand, the boundary shifts partially inwards and partially outwards, reducing the net result. It may be surprising that in this case the result is not identically zero, since as the mechanical displacement varies azimuthally with cos(2ϕ) and its integral over [0, 2π] is zero. Of course, the integral  Table I was zero. The reason this is not the case is a consequence of (i) the field discontinuity at the high-contrast interface, which makes the boundary integral somewhat unbalanced by the term n 2 glass /n 2 air , and (ii) the presence of a strong longitudinal field. Of course, if we make n glass → n air , all terms in Table I tend to zero as the boundary index difference is reduced. However, the moving boundary contribution tends to zero faster for the TR mode relative to AR. This is because moving toward a weakly guidance approximation, the unbalance caused by the field discontinuity tends to disappear as n 2 glass /n 2 air → 1 and |E ϕ | 2 → |Er| 2 .
Another instructive example is the moving boundary coupling induced by the fundamental flexural mechanical mode 161 F 11 , as illustrated in Fig. 7(a). For this mode, ur(r = a, ϕ) = urcos(ϕ), and so its perturbation is antisymmetric with respect to the vertical axis in Fig. 7(a). We analyze two cases, intramode scattering HE 11 → HE 11 and intermode scattering HE 11 → HE 21 ; in all cases, the modes are y-polarized. It is intuitive that an antisymmetric perturbation should not cause coupling between two symmetric modes, and it is, in fact, the result observed as the line integral ∫ f wg mb dl is zero for both forward and backward cases. As illustrated in Fig. 7(a), a simple lateral shift of the cross section causes an increase in the permittivity on one side and a decrease in the same magnitude on the other side. Since the optical modes involved are symmetric, the overlap integral is zero (note that, differently than the TR 21 case previously discussed, here the optical field is discontinuous along the vertical axis in Fig. 7, where the perturbation is zero, and therefore has no impact on the result). On the other hand, coupling between the even fundamental HE 11 mode and the odd high order mode HE 21 is highly efficient. Mathematically, the azimuthal dependence of the HE 21 mode is E 21,(r ,z ) (r = a, ϕ) = E 21,(r ,z ) sin(2ϕ) and E 21,ϕ (r = a, ϕ) = E 21,ϕ cos(2ϕ) and so performing the integral over [0, 2π] along the rod circumference, the result is where again the ± represents the forward and backward scattering, respectively. One way to think about this problem is in terms of local mode expansion as illustrated in Fig. 7(a). The local fundamental mode of the perturbed waveguide (i.e., the shifted rod) is represented by a combination of the HE 11 and HE 21 modes in the basis of the unperturbed waveguide. The results of the overlap integral Eq. (16) for F 11 are shown in Fig. 7(b).

Photoelastic effect
The overlap integral we discussed so far for the moving boundary effect is somewhat straightforward in the sense that it involves directly the mechanical displacement. On the other hand, the overlap integral for the photoelastic effect ∫ f wg pe dA, where f wg pe is given by Eq. (14), is more complicated because one needs to first understand the strain fields caused by the mechanical displacements and then combine the different strains to calculate the permittivity perturbation tensor δεpe. This is straightforward for bulk since the longitudinal mechanical plane wave only creates one strain component Szz, and thus, all terms of the permittivity tensor are proportional to Szz. As we mentioned earlier, generally speaking, a mechanical mode may have all displacement components ur, u ϕ , and uz and each one depending not only on z through e −iβ m z but also on the transverse components (r, ϕ). Before going to more complex cases, it is instructive to treat intramode forward scattering involving both incident and scattered optical fields in the HE 11 mode and consider only the azimuthally symmetric mechanical modes AR 0m . We can then compare the results of the photoelastic perturbation with the results obtained in Sec. III C 1 for the moving boundary for the same mechanical modes. The phase-matching condition requires the mechanical mode to be at the cut-off point, i.e., βm = 0 for the forward scattering. Generally, the azimuthally symmetric mechanical modes exhibit a coupled radial (ur) and axial (uz) motion; however, at the cut-off point, one branch becomes purely radial [only ur(r) is nonzero]. In this case, there are only two nonzero strain components Srr = ∂ur/∂r and S ϕϕ = ur/r. Figures 8(a) and 8(b) show the mechanical displacement profiles as well as the strain fields for the fundamental AR 02 and second order AR 05 . The fundamental mode exhibits positive displacement in the entire cross section, and the strain fields are almost entirely positive (except for a small radial compression in the outer region). On the other hand, the AR 02 modes have one node, with positive displacement in the inner core region and negative displacement in the outer region. This leads to positive and negative strain regions.
In the presence of only Srr and Szz, the elements of the permittivity perturbation tensor are δεrr = −ε 0 n 4 glass (p 11 Srr + p 12 S ϕϕ ), δε ϕϕ = −ε 0 n 4 glass (p 12 Srr + p 11 S ϕϕ ), δεzz = −ε 0 n 4 glass (p 12 Srr + p 12 S ϕϕ ), where p 11 and p 12 are the photoelastic coefficients for silica (see Table II). Figures 8(c) and 8(d) show all three elements δεrr, δε ϕϕ , and δεzz for both modes. As the photoelastic coefficients are positive in silica, a region with positive strain (tension) experiences a decrease in the permittivity, while a region with negative strain (compression) experiences an increase. The AR 02 mode results in negative permittivity perturbation throughout the entire glass cross section; however, the AR 05 shows regions with positive and negative perturbation. These examples illustrate well that confined mechanical modes exhibit spatial variations in the transverse plane that, in turn, dictates the spatial profile of the permittivity perturbation. One might expect that in the AR 05 case, these regions of positive and negative perturbation might result in a much lower average perturbation for the AR 02 mode. Although this intuition is correct, the exact result still depends on the spatial overlap between the perturbation profile and the optical mode. This overlap provides a sort of weighting for the positive and negative contributions. In cylindrical coordinates, the PE overlap integral for azimuthal symmetric mechanical modes at the cutoff is

