- #1
a2009
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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: [tex]\psi_1+\psi_2=\alpha[/tex]
2) Snell's law: [tex] \sin \phi_i = n \sin \psi_i [/tex]
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
[tex] d = \left( 1- \frac{\sin \alpha \sin \psi_2}{\cos \psi_1} \right) D [/tex]
where d is the incoming width, and D is the outgoing width. If I add an upside-down prism at angle [tex]\delta[/tex], I can simply do the transformation again and get the outgoing beam width
[tex] 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 [/tex]
where [tex]\delta[/tex] relates the outgoing angle from the first prism to the incoming angle of the second prism
[tex] \delta = \phi_2 + \phi_1' [/tex]
the primed quantities are related to the second prism.
My hope was that I could find a condition on [tex]\delta[/tex] 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 [tex]\delta[/tex] is [tex]\delta=0[/tex], 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!