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I am in the middle of focusing a blue (400nm) laser beam through a sapphire substrate, of thickness 430 microns. To do this, I am using a LD-Plan Neofluar 63x objective lens, which is equipped with a coverslip correction collar.

If this were glass, I would merely adjust the coverslip correction to 430 microns, and the thickness would be compensated for.

However, how should I adjust the correction for a substrate of different refractive index? (Assuming for simplicity sapphire is optically isotropic)

Is there any equation which allows you to find the optimum correction for such a system?

I have tried using a correction of 430 microns, and the optical spot was outside the sample completely. In contrast, a correction of 170 microns (which I tried first by mistake) gives a relatively good focus, so the change in refractive index is obviously very important!

Many thanks in advance!

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Well, what is the index for sapphire? ... http://refractiveindex.info/?shelf=main&book=Al2O3&page=Malitson says it's 1.7866 at 400 nm. BK-7 glass is 1.53, so if the correction collar were linear, you'd adjust the correction by at $ \frac{1.7866}{1.53} $ .

However, experiment trumps (not the comb-over offspring of an orang-utan!) theory, so I suspect the correction collar's settings are not linear. For one thing, they probably take into account the rather steep cone angle of the light, so Snell's law gets involved, meaning the optical path length is significantly different for sapphire vs glass.

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