APL Photonics
All terms in Eq. (17) are dependent on the transverse coordinate, i.e., Ei = Ei(r, ϕ) for i = r, ϕ and z and δεi = δεi(r) for i = rr, ϕϕ, and zz. Figures 8(e) and 8(f) show the spatial profile of the integrand in Eq. (17) for both modes. As expected, the AR 02 mode leads to negative permittivity perturbation throughout the entire rod. For the AR 05 mode, the optical mode being more intense in the center of the rod somewhat favors the negative perturbation region over the positive one, but still a significant positive outer region reduces the net photoelastic effect. The final results of the photoelastic overlap integral for the AR 02 and AR 05 modes are shown in Fig. 6(a) (solid and dashed red lines, respectively). For completeness, the results for the TR 21 and TR 23 modes are also plotted in Fig. 6(b). Similar to the moving boundary case, the photoelastic overlap integrals increase (in absolute value) as the diameter is reduced until the diffraction limit and decreases beyond it. As expected, the presence of a node in the AR 05 mode results in a much weaker photoelastic contribution relative to AR 02 . Having discussed individually the contributions from moving boundary and photoelastic effects, we can now compare the results shown in Figs. 6(a) and 6(b) for forward scattering. As mentioned previously, the moving boundary contribution for azimuthal symmetric AR 0m modes is always positive. On the other hand, the photoelastic is mostly negative for both modes. The total representing the direct sum of the overlap integrals show that the moving boundary dominates for AR 05 . For the AR 02 , the photoelastic dominates for larger diameters while the moving boundary dominates for smaller diameters. Interestingly, one can see that for a diameter of 0.72 μm, the total perturbation is zero as the negative photoelastic contributions exactly cancel out the positive moving boundary contribution. We experimentally demonstrated this Brillouin selfcancellation effect for the case of backward scattering 74 in which the same mechanical wave causes two opposing photoelastic and moving boundary perturbations that cancel out each other. The numerical results show here that this cancellation also occurs for forward scattering. The results for the TR 21 and TR 23 modes are shown in Fig. 6(b). For TR 21 , the two effects add constructively, while for TR 23 , destructively, including a point of Brillouin self-cancellation around 1 μm. To conclude this discussion on forward scattering, a map of the total intramode HE 11 → HE 11 forward Brillouin gain spectrum (GB/Qm) as a function of the rod diameter is shown in Fig. 6(c). Clearly, only the AR 0,m and TR 2,m modes appear in the spectrum (only mechanical modes with azimuthal index 0 or 2 contribute to HE 11 → HE 11 ). Similarly, a map of the GB/Qm for intermode HE 11 → HE 21 forward Brillouin scattering induced by flexural modes is shown in Fig. 7(c), where only F 1,m and F 3,m modes appear due to selection rules. In both cases, gains on the order of GB/Qm ≈ 10 −2 W −1 m −1 are observed.
Turning to backward scattering, phase-matching determines that the mechanical propagation constant should be βm = 2βp and thus represents a propagating mechanical mode (i.e., not at the cutoff). For βm ≠ 0, we no longer can decouple ur and uz for AR 0m modes, as was the case for intramode forward scattering. This is an interesting example because ur and uz oscillate out of phase. In other words, an axial compression is usually accompanied by a radial expansion (and vice versa), as shown in Fig. 9(a) for the AR 01 mode. One can use intuition to expect that these out-of-phase components may lead to opposite photoelastic contributions. The dominant strain fields and their combination to obtain the permittivity perturbation δεrr are shown in Fig. 9(b). We can observe clearly that the contributions from the transverse strain components Srr and S ϕϕ are mostly opposite to the axial strain Szz. Specifically, note that there is an axial compression (negative strain) in most of the inner core region, but also tension (positive strain) in the outer region. On the other hand, both transverse components are positive (expansion) in the central region, although a bit more complicated at the border. The combination p 11 Srr + p 12 S ϕϕ + p 12 Szz results in an effective expansion in the inner core region, leading to negative permittivity perturbation, and an effective compression in the outer region, leading to positive perturbation. There are two interesting scenarios to discuss. First, depending on the weighing provided by the overlap with the optical mode, the positive region can indeed exactly cancel out the negative perturbation and the photoelastic contribution becomes zero. For a silica glass rod, this occurs exactly at 0.51 μm diameter as is shown in Fig. 9(c). The second interesting aspect is that the overlap weighting may favor either the positive or negative regions, and the net photoelastic effect can either reinforce the moving boundary effect or oppose it, as shown in Fig. 9(c). In the latter, it is possible that the negative net photoelastic perturbation exactly compensates for the positive moving boundary effect, which occurs for the AR 01 mode at a diameter of 1.1 μm, leading to the Brillouin self-cancellation effect for backward scattering. A map of the overall backward Brillouin gain spectrum as a function of the rod diameter is shown in Fig. 9(d).
As a final instructive aspect, we discuss how the mechanical mode hybridization, i.e., coupling between purely longitudinal and purely shear waves, impacts the Brillouin optomechanical coupling. The analytical solution for the axial displacement component for azimuthally symmetric modes can be written as the sum of two terms, uz = c 1 J 0 (k l r) + c 2 J 0 (ksr). The first term represents the contribution from the longitudinal wave to the mode structure and the second from the transverse (shear) wave. Here, c 1 and c 2 are constants, and k l = √ (Ω/V l ) 2 − β 2 m and ks = √ (Ω/Vt) 2 − β 2 m are the transverse wavevectors for the longitudinal and shear components, respectively. Figure 10 shows the plots of uz for three specific modes indicated as circles in the Brillouin gain map of Fig. 9(d), AR 02 , AR 03 , and AR 04 . These modes have their respective phasematching frequencies below, almost at, and above the dispersion APL Photonics TUTORIAL scitation.org/journal/app curve for a purely longitudinal wave in the bulk [dashed black curve in Fig. 9(d)]. These positions basically determine the character of the mode, in other words, the relative contributions of the longitudinal and shear terms to the total displacement field. This can be seen in Fig. 10, where the blue and red curves represent the longitudinal and shear components, respectively, which added together gives uz (black curve). The AR 02 sits below the bulk longitudinal dispersion curve, which means its transverse wavevector k l is imaginary while ks is real and quite large. As a result, uz exhibits slow and fast oscillating components in the radial direction. The total mechanical displacement is predominantly given by the shear component and its transverse oscillations reduce the photoelastic effect. On the other hand, the AR 03 mode sits slightly above the bulk longitudinal dispersion curve, and its transverse longitudinal wavevector k l is real and very small (its shear component is also real, but quite large). This leads to a mode that, although having both longitudinal and shear components of about the same magnitude, its longitudinal term has practically no transverse oscillation (almost like a plane bulk longitudinal wave!) and therefore exhibits a quite strong photoelastic effect. This is the reason why the mode with a maximum Brillouin gain at a given diameter in the map of Fig. 9(c) is always following the longitudinal plane wave dispersion relation. The next AR 04 mode has large k l and ks, and thus, both longitudinal and shear terms oscillate, thus reducing the photoelastic effect.

