Formal selection rules for Brillouin scattering in integrated waveguides and structured fibers

We derive formal selection rules for Stimulated Brillouin Scattering (SBS) in structured waveguides. Using a group-theoretical approach, we show how the waveguide symmetry determines which optical and acoustic modes interact for both forward and backward SBS. We present a general framework for determining this interaction and give important examples for SBS in waveguides with rectangular, triangular and hexagonal symmetry. The important role played by degeneracy of the optical modes is illustrated. These selection rules are important for SBS-based device design and for a full understanding the physics of SBS in structured waveguides.


Introduction
Stimulated Brillouin Scattering (SBS) is the coherent nonlinear interaction between light and sound in matter [1]. It was initially predicted by Brillouin [2] and subsequently observed in quartz shortly after the invention of the laser [3]. Although it is the strongest third-order nonlinear process, it is usually quite weak in practical situations, meaning that long interaction lengths are required to observe significant Brillouin gain at moderate light intensities. As a result, most existing or proposed applications such as Brillouin lasers [4,5], all-optical isolators [6] or in microwave photonics [7] are based on optical fibers or on-chip waveguides. The interaction in conventional fibers is dominated by electrostriction and the photoelastic effect, which, until relatively recently, were viewed as the only significant processes underlying SBS.
Due to the nonlinear nature of the process, however, the interaction can be dramatically increased in strength by confining the optical mode more tightly, thereby increasing the field intensity for the same transmitted power. Consequently, theoretical and experimental investigations of SBS in structured fibers and integrated photonic waveguides have recently received considerable attention [8][9][10][11][12][13] with new physical mechanisms [11,12,14] being found to contribute to the SBS gain. In addition to the intensity enhancement, such structured waveguides further differ from traditional systems such as step-index fibers in that they support a broad variety of acoustic and optical modes. As solutions to linear wave equations, both sets of modes must respect the waveguide's symmetries (rectangular for many integrated waveguides, triangular or hexagonal for many photonic crystal or suspended core fibers), as expressed by their corresponding point groups. These symmetries play a key role in the question of whether a given combination of optical and acoustic modes can have non-zero SBS-interaction. This fact has been recently studied for the specific case of a rectangular waveguide; Qiu et al. [15] numerically computed SBS gains for a range of possible interactions and found starkly different gains for different mode combinations.
In this paper, we approach SBS interactions on a formal level using group representation theory. We derive clear selection rules for SBS in rectangular, triangular and hexagonal waveguides, these being the most common point-groups arising in photonic waveguides. Although the formalism of group theory is, a well-established approach in studying physical interactions of all types, its use in multi-wave interactions in the context of photonic waveguides has been relatively limited. The definitive categorization of symmetry classes for optical modes was given by McIsaac in Ref. [16,17], however the acoustic modes have not been similarly analyzed, and a systematic tabulation of the different types of acoustic modes, including their symmetries and possible degeneracies, is one aim of the present study. More importantly, SBS involves interactions between optical and acoustic modes in nonlinear combination, and the Fig. 1. Illustration of three different waveguide groups, divided according to the waveguide symmetry; a) Rectangular waveguides (C 2v group); b) Triangular waveguides (C 3v group) and c) hexagonal waveguides (C 6v group).
correct conjunction of modes that will permit a non-zero contribution to the gain is a non-trivial problem. Here, we systematically show which acoustic modes can be used to drive SBS transitions between optical modes, and conversely, which optical modes can be used to generate desired acoustic vibrations. These selection rules give important physical insight into the dominant couplings in SBS-active experiments, and can be used to greatly reduce the number of mode combinations that have to be considered in gain calculations. For example, we show that for backwards SBS, which generally involves transistions between the same optical mode, no flexural or torsional acoustic modes can possibly contribute to the gain in rectangular or hexagonal structures, whereas flexural modes can contribute in triangular waveguides. Knowledge of the appropriate symmetries of the acoustic modes can also simplify the design of SBS-active waveguides, because once the symmetries of the interacting modes are known, the computational domains can be restricted by imposing appropriate boundary conditions along planes of high symmetry [16]. This of course significantly reduces the calculation speed, but more importantly it greatly simplifies the isolation of degenerate or near-degenerate modes.
The paper is organized as follows: In Sections 2 we recall some key results about optoacoustic coupling in SBS processes. In Section 3, we briefly review the relevant fundamentals of group representation theory. In Section 4, we formulate the acousto-optic interaction in a way that is convenient for a symmetry analysis, and thereby derive a necessary condition on the mode symmetries to permit non-zero SBS-gain. Up to this point, the paper applies to any waveguide symmetry, including circular waveguides and more exotic cases such as five-fold symmetry. In Section 5, we apply this to the aforementioned three important symmetry classes (rectangular, triangular, hexagonal) and provide examples for combinations of modes with nonvanishing interaction. Section 6 concludes our paper.

