International Tables for Crystallography (2006). Vol. D. ch. 1.7, pp. 178-219
https://doi.org/10.1107/97809553602060000634

Chapter 1.7. Nonlinear optical properties

Contents

1.7. Nonlinear optical properties (pp. 178-219) | html | pdf | chapter contents |
B. Boulanger and J. Zyss
- 1.7.1. Introduction (p. 178) | html | pdf |
- 1.7.2. Origin and symmetry of optical nonlinearities (pp. 178-183) | html | pdf |
  - 1.7.2.1. Induced polarization and susceptibility (pp. 178-181) | html | pdf |
    - 1.7.2.1.1. Linear and nonlinear responses (p. 179) | html | pdf |
      - 1.7.2.1.1.1. Linear response (p. 179) | html | pdf |
      - 1.7.2.1.1.2. Quadratic response (p. 179) | html | pdf |
      - 1.7.2.1.1.3. Higher-order response (p. 179) | html | pdf |
    - 1.7.2.1.2. Linear and nonlinear susceptibilities (pp. 179-180) | html | pdf |
      - 1.7.2.1.2.1. Linear susceptibility (p. 180) | html | pdf |
      - 1.7.2.1.2.2. Second-order susceptibility (p. 180) | html | pdf |
      - 1.7.2.1.2.3. nth-order susceptibility (p. 180) | html | pdf |
    - 1.7.2.1.3. Superposition of monochromatic waves (p. 180) | html | pdf |
    - 1.7.2.1.4. Conventions for nonlinear susceptibilities (pp. 180-181) | html | pdf |
      - 1.7.2.1.4.1. Classical convention (pp. 180-181) | html | pdf |
      - 1.7.2.1.4.2. Convention used in this chapter (p. 181) | html | pdf |
  - 1.7.2.2. Symmetry properties (pp. 181-183) | html | pdf |
    - 1.7.2.2.1. Intrinsic permutation symmetry (pp. 181-182) | html | pdf |
      - 1.7.2.2.1.1. ABDP and Kleinman symmetries (pp. 181-182) | html | pdf |
      - 1.7.2.2.1.2. Manley–Rowe relations (p. 182) | html | pdf |
      - 1.7.2.2.1.3. Contracted notation for susceptibility tensors (p. 182) | html | pdf |
    - 1.7.2.2.2. Implications of spatial symmetry on the susceptibility tensors (pp. 182-183) | html | pdf |
- 1.7.3. Propagation phenomena (pp. 183-212) | html | pdf |
  - 1.7.3.1. Crystalline linear optical properties (pp. 183-187) | html | pdf |
    - 1.7.3.1.1. Index surface and electric field vectors (pp. 183-185) | html | pdf |
    - 1.7.3.1.2. Isotropic class (p. 185) | html | pdf |
    - 1.7.3.1.3. Uniaxial class (pp. 185-186) | html | pdf |
    - 1.7.3.1.4. Biaxial class (pp. 186-187) | html | pdf |
      - 1.7.3.1.4.1. Propagation in the principal planes (pp. 186-187) | html | pdf |
      - 1.7.3.1.4.2. Propagation out of the principal planes (p. 187) | html | pdf |
  - 1.7.3.2. Equations of propagation of three-wave and four-wave interactions (pp. 187-196) | html | pdf |
    - 1.7.3.2.1. Coupled electric fields amplitudes equations (pp. 187-188) | html | pdf |
    - 1.7.3.2.2. Phase matching (pp. 188-192) | html | pdf |
      - 1.7.3.2.2.1. Cubic crystals (p. 189) | html | pdf |
      - 1.7.3.2.2.2. Uniaxial crystals (p. 189) | html | pdf |
      - 1.7.3.2.2.3. Biaxial crystals (pp. 189-192) | html | pdf |
    - 1.7.3.2.3. Quasi phase matching (pp. 192-193) | html | pdf |
    - 1.7.3.2.4. Effective coefficient and field tensor (pp. 193-196) | html | pdf |
      - 1.7.3.2.4.1. Definitions and symmetry properties (pp. 193-194) | html | pdf |
      - 1.7.3.2.4.2. Uniaxial class (pp. 194-196) | html | pdf |
      - 1.7.3.2.4.3. Biaxial class (p. 196) | html | pdf |
  - 1.7.3.3. Integration of the propagation equations (pp. 196-212) | html | pdf |
    - 1.7.3.3.1. Spatial and temporal profiles (pp. 196-197) | html | pdf |
    - 1.7.3.3.2. Second harmonic generation (SHG) (pp. 197-206) | html | pdf |
      - 1.7.3.3.2.1. Non-resonant SHG with undepleted pump in the parallel-beam limit with a Gaussian transverse profile (pp. 197-202) | html | pdf |
      - 1.7.3.3.2.2. Non-resonant SHG with undepleted pump and transverse and longitudinal Gaussian beams (pp. 202-203) | html | pdf |
      - 1.7.3.3.2.3. Non-resonant SHG with depleted pump in the parallel-beam limit (pp. 203-205) | html | pdf |
