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Effective Second-Harmonic Generation Coefficient and C-eigenvalue of Nonlinear Susceptibility Tensors

Die Xiao, Yisheng Song

Abstract

The effective second-harmonic generation (SHG) coefficient is a crucial data that quantifies the efficiency of transforming fundamental frequency light into its second harmonic. With the help of the symmetry of nonlinear optical susceptibility tensors, we mainly discuss the computability of such a effective SHG coefficient in uniaxial crystals. For one thing, the calculation of effective SHG coefficient is converted into the optimization models with some geometric constraints by means of the peculiarity of fundamental frequency light. Secondly, the number of variables of such maximum models are cutted in half to $2$ to calculate it easier, and a comparison between the effective SHG coefficient and C-eigenvalue of susceptibility tensor is given also. Finally, some examples of typical crystal classes are presented to verify the correctness and broader applicabilities of the theoretical results.

Effective Second-Harmonic Generation Coefficient and C-eigenvalue of Nonlinear Susceptibility Tensors

Abstract

The effective second-harmonic generation (SHG) coefficient is a crucial data that quantifies the efficiency of transforming fundamental frequency light into its second harmonic. With the help of the symmetry of nonlinear optical susceptibility tensors, we mainly discuss the computability of such a effective SHG coefficient in uniaxial crystals. For one thing, the calculation of effective SHG coefficient is converted into the optimization models with some geometric constraints by means of the peculiarity of fundamental frequency light. Secondly, the number of variables of such maximum models are cutted in half to to calculate it easier, and a comparison between the effective SHG coefficient and C-eigenvalue of susceptibility tensor is given also. Finally, some examples of typical crystal classes are presented to verify the correctness and broader applicabilities of the theoretical results.
Paper Structure (5 sections, 5 theorems, 80 equations)

This paper contains 5 sections, 5 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Optimization Models of Effective SHG Coefficients
  3. The Largest Effective SHG Coefficient
  4. Examples in Uniaxial Crystals
  5. Conclusion

Key Result

Theorem 2.1

CJQ2023 Let $\mathscr{T}= (T_{ijk})$ be a piezoelectric-type tensor. Then the following conclusions holds.

Theorems & Definitions (11)

  • Definition 1.1: Piezoelectric-type Tensor
  • Theorem 2.1: C-eigenvalues of Piezoelectric-type Tensor
  • Lemma 2.1
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3: C-Eigenvalue Inequality
  • Example 4.1: $\bar{4}2$m Crystal Class (KH$_2$PO$_4$)
  • Example 4.2: $4$mm Crystal Class (LiNbO$_3$)
  • Example 4.3: $4$ Crystal Class (Urea)
  • Example 4.4: $62$m Crystal Class (Benitoite)
  • ...and 1 more