Although our graph of Frequency Separation vs. Cavity Length shows us that our experimental data is fairly close to predicted values, there is always a 30 to 40 MHz difference between the two. We contribute this constant source of error to two factors. First, our measurements of the cavity length were not as accurate as we would have liked. We were unable to get an exact location of the mirror inside the lasing tube, thus our cavity length measurements were crude.
A second source of error that we have identified was our use of the Fabry-Perot Interferometer readouts. Many times, we were unable to get distinct modal peaks. Therefore, our measurements for frequency separation between modes were not perfectly accurate.
In both cases, a slight error in either cavity length or distance between modes would create error in our calculating the frequency separation.
The mathematics behind determining the radius of curvature for our concave mirror are simple and well documented. The only measurement required of us was the maximum cavity length at which the laser would lase. However, this measurement is far from exact. To take this measurement, we moved the mirror back and used our (soon to be patented) alignment method to align the mirrors and produce a laser beam. When we reached a cavity length that would not lase, we returned to the last length that did lase and increased the cavity length by small increments until we could not produce lasing.
The possible source of error is glaringly obvious. What if we hadn't reached a cavity length at which lasing would not occur, we simply didn't find the proper alignment? The only way to verify our determination of the radius of curvature at 55.05 cm would be from documentation of the specific radius of curvature to which the mirror was manufactured, which we did not have.
We are fairly confident that our alignment method was effective, although longer cavity lengths were slightly more difficult to align. Available texts1 indicate that the maximum cavity length for our hypothetical cavity with two curved mirrors of equal radius of curvature (called a symmetric concentric resonator) is INCREDIBLY sensitive to misalignment. By inference, the maximum cavity length for our semi-confocal arrangement is likely to be equally difficult to align, if not more so. Our measurement may be inaccurate, but we believe that it is somewhat close to the true maximum cavity length.
We were able to find a few pure modes, but many of the modal structures we found were not found in the texts we had available. These mode structures are not pure TEM modes, but combinations of several modes. We cannot readily identify which modes make up our structures, as there are an infinite number of possible combinations of modes.
1. Siegman, Anthony E. Lasers Mill Calley, California: University Science Books, 1986. p 755f