Resonance frequency of confocal resonator

[semc]

Homework Statement

A symmetrical confocal resonator has a length d=30cm, and the medium has refractive index n=1. Determine the frequency spacing $\nu$_F and the displacement frequency ($\Delta\varsigma$/$\pi$)$\nu$_F. Determine all the resonance frequencies that lie within the band 5*10¹⁴$\pm$2*10⁹Hz. $\Delta\varsigma$= $\varsigma$(z₁)-$\varsigma$(z₂) where z₁ and z₂ are the position of the mirrors

Homework Equations

$\nu$_F=c/2d

The Attempt at a Solution

I can only find the resonance frequency. How can I get the phase shift $\Delta\varsigma$ from the length of the resonator?

 

[mark726]

To determine the phase shift \Delta\varsigma, you will need to use the formula \Delta\varsigma = 2\pi \Delta n L/\lambda, where \Delta n is the change in refractive index between the two mirrors, L is the length of the resonator, and \lambda is the wavelength of the light. Since the resonator is symmetrical, \Delta n will be zero and therefore \Delta\varsigma will also be zero. This means that the displacement frequency will also be zero, since it is given by (\Delta\varsigma/\pi)\nuF.

To determine the frequency spacing \nuF, you can use the formula \nuF = c/2d, where c is the speed of light and d is the length of the resonator. Plugging in the given values, we get \nuF = 1.5*10^10 Hz.

To determine the resonance frequencies within the given band, you can use the formula \nu_n = n\nuF, where n is the mode number. Since the given band is 5*10^14±2*10^9 Hz, we can plug in different values for n to find the resonance frequencies that fall within this range. For example, if we plug in n=335, we get a resonance frequency of 5.025*10^14 Hz, which falls within the given band.