Neumann's principle based eigenvector approach for deriving non-vanishing tensor elements for nonlinear optics

Abstract

Physical properties are commonly represented by tensors, such as optical susceptibilities. The 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 that 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 procedure with clear physical intuition to derive non-vanishing tensor elements based on eigensystems. The validity of the approach is demonstrated by examples of commonly known second and third-order nonlinear susceptibilities of chiral/achiral surfaces, together with complicated scenarios involving symmetries such as D₆ and O_h symmetries. We then further applied this method to higher-rank tensors that are useful for 2D and high-order spectroscopy. We also extended our approach to derive nonlinear tensor elements with magnetization, which is critical for measuring spin polarization on surfaces for quantum information technologies. A Mathematica code based on this generalized approach is included that can be applied to any symmetry and higher order nonlinear processes.

FIG. 1.

Basics of non-vanishing tensor element derivation. (a) Flowchart of the NPEV 3-step approach. First, finding the generating operations of a point group. Second, building a mapping matrix M based on the Kronecker product of the generating operations. Third, computing the eigenvectors of M whose eigenvalues are 1, i.e., nullspace of (M − I), and finding the non-vanishing elements and their relationships. (b) An example of tensor vectorization of a third-rank tensor β_i,j,k and the calculation of the position index of a tensor element.

FIG. 2.

Symmetry operations of the C_3v point group in a right-handed xyz coordinate system.

FIG. 3.

Deriving non-vanishing third-rank tensor elements for the C_3v point group. (a) Following the 3-step NPEV approach, mapping matrix $M=\frac{M_1+M_2}{2}$ is created with two generating operations (3-fold rotation about principal axis z, reflection about zx plane) where $M_1=C_3^1\otimes C_3^1\otimes C_3^1$, and $M_2={\sigma }_{\nu }^a\otimes {\sigma }_{\nu }^a\otimes {\sigma }_{\nu }^a$. (b) Five eigenvectors (V₁ to V₅) of M are calculated with eigenvalues equal to 1 and a generalized form (a₁V₁ + a₂V₂…a₅V₅) of a vectorized 3rd-rank tensor that survives after $C_3^1$ rotation and ${\sigma }_{\nu }^a$ reflection is presented. The extracted non-vanishing elements (c) and the tabulated (d) surviving tensor representation. Numbering is for counting purposes only and is different from the vectorized tensor position indexing in Fig. 1.

FIG. 4.

Deriving non-vanishing fourth-rank tensor elements for C_∞v point group. (a) Following the 3-step NPEV approach, mapping matrix $M=1/2\left(\frac{{\int }_0^{2\pi }{M}_1\mathrm{d}\theta }{{\int }_0^{2\pi }\mathrm{d}\theta }+\frac{{\int }_0^{\pi }{M}_2\mathrm{d}\phi }{{\int }_0^{\pi }\mathrm{d}\phi }\right)$ is created with two set of generating operations (C_∞ rotation about the z axis and infinite number of vertical mirrors σ_v containing the C_∞ axis) where M₁ = C_∞z ⊗ C_∞z ⊗ C_∞z ⊗ C_∞z and M₂ = σ_v ⊗ σ_v ⊗ σ_v ⊗ σ_v. (b) Ten eigenvectors (V₁ to V₁₀) of M are calculated with eigenvalues equal to 1 and a generalized form (a₁V₁ + a₂V₂…a₁₀V₁₀) of a vectorized fourth-rank tensor that survives after C_∞ rotation and infinite σ_v reflections is presented. (c) The tabulated surviving tensor representation. Numbering is for counting purposes only and different from the vectorized tensor position indexing in Fig. 1.

FIG. 5.

NPEV approach to derive non-vanishing SHG hyperpolarizability tensor elements of graphene monolayer with D₆ symmetry (a) Definition of lab coordinate and graphene monolayer coordinate. (b) without and with magnetization along the (c) x axis, (d) y axis, and (e) z axis.

FIG. 6.

Deriving non-vanishing sixth-rank tensor elements for fifth-harmonic generation in a cubic system (O_h point group) with no z component in incident fields. (a) Following the 3-step NPEV approach, mapping matrix $M=\frac{M_1+M_2+M_3}{3}$ is created with three generating operations (inversion, 4-fold rotation about principal axis z, 3-fold rotation about cubic body-diagonal), where M₁ = Inv ⊗ Inv ⊗ Inv ⊗ Inv ⊗ Inv ⊗ Inv, $M_2=C_{4z}^1\otimes C_{4z}^1\otimes C_{4z}^1\otimes C_{4z}^1\otimes C_{4z}^1\otimes C_{4z}^1$, and $M_3=C_{3dia}^1\otimes C_{3dia}^1\otimes C_{3dia}^1\otimes C_{3dia}^1\otimes C_{3dia}^1\otimes C_{3dia}^1$. 122 eigenvectors (V₁ to V₁₁₂) of M are calculated with eigenvalues equal to 1 and the simplified surviving tensor representation without the z component is shown (b). The full list of eigenvectors with the z component being considered can be found in our Mathematica code (section χ⁽⁵⁾) and supple. Numbering is for counting purposes only and is different from the vectorized tensor position indexing in Fig. 1.