Neumann’s Principle Based Eigenvector Approach for Deriving Non-Vanishing Tensor Elements for Nonlinear Optics

Physical properties are commonly represented by tensors, such as optical susceptibilities. Conventional approach of deriving non-vanishing tensor elements of symmetric systems relies on the intuitive consideration of positive/negative sign flipping after symmetry operations, which could be tedious and prone to miscalculation. Here we present a matrix-based approach which gives a physical picture centered on Neumann’s principle. The principle states that symmetries in geometric systems are adopted by their physical properties. We mathematically apply the principle to the tensor expressions and show a clear mathematical procedure to derive non-vanishing tensor elements based on eigensystems. Examples on commonly known 2nd and 3rd-order nonlinear susceptibilities are shown, together with complicated scenarios involving uncommon symmetries such as C7 and octahedral symmetries, as well as higher-rank tensors such as 5th-order nonlinear signals. This generalized approach can be applied to any symmetry and higher order nonlinear processes, useful for developing and understanding higher order nonlinear optical signals.