Opto-acoustic coupling in SBS
We consider longitudinally invariant opto-acoustic waveguides oriented along the z-axis, and possessing some degree of rotation or reflection symmetry in the x-y-plane (see Fig. 1). The SBS process can be described either in terms of field perturbations or in terms of optical forces and the mechanical displacement field. The descriptions are equivalent [14]; here we choose to work with the optical forces. The coupling between optical and acoustic modes is proportional to the overlap integral where u is the displacement field pattern of the relevant acoustic mode and f is the contribution of the optical force field (due to the two interacting optical modes) that is phase-matched to the acoustic mode. The integration is carried out over the cross section of the waveguide. Most key properties of the SBS interaction follow from knowledge of Q. For example, in long waveguides at steady state, the conventional SBS power gain parameter Γ is given by where κ is a detuning from phase-matching, α is the attenuation coefficient of the phonon mode, and the optical and acoustic modes have been assumed to carry one unit P of power. Thus, maximising the efficiency of calculation and understanding of the overlap Q underlies effective manipulation of SBS as a nonlinear tool. As discussed in a number of recent works [11,12,14] the force field f has two contributions: electrostriction and boundary radiation pressure, which lead to separate contributions to the coupling density In the first term, which describes electrostriction, ε 0 is the permittivity of free space, ε (a,b) r is the relative dielectric constant of the core (a) and cladding (b) media, e (1,2) are the complex mode amplitudes of the two optical fields and p is the fourth rank photoelastic tensor. The second term, describing radiation pressure, is expressed in terms of field value limits on the interior waveguide boundary through the waveguide local normal vector n and the fully invariant tensor [11,14] Here δ i j is the Kronecker symbol and δ (r ∈ B) is a Dirac-like distribution that is non-zero only on the waveguide boundary B. A third term is caused by the interaction between the magnetic optical fields and a dynamic magnetization caused by the motion of the material [14]. This term can be effectively treated as an antisymmetric part of the photoelastic tensor p [18] and is typically weak. The primary goal of this paper is to use symmetry arguments to provide a systematic and efficient procedure for determining whether the integral in Eq. (1) is non-vanishing or not.

