1. Introduction

Stimulated Brillouin scatterings (SBSs) [1–3], in which the pump, Stokes optical waves and associated acoustic or elastic waves at hypersonic frequency collectively interact, are generally the most significant nonlinear optical phenomena in waveguides such as the optical fibers. (As we just consider the SBSs in solids in the present study, hereinafter we will no longer mention acoustic waves, but will make reference to elastic waves only). By inducing optical transmission noise, SBSs are not desirable in some scenarios. On the other hand, it can also be utilized to develop many practical applications, such as the Brillouin sensors and lasers [4,5], slow and fast lights [6,7], optical pulse compression [8,9] and optical frequency comb generation [10]. At large scales, the SBSs in solids are generally mediated by the photoelastic (PE) and electrostrictive (ES) effects. In recent years, with the development of integrated photonics, the on-chip SBS phenomena and their novel applications have attracted widespread interests [11–19]. It has been well established that the on-chip SBSs can also be mediated by two surface/interface effects: the optical radiation pressure (RP) [20,21] and moving interface (MI) [13,22] effects (or referred to as “moving boundary effect”), which become physically relevant at the sub-wavelength/nano scales and coherently interfere with the ES and PE effects [23], respectively. Consequently, the on-chip SBSs may exhibit some extraordinary behaviors, such as the giant enhancement of SBS gains and their strong dependence on the waveguide scales and geometries [12,24], based on which we can develop many states-of-the-arts SBS-based applications such as the photonic-phononic memories [25–27], microwave filters and circuits [28–30].

For the optical and elastic waves involved in an SBS process, the phase-matching conditions must be strictly satisfied between their frequencies or wavenumbers, as a result of the conservation of their total energy and momentum. In addition, if they are able to interact effectively to activate an SBS process (i.e., the opto-mechanical couplings between them are non-vanishing), the spatial symmetries of their field distributions must also satisfy certain quantitative relations (i.e., the symmetry selection rules). Analysis of the impacts of the waveguide spatial symmetries has been raised as an important issue in the theoretical research on the on-chip SBSs [31–34]. For a longitudinally invariant waveguide, if the constituent material is isotropic or if the constituent material is anisotropic but its principal axes are perfectly aligned with the symmetry axes of the waveguide structure, the symmetry analysis of the physical fields in the waveguide can be relatively simplified. That is, only the symmetries within the waveguide’s two-dimensional cross-sectional plane need to be considered. Qiu et al. [31] conducted the theoretical analysis and simulation of the SBS in a rectangular silicon waveguide with aligned material and structural axes, demonstrating that the elastic modes excitable by the SBS process are determined by the in-plane symmetries of the optical force fields. Based on a systematic group-theoretical method, Wolff et al. [32] rigorously derived the symmetry selection rules for the SBS processes in several kinds of micro-structured waveguides with typical in-plane symmetries. Their work can be provided as a general theoretical reference for the symmetry analysis and optimal design of optical waveguides and other related SBS-active photonic devices.

The reporting studies mentioned above focus on discussing the impacts of the symmetries of the waveguide geometries on the physical fields and SBSs. However, typical materials for optical waveguides may be optically or mechanically anisotropic. For example, monocrystalline silicon, a ubiquitous material in modern electronics and photonics, possesses strong mechanical anisotropy, although it is optically isotropic. Therefore, the spatial symmetry of an optical waveguide may be jointly determined by the cross-sectional geometry and orientation of the principal axes of its constituent material. This means that in the SBS-related symmetry analysis studies, we may also need to take into account the impacts of material anisotropy. It has been shown numerically that in the waveguides made of materials such as germanium [35], silicon [36,37] and rutile [33,34], the SBS opto-mechanical coupling characteristics may be strongly impacted by the orientation of material axes. Su et al. even [33,34,37] found that a small arbitrary misalignment between the material and structural axes could engender some strong new resonance peaks in the SBS gain spectra. This extraordinary misalignment-sensitive phenomenon demonstrates not only the high sensitivity of SBS to symmetry breaking at the sub-wavelength/nano scales but also the possibility of utilizing material anisotropy to realize some sensitive tuning and control purposes in various applications of SBSs.

The assumption made by Su et al. [33,34] that the principal material axes are allowed to be arbitrarily orientated with respect to fixed waveguide geometry is essentially realizable, at least for silicon waveguides manufacturable on the current advanced etching platforms. There have been experimental reports of utilizing the mechanical anisotropy to control opto-mechanical couplings in a silicon photonic-phononic crystal cavity [38]. The present study is still based on such an assumption. First, via the group-theoretical method, a systematic point-group classification of suspended silicon waveguides is conducted according to their spatial symmetry features. Subsequently, based on the classification results, the impacts of spatial symmetries on the physical fields and SBS opto-mechanical couplings in silicon waveguides are further investigated, obtaining some general symmetry selection rules for SBSs in suspended silicon waveguides. To the present authors’ knowledge, in such related fields as the guided wave optics and nano opto-mechanics, there has been no reporting research regarding the classification and impacts of waveguide spatial symmetries in which the material anisotropies are fully considered.

The remaining part of the present paper is structured as follows. Section 2 introduces the related basic concepts and theories. Section 3 presents the point-group classification results for the suspended silicon waveguides. Section 4 presents the results of the theoretical and numerical analysis on the characteristics of the physical fields and opto-mechanical couplings in the silicon waveguides with different symmetry features. Section 5 concludes the whole study.

2. Fundamentals of the work

2.1 SBS theory

Let us consider the SBS process in an infinitely long waveguide with continuous translational invariance along the z-direction as shown in Fig. 1(a). In the figure, ω and k denote the angular frequency and wavenumber of the optical waves, respectively, while ${\varOmega}$ and q those of the associated elastic wave. (Note that in the present paper, the correspondence of a physical quantity to the pump or Stokes optical waves may be indicated by, respectively, the subscripts “p” or “s” if necessary.) According to whether the collinear Stokes and pump optical waves propagate in the same direction, SBSs can be further classified into the forward and backward SBSs (FSBS and BSBS). Additionally, according to whether the two interacting optical waves are the same guided mode, they can also be classified into the intramodal and intermodal SBSs. Based on the phase-matching conditions (i.e., ${\varOmega \;\ =\ \;\ }{{\omega }_\textrm{p}}\textrm{ - }{{\omega }_\textrm{s}}$ and ${q}\; = \; {{k}_\textrm{p}}\textrm{ - }{{k}_\textrm{s}}$) and the fact that the frequencies of the optical and elastic waves involved in the same SBS process generally differ by several orders of magnitude, we can have ${{\omega }_\textrm{s}}{\; } \approx {\; }{{\omega }_\textrm{p}}$, and for the intramodal scattering, we further have ${{k}_{\textrm{s}}}\; \approx \; {\ \pm }{{k}_\textrm{p}}$, with the symbols “+” and “-” applicable for the FSBS and BSBS, respectively. Therefore, ${q}\; \approx {\; 2}{{k}_\textrm{p}}\; $ and ${\; 0\; }$ for the intramodal BSBS and FSBS, respectively.

