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stimulated brillouin scattering ( sbs ) is a nonlinear and self - amplifying interaction between guided optical waves and hypersound waves in waveguides  @xcite . the scattering of light from mechanical vibrations was first predicted by brillouin in 1922  @xcite ; the stimulated version of this process was first experimentally demonstrated  @xcite shortly after the invention of the laser by chiao et al . and has since been used successfully to characterize materials at hypersonic frequencies  @xcite . following the general trend of miniaturization , sbs has been studied and applied in ever smaller structures such as nano - structured optical fibres  @xcite and even waveguides integrated on a chip  @xcite . in these systems sbs ceases to be a bulk effect and surface effects strongly come into play  most prominently radiation pressure appears as a second major interaction process in addition to the bulk photoelastic effect . due to this additional coupling as well as the very tight mode confinement , high sbs - gains can be achieved in integrated waveguides  @xcite , which makes them extremely interesting for signal processing applications  @xcite as well as narrow linewidth light sources  @xcite . recently the idea of coherent phonon generation in phonon - lasers has gained considerable interest  @xcite . however , all these potential applications require or at least greatly benefit from the extremely narrow linewidth of the sbs process ; line width broadening is highly detrimental in many cases  @xcite . this motivates our investigation of the inhomogeneous line broadening due to structural variations of the waveguide and potential connections to the sbs - gain itself . the finite phonon life time ( or equivalently in backward sbs : the phonon propagation length ) defines an intrinsic line width of the sbs - resonance . however , it has been known for some time that various imperfections of the sbs - system can distort the resonance and inhomogeneously broaden it . some of these imperfections such as insufficient acoustic guidance in combination with moderate optical mode confinement  @xcite ( one possible explanation for sbs broadening in optical fibres , but somewhat debated in this context ) or simply the finite length of suspended nanowires  @xcite are inherent to some waveguide designs and can not be avoided . in addition , local deviations from the intended ideal geometry can always lead to local variations in the stokes shift . as a result , the total sbs - response of a non - ideal waveguide is the convolution of the intrinsic resonance and the stokes frequency distribution along the waveguide . incidentally , the geometry - dependence of the local stokes shift has been used to measure the homogeneity of optical fibres  @xcite and to increase the sbs - threshold  @xcite . however , to date no connection has been made between the sbs - gain and the sensitivity of the stokes shift , although this is suggested by the close connection between sbs and opto - mechanics  @xcite , in which in turn the coupling strength is known to be closely related to the deformation - sensitivity of the optical subsystem  @xcite .    ) and an optical probe ( superscript @xmath0 ) excite a sound wave inside a waveguide that is assumed to be smooth , but whose geometry is smoothly and slowly perturbed along the axis of propagation , the @xmath1-axis . other cases studied in this manuscript include forward sbs and inhomogeneous strain fields ( due to temperature gradients or lattice mismatch ) instead of the sketched geometry variations . , scaledwidth=80.0% ]    the aim of this paper is to understand the impact of smooth and slowly varying perturbations of the waveguide structure ( as sketched in fig .  [ fig : system ] ) . we include variations of the waveguide geometry and internal strain fields that change smoothly along the waveguide and on a length scale greater than the acoustic decay length . this is opposed to surface roughness , which is usually a major concern in fabrication , since it causes linear optical propagation loss via rayleigh scattering  in backward sbs an analogous scattering of mechanical waves also contributes to the acoustic loss . in contrast , adiabatic structural variations do not scatter travelling waves and do not cause linear loss . however , they locally modify the dispersion relations and thereby influence the relative phase between the three waves participating in the sbs - process . here , we show that one mechanism for this is through the optical dispersion relation and the sensitivity is closely connected to the sbs - gain , whereas the second mechanism is via the mechanical dispersion relation and completely independent from the sbs - gain . we show that these two mechanisms influence forward and backward sbs quite differently and that increasing the sbs - gain either by increasing the acousto - optic coupling or reducing the mechanical damping leads to a regime where the line width is dominated by structural variations . the total sbs - response is then reduced to the square root of the intrinsic sbs - gain . we derive our results rigorously within the classical variant of a recent hamiltonian formulation of sbs  @xcite .    throughout this manuscript we adhere to the notation and conventions introduced in our earlier paper on coupled - mode theory of sbs  @xcite . in section  [ sec : qualitative ] , we qualitatively describe the relation between sbs and the sensitivity of the dispersion relations and the resulting impact on the sbs - resonance . although very useful for an intuitive understanding , this discussion lacks the quantitative precision of the formal derivation provided in section  [ sec : theory ] . the results of that section are then analytically studied in section  [ sec : discussion ] , focusing on the two important special cases of backward sbs and intra - mode forward sbs . next , we study the relative importance of the two broadening mechanisms in forward and backward sbs in section  [ sec : numerics ] using a family of suspended silicon nanowires as an example and find that they can compensate each other in appropriately engineered waveguides . section  [ sec : summary ] summarises and concludes the paper . the double - arrow highlights the opto - acoustic transition that fulfils the conservation of both energy and momentum . the solid lines represent the dispersion relations of the ideal system and the dashed lines those of the perturbed system . the light pink lines are parallels of the respective left side optical branches and indicate which acoustic wave number is phase matched . note that the magnitude of @xmath2 is grossly exaggerated compared to @xmath3 . , scaledwidth=50.0% ]    the acoustic properties of a waveguide are modified by static structural perturbations via mechanical nonlinearities , which are usually not relevant for the dynamics of optically generated sound waves . more precisely , there are separate contributions from internal strain fields and from variations of the cross section . strain fields enter via the nonlinear relation between stiffness and stress : @xmath4 where @xmath5 and @xmath6 are the strain and stress tensors , @xmath7 is the conventional linear stiffness tensor and @xmath8 is a second order nonlinear stiffness tensor ( see section  [ sec : theory ] ) . at the same time , internal strain fields also modify the mass density distribution , because the trace of the strain tensor corresponds to a volume dilatation . finally , changes in the cross section geometry cause a change in the boundary conditions ( to be more specific : in the position of the boundary , not in the type of condition imposed there ) of the acoustic helmholtz equation . as an example , a slight reduction of the waveguide width might increase or decrease the acoustic frequency @xmath9 depending on whether the acoustic mode is predominantly longitudinal or transversal and on whether the boundaries are approximately free or clamped .    