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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?

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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.

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