 

Fig. 1. (a) Schematic sketch of a waveguide with continuous translational invariance along the z-direction. The waveguide’s core and cladding are filled with materials $\textrm{I }$ and $\textrm{II }$, respectively. (b) Enlarged view of a small segment of the waveguide’s material interface ${\mathbb{C}}$. The small segment is assumed to undergo a displacement u, resulting in that the normal electric field ${{\bf e}_ \bot }$ and tangential electric displacement field ${{\bf d}_\parallel }$, which are discontinuous across the material interface, are changed by ${\Delta }{{\bf e}_ \bot }$ and ${\Delta }{{\bf d}_\parallel }$, respectively. This partially enlarged view illustrates the mechanism of the moving interface effect.

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The opto-mechanical coupling coefficients (OMCCs) [39–42] of an SBS process are generally evaluated as the overlap integrals of the interacting optical and elastic fields over the waveguide’s cross-sectional plane. The integral formula for evaluating the OMCC ${{{\cal C}}_{m}}$ corresponding to the ${{m}^{\textrm{th}}}$ elastic eigenmode of the waveguide can be written as [31,40]

Equation (1) conveys a clear physical meaning of the OMCC, which is essentially the total work done by the optical force on the elastic displacement within the waveguide per unit-length. The OMCC can be further expressed as the coherent sum of two parts corresponding to the ES and RP effects, respectively. Alternatively, we can also define the OMCCs corresponding to the PE and MI effects, and if the irreversible coupling effects and optical loss can be ignored, we can have the relations ${{\cal C}}$_ES= (${{\cal C}}$_PE)$^{\ast}$ and ${{\cal C}}$_RP= (${{\cal C}}$_MI)$^{\ast}$ [40]. (Note that here the subscript “m” labeling the elastic mode number has been omitted for convenience.) If the waveguide material is optically isotropic, the PE and MI OMCCs can be evaluated as follows [40]:

The SBS gain coefficient can be further evaluated based on OMCCs. According to Ref. [31], the total SBS gain coefficient ${{G}_\textrm{B}}$ as a function of the elastic angular frequency ${\varOmega}$ can be expressed as:

2.2 Related concepts of group theory

As indicated by Eq. (1), the strength of the opto-mechanical coupling of an SBS process is dependent on the correlation between the optical force field and the elastic field. If they possess exactly opposite symmetries, the OMCC should vanish, which means that the SBS process will not really be activated. The symmetric properties of the vectorial optical and elastic fields in a waveguide can be obtained in terms of the symmetry operations included in the point group to which the waveguide belongs. The waveguide appears unchanged if it is subject to these symmetry operations.

Let ${{\tilde{\bf g}}^{{(i)}}}\mathbf{(r)}$ denotes the position-dependent vectorial complex amplitude of a certain optical or elastic eigenmode of a waveguide, where the superscript i satisfies $\textrm{1} \le {i} \le {{N}_\textrm{d}}$ with ${{N}_\textrm{d}}$ being the degree of degeneracy of the eigenmode. If a symmetry operator ${\hat{R}}$ associated with the waveguide is applied to ${{\tilde{\bf g}}^{{(i)}}}\mathbf{(r)}$, then according to the group representation theory [43–49], we have

If the degree of degeneracy ${{N}_\textrm{d}}{\ > 1}$, then the representation matrix D corresponding to a certain symmetry operation ${\hat{R}}$ is not unique. It depends on the specific choice of the ${{N}_\textrm{d}}$ basis functions ${{\tilde{\bf g}}^{\textrm{(1)}}}\mathbf{(r)}$, ${{\tilde{\bf g}}^{\textrm{(2)}}}\mathbf{(r)}$, …, ${{\tilde{\bf g}}^{\textrm{(}{{N}_\textrm{d}}\textrm{)}}}{({\bf r})}$ of the subspace corresponding to the considered eigenmode. However, the trace (i.e., the algebraic sum of the diagonal elements) of the representation matrix D is an invariant irrelevant to such choice and specifically referred to as “character” in group theory [45–48]. Then, for any optical or elastic eigenmode of a waveguide, its symmetry property can be characterized by the series of characters obtained under all the symmetry operations of the associated point group, which define an irreducible representation of the point group. If the eigenmode is non-degenerate (i.e., ${{N}_\textrm{d}}{\; = \; 1}$), then the representation matrix D reduces to a scalar factor, which is essentially the character itself (either 1 or -1). Specifically, if all the characters equal to 1 (i.e., the mode field exhibits a monopole-like shape without any anti-symmetric pattern), the corresponding irreducible representation is generally denoted by the symbols A or A₁. Otherwise, it may be denoted by the symbols A₂, B_i (i ≥ 1), etc. If the eigenmode is degenerate (i.e., ${{N}_\textrm{d}}{\; } \ge {\; 1}$), the corresponding irreducible representation may be denoted by the symbols E, E_i, etc.

In addition to Eq. (6), we should also have [43,44,49]

Based on the related concepts of group theory introduced above, we can further formulate the symmetry selection rules for the SBS process in a waveguide. Here we only briefly introduce the general idea of it, while for the details the readers can refer to Ref. [32]. In the case that the interacting optical and elastic modes are all non-degenerate, if and only if the product of their characters equals 1 can the integrand functions of their OMCCs do not possess any odd symmetries. Then, the OMCC is possibly to be nonvanishing, which means that the SBS process may really be activated. In the case that at least one of the interacting optical and elastic modes is degenerate, we can first construct the direct product of their corresponding irreducible representations, which can further be rewritten as a direct sum of another set of irreducible representations. If and only if at least one of these resulting irreducible representations included in the direct sum is of A or A₁ type can the OMCC be non-vanishing.

3. Point-group classification of suspended silicon waveguides

Monocrystalline silicon belongs to the cubic crystal system featuring a unique set of mutually orthogonal principal material axes completely equivalent to one another, which result in the strong mechanical anisotropy of the material. It also manifests a relatively weak photoelastic anisotropy. In the present study, the physical parameters of this material are taken from Ref. [18,50] and listed in [52]. (Note that all the optical parameters correspond to a free-space wavelength of 1550 nm.) In Fig. 2 is defined the principal material coordinate system ξ-ζ-η of a z-invariant suspended silicon waveguide, which is allowed to rotate three-dimensionally and arbitrarily with respect to a fixed structural coordinate system x-y-z. The relative rotation of the principal material coordinate system is characterized by the Euler angles as also defined in Fig. 2.

 

Fig. 2. Definitions of principal material coordinate system ξ-ζ-η of a z-invariant suspended silicon waveguide and Euler angles (ψ, θ, φ) characterizing relative rotation between ξ-ζ-η and structural coordinate system x-y-z. The line of nodes used for defining the Euler angles is denoted by L.

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In terms of the mutual equivalence among the three principal material axes, we can identify the symmetry elements of the material coordinate system ξ-ζ-η defined above, some representative ones of which are shown in Fig. 3. Their detailed definitions are given in the figure caption. In the present paper, symmetry elements and point groups are denoted by the Schoenflies symbols [45–48].

