Helium-Neon (He-Ne) and other gas lasers contain a resonant optical cavity formed by two highly reflecting mirrors. Light amplified in the cavity reflects from the mirrors to form certain specific standing-wave patterns called modes. There are two distinct categories of such modes. Transverse modes are the spatial patterns of light seen when the laser beam is magnified and viewed on a screen; these patterns have the form of a Gaussian function times a Hermite-Gauss polynomial. Longitudinal modes are the standing wave patterns for which there is a node (zero) at each mirror. Only those modes that are amplified by the laser's gain medium (excited neon atoms in this case) will appear in the output beam of the laser. The wavelength λ of a longitudinal mode is related to the mirror separation (cavity length) L by λ = 2L/N, where N is a large integer. Typical HeNe lasers are 15 to 100 cm long and can produce 1 to 10 consecutive longitudinal modes. The frequency of the Nth mode is f=c/λ. It follows that the frequency separation of two adjacent modes is Δf = c/(2L). The cavity length L increases with time as the laser warms up; this causes the frequencies of the longitudinal modes to "sweep" across the gain curve.

We started this project by observing several types of HG transverse modes up to order 3. They were created in an "open cavity" laser by manipulating a hair inside the cavity and the alignment of the output coupler mirror. Longitudinal modes can't be observed directly because their frequency (over 400 THz) is far beyond the range of any instrument. What we observed instead are the frequency differences or "beats" between modes using a high-speed photodetector (Thorlabs DET-210) and a high-speed oscilloscope (Tektronix 485) or rf spectrum analyzer (HP 8566A). One unexpected result was that the intermode beats are not exact multiples of c/(2L). These variations produce dramatic "beat of beats" effects that slowly vary with time as the laser temperature drifts. In a 44 cm long laser with mode spacing of ~340 MHz the period of these low-frequency second-order beats cycled from ~15 micro-seconds to infinity and back over a few minutes. (Infinite period means that the mode spacings are then precisely equal.) Similar but even more complex varying beat effects were observed with the spectrum analyzer.

We hope to extend this project in the future by investigating the temporal coherence of HeNe laser beams with a long-arm interferometer.

We thank Dr. Sam Goldwasser for providing most of the lasers used in this project, and for invaluable technical advice.