IV. BRILLOUIN COUPLING IN AXISYMMETRIC MICROCAVITIES
An insight into Brillouin scattering in optical microcavities can be drawn by analyzing the optomechanical interaction in circular symmetric cavities. In this type of cavity, light confinement is well understood using the concept of whispering gallery modes (WGMs). 162 The high orbital angular momentum of the circulating optical field concentrates light at the very edge of the circular symmetric dielectric, where it can propagate with very low radiation losses. In contrast to other light-confinement approaches, such as photonic bandgaps, WGMs can be supported by structures with rather low refractive index contrast-provided the cavity radius is large enough. In microdisk and microring cavities, the association of the whispering gallery effect and the vertical confinement provides the necessary knobs to enhance the strength of the optomechanical interaction. Indeed, this kind of device has not only been the geometry of choice of many cavity nonlinear optics experiments but also the host for landmark experiments in cavity optomechanics and Brillouin scattering. 51,105,126,163 In the context of the optomechanical interaction, the spatial distribution of optical and mechanical modes is of primary importance. 164,165 As discussed in Sec. II A, the cylindrical symmetry ensures that the optical and mechanical fields will have a propagation phase in the form exp(−imϕ + iωt), where the azimuthal number m is an integer and defines the orbital angular momentum of a given mode and, therefore, its confinement toward the cavity boundary. Important characteristics of optical and mechanical WGMs can be drawn by considering an infinite cylinder cavity.

A. Optical and mechanical radial confinement
The radial confinement of optical WGMs in an infinite dielectric cylinder is a key factor determining the Brillouin optomechanical coupling and can be easily estimated as follows. The axial optical field (Ez, Hz) radial dependence is given by Bessel functions in the form Jm(kr), where k = ncorek 0 is the radial wavevector and ncore is the cavity dielectric refractive index. If we assume that an optical mode of pth radial order has a node at the cylinder boundary (r = R), the resonant wavevectors are given by k = zm,p/R with zm,p being the pth root of Jm(z). Due to their high angular momentum, these modes will penetrate only slightly into the cylinder body. The inner radial position ri at which the modes begins to fall-off [J ′ m (kr) = 0] will be given ri = z ′ m /k; z ′ m is the first nonzero root of J ′ m (z). Under these approximations, we can estimate that the pth radial order optical mode will be confined within the radial region of width w If we employ the mapping discussed in Sec. II A for the azimuthal numbers (m → k 0 n eff R), Eq. (18) predicts w opt r = 700 nm for fundamental TM-like mode (n eff ≈ 1.8, m ≈ 35) of a R = 5 μm disk radius silicon microdisk, whereas the TE-like mode (n eff ≈ 2.8, m ≈ 56) is predicted to have w opt r = 595 nm. In Fig. 11, we show the full vectorial modes calculated for a silicon microdisk. The simple prediction of Eq. (18) for the modes radial extents are indicated and agrees well with the actual mode profiles. However, the infinite cylinder picture will not capture the full vectorial distribution of the microdisk optical modes highlighted in Figs. 11(b) and 11(c). In analogy to the silica rod, where the out-of-plane z-component of the field is large with respect to the transverse components, the out-of-plane cavity field E ϕ is also large, relative to the in-plane (Er, Ez) components, in a tightly confining microcavity. This aspect will play a crucial role in the optomechanical coupling integrand as the tangential E ϕ explicitly appears in the overlap integral of Eqs. (13) and (14). A similar approach can be evoked to estimate the radial extent of the mechanical WGMs. As in the silica rod waveguide, the coupling of the shear and longitudinal waves at the cylinder's freesurface create hybrid mechanical modes. Each wave component contributes with distinct radial wavevectors k (s,l) = Ω/V (s,l) , and the hybrid mechanical modes will have two characteristic radial extensions, where the subscripts (s, l) indicate transverse (shear) and longitudinal waves. When deriving Eq. (19), we considered that Jm(k (s,l) R) = 0 and thus that k (s,l) R is a zero of the Bessel function. Inspecting Eqs. (18) and (19), we notice that optical and mechanical modes may have distinct radial extensions depending on their azimuthal numbers (m, M). Indeed, this simple formula correctly predicts that the phase-matched mechanical modes involved in intramode forward Brillouin scattering [M = 0, see Eq. (2)] will be distributed throughout the whole cavity radial extent, i.e., w (s,l) r ≈ R; this situation is typical of cavity optomechanics experiments. 51 Such radial spread of the M = 0 mechanical modes suggests that they will induce a rather weak radial strain, a simple explanation of why the MB effect is dominant in many WGM cavity optomechanics experiments. This also explains why mechanical radial gratings are needed to confine M = 0 mechanical modes toward the edge. 78 Meanwhile, the mechanical modes that are phase-matched for intramodal backward scattering [M = 2mp, see Eq. (2)] will be strongly concentrated at the disk edge. For instance, modes lying very close to the longitudinal or shear bulk velocity lines, Ω = MV (l,s) /R will be strongly concentrated toward the disk edge (w (l,s) r ≈ 0) and should weakly interact with the optical field.

B. Silicon microdisk cavity: An insightful cavity case
Although the rough estimates above help us to understand the role of the whispering gallery effect in the optomechanical overlap, they are only toy-models to the full picture. In order to show interesting properties of more realistic cylindrical symmetry devices, we investigate the Brillouin optomechanical interaction within a single silicon disk. As highlighted in Fig. 11, this device shares many of the whispering gallery characteristics of the infinite cylinder but also the strong vertical confinement typical of high-index contrast integrated photonic devices. We further explore this device as a pedagogical example on how Brillouin optomechanics can be affected in high-index contrast microcavities.