 

Fig. 3. Symmetry elements of principal material coordinate system of monocrystalline silicon. They can be divided into the following categories: the three ${{C}_\textrm{4}}\; $ (${{S}_\textrm{4}}$) axes coinciding with the principal material axes (along the crystallographic directions of [1 0 0], [0 1 0] and [0 0 1]); the six ${{C}_\textrm{2}}$ axes coinciding with the bisectors of the right angles formed by two principal material axes (along the [0 1 ± 1], [1 0 ± 1] and [1 ± 1 0] directions), among which only the one along the [0 1 1] direction is shown in this figure; the four ${{C}_{3}}$ (${{S}_\textrm{6}}$) axes coinciding with the body diagonals of the cubic unit cell of the crystal lattice (along the [1 ± 1 ± 1] directions), among which only the one along the [1 1 1] direction is shown in this figure; the three ${{\sigma }_\textrm{h}}$ symmetry planes perpendicular to the principal material axes, among which only the one perpendicular to the ζ axis is shown in this figure; the six ${{\sigma }_\textrm{d}}$ symmetry planes passing through a ${{C}_\textrm{2}}$ axis and the principal material axis perpendicular to the ${{C}_\textrm{2}}$ axis, among which only the one passing through the ${{C}_\textrm{2}}$ axis along the [0 1 1] direction is shown in this figure; and the inversion center i (i.e., the origin O of the coordinate system) as well.

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Besides the symmetry elements of the principal material coordinate system introduced above, the cross-sectional shape also determines another set of symmetry elements unique to the waveguide geometry. We finally identify the common symmetry elements of the whole silicon waveguide system as the ones shared by both the material and structural systems. Based on the highest order of any proper rotational axis included in these common symmetry elements, the silicon waveguides can be classified into four categories: the ones which possess, respectively, a ${{C}_\textrm{4}}$, ${{C}_\textrm{3}}$ and ${{C}_\textrm{2}}$ axes as the common proper rotational axis of the highest order as well as the ones do not possess any common ${{C}_{n}}$ axes with n ≥ 2. Each category can be further classified into different point groups via the symmetry classification method for molecules [45–48]. The decision tree used for the classification is given in Fig. 4. The point-group classification results obtained for the four categories are listed in Tables 1(a), (b), (c) and (d), respectively. For each point group, we give one or more specific waveguide configuration examples, in which the close boundary of a waveguide’s cross-section is depicted by the solid dark line; the principal axes and ${{C}_\textrm{2}}$ axes of the material coordinate system by the solid lines in orange and gray colors, respectively; and the projections of the principal material axes onto a waveguide’s cross-sectional plane by dashed orange lines.

 

Fig. 4. Decision tree used for point-group classification of suspended silicon waveguides.

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Table 1. Point-group classification results for suspended silicon waveguides

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4. Symmetry and coupling characteristics of physical fields

In this section, we examine the symmetry and coupling characteristics of the physical fields in suspended silicon waveguides according to the point-group classification results presented above. If the waveguides belong to the ${{C}_\textrm{1}}$ point group as listed in Table 1(d), the interacting optical and elastic fields should not possess any anti-symmetries which can result in that the OMCCs completely vanish. Thus, in a completely asymmetric waveguide the SBS interaction is generally very likely to be activated. In addition to such a trivial situation, we further find that the SBS opto-mechanical couplings can also exhibit relatively predictable behaviors (specifically, the OMCCs can be deterministically nonvanishing or vanishing under very few constraints), as long as the silicon waveguides belong to some point groups with certain symmetry features. In the remainder of this section, we will first focus on analyzing the physical fields and SBSs in these special kinds of silicon waveguides, followed by some analysis on other general situations. The numerical results presented below are obtained mainly based on the finite-element commercial software COMSOL Multiphysics.

4.1 FSBSs in waveguides with a mirror symmetry about cross-sectional plane

In the first kind of silicon waveguides we consider, the z-axis is aligned with either a principal axis or ${{C}_\textrm{2}}$ axis of the material system. Therefore, they possess mirror symmetries about their cross-sectional planes. They include all the waveguides belonging to the D_4h, ${{C}_{\textrm{4h}}}$, ${{D}_{\textrm{2h}}}$ or ${{C}_{\textrm{2v}}}$ groups (see Tables 1(a) and (c)), as well as all the type (i) waveguides belonging to the ${{C}_{\textrm{2h}}}$ and ${{C}_\textrm{s}}$ groups (see Tables 1(c) and (d)).

In an intramodal FSBS process, the elastic wave with a nearly vanishing wavenumber propagates almost parallelly with the waveguide’s cross-sectional plane. Then, if we further assume that the material is isotropic, the waveguide’s elastic eigenmodes can be divided into two kinds with mutually orthogonal polarizations: the out-of-plane and in-plane modes. Among them, an out-of-plane mode is purely shear and its displacement field includes the component ${{u}_{z}}$ only, while the displacement field of an in-plane mode includes the components ${{u}_{x}}$ and ${{u}_{y}}$. If the material is anisotropic, however, the elastic eigenmodes can also be classified into such two kinds, provided that they propagate parallelly with a symmetry plane of the principal material coordinate system. (It can be proved based on the Christoffel equation describing the elastic wave propagation in anisotropic material [51].) Specifically, for the mechanically anisotropic material monocrystalline silicon, this means that the waveguide’s cross-sectional plane coincides with the symmetry planes ${{\sigma }_\textrm{h}}$ or ${{\sigma }_\textrm{d}}$ as shown in Fig. 3. Equivalently, we can also say that the z-axis of the waveguide coincides with either a principal or ${{C}_\textrm{2}}$ axis of the material system.

It has been proven in Ref. [34] that for the intramodal FSBS process in a waveguide made of the tetragonal crystal material rutile, the two OMCCs (${{\cal C}}$_PE and ${{\cal C}}$_MI) both vanish, provided that the z-axis is aligned with a principal material axis and the involved elastic mode is out-of-plane polarized. Obviously, the statement remains valid if the waveguide material is replaced with monocrystalline silicon which possesses higher-level symmetries. Furthermore, we also show herein that for the silicon waveguides we considered, such a statement can still be valid even if the z-axis is not aligned with a principal axis but a ${{C}_\textrm{2}}$ axis of the material system. First, the proof that ${{\cal C}}$_MI= 0 is straightforward, as an out-of-plane polarized elastic mode cannot provide the in-plane displacement inducing the deformation of the material interface (i.e., we have ${{\tilde{\bf u}}^{\ast }} \cdot {{\hat{\bf n}}_ \bot } \equiv {\; 0}$ in the integral expression (3) of ${{\cal C}}$_MI). The proof that ${{\cal C}}$_PE= 0 is much more complex, and thus we put the detailed proving process into the Appendix of the present paper.

Next, a rectangular silicon waveguide belonging to the ${{D}_{\textrm{2h}}}$ point group, the z-axis of which is aligned with a C₂ axis of the material system, is exemplified to verify the conclusions obtained above. In Fig. 5 we present the gain spectrum of a FSBS process in this waveguide, detailed geometric parameters of which are given in the figure caption. Note that the pump and Stokes lights are both assumed to be the waveguide’s fundamental optical mode. We can find that each resonant peak present in the gain spectrum belongs to an in-plane polarized elastic mode.

