An optics problem - elliptical lens and plane waves

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SUMMARY

The discussion centers on demonstrating that a point source at one of the foci of an elliptical lens produces a plane wave upon refraction. The participant initially struggled with applying Fermat's theorem and the optical path length (OPL) but later identified that the OPL should be proportional to the distance traveled multiplied by the refractive index, rather than the inverse of the refractive index. This correction is crucial for satisfying the condition that the refractive index of the second medium (n2) is greater than that of the first (n1).

PREREQUISITES
  • Understanding of optical path length (OPL) in optics
  • Familiarity with Fermat's theorem in the context of light refraction
  • Knowledge of elliptical lens properties and focal points
  • Basic principles of wave optics and plane waves
NEXT STEPS
  • Study the derivation of optical path length in different media
  • Explore the properties of elliptical lenses and their applications in optics
  • Learn about the mathematical formulation of Fermat's theorem in optics
  • Investigate the behavior of plane waves in various refractive index scenarios
USEFUL FOR

Students and professionals in optics, physics educators, and anyone involved in optical engineering or lens design will benefit from this discussion.

pergradus
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Homework Statement



I must show that a point source placed at one of the focii will produce a plane wave (or vise versa) when the light is refracted through the interface. This problem is really kicking my ***, I've spent at least three hours trying to figure out different ways to do it...


Homework Equations



I tried writing the OPL such that y satisfies the equation of an ellipse centered at the origin. I apply Fermat's theorem, differentiate with respect to x and set this equal to zero.

The result should satisfy the initial condition that n2 > than n1 but I seem to get the opposite.

The Attempt at a Solution



See attached picture!
 

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I just figured out my mistake. The OPL should just be proportional to the distance traveled times the refractive index, not 1/refractive index. The letter will give 1/time rather than just time which is what I'm looking for.

I didn't get any help but I managed to follow the train of logic on my own :smile:
 
Pergradus did you ever solve this problem and if so how
 

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