Generation of phonons from electrostriction in small-core optical waveguides Vincent Laude1, a) 1, b)

We investigate the generation of acoustic phonons from electrostriction of optical waves in small core waveguides. We specifically consider simple step-index strip waveguides composed of silica or silicon in air, with sub-micron lateral dimensions. Such waveguides support one or a few optical modes, but a rich spectrum of acoustic phonons that becomes densely populated as the phonon frequency increases. We evaluate rigorously the phonon energy density that results from the electrostriction of two frequency detuned guided optical waves, that are either co- or contra-propagating, including phonon loss. Plotting this energy density as a function of frequency detuning reveals the phonon wave packets that are electrostrictively active and gives a quantitative estimation of the energy transfer from optical waves to particular phonons. Furthermore, in the backward interaction geometry, the dispersion relation of such phonons can be accessed directly by varying the optical wavelength.


I. INTRODUCTION
Nanostructured crystal waveguides have attracted attention recently in view of obtaining simultaneous confinement of elastic and optical waves and hence potentially strong interactions 1,2 . Optomechanical interactions involve optical surface forces such as radiation pressure 3 , which are predicted to scale to large values in nanoscale waveguides [4][5][6] . In addition, volume optical forces are induced in microscale waveguides via electrostriction, as exemplified by Brillouin scattering effects -nonlinear processes where two photons interact with a co-propagating acoustic phonon, both generating the phonon and being scattered by it 7 . This phenomenon is broadly documented for silica optical fibers, in both theory and experiment 8,9 . From the computation of phonon dispersion in optical fibers or waveguides, conventional stimulated Brillouin scattering (SBS) models determine the Brillouin gain spectrum from an elasto-optic overlap integral 10,11 . The precise dynamics of the generation of phonon wave packets (or elastic waves) by optical waves through electrostriction, however, has not been treated in detail until recently 12 .
In this paper, we consider the electrostriction of phonon wave packets from guided optical waves in a simple model system: the strip waveguide exemplified in Figure 1. The waveguide has a rectangular cross-section with width 1.5 µm and thickness 1 µm, and is composed af a single homogeneous dielectric material surrounded by air. The dimensions were chosen to ensure single mode optical guidance at a wavelength of 1.55 µm in case the waveguide is made of silica, but the case of silicon is considered as well in order to hold a comparison.
For both materials, the waveguide can be fabricated by microelectronic technologies, so that the corresponding experiments are in principle feasible. The main objective of the paper is to expose the rich dynamics of phonon generation, even within the simpliest of optical waveguides, and to obtain the energy transferred from optical waves to particular phonons and their polarization. It is found that these properties strongly depend on material properties, including the photoelastic tensor and phonon losses. Our numerical model is based on a finite element formulation of the elastodynamic wave equation subject to an arbitrary volume optical force; it is rigorous in that it naturally takes into account all anisotropic and structural details of the wave propagation problem. Significantly, there is no need to resort to a full phononic band structure computation 13 , followed by the computation of overlap integrals. In contrast, the spatial distribution of phonon wavepackets is obtained directly as a result of the applied optical force, and thus never has to be assumed to arise from a given normal elastic mode of the waveguide.
The paper is organized as follows. We first summarize the finite element elastodynamic model and then proceed to show electrostriction gain spectra for both silica and silicon strip waveguides. Such spectra show a series of Lorentzian resonances, the width of which is dictated by phonon losses, introduced in the model via a frequency-dependent viscoelastic tensor. We then proceed to discuss the effect of the dimensions of the waveguide and the differences between silica and silicon strip waveguides. The importance of the shear components of the displacements comes out of this discussion. We end by discussing the possibility of measuring phonon dispersion via all-optical means.