 

Fig. 5. Calculated FSBS gain spectrum of a rectangular ${{D}_{\textrm{2h}}}$ silicon waveguide. The waveguide’s z-axis is aligned with a C₂ axis of the material system. The length and width of the waveguide’s rectangular cross-section are 685.6 nm and 440.8 nm, respectively. As stated in the appendix part, the Euler angles defined in Fig. 2 are taken as: $({{\psi ,\;\ \theta ,\;\ \varphi }} ){\; = }\left( {\frac{\textrm{1}}{\textrm{2}}{\pi ,\;\ }\frac{\textrm{1}}{\textrm{4}}{\pi ,\;\ }\frac{\textrm{3}}{\textrm{2}}{\pi }} \right)$.

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Let us examine the resonant peak in the dashed box as shown in Fig. 5. The spatial profiles of the displacement fields ${{u}_{x}}$, ${{u}_{y}}$, and ${{u}_{z}}$ of the elastic mode corresponding to this resonant peak are presented in Figs. 6(a), (b) and(c), respectively, indicating clearly that the out-of-plane displacement field ${{u}_{z}}$ vanish everywhere. In Fig. 6(d) is presented the spatial profile of the integrand function of the PE OMCC ${{\cal C}}$_PE corresponding to this elastic mode. This spatial profile manifests a completely symmetric pattern, and consequently ${{\cal C}}$_PE does not vanish. (Note that as the elastic wavenumber is set to zero in the simulation, the elastic displacement fields along with the corresponding integrand function ${{\varTheta }_{\textrm{PE}}}$ can all be of real values if the initial phase of the considered elastic mode is taken properly [34]. In addition, we also note that as the integrand functions ${{\varTheta }_{\textrm{PE}}}$ and ${{\varTheta }_{\textrm{MI}}}$ possess the same symmetry features, we do not present the spatial profile of ${{\varTheta }_{\textrm{MI}}}$ in the present paper.)

 

Fig. 6. Calculated spatial profiles of elastic displacement fields [(a) ${{u}_{x}}$, (b) ${{u}_{y}}$ and (c) ${{u}_{z}}$] as well as that of integrand function ${{\varTheta }_{\textrm{PE}}}$ of PE OMCC (d) corresponding to resonance peak in a dashed box as shown in Fig. 5.

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For further verification, we present in Fig. 7 the spatial profiles of the displacement fields ${{u}_{x}}$, ${{u}_{y}}$ and ${{u}_{z}}$ of an out-of-plane polarized elastic mode of the considered waveguide as well as that of the integrand function ${{\varTheta }_{\textrm{PE}}}$ of the PE OMCC corresponding to the same elastic mode. [The frequency of this elastic mode (approximately 13.6 GHz) is marked by the symbol “x” as shown on the horizontal axis of Fig. 5]. Obviously, this elastic mode is out-of-plane polarized in that both ${{u}_{x}}$ and ${{u}_{y}}$ vanish everywhere, and ${{\varTheta }_{\textrm{PE}}}\; \equiv {\; 0}$ as predicted.

 

Fig. 7. Calculated spatial profiles of displacement fields [(a) ${{u}_{x}}$, (b) ${{u}_{y}}$ and (c) ${{u}_{z}}$] as well as that of integrand function ${{\varTheta }_{\textrm{PE}}}$ of PE OMCC (d) corresponding to an elastic mode marked by symbol “x” on horizontal axis of Fig. 5.

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4.2 BSBSs in waveguides belonging to C₂, D₃ and C_i point groups

In this subsection, we consider the intramodal BSBSs in the silicon waveguides belonging to the following three point groups: ${{D}_\textrm{3}}$ (Table 1(b)), ${{C}_\textrm{2}}$ (Table 1(c)) and ${{C}_\textrm{i}}$ (Table 1(d)). Among them, the D₃ and C₂ silicon waveguides possess the C₂ symmetry as the highest-order rotational symmetry about an axis inside the cross-sectional plane, while the C_i silicon waveguides possess no symmetry elements other than an inversion center. We will demonstrate that the OMCC of an intramodal BSBS in a silicon waveguide belonging to the three point groups generally does not vanish due to the symmetry properties of the physical fields in the waveguide.

According to their character tables, the ${{C}_\textrm{2}}$ and ${{C}_\textrm{i}}$ point groups do not have degenerate irreducible representations. The ${{D}_\textrm{3}}$ point group does have degenerate irreducible representation. However, numerical simulation shows that if the wavenumber is nonzero, the optical modes in the ${{D}_\textrm{3}}$ silicon waveguides can be degenerate, but all the elastic modes are nondegenerate. Therefore, for an intramodal BSBS process in the silicon waveguides belonging to the three point groups, we first assume that the involved optical and elastic modes with nonzero wavenumbers are all non-degenerate. The case that the involved optical modes are degenerate will be addressed later.

Let us assume that a symmetry operation ${\hat{R}}$ (either a 180° rotation about an in-plane ${{C}_\textrm{2}}$ axis or an inversion about the origin) is performed for a silicon waveguide belonging to the three point groups. Then, if a vectorial physical field g involved in an intramodal BSBS process in the waveguide is nondegenerate, we should have

Here, the subscripts “PE” or “MI” of ${\varTheta }$ are omitted for convenience. Equations (8) and (9) reveal the conjugate symmetry or anti-symmetry about the waveguide’s in-plane ${{C}_\textrm{2}}$ axis or inversion center possessed by the vectorial field g and scalar field ${\varTheta }$. As will be further proved by the following examples, the conjugate symmetry or anti-symmetry of an integrand function ${\varTheta }$ can result in that the corresponding integral (OMCC) does not vanish, and thus the BSBS process may be activated.

First, an isosceles trapezoidal ${{C}_\textrm{2}}$ silicon waveguide is exemplified to verify the symmetry properties predicted above. As sketched in the upper one of the two waveguide configuration examples (see the part corresponding to the ${{C}_\textrm{2}}$ point group in Table 1(c)), the in-plane structural symmetry axis (y-axis) coincides with a principal material axis, but both the other two principal material axes are not aligned with the longitudinal axis (z-axis) of the waveguide. Detailed geometric parameters of this waveguide will be given in the caption of Fig. 8.

 

Fig. 8. Calculated spatial profiles of electric field components (a) ${{e}_{x}}$, (b) ${{e}_{y}}$ and (c) ${\textrm{i}}{{e}_{z}}$ of fundamental optical mode of considered isosceles trapezoidal C₂ waveguide. As defined in the appendix of the present paper, i is the imaginary unit. For the trapezoid, the lengths of the top and lower bases and the height are taken to be 471.4 nm, 1010.1 nm and 538.8 nm, respectively. The Euler angles defined in Fig. 2 are taken as: $({{\psi ,\;\ \theta ,\;\ \varphi }} ){\; = }\left( {\frac{\textrm{1}}{\textrm{2}}{\pi ,\;\ }\frac{\textrm{5}}{{\textrm{12}}}{\pi ,\;\ }\frac{\textrm{3}}{\textrm{2}}{\pi }} \right)$.

