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I need to figure out the length of a crystal to move a light beam shining through it.

Sketch of the problem

The height is known, and so is the refractive index of the crystal and its length. What I can't figure out is the angle $\alpha$.

I came to a set of equation but even after using Mathematica, I can't get an answer. The equations are as follows: $$\sin(\alpha)=n\sin(\delta)$$ $$s=\frac{L_q}{\cos(\delta)}$$ $$\Delta x=s\sin(\alpha - \delta),$$ where $n$ is the refractive index, $\delta, \alpha$ are as shown in the picture, $L_q$ is the length of the crystal, and $s$ is the distance the beam has traveled in the crystal.

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  • $\begingroup$ The equations seem to be correct; but I wouldn't expect the resulting solution for $\alpha$ to be a "nice" equation, and it may be that it doesn't have a closed-form solution at all (a so-called transcendental equation). If you have specific numerical values in mind, FindRoot in Mathematica is probably your best bet (rather than trying to use Solve.) If you need help implementing that, rephrase your question and ask it on Mathematica. $\endgroup$
    – Michael Seifert
    Commented Oct 22, 2023 at 15:27
  • $\begingroup$ @MichaelSeifert okay thank you, using FindRoot it worked. $\endgroup$
    – 100xln2
    Commented Oct 22, 2023 at 16:04

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