at the same time , structural perturbations modify the optical properties of the waveguide . in contrast to the aforementioned acoustic frequency shift this modification is closely related to the sbs - process , which consists of two complementary nonlinear processes : the acoustic wave is excited by optical forces that are created by the beat of the optical pump and stokes waves . the sound wave spatio - temporally varies the waveguide s optical properties and modulates the pump wave such that the stokes side band is amplified . it is clear that the second process can only be efficient if the waveguide s dispersion relation is sensitive to small mechanical perturbations ; if it was insensitive the acoustic wave would not modulate the pump wave . this can not be compensated by a stronger excitation of the sound wave ( neglecting irreversible forces ) , because both processes are described by the same mode overlap integral , hence are equally strong . however , the optical dispersion relation can not distinguish between static and dynamic ( acoustic ) mechanical perturbations . therefore , the optical properties of any waveguide that exhibits strong sbs are intrinsically sensitive to small structural perturbations introduced in the fabrication process . more precisely , waveguides whose sbs is mainly caused by photoelasticity are predominantly sensitive to internal strain fields , caused by inhomogeneous cooling of the sample , whereas waveguides whose sbs is mainly caused by radiation pressure are predominantly sensitive to slight variations of the waveguide cross section . the variation of the optical dispersion relation occurs as a function of @xmath1 rather than time , breaking the conservation of optical momentum . therefore , this variation results in a change in the optical wave numbers @xmath10 and @xmath11 while leaving the optical frequencies unchanged . as a result , the pump frequency @xmath3 is exactly constant along the waveguide and the frequency @xmath12 of the perfectly phase - matched stokes wave varies only as a result of variations in the acoustic frequency @xmath9 . these optical wave number variations translate to variations of the acoustic wave number through the momentum matching condition @xmath13 therefore , the optical dispersion relation sensitivity indirectly affects the acoustic frequency . this effect has been practically applied in certain types of sbs - based fibre sensors  @xcite . the total local change in the acoustic frequency ( see also fig . [ fig : phase_matching ] ) is given by the combination of the optical wave number variations and the direct acoustic shift @xmath14 described earlier : @xmath15 it should be noted at this point that the two contributions to the total frequency change can have opposite sign and therefore cancel ( see sec .  [ sec : numerics ] for an example ) .        in a waveguide that is subject to inhomogeneous variations of the geometry and strain fields the acoustic frequency variation becomes a function of the position @xmath1 within the waveguide . it is instructive to imagine the waveguide to be composed of many short and approximately homogeneous sections ( see fig .  [ fig : broadening ] ) . each such section acts as an independent sbs - amplifier with a stokes frequency that is offset from the ideal by @xmath16 . all other properties such as intrinsic line width and differential sbs - gain ( the sbs gain per unit length of waveguide measured in units of @xmath17 ) can be assumed to be unchanged . the total sbs - spectrum of the complete waveguide is then simply the superposition of the individual sbs - spectra . in the limit of very short homogeneous sections , within an undepleted pump approximation and assuming a lorentzian intrinsic resonance shape , the stokes amplification on resonance is then the integral : @xmath18 ^ 2 } ,    \label{eqn : qualitative_superposition}\end{aligned}\ ] ] where @xmath19 represents the phonon life time and the waveguide is assumed to extend from @xmath20 to @xmath21 . within the integral , @xmath22 is the local ( differential ) sbs power gain measured in @xmath17 ( corresponding to the quantity @xmath23 in boyd s book  @xcite ) and @xmath24 is the total amplification accumulated over the total waveguide length @xmath25 ( denoted @xmath26 by boyd ) . if the distribution @xmath16 has a width of the order @xmath27 and especially if it is lopsided , the sbs - resonance is inhomogeneously broadened into a non - lorentzian shape as indicated in fig . [ fig : broadening ] . so far , we have qualitatively explained the two main effects of structural variations on the sbs - resonance : perturbations to the acoustic and the optical dispersion relation and we have furthermore argued that the magnitude of the latter effect must be closely related to the sbs - gain . for practical purposes , it is probably sufficient to compute @xmath14 and @xmath28 numerically and insert them into eqs . ( [ eqn : tot_shift],[eqn : qualitative_superposition ] ) . however , the exact relationship between optical sensitivity and sbs - gain as well as further consequences can only be elucidated by a formal derivation . this is therefore the topic of the next section . we assume sbs in a lossless optical waveguide that is oriented along the @xmath1-direction and provides confinement for the optical and all relevant acoustic modes , which are furthermore assumed to be only weakly damped . mostly out of convenience we restrict ourselves to mechanically isotropic materials . furthermore , we assume that the optical and acoustic envelopes as well as the deviations from the ideal waveguide geometry vary slowly along the @xmath1-axis compared to the acoustic decay length . we mostly follow the notation and conventions established in our earlier paper  @xcite : upper case letters refer to physical ( real - valued ) quantities , lower case letters to eigenmodes and envelope functions , and a tilde implies that the quantity is subjected to a phase transformation ( rotating frame ) . we furthermore introduce the symbol @xmath29 for the inverse