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As the y-axis is the sole in-plane ${{C}_\textrm{2}}$ axis of this waveguide, the second-order tensor R in Eq. (8) is of a diagonal form written as diag (-1, 1, -1). Substituting it into Eq. (8) and considering the independence on the coordinate z of the complex amplitude ${\tilde{\bf g}}$ of a vectorial physical field g, we obtain the following relations:

In Fig. 8 are presented the spatial profiles of the electric fields of the fundamental optical mode in the considered waveguide. (By taking a proper initial phase, the in-plane components ${{e}_{x}}$ and ${{e}_{y}}$ and out-of-plane component of the electric field ${{e}_{z}}$ as shown here are of purely real and imaginary values, respectively.) If the character c = -1, then it can be checked that Eqs. (10a)∼(10c) do describe the symmetries about the y-axis of the three electric field components as shown in Fig. 8, respectively.

Let us further consider the intramodal BSBS process in which the fundamental optical mode just considered of this trapezoidal waveguide is involved. The calculated gain spectrum is presented in Fig. 9, while in Fig. 10 are presented the spatial profiles of the real and imaginary parts of the elastic displacement fields corresponding to the resonance peak in the dashed box as shown in Fig. 9. If the character c also equals -1, then we can find that the symmetries of the complex-valued elastic field components ${{u}_{x}}$, ${{u}_{y}}$ and ${{u}_{z}}$ do satisfy Eqs. (10a)∼(10c), respectively. As a result of the predicted conjugate symmetry or anti-symmetries, the real and imaginary parts of each elastic displacement field exhibit opposite parities with respect to the y-axis.

 

Fig. 9. Calculated BSBS gain spectrum of considered isosceles trapezoidal C₂ silicon waveguide.

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Fig. 10. Calculated spatial profiles of real (upper) and imaginary (lower) parts of displacement fields (a) ${{u}_{x}}$, (b) ${{u}_{y}}$ and (c) ${{u}_{z}}$ of elastic modes corresponding to a resonance peak in dashed box as shown in Fig. 9.

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In Fig. 11 we present the spatial profiles of the real and imaginary parts of the complex-valued integrand function of the PE OMCC corresponding to the considered elastic mode. The real and imaginary parts of the integrand function are antisymmetric and symmetric about the y-axis, respectively, which is consistent with the theoretical prediction of Eq. (11) if the character c is still taken to be -1. The observed opposite parities of the real and imaginary parts of the integrand function cannot result in that the real and imaginary parts of the corresponding integral (the OMCC ${{\cal C}}$_PE) both vanish. In the present case, the real and imaginary parts of ${{\cal C}}$_PE is vanishing and non-vanishing, respectively. Thus, the complex-valued ${{\cal C}}$_PE does not vanish, as the corresponding significant resonant peak observed in the gain spectrum indicates.

 

Fig. 11. Calculated spatial profiles of (a) real and (b) imaginary parts of integrand ${{\varTheta }_{\textrm{PE}}}$ of PE OMCC corresponding to a resonance peak in dashed box as shown in Fig. 9.

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Next, we consider a regular-triangular D₃ silicon waveguide. In Fig. 12 we present the calculated spatial profiles of the real and imaginary parts of the integrand function of the PE OMCC of an intramodal BSBS process in which a nondegenerate optical mode of this waveguide is involved. The detailed geometric parameters of this waveguide are given in the figure caption. As theoretically predicted, the real and imaginary parts of the integrand function also exhibit opposite parities about the y-axis.

 

Fig. 12. Calculated spatial profiles of (a) real and (b) imaginary parts of integrand function ${{\varTheta }_{\textrm{PE}}}$ of PE OMCC of a BSBS process in which a nondegenerate optical mode of considered regular-triangular D₃ silicon waveguide is involved. The side length of the regular triangle is 775 nm. The Euler angles defined in Fig. 2 are taken as: $({{\psi ,\;\ \theta ,\;\ \varphi }} ) = \left( {\frac{{3}}{{4}}{\pi ,\;\ }{\textrm{cos}^{{ - 1}}}\frac{{\sqrt {3} }}{{3}}{,\; }\frac{{3}}{{2}}{\pi }} \right)$.

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As mentioned earlier, the optical modes of a silicon waveguide belonging to the D₃ point group may be degenerate. So far, we have not considered the case that the involved optical mode of an intramodal BSBS process is degenerate. Therefore, for the regular-triangular silicon waveguide just considered, we select a pair of mutually degenerate optical modes as the respective involved optical modes of two intramodal BSBS processes, but the involved elastic modes of the two BSBS processes are exactly the same one. The calculated spatial profiles of the absolute values of the integrand functions of the PE OMCCs corresponding to the two BSBS processes are presented in Figs. 13(a) and (b), respectively. Obviously, neither of the two spatial profiles possesses any symmetry or anti-symmetry, so the corresponding OMCCs does not vanish. Nevertheless, Figs. 13(a) and (b) are mirror images of each other, which reflects the ${{C}_\textrm{2}} $ symmetry of the two mutually degenerate optical modes as a whole. This example demonstrates that the OMCC of an intramodal BSBS process in a D₃ silicon waveguide generally does not vanish even if the involved optical mode is degenerate.

 

Fig. 13. Calculated spatial profiles of absolute values of integrand functions (${{\varTheta}_{\textrm{PE}}}$) of PE OMCCs of two intramodal BSBS processes whose involved optical modes are a pair of mutually degenerate optical modes of considered regular-triangular D₃ silicon waveguide. The subfigures (a) and (b) correspond to different ones of the two BSBS processes.

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4.3 BSBSs in waveguides belonging to C₃ point group

The last special kind of silicon waveguides belong to the ${{C}_\textrm{3}}$ point group only. We also consider the intramodal BSBSs in this kind of silicon waveguides. According to its character table [45–48], the C₃ point group only possesses two irreducible representations with the characters A and E, respectively. Numerical simulation shows that for the optical and elastic modes with nonzero wavenumbers involved in an intramodal BSBS process in a ${{C}_\textrm{3}}$ silicon waveguide, the optical modes can be either nondegenerate or degenerate, but the elastic mode can be nondegenerate only. If the involved optical and elastic modes are all nondegenerate, the symmetry properties of them can all be described by the irreducible representation with a character A. Then, their scalar product (the OMCC integrand function) will not exhibit any anti-symmetric pattern and consequently the OMCC generally does not vanish. In Fig. 14 is presented the gain spectrum of a BSBS process in a regular-triangular ${{C}_\textrm{3}}$ silicon waveguide. The detailed geometric parameters of this waveguide are given in the figure caption. The involved optical mode we select is the first nondegenerate optical mode of this waveguide. In Figs. 15(a) and (b), we present the respective calculated spatial profiles of the real and the imaginary parts of the integrand function of the PE OMCC corresponding to the resonance peak in the dashed box as shown in Fig. 14. As theoretically predicted, the 3-fold rotationally symmetric spatial profiles do not possess any anti-symmetries, so the corresponding integral (OMCC) does not vanish.

