The Circular Aperture: Transverse Modes

When the circular aperture is closed so that only the TEM₀₀ mode is visible, we see a graph similar to the one shown previously for the intermediate cavity length as is shown below.

As the aperture is opened, more and more transverse modes become visible. In the graph shown below, the original three axial modes are present, but the three smaller peaks which appear to the right of each of the three original peaks represent transverse modes that are now visible because of the increased size of the aperture. Using the equation for the eigenfrequencies of the transverse modes:

we can attempt to identify each new mode that appears. For instance, each of the three original modes are present in the graph shown below along with three additional modes. Using the above equation we can calculate the expected frequency difference between the TEM_00q mode and the TEM_01q, or TEM_10q modes. This frequency difference is 119,958,685 Hz. Assuming that the first peak in the graph below is the TEM_00q mode, we would expect the TEM _01q or the TEM_10q mode to appear 119,958,685 Hz from that mode. Knowing that the frequency difference between the TEM_00q and the TEM_00q+1 modes is 390,860,495 Hz, we can find the actual frequency difference between the first and second peak on the graph and compare it to the theoretical value of 119,958,685 Hz. The experimental value from the graph is 127,256,905 Hz. We would expect the same difference between the third and fourth peaks. However, the actual separation between peaks three and four is only 86,352,900 Hz. The fact that these two frequency differences are not remotely similar is disturbing. In fact, the error will not allow for the identification of all of the individual modes.

Opening the aperture farther allows still more of the beam to be detected and more modes to appear on the digital oscilloscope screen. The additional two "complex" modes appearing to the right of the second set of modes constitutes at least one other set of transverse modes, possibly two. The reason a third peak does not appear to make three sets of three peaks is that it is cut off by the Doppler envelope. It would be nice to be able to identify all of these peaks, however, a combination of factors does not allow for this identification process. The fact that the modes overlap causes major problems in identifying individual peaks. Furthermore, the spacing between the TEM_00q, TEM_00q+1, and the TEM_00q+1 modes should be equal, but it is not. Therefore, the task of identifying all of the individual modes seemed at best arbitrary, if not futile, so it was not undertaken.

If the aperture is opened completely, a graph containing many overlapping peaks, hardly distinguishable from one another results. In this graph all of the transverse modes along with the original axial modes are present.

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