Dispersion relation of a silicon microdisk
In order to identify the mechanical modes that can be phase matched with the optical modes in any of the Brillouin configurations described in Sec. II A, we show in Fig. 12(a)  dispersion relation of a 250 nm thick silicon-on-air microdisk cavity and some of its representative mechanical modes. Although the dispersion branches in Fig. 12(a) are quite complex, we can understand their behavior by joining the aspects of the mechanical WGMs in an infinite cylinder with the aspects of mechanical Lamb modes in a slab waveguide. There are three mode groups that can be clearly identified; they are the singleton Rayleigh edge (e-mode), dilatational (d-modes), and the whispering gallery (w-modes) and are shown in Figs. 12(d)-12(f). The singleton Rayleigh edge mode originates from the fundamental radial breathing mode (RBM) that concentrates toward the disk edge as the azimuthal number increases; this transition to edge mode is shown in Fig. 12(c). At M = 0, Eq. (19) correctly predicts the radial extent of the RBM mode, while it completely fails at large M with negative-valued predictions for the modal radial extent. The physics behind this failure is that the radial breathing mode becomes a surface mode as M increases and shear velocity would need to be replaced by the Rayleigh-wave velocity; this is also clear in Fig. 12(a) where it can be seen to lie below the shear velocity dispersion curve at low azimuthal numbers and move towards the Rayleigh velocity beyond M ≈ 20. Close to the phase-matching point for the TM-mode (M = 70), the RBM is transformed into an edge mode, as shown in Fig. 12(d). Despite the effects of vertical confinement, the e-mode dispersion has a good quantitative agreement with the cylinder surface mode, as highlighted in the zoomed-in view of Fig. 12(b) where the analytical infinite cylinder dispersion relation was overlaid with the disk's. Just above the shear-velocity line, there is also a fantastic agreement with the cylinder dispersion relation, and they are WGM-like (w-modes), shown in Fig. 12(e), with quasi pure radial-azimuthal (ur, u ϕ ) displacement profiles that are very uniform along the z-direction. The predictions of Eq. (19) for the radial extent of w-modes are very precise, as shown by the double-arrows in Fig. 12(e).
Finally, above the slab Lamb-mode dispersion relation (L 1 ), dilatational modes with large vertical displacement (uz) appear. These modes do not exist in an infinite cylinder, but resemble dilatational Lamb modes of a slab. The corresponding slab waveguide used in Figs. 13(a) and 13(b) has the same thickness of the disk and a mapped longitudinal wavevector βm = M/R. The first order dilatational mode in Fig. 12(f) is concentrated at the disk edge and move inwards as the frequency increases. The radial extent of these modes can also be predicted by Eq. (19), provided that the mechanical wave velocity is replaced the slab mode velocity-similar to the WGMs modes, dilatational modes lying near the L 1 -mode dispersion branch will be closer to the disk edge. This general behavior of modal spatial distribution will be crucial to understand the Brillouin optomechanical coupling which will be discussed next.

Brillouin optomechanical coupling in a silicon microdisk
In the intramodal backward scattering, the phase-matching condition requires M = 2mp; therefore, the modes tend to localize toward the edge. In Fig. 13, we show the calculated optomechanical coupling, as given by Eq. (11) for both the TE and TM optical modes shown in Fig. 11. The general behavior of the Brillouin active modes can be better visualized as the cavity thickness is varied, as shown in Figs. 13(a) and 13(b); we highlight the mode groups in Figs. 13(c) and 13(d) following the convention introduced in the previous paragraph (see Fig. 12). Despite the completely distinct geometry, many aspects of these coupling maps still resemble those observed in the silica cylinder waveguide [see Fig. 9(d)]. For instance, we can also roughly predict the position of the shear-like and longitudinal-like modes by overlaying the phase-matching relation Ω = 2k 0 n eff V (s,l) to these maps. Around these two curves, represented by white- dashed   FIG. 13. Optomechanical coupling in a silicon microdisk cavity. Optomechanical coupling rate for (a) TE and (b) TM optical modes as a function of the cavity thickness. In these log-scale maps, each mechanical mode was attributed a Lorentzian profile with a quality factor of 10 3 for aesthetic purposes. The dashed white (black) lines correspond to the expected phase-matching for shear (longitudinal) waves. [(c) and (d)] Optomechanical coupling rate (black bars) between the optical mode and the mechanical even modes [with respect to z = 0 plane in Fig. 11(a)] generated by the photoelastic g 0,pe /2π (red) and moving boundary g 0,mb /2π (green) contributions; (c) is for TE-mode, while (d) is for TM-mode, both for a 250 nm thick silicon disk; only modes with coupling rates g 0 /2π > 1 kHz are shown. In (c) and (d), the PE contribution is arbitrarily chosen to be positive.

