On a minima problem in optics

Question

I have trodding through a calculus textbook, more specifically — through a chapter on the methods of obtaining the extrema of functions using derivatives, including certain problems in optics (Fermat’s principle) as examples of their usage. To this chapter there is a problem set, and one of the problems is as follows:

A prism deflects a beam of light travelling in a plane perpendicular to the edge of the prism. What must the relative position of prism and beam be for the deflection to be a minimum.

To be fair, it has been more than a year since I last took an optics class; furthermore, I highly doubt the fact that we covered the notion of „deflection” of a beam of light passing through a refracting medium (a very basic high-school course in optics). I do not really need any hints or, God forbid, solutions to this problem. I simply want to be clear on the question. Do we assume that the prism is of some concrete shape? What exactly is meant by the „deflection” of the beam? How should I even picture this problem?

This is important, and though I have already said this, but I think it necessary to reiterate: I do not need solutions or hints, but a mere clarification of the formulation of the problem.

I shall be incredibly grateful to those who provide assistance. Thank you!

Answer 1

I would just like to complete @Puk's answer. The minimization is done on the total deviation angle ($-\theta_0+\pi+\theta_2$) with a fixed prism (ie fixed $\alpha$) by varying the incident angle ($\theta_0$).

The reason why it is interesting is that this angle correspond to a caustic. If you look at how the different rays are being deflected, you will notice that they backtrack exactly at this value. Experimentally, this translates into an accumulation of light intensity, and it is a common phenomenon that you may have noticed in daily life already (deflection through a glass, weird light patterns at the bottom of a pool, etc.). For a prism, it is especially relevant as you will notice that the this angle of minimum deviation depends on the index of refraction $n$, which in turn depends on wavelength due to dispersion. This dependence will explain the separation of colors through a prism, and applied to various geometries (spheres, hexagonal prisms, etc) is the classical explanation of rainbows.

Final word, there is a neat trick to check your calculations, that uses the reversibility of light rays. It turns out that the minimum of deviation corresponds to the case where $\theta_1'=\theta_0$ ie when you have a symmetric exiting and incident angle. Hope this helps and tell me if you find some mistakes.

Answer 2

For purposes of calculating deviation, you can take the prism to be a right triangular prism. Since the plane of incidence is specified to be parallel to the bases of the prism, the problem is essentially 2D: the incident ray gets refracted by two adjacent edges of a "triangle", which define the apex angle $\alpha$. The deviation angle $\delta$ simply refers to the angle by which the outgoing ray differs in direction relative to the incident ray. It will depend only on the angle of incidence, $\alpha$ and the refractive index of the prism.

The following figure from Wikipedia illustrates the geometry. The deviation angle is $\theta_0 + \theta_2$, and the article also gives an expression for $\delta$ although it sounds like you are not looking for that.