What is the relationship between the nonlinear susceptibility and the effective nonlinear coefficient?

Question

Oftentimes, I see equations for non-linear optics expressed in terms of the non-linear susceptibility tensor.

Other times, however, I see these same equations expressed in terms of an effective non-linear coefficient tensor.

What's the relationship between these two things? What am I missing here?

Answer 1

The nonlinear coefficient tensor $d_{il}$ is a contraction/approximation of the nonlinear susceptibility tensor $\chi_{ijk}$ under certain symmetries (here talking about second-order effects, $\chi^{(2)}$).

For a discussion, see e.g.

• Boyd, Nonlinear Optics, 2008, Chap. 1.5.6 - Contracted Notation
• Boyd and Kleinman, Parametric Interaction of Focused Gaussian Light Beams, Appl. Phys. 39, 3597 (1968), Appendix 2 - Definition of nonlinear coefficients, and Appendix 3 - Effective nonlinear coefficients.