TUTORIAL
scitation.org/journal/app (Vs) and black-dashed (V l ) lines, lie the mode groups with strongest optomechanical coupling. Below the shear-line dispersion relation, there is the lowest frequency edge mode [ Fig. 13(d)], which can be compared to the fundamental axial-radial mode in a cylinder that also becomes a surface mode at large wavenumbers. 75 The dilatational modes [ Fig. 12(f)] appear just above the shear-line, which is consistent with their dominantly transverse (shear) (uz ≫ u ϕ ) components. Toward higher frequencies, the trend observed for the whispering-gallery-like modes [ Fig. 12(e)] follows the longitudinal velocity phase-matching curve, showed in dashed-black. A sequence of avoided crossings, similar to the ones observed for the silica rod in Fig. 9(d), is a signature of the scattering being dominated by mechanical modes with strong shear-longitudinal hybridization. This sequence of avoided crossings reflects the mechanical dispersion relation where longitudinal and shear waves couple at free-boundaries of the cavity. The differences between the TE and TM dispersive properties of Figs. 13(a) and 13(b) are mostly due to the distinct optical dispersion relation of these two modes, the TM being steeper at lower thicknesses due its stronger vertical confinement.
In order to get a more quantitative insight on how the two optical modes actually couple to the various families of mechanical modes in a disk cavity, we investigate the components of photoelastic and moving boundary perturbations leading to the results shown in Fig. 13. As expected, due its edge localization, the emode barely couples to the TE or TM optical mode, despite of being the strongest mode at forward scattering (radial-breathing mode). The first remarkable feature comes with the dilatational (d-modes) as we may expect very strong coupling based on their vertical-breathing profile [ Fig. 12(f)]. However, when interacting either with TE or TM optical modes, they suffer from the complex interplay between photoelastic and moving boundary effects. In the TE case [ Fig. 13(a)], we notice that only the moving boundary effect participates in the overall coupling, whereas in the TM case [ Fig. 13(b)], both effects contribute, but with opposite signs. Indeed, at a thickness t = 360 nm, the fundamental dilatational mode d 1 suffers from Brillouin self-cancellation (BSC), as indicated in Fig. 13(b).
Since this kind of behavior can completely suppress Brillouin scattering in a given structure, it is instructive to look at all the individual components of the overlap integrals [Eqs. (13) and (14)] contributing to the net optomechanical coupling. We show in Fig. 14(a) the dominant strain components of the d 2 -mode (at 16.93 GHz); although this is the phase-matching frequency for the TM-mode, the mode has a similar profile for the TE phase-matching frequency at 20.54 GHz. This mode has not only a large vertical strain Szz but also comparable azimuthal S ϕϕ and shear Sr ϕ components, as shown in Fig. 14(a). While strain is an important figure, the actual optomechanical coupling still depends on the dielectric permittivity overlap with the optical fields. In order to understand how these components influence the coupling, it is instructive to decompose the PE coupling into components proportional to the tensor elements of the permittivity perturbation δε ij in Eq. (14), where ij = (rr, zz, ϕϕ, ϕz, rz, rϕ). We show the relative contribution of each photoelastic coupling component, g ij 0,pe /g 0 of Eq. (14) in Figs. 14(b1) and 14(b2). For the TE mode, the components with significant contribution are g rr 0,pe , g rϕ 0,pe . Using the backward propagating field relation for the backscattered field, 101 (Es,r, Es, ϕ , Es,z) = (Ep,r, −Ep, ϕ , Ep,z), we can write the following expression for the Stokes scattering coupling: g 0,pe ≈ g rr 0,pe + g rϕ 0,pe ∝ ∫ |Ep,r| 2 δε * rr − E * p,r E p,ϕ δε * rϕ dV, where the minus sign multiplying the azimuthal components is responsible for roughly canceling the photoelastic contribution, as shown in the spatial profile of the total photoelastic contribution in Fig. 14(c). Given this photoelastic effect self-cancellation, the moving boundary term is free to dominate the dilatational mode response, yielding a net coupling of roughly g 0 /2π ≈ 55 kHz for the TE mode. The equivalent waveguide gain obtained using Eq. (12) is It is also instructive to decompose the MB coupling, one inplane surface-normal contribution g 0,mb and two surface-tangential contributions, one in-plane g ∥ rz 0,mb and one out-of-plane g ∥ ϕ 0,mb involving only the azimuthal optical fields (E ϕ ). The relative moving boundary coupling components are shown in Figs. 14(d1) and 14(d2). As can be noticed in Fig. 14(d1), the moving boundary density contribution for the TE mode is essentially given by the tangential in-plane (∥rz) contributions, where δε mb = ε air − ε Si is the dielectric permittivity difference between air and silicon, and uz is the vertical displacement component evaluated the upper or lower disk boundaries. For the mechanical modal phase chosen in Fig. 14 perceives a lower effective refractive index as the width shrinks, leading to an increase in the optical frequency. In the TM mode case, the situation is quite different, and the net photoelastic contribution is dominated by two components of the same sign, g zz pe and g ϕϕ pe . However, the optical field concentration along the upper and lower boundaries also leads to a very strong negative moving boundary component, which compensates the photoelastic contribution. The physical reason of the negative moving boundary component is the strong azimuthal field along the boundaries, which has a negative sign due to the backscattering configuration, This competition results in rather small coupling of only g 0 /2π ≈ 27 kHz (GB/Qm = 8.6 × 10 −2 W −1 m −1 ). The analysis above further reinforces that the intricate interplay between all the photoelastic components may create unexpected cancellation effects, which could ultimately prevent the optomechanical interaction.
Finally, the mode branches with highest optomechanical coupling are the whispering-gallery modes (w-modes), peaking g 0 /2π ≈ 66 kHz at 24.5 GHz (GB/Qm = 3.4 × 10 −1 W −1 m −1 ) for the TM optical mode. Although the low transverse order w-modes lying close to the shear-velocity curve in Fig. 12(b) have only ur, u ϕ displacement components, at higher frequencies-above the slab Lamb mode (L 1 ) dispersion curve-they exhibit a significant vertical displacement component (uz). Nevertheless, their ur, u ϕ components can still be quantitatively compared to the true whispering gallery modes of an infinite cylinder, as shown in Fig. 15(a). The almost perfect agreement between the S ϕϕ strain in the two geometries allows us to explore the analytical expression for the modal displacement and understand how an oscillatory strain field can lead to such high optomechanical coupling. The azimuthal strain is given by S ϕϕ = ur/r − iMu ϕ /r, by breaking up ur, u ϕ into their shear and longitudinal components u (r,ϕ) = u  Fig. 15(b), extends deep into the cylinder, w s r ≈ 2750 nm [as predicted by Eq. (19)], but its fast radial oscillation (1/ks ≪ w opt r ) averages out its coupling contribution. The purely longitudinal component, represented by the green-dashed line in Fig. 15(b), does not oscillate significantly and has a shorter penetration w l r ≈ 610 nm predicted by Eq. (19). This is the physical origin of the "bump" seen in the azimuthal strain near the cavity edge, which can also be seen in the strain density maps of Fig. 15(a). This reasoning is confirmed in the relative contribution to the photoelastic optomechanical coupling shown in Fig. 15(c), which confirms that gpe ≈ g ϕϕ pe for the w-modes. The role of the uz displacement component was actually to slightly reduce the photoelastic coupling, through negative g ϕz pe , g zz pe , as shown in Fig. 15(c). As for the moving boundary relative contribution, shown in Fig. 15(e), although the longitudinal wave bump contributes to give a tiny tangential in-plane g and dominance of the g rr pe , g rϕ pe photoelastic components; in addition, the moving boundary is slightly enhanced by the tangential in-plane g ∥ rz mb contribution caused by the dominant radial optical field. Overall, the dominance of the azimuthal displacement contribution for the photoelastic effect in cavity w-modes is analogous to what has been discussed in a silica glass waveguide: the photoelastic effect for the mechanical modes lying close the longitudinal wave phase-matching curve (Ω = 2k 0 n eff V l ) is dominated by the out-of-plane displacement component-typical of longitudinal (pressurelike) mechanical modes.

