Solving Anamorphic Prism Homework Statement

[a2009]

Homework Statement

Consider a beam of light with a circular cross section passing through a triangular prism. The prism will change the propagation direction of the beam, but also, in general, its cross section will become elliptical. This effect can be used to convert elliptical beams to circular. However, if the light is not monochromatic, different colors will be dispersed to different angles. To avoid this in practice a pair of identical prisms is used. If a certain condition on the orientations of the two prisms is satisfied, chromatic angular dispersion is canceled for an arbitrary wavelength dependence of the refractive index of the prisms. Find this condition.

Homework Equations

The relevant equations are:

1) Sum of internal angles of prism equal head angle: $$\psi_1+\psi_2=\alpha$$
2) Snell's law: $$\sin \phi_i = n \sin \psi_i$$

The Attempt at a Solution

Using geometric considerations, I calculated the change in height of a beam going through one prism. I found that I can translate this into a condition of the width of the beam. Namely

$$d = \left( 1- \frac{\sin \alpha \sin \psi_2}{\cos \psi_1} \right) D$$

where d is the incoming width, and D is the outgoing width. If I add an upside-down prism at angle $$\delta$$, I can simply do the transformation again and get the outgoing beam width

$$D = \left( 1- \frac{\sin \alpha \sin \psi_2}{\cos \psi_1} \right)^{-1} \left( 1- \frac{\sin \alpha \sin \psi_2'}{\cos \psi_1'} \right)^{-1} d$$

where $$\delta$$ relates the outgoing angle from the first prism to the incoming angle of the second prism

$$\delta = \phi_2 + \phi_1'$$

the primed quantities are related to the second prism.

My hope was that I could find a condition on $$\delta$$ that would cancel the dependence on the refractive index. Obviously an approximation is necessary, however I'm not sure how to go about it. I tried a linear approximation, but the condition on $$\delta$$ is $$\delta=0$$, which is not so interesting.

The TA hinted that the problem could be solved using symmetry, however I don't see how to go about it.

Thanks for any help!

 

[mark726]

Thank you for bringing up this interesting problem. I am a scientist who specializes in optics and I would be happy to provide some insight on this topic.

First, let's consider the case where the light passing through the prism is monochromatic. In this case, the condition for canceling chromatic angular dispersion can be expressed as:

\delta = \phi_2 + \phi_1' = \pi

This means that the incoming angle of the second prism must be equal to the outgoing angle of the first prism, but in the opposite direction. This condition ensures that the dispersion caused by the first prism is canceled out by the second prism.

Now, let's consider the case where the light is not monochromatic. In this case, we can use the same condition as before, but we also need to take into account the different wavelengths of light. This can be done by introducing the dispersion equation, which relates the angle of refraction to the wavelength:

\sin \phi = n(\lambda) \sin \psi

where n(\lambda) is the wavelength-dependent refractive index. If we use this equation for both prisms, we can cancel out the dependence on n(\lambda) by choosing the angles \psi_1 and \psi_2 such that:

\frac{\sin \psi_1}{\sin \psi_2} = \frac{\sin \psi_1'}{\sin \psi_2'}

This condition ensures that the dispersion is canceled out for all wavelengths of light.

I hope this helps you in finding the solution to this problem. Good luck with your research!