II. THEORY
The elastodynamic equation describing the electrostriction of acoustic phonon wavepack- with ij is the relative dielectric tensor at optical frequencies, c klij is the elastic tensor, and p klmn is the photoelastic tensor. Tensorial notations are used to keep expressions short and summation over repeated indices is implied. Indices placed after a comma indicate partial derivates are taken, e.g., u k,l = ∂u k ∂x l . Note that the transverse variations of the two incident optical waves are fully taken into account when defining the electrostriction stress tensor T es ij = − 0 χ klij E k E * l . The electrostrictive (volume) force is the divergence of the electrostriction stress tensor, T es ij,j . Brillouin-type scattering processes are third-order parametric processes involving two photons and one phonon. Phase-matching must be achieved in order for the interaction to built constructively. For instance, a pump photon with center angular frequency ω 1 and wavenumber k 1 can produce a down-shifted Stokes photon characterized by center angular frequency ω 2 and wavenumber k 2 when scattered by an acoustic phonon of frequency ω = ω 1 − ω 2 and wavenumber k = k 1 − k 2 . Electrostriction of an acoustic phonon is the converse effect and is obtained from the interaction of two incident photons satisfying the previous phase-matching conditions. In this paper, we consider wave fields containing enough photons or phonons so that a classical picture can be used. The total incident optical field is the superposition of two guided monochromatic waves with frequencies ω 1 and ω 2 , with wavevectors k 1 = k(ω 1 ) and k 2 = k(ω 2 ) satisfying a dispersion relation for guided waves. Because we are considering acoustic phonons, ω is much smaller than ω 1 or ω 2 . Two cases can be separated in the case of optical waveguides. As depicted in Fig The only source term at frequency detuning ω = ω 1 −ω 2 is proportional to exp(i(ωt−kz)) with k = k 1 (ω 1 ) − k 2 (ω 2 ). Because of this source term, the following ansatz is assumed for the acoustic waves withū i (r) the transverse profile of the elastic phonon wave packet. We solve for the unknown displacementsū i (r) by setting k to a given value and scanning the detuning frequency ω.
Many different approximation methods could be used for that purpose. Since the displacement fieldsū i can be supposed continuous everywhere in the domain of definition and the exact boundaries enclosing the solid medium can be rather arbitrary, a finite element model (FEM) is a natural choice. Our FEM is detailed in the appendix.
Once the displacement fields have been obtained, the full solution is known accurately.
We find it useful to calculate the kinetic energy of the phonon wavepacket, as defined by with ρ the material density and σ denoting the cross-section of the fiber. The elastic or potential energy is If the solution were a normal mode, then the elastic (or potential) energy would equal the kinetic energy. As the internal energy is the sum of kinetic and elastic energies, we estimate the phonon wavepacket energy as with units J/m. We thus have access to both the Brillouin spectrum and the spatial distribution of three displacement components. Forward and backward interaction configurations can be considered, as well as any value of the k-vector. Contrary to the previous Brillouin models, the electrostriction force and hence the generated acoustic phonons are obtained right where the interaction takes place, without need to resort to a full band structure computation. As a consequence, our model can be universally applied to calculate the Brillouin gain characteristics with arbitrary refractive index and material compositions. In practice, and to allow for a fair comparison of deposited phonon wavepacket energies, the optical guided waves are normalized so that the flux of the z component of the Poynting vector is unity. This flux is