permittivity and the superscripts `` @xmath30 '' and `` @xmath31 '' to refer to the interior of the waveguide s cross section and its boundary , respectively . we use einstein s summation convention where it helps to simplify the notation . we formulate this problem within a classical hamiltonian framework . to this end , we transfer a recent comprehensive quantum - mechanical description of sbs  @xcite to the classical domain by replacing operators with functions and commutators with poisson brackets . one of the main results of ref . @xcite is the full acousto - optic hamiltonian , which reads in classical form : @xmath32    + \sum_{ij}\int { { \rm{d}}}^3 r \     \bigg [      \frac{b_i b_i}{2 \mu_0 } +       \frac{d_i \eta_{ij}({{\bf{r } } } ; { { \bf{u } } } ) d_j}{2 { \varepsilon}_0 }     \bigg ]      \quad = \int { { \rm{d}}}z \ { { \mathcal{h}}}(z ) \ ,    \label{eqn : initial_hamiltonian}\end{aligned}\ ] ] where @xmath33 is the hamiltonian @xmath1-density , the hamiltonian per unit length of waveguide . the mechanical displacement , strain and momentum density fields and the electric and magnetic induction fields are denoted with @xmath34 , @xmath35 , @xmath36 , @xmath37 and @xmath38 , respectively . the material properties are described by the mass density @xmath39 , the stiffness tensor @xmath40 , the inverse relative permittivity tensor @xmath29 ; the electromagnetic terms furthermore involve the permittivity @xmath41 and permeability @xmath42 of vacuum . all material constants are assumed to depend on the mechanical displacement field @xmath34 as well as its derivatives . the first integral describes the mechanical part of the system , whereas the second integral covers both the electromagnetic part and the acousto - optic interaction term , which is implicitly contained in the dependence of the inverse permittivity @xmath43 on the mechanical deformation field and its derivatives . the two main contributions  @xcite to this dependence are @xmath44 a further term of linear order in @xmath34  the moving polarization term described in ref . @xcite  is usually too weak to be observable and is neglected here . within this framework we describe structural perturbations as a static contribution to the total mechanical displacement field @xmath34 : @xmath45 where the superscripts `` @xmath46 '' and `` @xmath47 '' denote the static and acoustic contributions , @xmath48 is the permittivity distribution of the ideal ( perfectly fabricated , sound - free ) system and @xmath49 is the pockels tensor that describes the photoelastic effect . ( [ eqn : ao_perturb ] ) reflects the close connection between the acousto - optic interaction and the sensitivity of the optical dispersion relation regarding structural perturbations . similarly , the mechanical stiffness and the mass density depend on the mechanical displacement field : @xmath50 here , the trace of the strain tensor expresses the volume dilatation . the strain - dependent stiffness tensor presented in contains the leading order nonlinearity of mechanically isotropic bulk materials ; microscopically , the sixth rank tensor @xmath51 describes three - phonon scattering . this can also be derived within the classical framework of hyperelasticity  @xcite : in that approach , the stress is introduced as the strain - derivative of a free energy that is not quadratic in the strain variable . assuming a purely reversible mechanical response , this free energy is identical to the hamiltonian , whose strain - dependence can be expanded into a taylor series . the leading ( cubic ) anharmonic strain - term is created by the sixth rank tensor @xmath51 when inserting eq . ( [ eqn : nonlinear_stiffness ] ) into eq . ( [ eqn : initial_hamiltonian ] ) . in reality , this three - photon process might be forbidden by symmetry and a higher - order multi - phonon scattering would constitute the leader order of mechanical bulk nonlinearity . furthermore , we have no reliable information about the order of magnitude of static strain fields across the large number of realisable integrated waveguides . still , we have introduced this three - phonon term to illustrate how mechanical bulk nonlinearities would impact inhomogeneous broadening , although we neglect them in our analysis in section  [ sec : discussion ] . nonlinear mechanical wave processes usually can be neglected in typical sbs setups , so we only retain the static contributions to the mechanical nonlinearity : @xmath52 these terms describe the sensitivity of the acoustic dispersion relation to structural perturbations . we now decompose the total hamiltonian @xmath1-density into the unperturbed mechanical and electromagnetic parts ( @xmath53 , @xmath54 ) , the interaction term @xmath55 and the structural perturbations @xmath56 and @xmath57 : @xmath58 where the first three terms are derived in ref . @xcite : @xmath59 the integrals are carried out over the entire transverse plane . the fourth term is formed in direct analogy to @xmath55 : @xmath60 and the last term is @xmath61 the difference between @xmath55 and @xmath56 is that the former results in a nonlinear coupling term of the optical and acoustic equations of motion , whereas the latter causes a linear perturbation of the wave propagation within the optical fields . the fifth term @xmath57 does not have a counterpart , because we neglected nonlinear wave effects within the propagating mechanical waves . next , we expand the electromagnetic and mechanical fields into the eigenmode bases of the respective idealized waveguide problems  @xcite @xmath62 ^ 2 \tilde{{{\bf{e}}}}^{(i ) } ;    \label{eqn : ewp_opt }    \\    ( \nabla_\perp + { { \rm{i}}}q \hat z)_j c_{ijkl } ( \nabla_\perp + { { \rm{i}}}q \hat z)_k    \tilde u_l = & - \rho \omega^2 \delta_{il } \tilde u_l . \label{eqn : ewp_ac}\end{aligned}\ ] ] in both equations the wave numbers along the waveguide serve as parameters , where the optical wave number @xmath11 has to be chosen such that @xmath3 matches the frequency of the modelled pump laser and the acoustic wave number is subsequently found through eq . ( [ eqn : momentum_matching ] ) . in very extended systems such as conventional optical fibres , an infinite number of acoustic basis functions can contribute to the total acoustic sbs response . in this case the treatment by beugnot et al . @xcite is a viable alternative to our description . however , we here restricted ourselves to nano - scale waveguides , in which the spectral separation between acoustic modes is greater than the sbs linewidth , in which only one or very few acoustic eigenmodes can be simultaneously excited by the two optical waves . consequently , we express the optical and acoustic fields as modulations of single basis functions  @xcite @xmath63 here , @xmath64 and @xmath65 are the classical counterparts of the quickly rotating envelope functions in ref . @xcite and are related to the more common slowly varying envelope functions @xmath66 and @xmath67 via the relations @xmath68 they fulfil the poisson bracket relations @xmath69^\ast } , \ { \tilde a^{(j)}(z',t ' ) } \big\ } } = &     \frac{{{\rm{i}}}\omega^{(i)}}{{{\mathcal{e}}}^{(i ) } } \delta_{ij } \delta(z - z ' ) \delta(t - t ' ) ,    \\    { \big\ { { \tilde b^\ast(z , t ) } , \ { \tilde b(z',t ' ) } \big\ } } = &     \frac{{{\rm{i}}}\omega}{{{\mathcal{e}}}_b } \delta(z - z ' ) \delta(t - t').