Symmetry properties of optical and acoustic modes
To proceed, we must recall a number of properties of the representation theory of discrete point groups (see for example [19,20]). Formally, letR be a point symmetry operator, i. e. a rotation or mirror operation about the (longitudinal) z-axis, and let R be the corresponding 3 × 3-matrix that performs this operation in Cartesian space. The image of a scalar function φ (r) underR is found by transforming the underlying coordinate system r = (x, y, z): If the function is vector or tensor-valued, these values must also be transformed. For a first rank tensor (i. e. vector) field v(r) (such as the electric field or the mechanical displacement field) and a second rank tensor field T(r) (e. g. the stress tensor) the transformation becomes: Table 1. Relevant parts of the character tables for the groups C 2v , C 3v and C 6v . These tables are standard in many textbooks on group theory and we reproduce them here for convenience following the notation of Ref. [19].
Pseudotensorial quantities such as the magnetic field differ from this transformation by an additional factor det R = ±1.
If the waveguide in all its aspects (specifically including its tensorial material parameters,) is invariant with respect to a given set of point symmetry operations, these operations form a group G with respect to composition. Common examples are the C 2v , C 3v and C 6v groups that apply to waveguides with rectangular, triangular and hexagonal symmetry. The elements of the C nv groups consist of the identity operation (E), all multiples of the 2π/n rotation (C n ), and 2n reflection operations that are created by successively applying half the fundamental rotation operation, i. e. by applying the π/n-rotation to the x-mirror operation (σ x ), which maps any vector (x, y, z) to (−x, y, z). For each excitation frequency ω, the set of (optical or acoustic) modes { f (i) (r)} with the same wavenumber forms a linear subspace S f that is invariant under all operations in G . Each symmetry operationR ∈ G can therefore be expressed as an expansion in the basis of the mode functions f (i) (r): In general D (R) are matrices, however in the non-degenerate case, in which the set { f (i) (r)} consists of a single function, the matrices reduce to scalar phase factors. In the degenerate case, the matrices D (R) depend on the choice of basis functions f (i) (r) and are unitary for any ortho-normal basis. The set of matrices D (R) for all elementsR ∈ G is known as a (matrix) representation of the underlying symmetry group G , which means that these matrices fulfill the multiplication rules of the abstract group elements. Different mode types induce specific representations that are associated with the modal symmetry. As the matrices themselves depend on the choice of basis functions, they are ill-suited to characterize the symmetry of a degenerate subspace. Instead, the natural choice are the matrices' traces, which are invariant under similarity transformations and called the characters of the representation matrices. For a non-degenerate mode that is even with respect to all elementsR ∈ G , that is, it is unchanged under all operations in the group, the representation induced by the mode consists of D (R) = 1 for each elementR. This trivial representation, denoted by A or by A 1 exists for each group and corresponds to monopole-like fields. We should stress that the mere existence of this representation does not mean that the (optical or acoustic) wave equation will necessarily have a physical solution of this symmetry at any given frequency (indeed the fundamental mode of a dielectric waveguide is only rarely of A 1 type [21,22]). Representations that are characterized by sign changes under the various operations of the group (rotations and reflections) correspond to higher-order multipole fields or superpositions thereof and are denoted A 2 , B-type or E-type representations, the latter denoting degenerate representations. Table 1 lists the characters of all irreducible representations of the three point groups under consideration, for sets of operations that are sufficient to unambiguously identify the representations based on their characters. These tables are reproduced from Ref. [19] (and found in many other texts). The degenerate representations E and E 1 of the groups C 3v and C 6v , respectively, are particularly important since the fundamental optical mode of a dielectric waveguide with threefold or sixfold symmetry is of this type in practical situations involving index-guided waveguides.
The acousto-optic overlap integral involves a product of three modes (two optical and one acoustic), so we need the symmetry properties of products. The product of two modes f (i) (r) · g ( j) (r) from two invariant subspaces S f and S g lies inside the direct product subspace spanned by every possible product S f ⊗ S g = span{ f (i) (r)g ( j) (r)}. If S f is one-dimensional, the characters of S f ⊗ S g are obtained by simply multiplying the characters of S g with those of S f (and analogously if S g is one-dimensional). If both subspaces are degenerate (with dimensions n and m), the product space is n × m-dimensional and usually reducible into the few irreducible representations of the point group, i. e. it can be decomposed into a number of independent smaller subspaces. In the context of discrete symmetry groups for waveguides, this is always the case. The details of this decomposition depend on whether S f = S g (general case) or whether the functions f (i) (r) and g ( j) (r) are identical or orthogonal to each other within the same degenerate subspace S f = S g . For the two point groups under consideration that feature degenerate representations, the product decompositions are listed in Table 2. Again, these tables are reproduced from Ref. [19].

