How to Derive Resonant Frequencies for Planar Mirror Resonators?

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To derive resonant frequencies for planar mirror resonators, constructive interference requires the total phase accumulated during a round trip to equal an integer multiple of 2π. In a three-mirror ring configuration, the phase accumulation can be expressed as ϕ + ϕ + ϕ = 2πn, leading to the equation 3π = 2πn. The discussion highlights a challenge in further solving this equation, particularly when ϕ is set to π, suggesting that the distance d must equal zero for this condition to hold. The conversation emphasizes the need for clarity on how to proceed from this point in the derivation. Understanding these relationships is crucial for accurately determining resonant frequencies in such systems.
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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