 

Fig. 14. Calculated BSBS gain spectrum of considered regular-triangular ${{C}_\textrm{3}}$ silicon waveguide. The side length of the regular triangle is taken to be 775 nm. The Euler angles defined in Fig. 2 are taken as: $({{\psi ,\;\ \theta ,\;\ \varphi }} ){ = }\left( {\frac{{3}}{{4}}{\pi ,\;\ }{\textrm{cos}^{{ - 1}}}\frac{{\sqrt {3} }}{\textrm{3}}{,\; }\frac{\textrm{5}}{\textrm{4}}{\pi }} \right)$.

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Fig. 15. Calculated spatial profiles of (a) real and (b) imaginary parts of OMCC integrand function ${{\varTheta }_{\textrm{PE}}}$ corresponding to a resonance peak in dashed box as shown in Fig. 14.

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Just like the corresponding case considered for the intramodal BSBS in a D₃ silicon waveguide (see the part corresponding to Fig. 13 in Section 4.2), if the involved optical mode is degenerate, we also find numerically that the OMCC integrand function is generally asymmetric and thus does not possess any anti-symmetries which can result in that the OMCC vanish. For detailed theoretical analysis of this finding, we may resort to formal selection rules derivation based on the direct-product decomposition formulae for the considered point group [32]. Alternatively, we provide here a qualitative understanding as follows: For an intramodal BSBS process in a C₃ silicon waveguide, the optical force produced by the pump and Stokes optical modes with the same modal number should be decomposable as the superposition of several components including a one of A-symmetry, regardless of whether the two optical modes are degenerate or not. The A-symmetry component in the optical force field further interacts with the involved nondegenerate elastic mode of A-symmetry, producing nonvanishing OMCCs.

4.4 Other cases

Except the cases considered above, for the FSBS or BSBS in any given silicon waveguide, we generally cannot say for sure whether the OMCCs are vanishing or not. It depends on the specific symmetry properties of the optical and elastic modes involved in SBS. Let us consider, for example, the FSBS in a ${{C}_\textrm{3}}$ silicon waveguide. As mentioned earlier, the ${{C}_\textrm{3}}$ point group possesses only two irreducible representations with the characters A and E, respectively. Unlike the corresponding BSBS case considered earlier, the elastic mode involved in a FSBS process in a ${{C}_\textrm{3}}$ silicon waveguide may be degenerate due to the nearly vanishing wavenumber. If the involved optical and elastic modes are both nondegenerate, the A-symmetries possessed by them result in that the OMCC integrand functions should also be of an A-symmetry and thus the integrals (OMCCs) may not vanish. However, if the elastic mode is no longer nondegenerate, the optical mode’s A symmetry and the elastic mode’s E symmetry lead to the E symmetries of the OMCC integrand functions. Thus, the OMCCs are sure to vanish and the FSBS process cannot really be activated.

However, for the silicon waveguides belonging to the ${{D}_{\textrm{3d}}}$, ${{S}_\textrm{6}}$, and ${{C}_{\textrm{3v}}}$ point groups which also possess a ${{C}_\textrm{3}}$ axis, we are unable to obtain their SBS opto-mechanical coupling properties theoretically unless formal symmetry selection rules are rigorously derived [32]. Here, we demonstrate, numerically, that for any silicon waveguide belonging to these point groups, the symmetric and anti-symmetric structures are both possibly exhibited by the OMCC integrand functions of both the FSBS and BSBS processes. In Figs. 16–21 are presented the calculated spatial profiles of PE OMCC integrand functions of the FSBS or BSBS processes in specific examples of these kinds of silicon waveguides. (See the figure captions for detailed parameters of the exemplified waveguides. As a separate note, the choice for the optical and elastic modes involved in each simulated SBS process is not specified in the figure caption due to the relative unimportance of such information.) For both the FSBS and BSBSs in each exemplified waveguide, we first present a case that the spatial profiles of the real and imaginary parts of the OMCC integrand function both possess anti-symmetries, and thereafter present another case that the spatial profile of either the real or imaginary part of the OMCC integrand function does not possess any anti-symmetry. Only in the latter case, the corresponding integral (OMCC) is likely to be nonvanishing.

 

Fig. 16. Calculated (a) anti-symmetric and (b) symmetric spatial profiles of real-valued OMCC integrand functions ${{\varTheta }_{\textrm{PE}}}$ of FSBS processes in a regular-hexagonal ${{D}_{\textrm{3d}}}$ silicon waveguide. The side length of the regular hexagon is taken to be 258.3 nm. The Euler angles defined in Fig. 2 are taken as: $({{\psi ,\;\ \theta ,\;\ \varphi }} ){\; = }\left( {\frac{\textrm{3}}{\textrm{4}}{\pi ,\;\ }{\textrm{cos}^{{ - 1}}}\frac{{\sqrt {3} }}{\textrm{3}}{,\; }\frac{\textrm{4}}{\textrm{3}}{\pi }} \right)$.

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Fig. 17. Calculated spatial profiles of complex-valued OMCC integrand functions ${{\varTheta }_{\textrm{PE}}}$ with (a) real and imaginary parts being both anti-symmetric and (b) anti-symmetric real part and symmetric imaginary part of BSBS processes in a regular-hexagonal ${{D}_{\textrm{3d}}}$ silicon waveguide with geometric parameters specified in caption of Fig. 16.

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Fig. 18. Calculated (a) anti-symmetric and (b) symmetric spatial profiles of real-valued OMCC integrand functions ${{\varTheta }_{\textrm{PE}}}$ of FSBS processes in a regular-hexagonal ${{S}_\textrm{6}}$ silicon waveguide. The side length of the regular hexagon is taken to be 258.3 nm. The Euler angles defined in Fig. 2 are taken as: $({{\psi ,\;\ \theta ,\;\ \varphi }} ){\; = }\left( {\frac{\textrm{3}}{\textrm{4}}{\pi ,\;\ }{\textrm{cos}^{\textrm{ - 1}}}\frac{{\sqrt {3} }}{\textrm{3}}{,\; }\frac{\textrm{5}}{\textrm{4}}{\pi }} \right)$.

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Fig. 19. Calculated spatial profiles of complex-valued OMCC integrand functions ${{\varTheta }_{\textrm{PE}}}$ with real and imaginary parts being both (a) anti-symmetric and (b) symmetric of BSBS processes in a regular-hexagonal ${{S}_\textrm{6}}$ silicon waveguide with geometric parameters specified in caption of Fig. 18.

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Fig. 20. Calculated (a) anti-symmetric and (b) symmetric spatial profiles of real-valued OMCC integrand functions ${{\varTheta }_{\textrm{PE}}}$ of FSBS processes in a regular-triangular ${{C}_{\textrm{3v}}}$ silicon waveguide. The side length of the regular triangle is taken to be 775 nm. The Euler angles defined in Fig. 2 are taken as: $({{\psi ,\;\ \theta ,\;\ \varphi }} ){\; = }\left( {\frac{\textrm{3}}{\textrm{4}}{\pi ,\;\ }{\textrm{cos}^{\textrm{ - 1}}}\frac{{\sqrt {3} }}{\textrm{3}}{,\; }\frac{\textrm{4}}{\textrm{3}}{\pi }} \right)$.