\end{aligned}\ ] ] the symbols @xmath70 denote the respective modal energies per unit length of waveguide  @xcite : @xmath71^\ast { \varepsilon}^{{\text{ideal}}}_{kl } \tilde e^{(i)}_l ,    \\    { { \mathcal{e}}}_b = & 2 \omega^2 \int { { \rm{d}}}^2 r \ \rho^{{\text{ideal}}}|\tilde{{{\bf{u}}}}|^2.\end{aligned}\ ] ] within the expansions ( [ eqn : expansion_opt],[eqn : expansion_ac ] ) the unperturbed contributions to the hamiltonian @xmath1-density simply become @xmath72    using the fact that the acoustic eigenmodes ( indexed by superscript @xmath73 ) for a fixed wave number @xmath74 form a complete function set , we can expand the static deformations : @xmath75.\end{aligned}\ ] ] in this we have introduced separate expansions for the distribution of a static strain field and the deviation of the waveguide geometry , which contains both effects of the static strain field and variations of the geometry due to fabricational ( photolithography or etching ) imperfections . in analogy to refs . @xcite , we now introduce separate modal overlap integrals to capture the effects of photoelasticity and of boundary displacement : @xmath76^\ast e^{(j)}_l p_{klmn } \partial_m [ u^{(p)}_n]^\ast ;    \label{eqn : def_q_bulk }    \\ q^{{{\text{edge}}}}_{ij;p } = &    \int_{{{\text{edge } } } } { { \rm{d}}}{{\bf{r } } } \     \big [      ( { \varepsilon}_a - { \varepsilon}_b ) { \varepsilon}_0 ( \hat n \times { { \bf{e}}}^{(i)})^\ast ( \hat n \times { { \bf{e}}}^{(j ) } )      -      ( { \varepsilon}^{-1}_a - { \varepsilon}^{-1}_b ) { \varepsilon}^{-1}_0 ( \hat n \cdot { { \bf{d}}}^{(i)})^\ast ( \hat n \cdot { { \bf{d}}}^{(j ) } )    \big ]    ( \hat n \cdot { { \bf{u}}}^{(p)})^\ast . \label{eqn : def_q_edge}\end{aligned}\ ] ] now , by substituting the modal expansions in ( [ eqn : expansion_opt],[eqn : expansion_ac ] ) and ( [ eqn : perturb_expansion_u],[eqn : perturb_expansion_s ] ) into the interaction terms in ( [ eqn : perturb_int],[eqn : perturb_opt ] ) , and integrating over the transverse plane , we find the interactions can be reduced to the forms @xmath77^\ast b \underbrace { \big ( q^{{{\text{bulk}}}}_{12;n_b } + q^{{{\text{edge}}}}_{12;n_b } \big)}_{q_{{\text{sbs } } } } + { { \rm{c.c.}}}\label{eqn : modal_interact }    \\    { { \mathcal{v}}}^{{\text{opt}}}(z ) = & \sum_n |a^{(1)}|^2 \big ( \zeta^{(n ) } q^{{{\text{bulk}}}}_{11;n } + \xi^{(n ) } q^{{{\text{edge}}}}_{11;n } \big )     + |a^{(2)}|^2 \big ( \zeta^{(n ) } q^{{{\text{bulk}}}}_{22;n } + \xi^{(n ) } q^{{{\text{edge}}}}_{22;n } \big ) ,    \label{eqn : modal_perturb_opt}\end{aligned}\ ] ] where @xmath78 is the index of the phase - matched acoustic mode . the edge contribution @xmath79 to the perturbation term @xmath56 is of course identical to the expression by johnson  @xcite . finally , we analogously express the acoustic structural perturbation term @xmath80 in terms of a bulk overlap integral @xmath81^\ast [ \partial_k u^{(n_b)}_l ] f_{ijklmn }     + \frac{\omega^2 \big|u_i^{(n_b)}\big|^2}{2 \rho^{{\text{ideal } } } }    \bigg\ } \big(\partial_m u^{(p)}_n + \partial_n e^{(p)}_m \big),\end{aligned}\ ] ] which directly corresponds to the photoelastic perturbation overlap @xmath82 and an edge integral @xmath83 that corresponds to the electromagnetic boundary displacement overlap @xmath79 . the explicit expression for @xmath83 is slightly involved and requires a short derivation ; both can be found in appendix  [ appx : mech_edge ] . mathematically , this term contains the edge contributions from eq . ( [ eqn : perturb_ac ] ) , the spatial derivatives of the stiffness and mass density functions . physically , it describes the simple fact that the transverse acoustic pattern  which is a standing wave in the transversal plane  is detuned very much like a drumhead upon size variations of the transverse acoustic cavity .    using the poisson bracket relations , expanding the @xmath1-integral of the hamiltonian density into the appropriate taylor series  @xcite and truncating the expansion to first order , we find the equations of motion : @xmath84      we now have all tools to compute the spectrum of the steady state sbs response of an imperfect waveguide using the acoustic green function  @xcite . to this end , we assume a small signal approximation and a local acoustic response approximation , for which we assume that the modal envelopes @xmath85 , @xmath67 as well as the perturbation envelopes @xmath86 , @xmath87 vary slowly compared to the acoustic decay length , which is of the order @xmath88 in backward sbs and is negligible in intra - mode forward sbs . following ref . @xcite , we introduce the inverse acoustic decay length @xmath89 as well as an additional parameter @xmath90 that describes a wave number offset between the ideally phase - matched stokes mode and the probe wave that is injected in a seeded small - signal sbs - experiment . the poisson brackets involving the perturbation terms  ( [ eqn : modal_perturb_opt],[eqn : modal_perturb_ac ] ) are : @xmath91 expressed in terms of the structural detuning parameters @xmath92 the steady state equations then follow : @xmath93^\ast a^{(2)},\end{aligned}\ ] ] with the sbs coupling parameter @xmath94 from eq . ( [ eqn : modal_interact ] ) and the powers @xmath95 and @xmath96 of the eigenmodes as introduced in ref .  @xcite . we then solve the acoustic equation using its green function : @xmath97^\ast     a^{(2)}(z - z ' ) \exp\big(-[\alpha - { { \rm{i}}}\delta q^{{\text{dir}}}(z - z ' ) ] z'\big ) \big\}.    \label{eqn : b_convolution}\end{aligned}\ ] ] we assumed that the optical intensities as well as the structural envelopes vary slowly on the length scale @xmath98 , so we can assume that the @xmath99 , @xmath100 , @xmath101 and the absolute values @xmath102 and @xmath103 are constant within the convolution ; @xmath90 is simply a constant . we may therefore approximate the product of optical envelopes at the position @xmath104 as that product at position @xmath1 modulo a phase factor that expresses the relative beat @xmath105 : @xmath106^\ast a^{(2)}(z - z ' )     \approx &    [ a^{(1)}(z)]^\ast a^{(2)}(z )     \exp \big ( { { \rm{i}}}[\delta \beta^{(2)}(z ) - \delta \beta^{(1)}(z ) - \kappa ] z ' \big ) ,    \\    \rightarrow \quad b(z ) \approx & - \frac{{{\rm{i}}}\omega q_{{\text{sbs}}}}{{{\mathcal{p}}}_b }     [ a^{(1)}(z)]^\ast a^{(2)}(z ) l(z ) , \intertext{\centering with }    l(z ) = & \frac{1}{\alpha - { { \rm{i}}}[\delta q^{{\text{tot}}}(z ) - \kappa ] }    \\ \delta q^{{\text{tot}}}= &    \delta q^{{\text{dir}}}- \delta \beta^{(1 ) } + \delta \beta^{(2)}.\end{aligned}\ ] ] we can now insert this expression for the acoustic field into the optical equations . along the lines of ref . @xcite , we then transform them into a pair of equations for the optical powers @xmath107 : @xmath108 the local sbs - gain function is given by @xmath109 ^ 2}.    \label{eqn : local_gain}\end{aligned}\ ] ] within a small signal approximation , @xmath110 , this leads to a separable equation for the stokes power @xmath111 : @xmath112,\end{aligned}\ ] ] and an expression for the total stokes amplification of the waveguide of length @xmath25 : @xmath113 ^ 2 }    }     \label{eqn : theory_result}\ ] ] in units of db .    in the simple case of @xmath1-independent aberration coefficients @xmath86 and @xmath87 the integral in eq . ( [ eqn : theory_result ] ) becomes equal to one if the detuning @xmath90 is chosen to be @xmath114 . this means that to an excellent approximation the acoustic frequency and thereby the sbs - resonance is simply shifted by the frequency @xmath115 in agreement with eq . ( [ eqn : tot_shift ] ) . in the case of inhomogeneous coefficients , the integral in eq . ( [ eqn : theory_result ] ) expresses the linear superposition of individual sbs - resonances and therefore the inhomogeneous resonance broadening in agreement with eq . ( [ eqn : qualitative_superposition ] ) . we will now study the two practically most relevant special cases : intra - mode forward sbs and backward sbs . qualitatively , inter - mode scattering behaves similarly to backward sbs . first , we study the case of forward intra - mode sbs , we assume @xmath116 in this case , the sbs coupling coefficient is @xmath117 furthermore , the optical wave number perturbations are identical : @xmath118 as a result , @xmath119 , the indirect resonance broadening via the optical dispersion relation is absent in intra - mode forward sbs . strictly speaking , this result is exact only if the optical group velocity is identical at @xmath120 and @xmath3 ; in practice this is irrelevant except for extremely dispersive optical modes , in the slow light regime near band edges . the remaining source of structural resonance broadening is the direct mechanical contribution @xmath121 if we assume that internal strain fields are too weak to cause noticeable resonance broadening , we may assume @xmath122 that the resonance detuning can be regarded as the simple geometric effect of changing the size of a transversal acoustic cavity . the mechanical perturbation overlaps @xmath123 and the acousto - optic overlaps @xmath124 are in principle completely independent , so there is no fundamental reason why an increase in the sbs - gain would increase the inhomogeneous broadening in the case of intra - mode forward sbs . this is fundamentally different in the case of backward sbs and ( to a lesser extent ) in the case of general inter - mode sbs . the optical modes that participate in conventional backward sbs are the counter - propagating partners of each other and therefore related via complex conjugation : @xmath125^\ast ;    \quad    { { \mathcal{p}}}^{(1 ) } = - { { \mathcal{p}}}^{(2)}.    \label{eqn : modes_bsbs}\end{aligned}\ ] ] therefore , the sbs coupling coefficient and the optical wave number perturbations are : @xmath126 while the expression for the direct acoustic detuning is identical to the case of forward sbs . the subscript indices of the individual overlap integrals @xmath82 and @xmath79 are the mode labels of the involved two optical and the acoustic mode [ see eqs . ( [ eqn : def_q_bulk][eqn : def_q_edge ] ) ] . it is in fact the acousto - optic forward - sbs coupling of the respective acoustic and optical modes , basically the square root of the forwards sbs gain for that combination of modes . this time , the optical wave number perturbations add up and the total stokes shift becomes : @xmath127.\end{aligned}\ ] ] the direct mechanical contribution is qualitatively identical to the previous case . furthermore , we can assume that the predominantly longitudinal acoustic mode contributing to backward sbs is rather insensitive to slight geometry variations . this suggests to focus on the indirect ( optical ) contribution to the broadening and ignore the direct ( acoustic ) contribution . as a result , the general result eq . ( [ eqn : theory_result ] ) then has the explicit form ( @xmath128 ) : @xmath129 the mode coefficients @xmath130 and @xmath131 in the denominator are mode overlaps that define the forward sbs gain and are identical to the coefficients @xmath132 and @xmath133 in the numerator except for a complex conjugation of one of the optical modes courtesy of eq . ( [ eqn : modes_bsbs ] ) .      the general bsbs result eq . ( [ eqn : bsbs_general_result ] ) is too convoluted to interpret directly . we therefore restrict ourselves to a very specific limiting case that nonetheless provides some insight into the general impact of geometrical variations in bsbs in nanoscale waveguides . we assume that the acousto - optic interaction is dominated by the edge term @xmath134 , by radiation pressure . we furthermore assume that the contributions @xmath135 of the phase matched acoustic mode to the geometry perturbations provide a good estimate for the total optical sensitivity @xmath136 . we finally assume that the geometry variations are completely random within a certain interval . within this approximation @xmath137 is uniformly distributed in an interval @xmath138 $ ] ; other distributions of @xmath137 clearly lead to different broadened resonance shapes , but also exhibit the general trends described below . these assumptions reduce eq . ( [ eqn : bsbs_general_result ] ) to the much simpler form @xmath139 for the case of strong broadening ( @xmath140 ) the integral approaches the limit @xmath141 and therefore the total amplification approaches the limit @xmath142 this result consists of a combination of eigenmode constants such as frequencies and mode powers , the ratio of the intrinsic forward and backward sbs gains and a term defining how the broadened sbs amplification scales with various design parameters , especially the fabrication tolerance @xmath143 . the gain ratio is always slightly below one , as we show in appendix  [ appx : gain_ratio ] , and we can simply assume it to be a constant . the last term finally scales with the inverse of the acoustic damping , the fabrication tolerance and @xmath144 , which can be identified as being proportional to the square root of the fsbs power gain using eq . ( [ eqn : local_gain ] ) and eq . ( [ eqn : fsbs_gain ] ) . alternatively , we can also formulate the result as scaling with the square root of the bsbs power gain by using @xmath145 both interpretations are basically equivalent , because fsbs and bsbs gain are closely related ( appendix  [ appx : gain_ratio ] ) . if we assume a waveguide with a significant sbs contribution due to radiation pressure and we assume the tolerance @xmath143 to be given , the reduction of @xmath146 either by reducing the acoustic damping parameter @xmath89 or by increasing the fsbs acousto - optic coupling @xmath144 will lead into a regime where the bsbs amplification is defined by @xmath144 . in practice , the acoustic loss can be reduced by reducing the contact area of