      - 1.7.3.3.2.4. Resonant SHG (pp. 205-206) | html | pdf |
    - 1.7.3.3.3. Third harmonic generation (THG) (pp. 206-207) | html | pdf |
      - 1.7.3.3.3.1. SHG ( $[\omega+\omega=2\omega]$ ) and SFG ( $[\omega+2\omega=3\omega]$ ) in different crystals (pp. 206-207) | html | pdf |
      - 1.7.3.3.3.2. SHG ( $[\omega+\omega=2\omega]$ ) and SFG ( $[\omega+2\omega=3\omega]$ ) in the same crystal (p. 207) | html | pdf |
      - 1.7.3.3.3.3. Direct THG ( $[\omega+\omega+\omega=3\omega]$ ) (p. 207) | html | pdf |
    - 1.7.3.3.4. Sum-frequency generation (SFG) (pp. 207-208) | html | pdf |
      - 1.7.3.3.4.1. SFG ( $[\omega_1+\omega_2=\omega_3]$ ) with undepletion at $[\omega_1]$ and $[\omega_2]$ (p. 208) | html | pdf |
      - 1.7.3.3.4.2. SFG ( $[\omega_s+\omega_p=\omega_i]$ ) with undepletion at $[\omega_p]$ (p. 208) | html | pdf |
    - 1.7.3.3.5. Difference-frequency generation (DFG) (pp. 208-212) | html | pdf |
      - 1.7.3.3.5.1. DFG ( $[\omega_p-\omega_s=\omega_i]$ ) with undepletion at $[\omega_p]$ and $[\omega_s]$ (p. 208) | html | pdf |
      - 1.7.3.3.5.2. DFG ( $[\omega_s-\omega_p=\omega_i]$ ) with undepletion at $[\omega_p]$ (p. 208) | html | pdf |
      - 1.7.3.3.5.3. DFG ( $[\omega_p-\omega_s=\omega_i]$ ) with undepletion at $[\omega_p]$ – optical parametric amplification (OPA), optical parametric oscillation (OPO) (pp. 208-212) | html | pdf |
- 1.7.4. Determination of basic nonlinear parameters (pp. 212-214) | html | pdf |
  - 1.7.4.1. Phase-matching directions and associated acceptance bandwidths (p. 212) | html | pdf |
  - 1.7.4.2. Nonlinear coefficients (pp. 212-214) | html | pdf |
    - 1.7.4.2.1. Non-phase-matched interaction method (pp. 212-214) | html | pdf |
    - 1.7.4.2.2. Phase-matched interaction method (p. 214) | html | pdf |
- 1.7.5. The main nonlinear crystals (pp. 214-216) | html | pdf |
- 1.7.6. Glossary (p. 216) | html | pdf |
- References | html | pdf |
- Figures
  - Fig. 1.7.3.1. Field vectors of a plane wave propagating in an anisotropic medium (p. 183) | html | pdf |
  - Fig. 1.7.3.2. Index surfaces of the negative and positive uniaxial classes (p. 186) | html | pdf |
  - Fig. 1.7.3.3. Index surfaces of the negative and positive biaxial classes (p. 186) | html | pdf |
  - Fig. 1.7.3.4. Index surface sections in a plane containing the optic axis z of a negative uniaxial crystal (p. 189) | html | pdf |
  - Fig. 1.7.3.5. Stereographic projection on the optical frame of the possible loci of phase-matching directions in the principal planes of a biaxial crystal (p. 193) | html | pdf |
  - Fig. 1.7.3.6. Schematic configurations for second harmonic generation (p. 196) | html | pdf |
  - Fig. 1.7.3.7. Spatial growth evolution of second harmonic conversion efficiency (p. 198) | html | pdf |
  - Fig. 1.7.3.8. Conversion efficiency evolution as a function of ξ for a given crystal length (p. 198) | html | pdf |
  - Fig. 1.7.3.9. Beam separation in the particular case of type-II (oeo) SHG out of the xy plane of a positive uniaxial crystal or in the xz and yz planes of a positive biaxial crystal (p. 201) | html | pdf |
  - Fig. 1.7.3.10. Twin-crystal device allowing walk-off compensation for a direction of propagation θ_PM in the yz plane of a positive uniaxial crystal (p. 201) | html | pdf |
  - Fig. 1.7.3.11. Comparison between (a) collinear and (b) special non-collinear phase matching for (oeo) type-II SHG (p. 202) | html | pdf |
  - Fig. 1.7.3.12. Position f_opt of the beam waist for different values of walk-off angles and , leading to an optimum SHG conversion efficiency (p. 203) | html | pdf |
  - Fig. 1.7.3.13. Optimum walk-off function as a function of for various values of $[B=(1/2)\rho\{[(k_o^\omega+k_e^\omega)/2]L\}^{1/2}]$ (p. 203) | html | pdf |