C. Backward Brillouin scattering in other microcavities
Although we focused our microcavity discussion on a silicon microdisk device, we highlighted the structural similarities between the microcavity scattering and the simple silica rod of Sec. III. To further emphasize that the general structure of Brillouin scattering is shared among a variety of structures, we briefly discuss in this section two other popular suspended whispering gallery microcavities, a silica microdisk 168 and a silica microtoroid. The optomechanical coupling evolution, with respect to the disk thickness or toroid minor diameter, is shown in Fig. 16. Both resemble many aspects already discussed in backward scattering within the silica rod waveguide (Sec. III) and the silicon microdisk (Sec. IV). Lower frequency branches appear close to the shear-velocity dispersion relation, which tend to concentrate around the dielectric edge, and a higher frequency branch tends to concentrate around the longitudinal velocity dispersion. While the former exhibit a reasonable-or even dominant-contribution from the moving boundary effect, the latter couples essentially due to photoelasticity. In the silica microdisk, shown in Figs. 16(a1), 16(a2), 16(b), and 16(c), the coupling dependence on the thickness is remarkably similar to the silicon microdisk case of Fig. 13. In the silica microtoroid, shown in Figs. 16(d1), 16(d2), 16(e), 16(f1), and 16(f2), we restricted to analyze minor diameters larger than 2 μm, closer to experimentally demonstrated devices. 77 We verified that the mechanical modes responsible for the scattering peaks around 11 GHz originate from higher order axial-radial modes (similar to those in the silica rod) that were shifted toward the disk edge due their high orbital angular momentum (M). We show in Fig. 16(d2) the mechanical mode profile with largest coupling for a real toroid and an idealized one, where only the circular toroidal region was simulated. Not only there is a remarkable agreement between the two profiles but also the actual coupling spectra, shown in Figs. 16(f1) and 16(f2), are very similar. Yet, the real toroid exhibits a slightly smaller coupling as the mechanical mode hybridizes with displacement along the supporting disk that increases the mode effective mass. As shown in the line-cut plots of Fig. 16(d2), similar to the silicon disk whispering gallery modes [ Fig. 15(a)], on top of the oscillatory azimuthal displacement, the dominant mechanical mode has a slowly varying component due to the longitudinal velocity modal contribution superposed with an oscillatory shear-component. Like in the silicon w-modes of Fig. 15, it is the slowly varying longitudinal component that ensures a net optomechanical coupling. For the sake of shorter computational times, the minor diameter dependence of the optomechanical coupling, shown in Fig. 16(e), was calculated using the idealized toroid. The minor diameter and its range, calculated in Fig. 16(e), are much larger than the microdisk thickness, which is the major reason why it differs from the microdisk coupling map of Fig. 16(b). The larger toroid cross section (shorter frequency interval between mechanical modes) causes the sequence of avoided crossings to appear more frequently in the toroid case. We emphasize that the experimental observation of backward Brillouin scattering within a microtoroid has been recently reported, 77 although the mechanical modes shown here were not identified.
The remarkable similarities of the Brillouin backscattering between the silicon microdisk and microtoroid cavities confirm our initial assertion of the general properties of Brillouin optomechanical coupling in axisymmetric cavities.

D. Bullseye cavity
In forward intramode Brillouin scattering on axisymmetric cavities, the phase match condition is satisfied for M = 0 [see Eq. (2)] and leads to a mechanical mode that extends over the entire disk structure, as predicted by Eq. (19). Therefore, initial experiments 163 used the natural confinement of tiny posts on circular and diskshaped cavities to prevent coupling of the mechanical modes to the necessary bulky and lossy substrates. Nonetheless, optomechanical experiments using many types and shapes of optical and mechanical cavities have been carried out in a large span of applications, from high precision measurements of acceleration 59 and displacement 89 to measurement and generation of nonclassical mechanical states. 62,63,65 Most of these experiments benefit from the simultaneous control of the optical and mechanical fields. In what follows, we will discuss one type of optomechanical cavity, the bullseye optomechanical cavity, 78 which confines optical and mechanical modes through distinct physical mechanisms and allows for independent control of the mechanical and optical properties. As in Sec. IV A, the radial confinement of the optical mode can be understood through WGMs, while the mechanical confinement is achieved by defining an elastic grating along the radial direction in order to create a phononic quasibandgap.
Due to these two edge confinement mechanisms, the bullseye resonator can be understood by analyzing a floating ring structure that is analogous to a bent rectangular waveguide in many aspects. Due to the large refractive index of silicon at 1550 nm, the floating ring structure can have lateral and thickness dimensions of a few hundred nanometers. Like in the silica nanorod, the strong confinement for the optical and elastic waves also increases the photoelastic and moving boundary perturbations. Figure 17(a) shows the dispersion relation for all the mechanical modes that satisfies the intramodal forward scattering (M = 0) phase-matching condition of Eq. (2). For each w ring , we calculate all the mechanical modes that couple to the TE optical mode due to the PE and MB perturbations using Eq. (11). The corresponding spectra are normalized such that its peak height is proportional to the total optomechanical coupling rate. Clearly only one mode family, the breathing mode, is dominant in this case. The breathing mode frequency can be estimated assuming that the mechanical wave is confined along the ring width with the velocity of longitudinal waves V l in bulk silicon. Multiple orders of breathing mode are given by Ωp = pV l /(2w ring ), where V l = 9660 m/s and p is an integer representing the mode order. The dashed white lines in Fig. 17(a) represent these estimated mechanical frequency dispersion curves as a function of the ring width for each mode order p. The correct design of the ring and mechanical mode order can be used to either enhance the optomechanical coupling or for self-canceling it. Both situations are analyzed in Figs. 17(b)-17(k). The enhancement of the optomechanical coupling rate between the first order mechanical breathing mode and

TUTORIAL
scitation.org/journal/app the TE optical mode is due to the fact that both PE and MB contributions have the same sign for w ring smaller than 750 nm. The large optomechanical coupling rate is achieved in this case since the largest strain component (Srr) has almost unit overlap with the optical mode. On the other hand, self-cancellation is achieved for the coupling between the third order breathing mode and the TE mode around w ring ≈ 1380 nm, which can be noticed in Fig. 17(a). In this case, like in the silica rod, there is a competition between the photoelastic term (related once again with Srr) and the moving boundary term, as shown in Figs. 17(j) and 17(k). The boundary and volume integrands have exactly opposite phase for this geometry, thus causing the self-cancellation effect. Despite being a very intuitive geometry, the floating ring structure is not feasible. In addition, as we already mentioned, a simple disk is not sufficient to confine the elastic waves towards the disk edge in the intramodal forward scattering regime. Santos et al. 78 discussed in detail a strategy to confine the mechanical waves toward the disk edge using a bullseye type structure. This strategy is very successful and is represented in Fig. 17(l), where both the dispersion relation of the bullseye disk mechanical modes (red and blue shades) and the mechanical density of states (MDOS, gray-shaded) of a 1D-grating are overlaid. Darker gray regions are to be interpreted as higher MDOS within the mechanical grating. The red shaded dispersion relation is for w ring = 500 nm, while the blue shaded dispersion relations are for a w ring = 1000 nm. In both cases, the darker colors represent larger optomechanical coupling (white means zero optomechanical coupling). In both cases, we notice that the existence of the quasibandgap (mechanical frequency range with very high reflectivity) not only confines the mechanical modes toward the disk edge but, by doing so, also increases the optomechanical coupling due to the larger overlap between the δεrr and the optical mode. In Fig. 17(m), we also show the spatial profiles of the major photoelastic and moving boundary components of the radial breathinglike mechanical mode in the full bullseye structure. We notice that despite of the symmetry break in the z-direction (the grating struts are along the bottom part of the disk), which couples the radial ur and vertical uz displacement, there is no noticeable difference regarding the origin of the optomechanical coupling between the bullseye disk and the floating ring. These analyses could be easily adapted for other crystalline and compound materials that could be used to form bullseye cavities.

