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I know that in general, $\epsilon(r,\omega)$ and $\mu(r,\omega)$ are complex tensors, so now how do you resolve maxwell equations?

I don't understand why there are techniques to measure them if you could know these parameters from knowledge of the crystal structure of materials.

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  • $\begingroup$ This question is equally applicable to any branch of physics. The point is that theories should be guided and validated by experimental measurements. A lot of materials are not crystalline (silica etc), or non single-crystal. So crystal selection rules techniques will not help there. A lot depends material surface roughness, a lot depends on deposition method, abundance of dopands, annealing/non-annealing etc. etc. There is only one material the optical properties of which I would trust without re-measuring them - crystalline silicon in wafers, and only because it is an industry standard. $\endgroup$ – Cryo Apr 1 at 3:09
  • $\begingroup$ Even then, as soon as you pattern this silicon, all trust is gone. $\endgroup$ – Cryo Apr 1 at 3:10
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If a theory describing the crystal lattice predicts a certain value of $\epsilon$ or $\mu$ (complex or otherwise) conducting an experiment to measure the actual value will tell you if your model is correct. All models need to be compared to experiment.

The complex parts of these two parameters describe how much the material absorbs energy from the field. Describing how the complex part affects MWE had been done in numerous places. Just google "complex dielectric solution maxwell" and take your pick :-)

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The Maxwell equations can be solved numerically.

Theoretical calculation of permittivity and permeability can be difficult or impossible, and sometimes one does not know the exact composition and structure of the material, so it can be easier to measure its properties.

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