How to Derive Resonant Frequencies for Planar Mirror Resonators?

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  • #1
Marco Oliveira
1
0
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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  • #2
Marco Oliveira said:
Since ϕ = π,
Only if d=0.
 

FAQ: How to Derive Resonant Frequencies for Planar Mirror Resonators?

What are planar mirror resonators?

Planar mirror resonators are optical cavities typically consisting of two parallel mirrors facing each other. They are used to confine and reflect light between the mirrors, allowing for the formation of standing waves at specific resonant frequencies.

How do you derive the resonant frequencies of a planar mirror resonator?

The resonant frequencies can be derived by considering the condition for constructive interference of the standing waves formed between the mirrors. The path length between the mirrors must be an integer multiple of the wavelength of the light. Mathematically, this is given by the equation: \( L = m \frac{\lambda}{2} \), where \( L \) is the distance between the mirrors, \( \lambda \) is the wavelength, and \( m \) is an integer. The resonant frequency \( f \) can then be expressed as \( f = \frac{m c}{2L} \), where \( c \) is the speed of light.

What factors affect the resonant frequencies of planar mirror resonators?

The resonant frequencies are primarily affected by the distance between the mirrors \( L \) and the speed of light \( c \). Changes in the refractive index of the medium between the mirrors can also affect the resonant frequencies, as the effective wavelength of light changes with the refractive index.

What is the significance of the integer \( m \) in the resonant frequency equation?

The integer \( m \) represents the mode number of the resonant frequency. Each value of \( m \) corresponds to a different standing wave pattern or mode within the resonator. Higher values of \( m \) correspond to higher resonant frequencies and more nodes in the standing wave pattern.

Can planar mirror resonators support multiple resonant frequencies simultaneously?

Yes, planar mirror resonators can support multiple resonant frequencies simultaneously. Each mode number \( m \) corresponds to a different resonant frequency, and multiple modes can coexist within the resonator depending on the coherence and bandwidth of the light source used.

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