V. PERSPECTIVES AND CONCLUSIONS
Despite remarkable progress in formalizing and unifying the theoretical basis underlying the Brillouin optomechanical interaction in nanophotonics, fundamental and technical aspects are yet to be fully developed, with many opportunities for novel research in this field. In the context of this tutorial, which focused on the two main fundamental interaction mechanisms and the ability to design structures to control both mechanical and optical waves at the nanoscale, there are several new developments still in their early days. In analogy to many unique properties obtained in optical metamaterials, as an example, mechanical metamaterials may offer opportunities to achieve novel properties not available in constituent bulk materials. [169][170][171] For example, mechanical metamaterials with photoelasticity enhanced beyond that of its constituent materials or metamaterials with complete suppression of photoelasticity. As in optics, mechanical metamaterials may also offer the ability to further design the dispersion relation of mechanical waves.
Anisotropy is also a topic that we barely touched in this tutorial; however, recent progress in using materials such as LiNBO 3 , [172][173][174][175] AlN, [176][177][178] diamond, 37,179,180 GaAs, 181 and ionic crystals (e.g., CaF 2 ) for optomechanics shows that new design and fundamental challenges are open. Optical anisotropy could significantly modify the photoelastic effect 146 as well as backward effects (forces and torques) accompanying it. Within optical cavities, optical or mechanical anisotropy adds significant constraints in the optomechanical design, for instance, mechanical anisotropy in silicon hazes otherwise simple designs. 78 As a consequence of a lower crystal symmetry, anisotropy may also be accompanied by other physical effects, such as electro-optic and piezoelectric effects, and enable full electro-optical-mechanical interaction, 90,93,[182][183][184] which shares many of the physical similarities of the discussed devices, displaying both electrostrictive and radiation pressure contributions. Incorporating other nonlinear effects present in semiconductor materials, such as free-carrier absorption, photothermal and photorefractive effects have been pursued theoretically; 185,186 however their possible mitigation or benefit 187,188 for the Brillouin optomechanical interaction is still not fully developed. Despite the large knowledge-base of this field due to technological relevance of electro-optic and acousto-optic modulators, developing effective devices at the subwavelength scale to enable large Brillouin coupling is still underway.
Beyond just the material itself, taking full advantage of periodic structures such as photonic and phononic crystals waveguides and cavities is another area of intense development, both experimentally and theoretically. 45,46,56,57,66,102,103,189 One key aspect of these structures is that they may lead to full confinement of both optical and mechanical defect modes as well as offer mechanisms to mitigate dissipation. In particular, mechanical dissipation 190 is still hindered by the large number of mechanical modes and their inherent coupling caused by geometrical imperfections or residual stress in fabricated structures. Along the same lines, opportunities for designing defect-robust structures-such as by exploring mechanical topological insulators-is an exciting new area. [191][192][193] A whole new quantum horizon for Brillouin optomechanics may arise when feasible device designs and improvement on microfabrication processes enable experimental demonstrations within the strong couple interaction limit, at which light and sound hybridize to form a coupled entity, 90 with initial works focusing on reducing optical 194 and mechanical dissipation 183,[195][196][197] or enhancement of the effective coupling strength. 87,194,198 Finally, multimode Brillouin optomechanical interaction 64,199 can lead to exquisite results where interference between multiple mechanical modes coupled to an optical mode can mitigate extrinsic loss mechanisms, 200 extending optical delay through electromagnetically induced transparency 201 and creating highly coherent mechanical modes. 202 In its simpler form, where distinct optical modes couples a single mechanical mode, 203 cooling and amplification in continuous media 204 can be explored and mediate coupling between distinct optical and mechanical modes. 205

TUTORIAL scitation.org/journal/app
Although we described the Brillouin interaction using energy arguments-instead of force densities-it is worth emphasizing how such force densities depend on the electromagnetic stresstensor and momentum density, and their impact in the Brillouin optomechanical coupling described in this tutorial. These choices have been at the core of the century-old Abraham-Minkowski controversy, [206][207][208] and while major progress has been made on this controversy, 209-211 some aspects of the debate are still alive [212][213][214] and it was not the goal of this tutorial to resolve them. In the context of cavity optomechanics, for example, there are several articles that derive the optical force, which acts upon the cavity mechanical degree of freedom, following the convention 51,215,216 of photon momentum given by ̵ hk (k being the wavevector within the dielectric defining the optical cavity). Although not explicitly mentioned in most articles, this convention is consistent with the Minkowski approach 217 and ensures a match between the force calculated using a momentum picture 217 and the more widespread energy picture, which uses a virtual-work argument to derive the optical force for an adiabatic (slowly varying) mechanical degree of freedom. 217-221 Those descriptions and their success in predicting experimental results (such as optical spring effect and cooling 89 ) may suggest that the optomechanics community has taken the Minkowski side. However, it is worth mentioning Nelson's perspective 222 where he argues that although the light momentum is partially carried by the vacuum and material response fields, it is the sum of both-coined by him as wave momentum-that is given by ̵ hk per photon in the quantum regime. It is also mentioned that apart from material dispersion, the wave momentum represents the Minkowski one, a perspective that has been previously pointed out by Van Laer. 223 Finally, from a macroscopic electrodynamics perspective, the difference between the force densities calculated using either Abraham or Minkowski formalisms should differ by a time-oscillatory term 208,224,225 (known as the Abraham force) that averages out with optical monochromatic fields. It is precisely this term that explicitly appears in Wolff's coupled-mode theory derivation 96 as the "moving polarization" effect. Similar to the history of the moving boundary perturbation, first proposed by Johnson, 100 and its connection to radiation pressure boundary forces, the discovery of the "moving polarization" term as an additional force strengthens the idea that approaching the problem from different perspectives may grasp us into new physics; however, due to its scaling as the ratio of mechanical to optical frequencies, 96 it has yet to find an experiment that could probe it.
In summary, we discussed in detail how the photoelastic and moving boundary effects interplay to enhance or suppress the Brillouin optomechanical interaction in nanophotonic structures. Its general structure in both forward and backward scattering configurations was discussed based on pedagogical case studies. Albeit emphasis was given to specific examples, we highlighted the underlying general structure of the Brillouin coupling. This investigation also revealed that the Brillouin self-cancellation effect is present in many structures in both forward and backward interaction and should be carefully considered in nanophotonic device design. Also our numerical simulations identified the mechanical modes responsible for backward scattering in a silica microtoroid. By providing the simulation files used in preparing this tutorial 80 -developed in a widely used commercial solver-we hope to encourage and shorten the path for students and researchers initiating their investigation in this exciting field.

