Type-III Shubnikov Point Groups for Guided-Wave Stimulated Brillouin Scattering: Conjugate Symmetry and Selection Rules

In guided-wave stimulated Brillouin scattering (SBS), the opto-mechanical coupling strength is determined by the spatial overlap of optical and elastic fields, fundamentally constrained by symmetry. Conventional analyses based on ordinary point groups assume that fields share the waveguide’s symmetry, which is valid for standing-wave modes with zero longitudinal wavenumber. However, in waveguides with longitudinal-axis-reversing operations, such operations flip the wavenumber sign for traveling-wave modes, making the conventional framework insufficient—a limitation not addressed before. Here, we introduce the type-III Shubnikov (magnetic) point groups, combining time reversal with axis-reversing spatial operations, and establish a co-representation theory for such traveling-wave modes. We prove that these modes obey a conjugate symmetry derived from the antiunitary elements of the magnetic point group. From this, we derive a general selection rule for backward SBS: if the waveguide possesses only one nontrivial axis-reversing operation (and no other independent symmetry), the conjugate symmetry allows the coupling to be nonzero. Numerical validations on single-crystal lithium niobate, fused silica, and single-crystal silicon waveguides of a trapezoidal cross-section confirm the predicted conjugate symmetry and show that materials with lower intrinsic symmetry more favorably realize such symmetry-enabled backward SBS. This work represents a systematic introduction of magnetic group theory to nonmagnetic waveguides, offering new insights for material selection and coupling control in guided-wave SBS.