Examples for optical and acoustic modes
The symmetry properties of optical waveguides are well-known [16]. Here we show some specific examples to illustrate how the group-theoretical representation relates to the more widelyused categories of waveguide modes. In Fig. 2 we show examples of optical modes of a photonic crystal fiber, with a structure belonging to the C 6v point group. The two fundamental modes have E 1 -symmetry, and correspond to the fundamental linearly x-and y-polarized modes (often Fig. 3. Examples for the symmetries of acoustic modes in a structured silica fiber with hexagonal symmetry. The arrows and the deformation plot indicate the in-plane components u ⊥ = (u x , u y , 0) of the mechanical displacement field. The color represents the absolute value of the total displacement field |u| = u 2 x + u 2 y + u 2 z . The color scale is in arbitrary units.
classified as HE 01 modes, by way of analogy with the step-index fiber). These modes are the most relevant for optical experiments, since they are readily coupled into from free space or other waveguides. Higher order modes are also possible in this structure; in this specific configuration and at this frequency these are all cladding modes, however these mode types can become important for large-core structures. The first higher-order x and y polarized modes are of E 2 type and are also degenerate. The A 1 type mode in Fig. 2 has an electric field pointing radially outward, this mode possessing even vectorial symmetry both under all reflections and rotations, in accordance with Table 1. The A 2 -type mode has odd symmetry under all reflections and even symmetry under all rotations; this "donut" mode rotates its polarization with respect to angle and possess a phase singularity at the origin, and is analogous to the TE 01 mode of the step-index fiber. The B 1 and B 2 type modes arise rarely in practical situations and so we have omitted these in the figure; a sketch of the properties of these modes can be found in Fig. 5. In all cases we note that it is the transformation properties of the electric field which determine the classification, the magnetic field being a pseudo-vector.
The symmetry properties of acoustic modes are perhaps less known within the photonics community. In Fig. 3 we show examples of the different types of acoustic modes, again for the C 6v point group. The A 1 type mode corresponds to a longitudinal-like acoustic mode, because the dominant motion is in the waveguide direction. The A 2 -type mode is a torsional-like mode, and the degenerate pair of E 1 modes are the orthogonally-polarized fundamental flexural modes. The additional degenerate pair of E 2 -type acoustic modes either stretches or shears the core in the horizontal direction, rather than displacing the core as a whole. This representation covers e. g. accordion-like "breathing modes", which simultaneously stretch the core horizontally while compressing it vertically. The acoustic B 1 and B 2 modes describe non-degenerate higher-order vibrations. The B 1 modes displace the material either inwards or outwards along symmetry planes separated by 120 degrees, while displacing material in the opposite direction along the alternate symmetry planes. The B 2 modes act in a similar way to the B 1 modes, but with the symmetry planes rotated by 30 degrees. Table 2. Relevant parts of the product decomposition tables for C 3v and C 6v . The abbreviations "i.d.m." and "o.d.m." in the decomposition tables stand for "identical degenerate modes" and "orthogonal degenerate modes", respectively. Again, these tables are standard in many textbooks on group theory such as Ref. [19].

