Fabry Perot Free Spectral Range

[marco1235]

Hi all,

the question which I'd like to share is the following: if you look at the formal theory of any random optics book for FP interferometer you will get to the formula of the FSR (Free Spectral Range), defined as "The wavelength separation between adjacent transmission peaks" (Wikipedia) and the formula is given by:

Δλ ≈ Λ²₀ / 2nlcos(θ) where n is the refractive index inside the mirrors and l is the mirror separation. Now the question is, if I shine a monochromatic laser inside such a device I'll obtain ring fringes of the same color (suppose I shine the cavity with a red laser). So red circles! why should I have a FSR defined as a Δλ since the λ of my source is always the same?? I really don't understand this!

Thanks for your help!

 

[blue_leaf77]

FSR is relevant when the FP resonator is used as spectrometer, assuming there are more than one frequency components contained by the light. If you have been sure your laser only has one frequency and know its value, why send it into a spectrometer. Anyway FSR is more about the specification of a FP resonator used as spectrometer, whether the incoming light has one or more frequencies doesn't affect the property of the spectrometer, does it?

 

[marco1235]

I understand your reply and it's correct. You're right! But the problem is that the general theory doesn't specify that the field impinging on the cavity is made of a frequency range. Probably I'm making a lot of confusion in my brain, and also I may not fully understand interference in general. I guess that FSR with a monochromatic field in a FP could be the separation (in cm, μm or whatever) between to orders of interference, say the first and the second bright ring. What I think is that the theory is misleading.. I mean, the only difference in two rings is the angular one, so there's a Δθ rather than a Δλ!

 

[blue_leaf77]

But the problem is that the general theory doesn't specify that the field impinging on the cavity is made of a frequency range.

Indeed, that's why the theory should prevail both for monochromatic or polychromatic light.

I guess that FSR with a monochromatic field in a FP could be the separation (in cm, μm or whatever) between to orders of interference, say the first and the second bright ring.

Again, FSR which stands for free spectral range is a kind of ability of a FP resonator related to its application as a spectrometer. The important point to note is that, FSR was derived from the general expression of the phase separation between two successive maxima in the transmitted intensity. The requirement for the phase difference between each outgoing rays in an FP resonator so that they interfere constructively is $\delta = (2\pi/\lambda)2nl\cos{\theta} = 2m\pi$ where $m$ is an integer. Hence you can vary either wavelength, refractive index, resonator length, or angle of incidence to move from one maximum to the next maximum, or in general to scan the whole range of transmitted intensity curve. As for FSR, it's assumed that the incoming light has broad spectrum coming at equal incidence for all frequencies (the other quantities in $\delta$ are also fixed) so that you can plot the transmitted intensity as a function of frequency and you come to the idea of FSR, which is defined as the separation between two wavelengths which correspond to adjacent maxima. On the other hand if the incoming light is monochromatic but has no defined direction, for example a gas lamp, the transmitted intensity profile can be expressed as a function of incidence angle, and if you use a lens behind the resonator you get a collection of concentric rings as you said above.

 

[marco1235]

Now makes more sense! Thank you for this enlightening discussion!