  - Fig. 1.7.3.14. Type-I and -II conversion efficiencies calculated as a function of for different typical walk-off angles ρ (p. 204) | html | pdf |
  - Fig. 1.7.3.15. Schematic SHG conversion efficiency for different situations of pump depletion and dephasing (p. 205) | html | pdf |
  - Fig. 1.7.3.16. Configurations for third harmonic generation (p. 206) | html | pdf |
  - Fig. 1.7.3.17. Frequency up-conversion process $[\omega_1+\omega_2=\omega_3]$ (p. 207) | html | pdf |
  - Fig. 1.7.3.18. Schematic OPO configurations (p. 209) | html | pdf |
  - Fig. 1.7.3.19. Calculated angular tuning curves (p. 211) | html | pdf |
  - Fig. 1.7.3.20. Calculated pump wavelength tuning curve (p. 211) | html | pdf |
  - Fig. 1.7.4.1. (a) The Maker-fringes technique; (b) the wedge-fringes technique (p. 214) | html | pdf |
- Tables
  - Table 1.7.2.1. The most common nonlinear effects and the corresponding susceptibility tensors in the frequency domain (p. 181) | html | pdf |
  - Table 1.7.2.2. Nonzero χ⁽²⁾ coefficients and equalities between them in the general case (p. 182) | html | pdf |
  - Table 1.7.2.3. Nonzero χ⁽²⁾ coefficients and equalities between them under the Kleinman symmetry assumption (p. 183) | html | pdf |
  - Table 1.7.2.4. Nonzero χ⁽³⁾ coefficients and equalities between them in the general case (p. 184) | html | pdf |
  - Table 1.7.2.5. Nonzero χ⁽³⁾ coefficients and equalities between them under the Kleinman symmetry assumption (p. 185) | html | pdf |
  - Table 1.7.3.1. Correspondence between the phase-matching relations, the configurations of polarization and the types according to the sum- and difference-frequency generation processes SFG ( $[\omega_3=\omega_1+\omega_2]$ ), DFG ( $[\omega_1=\omega_3-\omega_2]$ ) and DFG ( $[\omega_2=\omega_3-\omega_1]$ ) (p. 188) | html | pdf |
  - Table 1.7.3.2. Correspondence between the phase-matching relations, the configurations of polarization and the types according to SFG ( $[\omega_4=\omega_1+\omega_2+\omega_3]$ ), DFG ( $[\omega_1=\omega_4-\omega_2-\omega_3]$ ), DFG ( $[\omega_2=\omega_4-\omega_1-\omega_3]$ ) and DFG ( $[\omega_3=\omega_4-\omega_1-\omega_2]$ ) (Boulanger et al., 1993) (p. 189) | html | pdf |
  - Table 1.7.3.3. Classes of refractive-index inequalities for collinear phase matching of three-wave interactions in positive and negative uniaxial crystals (p. 190) | html | pdf |
  - Table 1.7.3.4. Classes of refractive-index inequalities for collinear phase matching of four-wave interactions in positive () and negative () uniaxial crystals with $[(n_{b4}/\lambda_4)\,\lt\,(n_{a1}/\lambda_1)+(n_{a2}/\lambda_2)+(n_{a3}/\lambda_3)]$ (p. 190) | html | pdf |
  - Table 1.7.3.5. Refractive-index conditions that determine collinear phase-matching loci in the principal planes of positive and negative biaxial crystals for three-wave SFG (p. 191) | html | pdf |
  - Table 1.7.3.6. Refractive-index conditions that determine collinear phase-matching loci in the principal planes of positive and negative biaxial crystals for four-wave SFG (pp. 192-193) | html | pdf |
  - Table 1.7.3.7. Matrix representations of the (oee) and (eoo) field tensors of the uniaxial class and of the biaxial class in the principal planes xz and yz, with $[\omega_1\ne \omega_2]$ (Boulanger & Marnier, 1991) (p. 195) | html | pdf |
  - Table 1.7.3.8. Matrix representations of the (oeee), (eooo) and (ooee) field tensors of the uniaxial class and of the biaxial class in the principal planes xz and yz, with $[\omega_1\ne\omega_2\ne\omega_3]$ (Boulanger et al., 1993) (p. 195) | html | pdf |
  - Table 1.7.3.9. Field-tensor components specifically nil in the principal planes of uniaxial and biaxial crystals for three-wave and four-wave interactions (p. 196) | html | pdf |
  - Table 1.7.5.1. Mineral nonlinear crystals (pp. 213-214) | html | pdf |
  - Table 1.7.5.2. Organic and organo-mineral crystals (pp. 215-216) | html | pdf |