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Optical axis

 

The physical properties of one optical axis are determined by the crystal dielectric tensor $ \varepsilon$ which is hermitian. Here is considered only the case of symmetric tensor, i.e. optical axes without optical activity. (For some crystal point groups it is possible the lack of optical activity even for crystal structures without inversion center.)

Definition: An optical axis is determined by the unit vectors $ \mathbf{O_{1}}$ and $ \mathbf{O_{2}}$ of the axial representation of the inverse dielectric tensor:

   $\displaystyle \mbox{\boldmath$\varepsilon$}$$\displaystyle ^{-1} = \frac{1}{\varepsilon_{2}}\mathbf{I}+ \frac{1}{2} \Biggl( ... ...{O_{1}} \otimes \mathbf{O_{2}} + \mathbf{O_{2}} \otimes \mathbf{O_{1}} \biggr) $

where $ \varepsilon_{1} \leq \varepsilon_{2} \leq \varepsilon_{3}$ are the eigenvalues of the dielectric tensor $ \varepsilon$ , $ \mathbf{I}$ is rank-3 unit tensor, and $ \otimes$ is a tensor product. (Feodorov, 1958)