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Fig. 21. Calculated spatial profiles of complex-valued OMCC integrand functions ${{\varTheta }_{\textrm{PE}}}$ with real and imaginary parts being both (a) anti-symmetric and (b) symmetric of BSBS processes in a regular-triangular ${{C}_{\textrm{3v}}}$ silicon waveguide with geometric parameters specified in caption of Fig. 20.

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5. Concluding remarks

Based on group theory, we conduct a theoretical and numerical study on the impacts of spatial symmetries on the physical fields and SBS opto-mechanical couplings in nanoscale suspended silicon waveguides, in which the material anisotropies of monocrystalline silicon are fully considered. We first systematically classify the suspended silicon waveguides into a number of point groups according to the spatial symmetry features possessed by them. Based on the detailed classification results [see Tables 1(a)∼(d)], the symmetry characteristics of physical fields and SBS opto-mechanical coupling characteristics in the silicon waveguides belonging to different point groups are further examined, and for the sake of clarity, these characteristics have been tabulated altogether into Table 2 of the present paper. Among them, we can summarize the major new findings as follows: The SBS opto-mechanical couplings in several kinds of silicon waveguides with certain nontrivial symmetry features exhibit relatively predictable behaviors in that the OMCCs can be deterministically vanishing or nonvanishing under very few constraints, which can thus serve as general symmetry selection rules for SBSs in suspended silicon waveguides. The results obtained in the present study may be provided as theoretical references for the design and application of SBS-active silicon photonic devices, and may also be helpful for us to gain some deeper insights into the sub-wavelength opto-mechanics in anisotropic media.

Table 2. Symmetry characteristics of physical fields and SBS opto-mechanical coupling characteristics in suspended silicon waveguides

View Table | View all tables in this article

APPENDIX: Proof that the PE OMCC of an out-of-plane polarized elastic mode vanishes if the waveguide’s z-axis is aligned with a ${{C}_\textrm{2}}$ axis of the material system

According to Table 1 (see the waveguide configuration examples for the D_2h, C_2h, C_2v and C_s groups), if the z-axis is aligned with a ${{C}_\textrm{2}}$ axis of the material system, then another ${{C}_\textrm{2}}$ axis along with a principal material axis should locate inside the waveguide’s cross-sectional plane. We can take them as the x-axis and y-axis of the structural coordinate system, respectively, as shown in Fig. 22. Then, it is not difficult to obtain the Euler angles defined in Fig. 2 as: $({{\psi ,\;\ \theta ,\;\ \varphi }} )\textrm{ = }\left( {\frac{\textrm{1}}{\textrm{2}}{\pi ,\;\ }\frac{\textrm{1}}{\textrm{4}}{\pi ,\;\ }\frac{\textrm{3}}{\textrm{2}}{\pi }} \right)$.

 

Fig. 22. Structural and material coordinate systems of a silicon waveguide whose z-axis is aligned with a ${{C}_\textrm{2}}$ axis of material system.

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For the subsequent derivation, we first transform the material property tensors from the principal coordinate system ξ-ζ-η to the structural coordinate system x-y-z. Silicon belongs to the cubic crystal system and is optically isotropic. Its second-order dielectric permittivity tensor is thus a spherical tensor which remains unchanged after the coordinate transformation. However, the material is photoelastically anisotropic. Despite that the anisotropy is not very strong, accurate evaluation of the components of the fourth-order photoelastic coefficients tensor in any given coordinate system should be accomplished via a rigorous coordinate transformation operation. Let us assume that such photoelastic coefficients tensors are ${\mathbf{p}_\textrm{0}}$ and p in the coordinate systems ξ-ζ-η and x-y-z, respectively. In Voigt notation, they can be expressed as ${6\ \times 6}$ matrices. The tensor ${\mathbf{p}_\textrm{0}}$ contains only three independent nonzero elements ${p}_{\textrm{11}}^{\textrm{(0)}}$, ${p}_{\textrm{12}}^{\textrm{(0)}}$ and ${p}_{\textrm{44}}^{\textrm{(0)}}$ distributing inside the Voigt matrix as follows:

(12)$$[{{{\bf p}_0}} ]= \left[ {\begin{array}{{cccccc}} {p_{11}^{(0)}}&{p_{12}^{(0)}}&{p_{12}^{(0)}}&{}&{}&{}\\ {p_{12}^{(0)}}&{p_{11}^{(0)}}&{p_{12}^{(0)}}&{}&{}&{}\\ {p_{12}^{(0)}}&{p_{12}^{(0)}}&{p_{11}^{(0)}}&{}&{}&{}\\ {}&{}&{}&{p_{44}^{(0)}}&{}&{}\\ {}&{}&{}&{}&{p_{44}^{(0)}}&{}\\ {}&{}&{}&{}&{}&{p_{44}^{(0)}} \end{array}} \right]\textrm{ }{\kern 1pt} \textrm{,}$$

where the zero elements are displayed as blanks. The Voigt matrix of p is related to that of ${\mathbf{p}_\textrm{0}}$ as

(13)$$[{\bf p} ]\textrm{ = }{\bf A}[{{{\bf p}_\textrm{0}}} ]{{\bf A}^\textrm{T}}{\kern 1pt} ,$$

where the superscript T denotes the matrix transposition operation, and A is a ${6\ \times 6}$ matrix defined as [51]

(14)$${\bf A} = \left[ {\begin{array}{{cccccc}} {a_{11}^2}&{a_{12}^2}&{a_{13}^2}&{2{a_{12}}{a_{13}}}&{2{a_{13}}{a_{11}}}&{2{a_{11}}{a_{12}}}\\ {a_{21}^2}&{a_{22}^2}&{a_{23}^2}&{2{a_{22}}{a_{23}}}&{2{a_{23}}{a_{21}}}&{2{a_{21}}{a_{22}}}\\ {a_{31}^2}&{a_{32}^2}&{a_{33}^2}&{2{a_{32}}{a_{33}}}&{2{a_{33}}{a_{31}}}&{2{a_{31}}{a_{32}}}\\ {{a_{21}}{a_{31}}}&{{a_{22}}{a_{32}}}&{{a_{23}}{a_{33}}}&{{a_{22}}{a_{33}} + {a_{23}}{a_{32}}}&{{a_{23}}{a_{31}} + {a_{21}}{a_{33}}}&{{a_{21}}{a_{32}} + {a_{22}}{a_{31}}}\\ {{a_{31}}{a_{11}}}&{{a_{32}}{a_{12}}}&{{a_{33}}{a_{13}}}&{{a_{32}}{a_{13}} + {a_{33}}{a_{12}}}&{{a_{33}}{a_{11}} + {a_{31}}{a_{13}}}&{{a_{31}}{a_{12}} + {a_{32}}{a_{11}}}\\ {{a_{11}}{a_{21}}}&{{a_{12}}{a_{22}}}&{{a_{13}}{a_{23}}}&{{a_{12}}{a_{23}} + {a_{13}}{a_{22}}}&{{a_{13}}{a_{21}} + {a_{11}}{a_{23}}}&{{a_{11}}{a_{22}} + {a_{12}}{a_{21}}} \end{array}} \right]\textrm{ }{\kern 1pt} \textrm{,}$$

with ${{a}_{{ij}}}$ being the element of the coordinate transformation matrix a defined according to the Euler angles as

