Fabry-Perot Etalon Homework: Determining Wavenumber Separation

[rabbit44]

Homework Statement

A certain spectral line is known to consist of two equally intense components
with a wavenumber separation less than 20 m^-1: The Fabry-Perot fringes
produced by this line are photographed, using a plate separation of 25 mm. The
diameters of the smallest rings are found to be, in mm :
1:82; 3:30; 4:84; 5:57; 6:60; 7:15:
Explain why this experiment does not allow the wavenumber separation to be determined uniquely, but gives two possible values. What are they? Suggest a further experiment
which could be carried out to resolve the ambiguity.

Homework Equations

Wavenumber FSR=1/2nd

The Attempt at a Solution

So I think this has something to do with the free spectral range? If you work that out with the above equation, assuming the etalon is in air, you get 20m^01. So as the wavenumber separation is less than this, fringes from the two different wavenumbers will overlap. But then I don't know how to get the two possible answers.

Any help? Please?

 

[Jhero]

You are correct in thinking that the issue lies with the free spectral range (FSR). The FSR is the spacing between two consecutive interference fringes in an etalon. In this case, the FSR is 20 m^-1, which means that the two components of the spectral line are too close together to be resolved by the etalon. This is why the smallest rings in the photograph have overlapping diameters, as you have observed.

To determine the two possible values for the wavenumber separation, we need to consider the ratio of the FSR to the plate separation (d). This ratio is equal to 1/2n, where n is the number of fringes between the two components. In this case, we have n=1, so the ratio is 1/2. This means that the two possible values for the wavenumber separation are 1/2 of the FSR and 3/2 of the FSR. In other words, the wavenumber separation could be either 10 m^-1 or 30 m^-1.

To resolve this ambiguity, we could carry out a further experiment using a different etalon with a larger FSR. This would allow us to clearly distinguish between the two components of the spectral line and determine the wavenumber separation more accurately. Alternatively, we could use a narrowband filter to isolate one of the components and measure its wavenumber directly. This would also give us a more precise value for the wavenumber separation.

I hope this helps to clarify the situation. Keep up the good work in your scientific investigations!