a nearly suspended waveguide to its substrate  @xcite ; certain slot waveguide designs  @xcite on the other hand allow control of the radiation pressure coupling by varying the gap size . crystal direction . we study the interaction between the fundamental optical mode and two symmetry - permitted acoustic modes with lowest frequency . the acoustic mode patterns show the in - plane displacement components as arrows and the axial displacement coordinate via a color map . the first acoustic mode ( orange frame and graph lines ) does not have any axial contributions , whereas the second acoustic mode ( blue frame and lines ) has no in - plane contributions to the displacement field . the relative detunings are displayed in percent for a change of the waveguide width by @xmath147 . due to the extremely small acoustic wave number in fsbs only the direct acoustic perturbation process contributes to the resonance shift , whose relative magnitude follows the relative width change exactly for mode 2 and qualitatively for mode 1 . this demonstrates the direct acoustic response to width changes is basically given by the detuning of the transverse mechanical cavity . the annotations in the top left panel specify the computed sbs power gains of the two acoustic modes for the extreme waveguide widths assuming a mechanical quality factor of @xmath148 . ]    as the last part of our study , we performed numerical experiments on the sensitivity of the sbs frequency shift to variations of a waveguide s geometry . we selected a family of silicon nanowires with a height of @xmath149 and a width varying between @xmath150 and @xmath151 in steps of @xmath152 . for each such waveguide , we characterized forward and backward sbs between the fundamental optical mode and the two lowest symmetry - permitted acoustic modes  @xcite using the finite - element solver comsol with an element size of @xmath153 inside the waveguide . for each step of @xmath154 , we estimated the width sensitivity by increasing the width by @xmath147 and repeating the calculation . the resulting relative frequency shifts are plotted as solid lines in the lower left panels of fig . [ fig : example_fsbs ] and fig . [ fig : example_bsbs ] . finally , we added computed sbs power gains of the respective acoustic modes as annotations to the sbs frequency graphs ; they are intended to give a rough idea of the respective acousto - optic coupling .    in the case of fsbs ( fig . [ fig : example_fsbs ] ) , the difference is entirely due to the direct ( acoustic ) perturbation effect , because the acoustic wave number is basically zero . both acoustic modes are standing waves in the acoustic resonator , formed perpendicular to the axis of light propagation , where the first mode has contributions from both in - plane components of the mechanical displacement field and the second mode only contributions from the out - of - plane component . as a result , the second mode is a pure standing s - wave and consequently its eigenfrequency is entirely determined by the waveguide width and the transverse speed of sound . this leads to the exact correspondence of the variation of @xmath9 and the relative change of the waveguide width ( red crosses in fig . [ fig : example_fsbs ] ) . in contrast , the first mode is a hybrid of an s - wave and a p - wave , because both in - plane components contribute . as a result , the relationship between waveguide width and @xmath9 is not as straightforward . this mode for a waveguide width of @xmath151 was studied experimentally in ref . @xcite and its sensitivity has been determined to be @xmath155 at a measured brillouin frequency of @xmath156 equivalent to a sensitivity of @xmath157 in excellent agreement with fig .  [ fig : example_fsbs ] . still , fig . [ fig : example_fsbs ] demonstrates that the geometry sensitivity of fsbs is only a response to size variations of a transverse acoustic resonator . in addition , we have also computed broadened fsbs spectra of the first mode ( computed frequency shift : @xmath158 at a width of @xmath151 ) . [ fig : num_resonance_shapes ] shows the impact of a width variation of @xmath159 . this is a little less than two lattice constants of the silicon crystal , it corresponds roughly to two monolayers of atoms . [ fig : num_resonance_shapes ] also illustrates the effect of different distributions of width variations along the waveguide . the leftmost panel shows the unperturbed spectrum assuming a mechanical quality factor of @xmath148 , which is a realistic value for certain fsbs experiments  @xcite where mechanical loss is dominated by acoustic leakage . the center panel shows the broadened spectrum assuming that the waveguide length linearly changes along the total waveguide length ( typically several millimetres to centimetres ) ; as a result the quality factor significantly drops to @xmath160 . the rightmost panel finally shows the broadened spectrum assuming that the waveguide width fluctuates sinusoidally along the waveguide with a period that is large compared to the acoustic decay length and the optical wave length ( the exact periodicity does not matter ) . in this case , not only is the quality factor further reduced to @xmath161 , but the resonance is also deformed into a highly non - lorentzian shape with two distinct peaks . this example shows that high effective quality factors in integrated sbs require very consistent fabrication down to the atom level .     at a mechanical quality factor of @xmath148 , the system studied in fig  .[fig : example_fsbs ] and in ref . the three panels show the unperturbed resonance ( left panel ) , the line for the case that the width linearly varies from @xmath162 to @xmath163 over the waveguide length ( center panel ) and for the case that the waveguide width is sinusoidally undulated between the values @xmath162 and @xmath163 . in both cases , the perturbation reduces the quality factor by about a factor of two . , scaledwidth=70.0% ]    for bsbs , the situation is slightly more complicated , because the indirect ( optical ) perturbation contributes to the total sensitivity . therefore , we additionally determined the impact of the direct contribution by solving the acoustic problem for the perturbed geometry and the wave number derived from the unperturbed problem ( circles in fig . [ fig : example_bsbs ] ) ; the indirect ( optical ) contribution ( squares in fig . [ fig : example_bsbs ] ) then follows as the difference . both studied acoustic modes are hybrid and have an axial wave length that is comparable to the waveguide s dimensions . therefore , the acoustic frequency is roughly determined by the bulk speed of sound and the wave number . the displacement field of the first acoustic mode has appreciable @xmath164-components , which leads to a noticeable direct ( acoustic ) sensitivity . in contrast , the second mode is mostly insensitive to the direct perturbation effect . both acoustic modes are very susceptible to the indirect ( optical ) perturbation effect . for example , the second acoustic mode responds to a width variation of @xmath147 with a resonance shift of up to @xmath165 . correspondingly , a waveguide with a width of @xmath150 would have to be manufactured to better than @xmath166 ( less than a monolayer of atoms ) along its entire length in order to maintain an overall quality factor of @xmath167 . evidence for this problem has been observed experimentally  @xcite .    