Formal selection rule
The term "selection rule" usually refers to a set of constraints on the symmetry properties of physical states to allow for interaction. The best known example of these are the conditions for the atomic states between which optical transitions can occur. For that problem, the point group is the group of continuous rotations and the irreducible representations are distinguished by the two angular momentum quantum numbers. Transitions can occur if the product of the two electronic representations and the representation decomposition of the electric dipole operator has a trivial (i. e. with total angular momentum L = 0) contribution, which is just another formulation of the well known optical selection rules ∆L = ±1 and −1 ≤ ∆m ≤ 1. Similar rules for discrete symmetry groups of atoms in solids are known from the context of crystal field theory.
In this section, we formulate the problem of acousto-optic interaction in a very similar picture, where the optical waveguide modes are analogous to the electronic eigenstates and the acoustic wave corresponds to the electric dipole operator. Using this, we formulate the optoacoustic selection rule in its most general form in analogy to the problem of optical transitions in atoms. To this end, we now analyse the symmetry properties of the overlap integral Eq. (3). The photoelastic tensor p is invariant under the symmetry operations, because we assumed the waveguide (including all material parameters) to be so. This is strictly true for cubic materials in rectangular geometries and for isotropic materials such as silica or various glasses under all circumstances. The tensor M is invariant by construction. The partial derivative and the normal vector distribution n induce the trivial group representations. Thus, we can formally write Eq. (3) in the form where the third rank tensor operator N is invariant under all waveguide symmetries, and induces the trivial representation A or A 1 , depending on the precise group.
To formulate the selection rule for SBS, we first write the integrand as the superposition of symmetric functions w = ∑ α w α , where each index α corresponds to one irreducible representation. The integrals over all symmetric contributions are zero except for the trivial representation [19]. That is, the coupling integral Q can be only non-zero for modes whose product contains the trivial representation (note however that this does not imply that the integral over a trivial representation is always non-zero). To this end, we first determine the symmetry properties of the product between the two optical modes, either by simply multiplying the characters from Table 1 in the case of at least one non-degenerate mode or by using Table 2 if both modes are degenerate. In the same way, we then combine every irreducible representation from this first product with the acoustic representation. If this yields a trivial representation, coupling is possible.
To illustrate this evaluation of representation products, we give two examples: First assume that in a rectangular waveguide the optical representations are B 1 and B 3 and the acoustic representation is also B 3 . The product of the characters of B 1 and B 3 result in the characters of B 2 , thus the product of the two optical modes has B 2 -symmetry. The product of these characters and those of the acoustic B 3 -representation indicate that the product of all three modes has B 1 -symmetry, and hence that the overlap integral vanishes.
Second, assume that in a hexagonal waveguide the optical representations are E 1 and E 2 and the acoustic representation is B 1 . From Table 2 we can see that the product of the optical modes is a superposition of a B 1 -, a B 2 -and a degenerate E 1 -representation. Each of these representations are now individually combined with the acoustic B 1 -representation to find the decomposition of the total integrand: it is a superposition of an A 1 -, an A 2 -and an E 2 -representation. The integral over the latter two contributions vanishes, but the integral over the first term can be non-zero. Thus coupling is possible but may depend on the exact choice of the optical modes within their respective E 1 -and E 2 -subspaces.

Examples for SBS-active mode combinations
We now examine the three most important cases in more detail.

Rectangular group C 2v
The rectangular group does not exhibit any degenerate representations and is therefore simple to discuss. The symmetry of the integrand in Eq. (8) can be easily established by multiplying the characters of the constituent representations. We find two classes of combinations that lead to w having trivial symmetry. The first class is if two modes have identical symmetry and the third one has A-symmetry. This is the case in backward SBS, where the two optical modes have identical symmetry (but opposite propagation direction) and is only possible with an acoustic mode of A-type, i. e. not with flexural or torsional modes. Another example for this class is the conversion between a fundamental, linearly polarized optical mode (B 1 -type or B 3 -type) and a higher order mode (A-type). In this case, the acoustic mode must be of flexural type. The second class is if the optical and acoustic modes cover all three non-trivial representations. This is for example the case for conversion between orthogonally polarized fundamental optical modes (B 1 -type and B 3 -type) and is only possible with an acoustic mode of B 2 -type, i. e. a torsional mode. These combinations are sketched in the left hand part of Fig. 4.

Triangular group C 3v
The triangular point group supports one degenerate and two non-degenerate representations. The degenerate representations describe linearly polarized optical and flexural acoustic modes. This degeneracy slightly complicates the problem of identifying gain combinations. The reason for this is that the symmetry of the product of degenerate modes depends on the degree of overlap between them. To resolve this, we have to treat the cases of orthogonal eigenmodes (orthogonal polarizations) separately from the case where the two modes have maximally possible overlap (identical polarization). The product decomposition table for the group C 3v in Table 1 shows that the former case always results in a mode of A 2 -symmetry, whereas the latter case leads to a superposition of a trivial and a degenerate representation. The general case of partially overlapping eigenmodes (skew polarizations) is a superposition of these special cases. The same holds if both relevant eigenmodes stem from different degenerate subspaces (e. g. an optical and an acoustic mode of E-symmetry).
As before, we find two classes of gain-supporting combinations. The first case is again the combination of one A 1 -type eigenmodes and two other modes of identical symmetry properties, i. e. either two modes of the same non-degenerate type or a pair of degenerate modes with the same polarization. Again, this implies that backward-SBS requires an A 1 -type, i. e. a longitudinal acoustic mode. The second case is again a combination of all three non-trivial symmetry properties, i. e. one A 2 -type mode and two orthogonal modes of E-type. As in the rectangular case, this means that SBS between orthogonally polarized optical modes requires a torsional mode. The product between a fundamental optical mode and a flexural acoustic mode is the general case of the product decomposition and may contain any representation. This means that a flexural acoustic mode can cause SBS between a fundamental mode and any other optical band, not only between the fundamental and the corresponding higher-order mode. We provide some examples for combinations of modes with non-zero overlap in the right hand part of Fig. 4.

