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# Optical Isotropy: Nonlinear Interactions are Different!

Posted on 2007-05-26 as a part of the Photonics Spotlight (available as e-mail newsletter!)

Abstract: It is instructive to look at nonlinear crystals such as BBO which can e.g. have a threefold rotational symmetry in terms of nonlinear interactions, while being isotropic in terms of nonlinear optics. The article discusses how that can be.

Ref.: encyclopedia articles on nonlinearities, birefringence, nonlinear crystal materials

Consider a laser beam propagating through a BBO crystal along its c axis. BBO is uniaxial, so the refractive index will not depend on the polarization direction of that beam. There is no birefringence.

Now consider the nonlinear properties. Here it does matter quite substantially how you rotate the crystal around its optical axis: for certain orientations, there is no χ(2) nonlinearity at all, for others it has a maximum. (You can't use that for nonlinear frequency conversion, as phase matching is not possible for that propagation direction, but let's ignore that here.) You find a threefold symmetry, corresponding to the trigonal crystal symmetry. No isotropy in this respect!

In fact, the linear optical properties often exhibit a higher symmetry than the nonlinear properties. Why is that? Consider the relation between the polarization of the medium and the electric field of a light beam. A linear relation is described by the linear susceptibility tensor of rank 2. Considering only two dimensions, we have only four tensor components. If we have a crystal exhibiting the same refractive index in two orthogonal directions, this tensor must be diagonal, with the two diagonal components being equal, and it is mathematically impossible that the refractive index is different for any other polarization direction. Also, you can never have e.g. a threefold symmetry, even if the crystal is trigonal. The higher-order tensors corresponding to nonlinear interactions, however, have many more components, and there is a lot of freedom for more complicated angular dependencies.

So we should keep in mind: optical isotropy concerning linear properties does by no means imply that all directions are physically equal. This is revealed particularly by nonlinear interactions. And of course it is obvious when looking at the crystal structure: have you ever seen a crystal with rotational symmetry?

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