How to Derive Resonant Frequencies for Planar Mirror Resonators?

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SUMMARY

The discussion focuses on deriving resonant frequencies for planar mirror resonators, specifically in the context of a three-mirror ring configuration. The key equation presented is ϕ + ϕ + ϕ = 2πn, leading to the conclusion that for constructive interference, the phase ϕ must equal π, resulting in the equation 3π = 2πn. The participant expresses uncertainty about the next steps in the derivation process, particularly when the distance d equals zero.

PREREQUISITES
  • Understanding of optical resonators and their configurations
  • Familiarity with phase accumulation in wave interference
  • Knowledge of Saleh's book on optics
  • Basic principles of constructive interference in physics
NEXT STEPS
  • Study the derivation of resonant frequencies in optical resonators
  • Explore the implications of varying the distance d in resonator configurations
  • Review examples of three-mirror ring resonators in optical physics
  • Learn about the mathematical treatment of phase conditions in wave optics
USEFUL FOR

Students and researchers in optics, physicists working with resonant systems, and anyone interested in the mathematical foundations of optical resonators.

Marco Oliveira
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Homework Statement
Derive expressions for the resonance frequencies and their frequency spacing for the three-mirror ring and the four-mirror ring bow-tie resonator shown below. Assume that each mirror reflection introduces a phase shift of π.
Relevant Equations
Resonator
I was trying to do the exercise from Saleh's book, but I had some doubts. Any tips on how to resolve it?

My partial solution for the three-mirror ring:

For constructive interference to occur, the total phase accumulated in a round trip must be an integer multiple of 2π. Let's denote the phase accumulated in each arm as ϕ. We have: ϕ + ϕ + ϕ = 2πn, where n is an integer representing the mode number. Since ϕ = π, then: 3π=2πn.

Now i don't know what to do.
resonator.png
 
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Marco Oliveira said:
Since ϕ = π,
Only if d=0.
 

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