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I am studying nonlinear optics, specifically the wave equation of nonlinear optical media. During the derivation I came across a point where the divergence of electric field was zero for a linear isotropic medium. I couldn't understand the reason behind it.

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    $\begingroup$ Hint: which of Maxwell’s equations tells you about divergence of E field, and what needs to be there for it to be nonzero? $\endgroup$
    – The Photon
    Aug 24 '20 at 14:24
  • $\begingroup$ In English, only capitalize the first word of each sentence, names, and the word "I". English is not like German. $\endgroup$
    – DanielSank
    Aug 24 '20 at 15:34
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Usually, the derivations for wave equation of non linear optical media are done assuming, charge-free and current-free conditions, in which case $\nabla \cdot \mathbf{D} = 0$.

I am sure the general wave equation will have source term accounting for the presence of charge or current:

$$ \nabla^2 \mathbf{E} - \frac{1}{c^2}\frac{\partial^2 \mathbf{E}}{\partial t^2} = \mu_0\frac{\partial^2 \mathbf{P}_{NL}}{\partial t^2},$$ where the $\mathbf{P}_{NL}$ is a non-linear function of $\mathbf{E}$. The term on RHS corresponds to source.

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