Birefringent materials have two main orthogonal axes of symmetry along which the index of refraction differs. When a wave of light passes through a material like this, one part of it is phase shifted from the other so we can achieve different sorts of polarizations.

We align a birefringent material such that the axis with a lower index of refraction is along the x-axis and the axis with a higher index of refraction is along the y-axis. We shine onto the crystal a plane wave of light traveling along the z-axis and polarized at an angle \(\theta\) with respect to the x-axis.

How long should the sample of the birefringent material be **in micrometers** in order for the polarization of the input light to be rotated to where it makes an angle of \(-\theta\) with the x-axis?

**Details and assumptions**

- Input light is green, i.e. its wavelength is \(510~\mbox{nm}\)
- The difference between the refraction indexes is \(0.001\)

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