APPENDIX A: COUPLED MODE EQUATIONS ANSATZ AND RELATION BETWEEN GAIN EXPRESSIONS
The expression for the waveguide Brillouin gain in Eq. (9) was derived by Wolff et al. in Ref. 96 using the coupled-wave theory formalism. Their ansatz for the waveguide slowly varying fields is such that the slowly varying envelopes are dimensionless and the optical and mechanical modal fields carry on their respective SI units. In contrast to Ref. 96, we choose to write the waveguide coupledwave equations in Eq. (7) following the normalization adopted in Refs. 69 and 90 and the cavity equations in Eq. (10) following their usual normalization; 79,89,90 these choices not only keep the optical and mechanical equations symmetric but they also provide a connection between the waveguide and cavity couplings. However, if one wants to keep the actual physical dimensions of the optical and mechanical fields, which is the case when using a commercial mode solver such as COMSOL Multiphysics, the choice in Ref. 96 where Q SBS is the total SBS coupling coefficient ([Q SBS ] = J m −1 ) and can be interpreted as work per unit length; 96 αm = γm/(2υm) is the mechanical displacement attenuation ([αm] = m −1 ) (the factor 2 is due to the connection between displacement and energy dissipation); δβ = βm − (βs − βp) is the wavevector mismatch; Pp,s,m represents the power carried in the pump, Stokes, or mechanical modes. QSBS in Eq. (A1) is calculated from the field profiles [see Eqs. (33) and (41) in Ref. 96] and is subject to whatever normalizations are carried out in the mode solver of choice; for instance, in COMSOL, the fields are normalized either through the finiteelement mass matrix or by their norm maximum. Since we wanted to compare the coupling integrals among different geometries in this tutorial, we found instructive to incorporate the normalizing powers appearing in the denominator of Eq. (A1) into the coupling expression. Furthermore, since the mechanical Poynting theorem 149 ensures the relation Pm = Emvm between the power, the linear energy density (Em), and the group velocity (vm) of the mechanical wave, (γm/2) 2 (γm/2) 2 + (Δm) 2 ( |Q SBS | 2 PsPpEm ), (A2) scitation.org/journal/app where we converted the spatial attenuation and wavevector mismatch to the dissipation rate and frequency detuning by employing αm = γm/(2υm) and Δm = Ω − Ωm = υmδβ, together with γm = Ωm/Qm. The rightmost quantity in parentheses in Eq. (A2) is independent of normalization and could be directly used to compare different structures. However, because the mechanical energy scales as the frequency squared (Em = 2Ω 2 m ∫ ρ|um| 2 dA), such a quantity would favor modes with low-frequency and hinder the actual spatial optomechanical overlap that is solely dependent on the mode field profiles. To remove this frequency dependence, we introduce the effective mass linear densitym eff = ∫ ρ|um| 2 / max(|um| 2 )dA so that where the Lorentzian lineshape defined in Eq. (8)  Finally, it important to mention that the finite-element solver employed throughout this tutorial adopts the exp(−iβz + iωt) ansatz in the RF Electromagnetic Waves and Structural Mechanics modules. This is very important as the sign of all the expressions involving the z-derivative of mechanical fields, e.g., strain tensor, depend on this convention. Therefore, the modal expansion leading to Eq. (A1) and consistent with COMSOL's ansatz is given by Fwg(r, t) =ã(z)F(rt) exp(−iβz + iωt) + c.c., (A5a) Fcav(r, t) = a(t)F(rt) exp(−imϕ + iωt) + c.c., Uwg(r, t) =b(z)u(rt) exp(−iβmz + Ωt) + c.c., (A5c) Ucav(r, t) = b(t)u(rt) exp(−iMϕ + Ωt) + c.c., where F denotes either an electric or magnetic field and F(rt) denotes their modal vector fields depending only on the transverse coordinates; U denotes the mechanical displacement field, while u(rt) is the modal vector field; a(z) and a(t) [b(z) and b(t)] are the dimensionless slowly varying optical (mechanical) field amplitudes. Since COMSOL preserves the SI units of the modal fields, the normalization employed in the discussion of the coupled-mode equations [Eq. (7)] is distinct from Eq. (A5). The advantage of COMSOL normalization is that one can keep track of the dimensions of each quantity in the gain expressions [Eqs. (9), (11), (13), and (14)].

APPENDIX B: MATERIAL PROPERTIES
In our calculations, the material properties shown in Table II were used. 227,228 Despite silicon being anisotropic and there are different directions, the purpose of this tutorial was worth considering an isotropic approximation. This particular choice of parameters does not affect the general behavior of the Brillouin optomechanical interaction discussed here. One may expect, however, that genuine  78 Nonetheless, in order to provide the reader with an example of the use of anisotropic material and to better compare our simulations with Refs. 71 and 125, we simulate the under-etched waveguide shown in Fig. 1(e) aligned along a ⟨110⟩ axis. In this case, the elasticity (c 11 , c 12 , c 44 ) = (166,64,79) GPa and photoelasticity (p 11 , p 12 , p 44 ) = (−0.09, 0.017, −0.051) matrices were rotated by π/4 around the crystal z-axis (100 direction). 149 The photoelastic contribution to the Brillouin gain has a strong dependence on the properties of several materials. In order to compare the Brillouin gain performance of different materials using Eq. (9), it is worth making a few approximations. We assume (i) intramodal scattering so N wg p = N wg s , (ii) the mechanical mode is uniform and perfectly confined to a spatial region of area Am such thatm eff = ∫ ρ|um| 2 / max |um| 2 dA ≈ ρAm. (B1) Within these approximations, the overlap integrands in Eq. (14) can also be simplified and written as wherep is an effective photoelastic constant associated with the average normalized strain fieldS = S/ max(|um|), and the relation (N wg p ) 2 = 2R(∫ Ep × H * p ⋅ zdA) ≈ 2cnε 0 |Ep| 2 was used, implicitly assuming a plane-wave-like optical mode. The numerator integral in Eq. (B2) is carried out only in the "guiding" region where the optical mode is confined. Further assuming that the optical field is uniform and distributed in the guiding region of refractive index n, Eq. (B2) simplifies to ∫ f wg pe dA ≈ −n 3pS /(2c). Using these approximations, Eq. (9) can be written as