one interesting detail of fig . [ fig : example_bsbs ] is the fact that the direct and indirect contributions to the inhomogeneous broadening can be of comparable magnitude . furthermore , they will tend to have opposite signs , because an increase in waveguide width tunes the transverse acoustic resonator to lower frequencies via the direct ( acoustic ) perturbation mechanism , while an increase in the width also leads to higher optical mode indices , corresponding to greater wave numbers and eventually higher acoustic frequencies via the indirect ( optical ) perturbation mechanism . this suggests the very promising idea to design a waveguide structure with a deliberately _ strong _ direct sensitivity in order to cancel the inevitable indirect sensitivity . the result would be a waveguide design with greatly relaxed fabrication tolerances with respect to the waveguide width . the general applicability of this concept to minimise the sensitivity with respect to one design parameter ( in this case the width ) is proven by the behavior of mode 1 in fig . [ fig : example_bsbs ] . there , the direct and indirect contributions for width variations nearly cancel for small waveguide widths . however , the presented mode is only of academic interest because of its very low sbs power gain . finally , it should be noted that only one specific type of perturbations ( e.g. width variations , wall angle variations or height variations ) can be compensated with this technique while potentially increasing the sensitivity of the other perturbation types .     crystal direction . in contrast to fsbs ( fig . [ fig : example_fsbs ] ) , both studied acoustic modes involve axial and in - plane components . furthermore , bsbs features both the direct ( acoustic ) process ( with negative sign as for fsbs ) and in addition the indirect ( optical ) process with the opposite sign . the former can be neglected for wider waveguides ; however , it is possible that both contributions cancel each other . as in fig . [ fig : example_fsbs ] we annotated the power gain , this time assuming a quality factor of @xmath167 . the extremely low power gain of mode 1 is caused by nearly exact u cancellation of the photoelastic and moving - boundary related coupling in this structure . ] in conclusion , we have analysed the sensitivity of the optical and acoustic dispersion relations on structural variations of a waveguide and related this to the sensitivity of the stokes shift @xmath9 . we have shown that variations of the optical dispersion relation are a major source of inhomogeneous broadening in backward sbs and that they are intimately related to the forward sbs coupling , because slowly and smoothly varying perturbations mainly modify the phase of an optical mode , and induce negligible back - scattering . this leads to a counter - intuitive result in sbs dominated by radiation pressure ( see section  [ sec : rad_bsbs ] ) : since the inhomogeneous broadening is proportional to the forward sbs coupling ( square root of fsbs - gain ) , the maximum of a strongly broadened sbs spectrum is proportional to the ratio of the backward gain and the square root of the forward gain . as a result the maximum of the strongly broadened response of a backward sbs system scales only with the square root of the naively expected sbs - gain . this means that high - gain backward sbs waveguides are intrinsically sensitive to fabricational imperfections and require fabricational tolerances matched not only to the intrinsic acoustic line width but also to the opto - acoustic coupling strength . therefore , it may be advisable in practice to sacrifice some intrinsic sbs - gain to avoid the tolerance - dominated regime , especially since this might reduce the linear or nonlinear optical loss at the same time and thereby increase the overall figure of merit . we have furthermore shown that the indirect resonance sensitivity is absent in forward sbs and that therefore no fundamental link between the resonance sensitivity and the sbs - gain exists in this case . nonetheless , the direct ( acoustic ) resonance sensitivity can be appreciable and has been observed in at least one nanophotonic realisation of forward sbs  @xcite . our numerical example for fsbs is in excellent agreement with this experiment . finally , we have argued that the direct ( acoustic ) and indirect ( optical ) contributions to the resonance sensitivity tend to have opposite sign . this allows one to design comparatively insensitive high - gain waveguides with a strong direct sensitivity carefully engineered to compensate the intrinsic indirect sensitivity . c.w . , m.j.s . , b.j.e . and c.g.p . acknowledge financial support from the australian research council ( arc ) via the discovery grant dp130100832 , its laureate fellowship ( b.j.e . , fl120100029 ) program , and the arc center of excellence cudos ( ce110001018 ) . r.v.l . acknowledges the agency for innovation by science and technology in flanders ( iwt ) for a phd grant . in this appendix , we derive the edge contribution to the mechanical perturbation term . this is analogous to the perturbation theory developed by johnson et al . for the perturbation of the electromagnetic dispersion relation  @xcite , which is based on the continuity of the transversal electric field and of the normal electric induction field across a material boundary . in the context of continuum mechanics , the quantities that are continuous across a material boundary are the normal projection of the stress tensor @xmath168 and the in - plane projection of the strain tensor @xmath169 where @xmath170 is the surface normal . in analogy to the electromagnetic case , the perturbation to the hamiltonian is the normal projection of the surface displacement multiplied by the mechanical energy density expressed in terms of the conserved quantities , where we restrict ourselves to isotropic materials such as glasses or homogenised polycrystalline cubic materials ( the expressions for anisotropic materials are more complex , but of the same general form ) . we introduce an auxiliary strain quantity @xmath171_{kjlm } t^\perp_{lm } ,    \label{eqn : s_eff}\end{aligned}\ ] ] where @xmath172 $ ] is the mechanical compliance tensor . this allows us to write the acoustic perturbation overlap caused by boundary displacements in a concise form : @xmath173 ( \hat n \cdot { { \bf{u}}}).\end{aligned}\ ] ] within the modal expansion employed in section  [ sec : theory ] , the edge - effect of the @xmath174-th acoustic basis function of the phase matched acoustic mode @xmath78 becomes : @xmath175     ( \hat n \cdot { { \bf{u}}}^{(p)})^\ast ,    \intertext{with the modal ( note the lower case ) generalisation of { eq.