III. STRIP WAVEGUIDE
We consider a strip waveguide with lateral dimensions equal to 1.5 µm by 1 µm, as depicted in Fig. 1(a). The solid core, made either of silicon (Si) or silica (SiO 2 ), is surrounded by air (n = 1). With these dimensions, the effects of radiation pressure are in principle almost negligible 6 . The different material constants used in the calculations are presented in Tab. I.  The photoelastic coefficients for silicon at 1.5 µm have been extracted from Refs. 16,17 . Here, p IJ and c IJ are the photoelastic and the elastic tensor presented in contracted notation 18 .
We first compute the optical guided waves using a 2D FEM model 19 . This loss model is compatible with the usual assumption that the Qf product is constant for a given material. The symmetry of the viscosity tensor is the same as that of the elastic tensor. Viscosity constants are hardly available in the literature for silicon and silica.
For silicon wafers, we estimated Qf = 5.10 13 Hz from independent experiments performed with bulk acoustic wave resonators. From measurements of the SBS linewidth in standard optical silica fiber, we estimated Qf = 5.10 12 Hz. So far, we have assumed that the loss factor for shear and longitudinal elastic waves is identical, but the model can consider a fully anisotropic viscosity tensor.
The elastic energy spectrum for forward and backward interaction in silica are presented in Fig. 3 and Fig. 4, and for silicon in Fig. 5 and Fig. 6. We superimpose on the elastic energy spectrum the discrete set of eigenfrequencies obtained without the optical force term, i.e.,

A. Dependence of electrostriction efficiency with size
It was a major conclusion of Ref. 6 that the gain of stimulated Brillouin scattering in silicon is strongly enhanced as the lateral size of the waveguide reduces down to approximately half a micron. We can derive from our model how the electrostriction gain scales with this size. Quite naturally, a first source of enhancement comes from the choice to normalize the incident optical fields for unit transported power. As the same power has to be carried by a smaller cross-section, indeed, the electric field magnitude scales inversely with the size of the waveguide. As a result, the optical stress density scales with the inverse of the square of the size of the waveguide, as do the induced mechanical displacements, following Eq. (1).
When the mechanical energy is integrated over the cross-section, the same square-inverse dependence is found for the electrostriction gain.
Apart from the straightforward normalization effect just discussed, it is natural to wonder if there is an implied scale-dependence in the elastodynamic equation. Let us derive the corresponding energy statement. We denote the time-derivative of u i byu i and consider a virtual displacement δu i . Multiplying the equation by δu i , integrating over a volume V , and using Green's theorem (performing integration by parts), we arrive at This relation is analog to the theorem of virtual work in mechanics, as applied to work of the electrostriction stress. It is also equivalent to Poynting's theorem for elastic waves (when i.e., when the strain distribution u i,j matches the electrostriction stress T es ij pointwise. It is thus clear that there is some scale factor maximizing the electrostriction gain 6 .

B. Material constants
It comes as a surprise that the case of silicon seems to be favored over the case of silica in terms of electrostriction gain. Brillouin scattering in silica optical fibers is broadly documented and not too difficult to observe. With the photoelastic tensor written p IJ in contracted notation, I and J being integers with values between 1 and 6, the literature generally attributes this fact to the rather large p 12 = 0.27 value, that is favorable for longitudinal acoustic waves. In contrast, we are not aware of experimental results for silicon waveguides. In large core waveguides, the electric field is very nearly transverse. In small core waveguides as we consider, this assertion is not true anymore, but transverse components of the polarization still dominate. As a result, only values J = 1, 2, or 6 have to be regarded among tensor components p IJ . For both silica and silicon, there are 3 independent sets of constants. Let us discuss them separately. waves if we were considering plane waves; as above, however, the transverse dependence of u 1 (r) and u 2 (r) generally allows coupling to axial shear elastic waves.
Third, of the set p 44 = p 55 = p 66 , only p 66 is active and creates a possibility for orthogonally polarized optical waves to generate phonons with non vanishing S 6 , corresponding to transverse shear waves only in the case of plane waves, but in addition to axial shear waves thanks to the transverse dependence of u 1 (r) and u 2 (r). Table II summarizes the polarization of the phonon wavepackets that can be excited by each independent set of photoelastic constants. In the forward case, the axial wavevector k is always zero, meaning that elastic waves propagate in the (x 1 , x 2 ) transverse plane. As a result, waves with I = 1, 2 are longitudinal (L) waves and waves with I = 6 are shear (S) waves 23 . In the backward case, k = 0, and propagation is axial. As a result, waves with I = 1, 2, 6 are S waves and waves with I = 3 are L waves.
It is interesting to notice that the transverse dependence of axial displacement u 3 never enters the picture. Actually, this would require strains S 4 or S 5 to be active, through p 44 or p 55 , but these are excluded by the transverse hypothetis for the optical electric field. For a fair comparison of silica and silicon, two additional facts have to be taken into account.
First, χ ijkl is n 4 times p ijkl , which is a clear advance for silicon. Second, the electrostriction gain is proportional to the Q-factor of the material 12 ; since phonon loss is lower in silicon than in silica, this is again a clear advantage for silicon.

