Using the FabryPerot interferometer, we were able to learn a great deal about the longitudinal modes of the HeNe laser. We were able to see the frequency separation between modes, and could investigate how this separation changed due to cavity length.
The key to measuring the frequency separation of the HeNe laser was understanding the concept of Free Spectral Range. This value describes the distance between two resonant frequencies of the FabryPerot interferometer and is known to be 2 GHz. Therefore, in looking at a readout from the FabryPerot, we know that the frequency separation between any identical peaks is 2 Ghz. Knowing this value, we were able to calculate the frequency separation between any two peaks. We could then compare this experimental value to the known value of frequency separation, c/2L.
Knowing that c= f l , we can take the derivative and derive the equation, df =  (c / l*l ) dl. With this equation, we can find the wavelength difference between the longitudinal modes. Using the Wavemeter Jr., we measured the laser's wavelength to be 632.83 nm. This value is extremely close to theoretical values.
Below we show a few sample readouts and the theoretical and experimental frequency separation.
Cavity Length = 39.45 cm
Experimental Frequency Separation: 419.2 MHz
Theoretical Frequency Separation: 380.2 MHz Wavelength Separation: 0.00056 nm

Cavity Length = 41.55 cm
Experimental Frequency Separation: 327.6 MHz
Theoretical Frequency Separation: 361.0 MHz Wavelength Separation: 0.00048 nm

Using the data taken from all trials, we were able to create a graph of Frequency Separation vs. Cavity Length. Plotting our data along with the theoretical equation, df =c /2L, we can check the accuracy of our measurements..
As the graph above shows, our experimental data correlates very nicely with the theoretical equation. As expected, the frequency separation decreases as the cavity length increases.
A HeNe laser usually uses one flat mirror and one curved mirror, called a "semiconfocal" arrangement. Curved mirrors make aligning the laser somewhat easier, but are much more expensive to produce. Using one flat and one curved mirror makes for a good trade between cost and usability. The curvature of the transmission mirror on the outside (not the reflective side inside the cavity) is slightly different from the curvature on the inside of the mirror, so the beam is as concise as possible.
In order for the laser to lase at all, there must be optical resonance between the two mirrors. For two curved mirrors with the same radius of curvature, a "waist" will occur halfway between them. The setup will lase at any cavity length less than or equal to twice the radius of curvature for the mirrors.
Looking at the representation of the Gaussian beam propagating, the positions marking the phase fronts represent possible positions for reflectors to cause resonance. Notice that the flat phase front (z=0) is positioned exactly halfway between either pair of symmetrical phase fronts. So, if two symmetrical mirrors will lase at any cavity length less than or equal to twice the radius of curvature, the semiconfocal configuration will lase at any cavity length less than or equal to the radius of curvature.
We found a maximum cavity length at which the laser still lased of 55.05 cm. This translates to a radius of curvature for our mirror of 55.05 cm.
References:
Kallard, T. Exploring Laser Light Stony Brook NY: American Association of Physics Teachers, 1982. (Reprinted with permission from Optosonic Press ©1977)
Yariv, Amnon. Optical Electronics 3rd Edition. New York: Holt, Rinehart, and Winston Inc., 1985.