~(\ref{eqn : s_eff } ) } }    s^{{\text{eff}}}_{ij } = &    \frac{1}{2 } \bigg [    \partial_i u^{(n_b)}_j + \partial_j u^{(n_b)}_i    - n_i \sum_k n_k \big ( \partial_k u^{(n_b)}_j + \partial_j u^{(n_b)}_k \big )    \\ \nonumber & \quad    + n_i \sum_{klpq\alpha\beta } n_k     [ c^{-1}]_{kjlp } n_l n_q c_{pb\alpha\beta }     \big(\partial_\alpha u^{(n_b)}_\beta + \partial_\beta u^{(n_b)}_\alpha \big )    \bigg].\end{aligned}\ ] ] in this appendix we investigate the ratio between the acousto - optic coupling terms of forward and backward sbs . we find that they are always of the same order and that the radiation pressure coupling is always greater for forward sbs . inserting @xmath176^\ast = { { \bf{e}}}$ ] into the explicit definitions eqs . ( [ eqn : def_q_bulk],[eqn : def_q_edge ] ) the forward and backward coupling terms are : @xmath177^\ast     \\ \nonumber & \quad    +    \int_{{{\text{edge } } } } { { \rm{d}}}{{\bf{r } } } \     \big [      ( { \varepsilon}_a - { \varepsilon}_b ) { \varepsilon}_0 ( \hat n \times { { \bf{e}}}^\ast ( \hat n \times { { \bf{e } } } )      -      ( { \varepsilon}^{-1}_a - { \varepsilon}^{-1}_b ) { \varepsilon}^{-1}_0 ( \hat n \cdot { { \bf{d}}})^\ast ( \hat n \cdot { { \bf{d } } } )    \big ]    ( \hat n \cdot { { \bf{u}}}^{(p)})^\ast ,    \\    q^{{\text{bsbs}}}_p = & q^{{{\text{bulk}}},{{\text{bsbs}}}}_{p } + q^{{{\text{edge}}},{{\text{bsbs}}}}_{p }    = q^{{\text{bulk}}}_{12;p } + q^{{\text{edge}}}_{12;p }    \\    = &     { \varepsilon}_0 \int_{{\text{bulk}}}{{\rm{d}}}^2 r \ \sum_{klmn }    { \varepsilon}_r^2 e_k^\ast e_l^\ast p_{klmn } \partial_m [ u^{(p)}_n]^\ast     \\    \nonumber & \quad    +    \int_{{{\text{edge } } } } { { \rm{d}}}{{\bf{r } } } \     \big [      ( { \varepsilon}_a - { \varepsilon}_b ) { \varepsilon}_0 ( \hat n \times { { \bf{e}}}^\ast ( \hat n \times { { \bf{e}}}^\ast )      -      ( { \varepsilon}^{-1}_a - { \varepsilon}^{-1}_b ) { \varepsilon}^{-1}_0 ( \hat n \cdot { { \bf{d}}})^\ast       ( \hat n \cdot { { \bf{d}}}^\ast )    \big ]    ( \hat n \cdot { { \bf{u}}}^{(p)})^\ast .\end{aligned}\ ] ] the only difference is the complex conjugation of the second optical mode . due to the symmetry of eqs .  ( [ eqn : ewp_opt],[eqn : ewp_ac ] ) the phases of the optical as well as the acoustic modes can be adjusted such that the in - plane field components are purely real - valued and the axial components are purely imaginary . thus , the main effect of the complex conjugation is a sign reversal in the @xmath1-component of the second optical mode . in the photoelastic coupling term @xmath82 the axial components of one mode and the transversal components of another mode can be nontrivially combined depending on the orientation of the principal axes of @xmath49 relative to the waveguide geometry . however , the edge term @xmath79 can be easily analysed . first we notice that the axial component of @xmath178 is irrelevant . next , we decompose the transversal electromagnetic fields into components normal and parallel to the waveguide surface : @xmath179 within this notation we find : @xmath180 ( \hat n \cdot { { \bf{u}}}^{(p)})^\ast ,    \\    q^{{{\text{edge}}},{{\text{bsbs}}}}_p = &     \int_{{{\text{edge } } } } { { \rm{d}}}{{\bf{r } } } \     \big [      ( { \varepsilon}_a - { \varepsilon}_b ) { \varepsilon}_0 ( |e_\parallel|^2 - |e_z|^2 )      +      ( { \varepsilon}^{-1}_b - { \varepsilon}^{-1}_a ) { \varepsilon}^{-1}_0 |d_\perp|^2     \big ] ( \hat n \cdot { { \bf{u}}}^{(p)})^\ast.\end{aligned}\ ] ] the factors @xmath181 and @xmath182 always have the same sign ( note the reversal of subscripts ) . therefore , the radiation pressure contribution to the intra - mode forward sbs coupling is always greater than the corresponding contribution to the backward sbs coupling . this result does not apply to the photoelastic coupling . p.  dainese , p.  st .  j. russell , n.  joly , j.  c. knight , g.  s. wiederhecker , h.  l. fragnito , v.  laude , and a.  khelif , `` stimulated brillouin scattering from multi - ghz - guided acoustic phonons in nanostructured photonic crystal fibres , '' nature phys . * 2 , * 388392 ( 2006 ) . r. pant , c.  g. poulton , d .- y . choi , h.  mcfarlane , s.  hile , e.  li , l.  thvenaz , b.  luther - davies , s.  j. madden , and b.  j. eggleton , `` on - chip stimulated brillouin scattering , '' opt . express * 19 , * 82858290 ( 2011 ) . h.  shin , w.  qiu , r.  jarecki , j.  a. cox , r.  h. o.  ill , a.  starbuck , z.  wang , and p.  t. rakich , `` tailorable stimulated brillouin scattering in nanoscale silicon waveguides , '' nature commun . * 4 , * 1944 ( 2013 ) . s. chin , l. thvenaz , j. sancho , s. sales , j. capmany , p. berger , j. bourderionnet , and d. dolfi , `` broadband true time delay for microwave signal processing , using slow light based on stimulated brillouin scattering in optical fibers , '' opt . express * 18 , * 2259922613 ( 2010 ) . b.  stiller , a.  kudlinski , min won lee , g.  bouwmans , m.  delqu , j .- c . beugnot , h.  maillotte , and t.  sylvestre , `` sbs mitigation in a microstructured optical fiber by periodically varying the core diameter , '' ieee photon . * 24 , * 667669 ( 2012 ) . c.  wolff , m.  j. steel , b.  j. eggleton , and c.  g. poulton , `` stimulated brillouin scattering in integrated photonic waveguides : forces , scattering mechanisms and coupled mode analysis , '' phys . a * 92 * , 013836 ( 2015 ) . s.  g. johnson , m. ibanescu , m.  a. skorobagotiy , o. weisberg , j.  d. joannopoulos , y. fink , `` perturbation theory for maxwell s equations with shifting material boundaries , '' phys . e * 65 , * 066611 ( 2002 ) .
we study the impact of structural variations ( that is slowly varying geometry aberrations and internal strain fields ) on the width and shape of the stimulated brillouin scattering ( sbs ) resonance in nanoscale waveguides . we find that they lead to an inhomogeneous resonance broadening through two distinct mechanisms : firstly , the acoustic frequency is directly influenced via mechanical nonlinearities ; secondly , the optical wave numbers are influenced via the opto - mechanical nonlinearity leading to an additional acoustic frequency shift via the phase - matching condition . 
 we find that this second mechanism is proportional to the opto - mechanical coupling and , hence , related to the sbs - gain itself . 
 it is absent in intra - mode forward sbs , while it plays a significant role in backward scattering . 
 in backward sbs increasing the opto - acoustic overlap beyond a threshold defined by the fabrication tolerances will therefore no longer yield the expected quadratic increase in overall stokes amplification . 
 finally , we illustrate in a numerical example that in backward sbs and inter - mode forward sbs the existence of two broadening mechanisms with opposite sign also opens the possibility to compensate the effect of geometry - induced broadening . 
 our results can be transferred to other micro- and nano - structured waveguide geometries such as photonic crystal fibres .