Hexagonal group C 6v
The hexagonal group C 6v is a supergroup of both C 2v and C 3v , and supports two degenerate and four non-degenerate representations. They emerge from the representations of the C 3v group by including the C 2 operation as a new generator. The E 1 representations include linear polarized (dipole-like) modes, which are the fundamental modes in the optical case. The modes with E 2 symmetry have quadrupolar character, as can be seen in the second-order optical mode provided this is above cut-off (see Fig. 2). The B 1 and B 2 modes have hexapolar character-in contrast to Examples for group C 6v the dipolar and quadrupolar modes these modes are not degenerate-and the A 2 representation correspond to modes with dodecapolar symmetry such as torsional modes in the acoustic case and azimuthally polarized optical modes. As in the rectangular and triangular cases, the A 1 representation corresponds to longitudinal or breathing acoustic modes and radially polarized optical modes.
The analysis of possible gain combinations is greatly simplified by the prior discussion of the two subgroups C 2v and C 3v . First we note that the character table for the non-degenerate representation is very similar to that of C 2v . The only difference are the rotation operators C 3 and C 6 , whose characters are identical to those of E and C 2 in the non-degenerate case. As a consequence, the discussion of Section 5.1 applies to the gain combinations of any nondegenerate representations. Furthermore we notice that the representation table reduces to two separate instances of the C 3v table, which differ only by the sign of the character of the C 2 rotation. As a consequence the discussion of Section 5.2 applies, with the additional constraint that the number of representations that contain a negative character for C 2 must be even in order to obtain the trivial representation.
The final case is unique to hexagonal structures, and consists of combinations of E 1 together with E 2 modes. This type of interaction must be handled with the help of the product decomposition table (see Table 2). According to this table, two such modes can be combined with either a B-type non-degenerate mode or with an E 1 -type mode in order to obtain non-zero gain. As an example, an optical fundamental mode can couple to the corresponding second order mode via an acoustic flexural mode. Alternatively, an acoustic accordion-like acoustic mode can facilitate the coupling between two optical modes of the same type. A complete discussion of all the possible mode combinations would be excessive without being particularly enlighting; a representative selection of examples that might be important is given in Fig. 5.

Conclusion
We have formulated the selection rules for SBS for three important waveguide symmetries. If the symmetry is only weakly broken, for example by anisotropy in the mechanical properties induced by the cubic nature of some materials, we can expect these selection rules to hold to a good approximation. If the symmetry is strongly broken, these rules are readily adapted to waveguides with reduced symmetry-for example, the selection rules for rectangular waveguides on a substrate will reduce to the C v group, combinations of which will follow the character table with the σ y and C 2 operations removed.
Apart from its use in the understanding and design of SBS-active waveguides, our results are very useful for the development and testing of numerical tools. Firstly, they provide sets of modes that should not be able to couple for symmetry reasons. This is an excellent way to indentify convergence and discretization issues in the evaluation of the coupling integrals. Secondly, they eliminate complete symmetry classes of acoustic solutions for the interaction between any two given optical eigenmodes. These symmetry constraints can be exploited in numerical calculations by reducing the computational domain and imposing approriate boundary conditions along the mirror planes, thereby greatly reducing the computational effort. This can be significant in automatized design optimzation problems or large-scale geometries.