C. Phonon dispersion
The fact that the frequency detuning between the two optical waves can be scanned, on the one hand, and that the acoustic wavevector k is given by the difference of optical wavevectors in the backward interaction case, on the other hand, suggest that phonon dispersion can be probed all-optically. In this case, indeed, k ≈ 2k 1 is almost independent of ω. Suppose that the frequency of a continuous wave laser can be adjusted to create a variable optical wavevector k 1 = 2πn eff (λ)/λ, and that an electro-optical modulator is used to shift the angular frequency of the second optical wave by ω. We have explicitely written n eff (λ) as a function of the wavelength, because of the dispersion of the optical waveguide.
As a result, we would have access to any point (ω, k = k 1 − k 2 ) of the dispersion diagram for phonons. This is illustrated by Fig. 7. The central panels show the phononic band structure for silicon and silica. As we commented before, these band structures are densely populated.
The corresponding elastic energy spectra at k = 0 (forward interaction) are shown in the left panels, and are essentially identical to the spectra shown in Fig. 3 and

V. CONCLUSION
We have presented an electrostriction model for the generation of acoustic phonons in optical waveguides that is valid for an arbitrary spatial dependence of the optical field.
We have identified the phonons that are exited in the forward and the backward interaction geometries and detailed their characteristics. Observing the spatial distribution of the generated phonon wave packets is of paramount importance in order to understand optoacoustic interactions under strong spatial confinement. It is found that small-core silicon waveguides seem to be hold more potential than similar silica waveguides. withT The displacement fieldsū i are assigned to live in a finite element space of Lagrange-type defined over the mesh of the waveguide cross-section. Free boundary conditions are applied.
The elastic equation formulating a variational problem on the finite element space is obtained by left-multiplying by virtual displacement (v i e −ıkz ) * and integrating over the cross-section of the waveguide.
Next Green's theorem is used to transform integrals involving a divergence operator to integrals involving the gradient of the test function The boundary integral is along δσ and n is the outward normal. The natural, or free, boundary condition isT ij n j =T es ij n j along δσ, meaning that the total normal stress vanishes on the boundary of the waveguide. Under this condition, the boundary integral vanishes.
This weak formulation of the elastodynamic equation is all that is necessary for coding with a FEM language such as FreeFem++ 25 . Let us, however, provide more insight inside the structure of the FEM equations by expliciting the dependence with k σv * i,j c ijklūk,l + ık Writing U and T es the vectors of nodal values of theū i and theT es J (in contracted notation), the usual Galerkin FEM procedure leads to the linear equations (K + ıkK 1 + k 2 K 2 − ω 2 M )U = (X 0 + ıkX 1 )T es . (A6) Each of the matrices come from one of the integrals in Eq. (A5). The total stiffness matrix, K = K + ıkK 1 + k 2 K 2 is a polynomial of degree 2 in k. The mass matrix M is independent of the wavevector, as usual. The right-hand-side matrix X = X 0 + ıkX 1 , relating the optical force to the electrostriction stress, is a polynomial of degree 1 in k. In the forward interaction case, k ≈ 0, and Eq. (A6) reduces to In the backward interaction case, Eq. (A6) never simplifies except for large core waveguides: transversal strains can be neglected in this case and as a result. This is the short wavelength approximation that describes well SBS in classical silica optical fibers 12 .