(15)$$\scalebox{0.96}{$\begin{array}{@{}c@{}} {\bf a}({\psi ,\theta ,\phi } )= \\ \left[ {\begin{array}{@{}ccc@{}} {\cos (\phi )\cos (\psi )- \sin (\phi )\cos (\theta )\sin (\psi )}&{\cos (\phi )\sin (\psi )+ \sin (\phi )\cos (\theta )\cos (\psi )}&{\sin (\phi )\sin (\theta )}\\ { - \sin (\phi )\cos (\psi )- \cos (\phi )\cos (\theta )\sin (\psi )}&{ - \sin (\phi )\sin (\psi )+ \cos (\phi )\cos (\theta )\cos (\psi )}&{\cos (\phi )\sin (\theta )}\\ {\sin (\theta )\sin (\psi )}&{ - \sin (\theta )\cos (\psi )}&{\cos (\theta )} \end{array}} \right].\end{array}$}$$

With the Euler angles given above, we can have

(16)$${\bf a}({\psi ,\theta ,\phi } )= \left[ {\begin{array}{{ccc}} {\sqrt 2 /2}&0&{ - \sqrt 2 /2}\\ 0&1&0\\ {\sqrt 2 /2}&0&{\sqrt 2 /2} \end{array}} \right]{\kern 1pt} {\kern 1pt} \textrm{,}$$

And

(17)$${\bf A} = \left[ {\begin{array}{{cccccc}} {1/2}&0&{1/2}&0&{ - 1}&0\\ 0&1&0&0&0&0\\ {1/2}&0&{1/2}&0&1&0\\ 0&0&0&{\sqrt 2 /2}&0&{\sqrt 2 /2}\\ {1/2}&0&{ - 1/2}&0&0&0\\ 0&0&0&{ - \sqrt 2 /2}&0&{\sqrt 2 /2} \end{array}} \right]{\kern 1pt} {\kern 1pt} .$$

Then, based on Eq. (13) along with Eqs. (12) and (17), we obtain that the $\mathrm{6\ \times 6}$ Voigt matrix of the photoelastic coefficients tensor p can be expressed as:

(18)$$[{\bf p} ]= \left[ {\begin{array}{{cccccc}} {{p_{11}}}&{{p_{12}}}&{{p_{13}}}&{}&{}&{}\\ {{p_{12}}}&{{p_{22}}}&{{p_{12}}}&{}&{}&{}\\ {{p_{13}}}&{{p_{12}}}&{{p_{11}}}&{}&{}&{}\\ {}&{}&{}&{{p_{44}}}&{}&{}\\ {}&{}&{}&{}&{{p_{55}}}&{}\\ {}&{}&{}&{}&{}&{{p_{44}}} \end{array}} \right]\textrm{ }{\kern 1pt} {\kern 1pt} \textrm{,}$$

Where

(19)$${p_{11}} = \frac{1}{2}p_{11}^{(0)} + \frac{1}{2}p_{12}^{(0)} + p_{44}^{(0)}{\kern 1pt} ,$$

(20)$${p_{12}} = p_{12}^{(0)}{\kern 1pt} ,$$

(21)$${p_{13}} = \frac{1}{2}p_{11}^{(0)} + \frac{1}{2}p_{12}^{(0)} - p_{44}^{(0)}{\kern 1pt} ,$$

(22)$${p_{22}} = p_{11}^{(0)}{\kern 1pt} ,$$

(23)$${p_{44}} = p_{44}^{(0)}{\kern 1pt} ,$$

(24)$${p_{55}} = \frac{1}{2}p_{11}^{(0)} - \frac{1}{2}p_{12}^{(0)}{\kern 1pt} ,$$

Based on Eq. (18) along with the nonzero elements of p defined above, it is not difficult to prove that for the intramodal FSBS process of a suspended silicon waveguide whose z-axis is aligned with a ${{C}_\textrm{2}}$ axis of the material system, the integrand ${{\varTheta }_{\textrm{PE}}}$ of the PE OMCC corresponding to an out-of-plane polarized elastic mode can be expressed as follows:

(25)$${\varTheta _{\textrm{PE}}} = 2{\varepsilon _0}\varepsilon _r^2\left\{ \begin{array}{l} {[{\widetilde e_x^{(\textrm{s})}} ]^\ast }{p_{55}}\widetilde s_{zx}^\ast \widetilde e_z^{(\textrm{p})} + {[{\widetilde e_z^{(\textrm{s})}} ]^\ast }{p_{55}}\widetilde s_{zx}^\ast \widetilde e_x^{(\textrm{p})} + \\ {[{\widetilde e_y^{(\textrm{s})}} ]^\ast }{p_{44}}\widetilde s_{yz}^\ast \widetilde e_z^{(\textrm{p})} + {[{\widetilde e_z^{(\textrm{s})}} ]^\ast }{p_{44}}\widetilde s_{yz}^\ast \widetilde e_y^{(\textrm{p})} \end{array} \right\}{\kern 1pt} {\kern 1pt} .$$

In the derivation leading to Eq. (25), we make use of the fact that a single displacement component ${{u}_{z}}$ can only produce non-zero shear strains ${{s}_{{zx}}}$ and ${{s}_{{yz}}}$, as the normal strain ${{s}_{{zz}}}{ = }\frac{{\partial {{u}_{z}}}}{{\partial {z}}}{ = {\textrm{i}}q}{{u}_{z}}{\;} \approx {\; 0}$ (with i being the imaginary unit). Then, based on the general phase difference ${\pm} \frac{\pi }{2}$ between the out-of-plane electric field ${{e}_{z}}$ and the in-plane electric fields ${{e}_{x}}$ and ${{e}_{y}}$ in a z-invariant optically isotropic waveguide [34] as well as the fact that the pump and Stokes lights propagate in the same direction in an intramodal FSBS process, we can finally obtain that ${{\varTheta }_{\textrm{PE}}}{\; = \; 0}$ and thus ${{\cal C}}$_PE= 0.

Funding

National Natural Science Foundation of China (12272036); State Key Laboratory of Acoustics, Chinese Academy of Sciences; Youth Innovation Promotion Association of the Chinese Academy of Sciences (2021023).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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52. The material parameters of silicon are listed as follows. The optical refractive index n=3.4777; mass density ρ=2329 kg/m3; elastic constants (in units of GPa): C₁₁=165.78, C₁₂=63.94 and C₄₄=79.62; and photoelastic coefficients: p₁₁= -0.094, p₁₂= 0.017 and p₄₄= -0.051.