Schawlow-Townes Linewidth | previous | next | feedback |
Definition: linewidth of a single-frequency laser with quantum noise only
Even before the first laser was experimentally demonstrated, A. L. Schawlow and C. H. Townes calculated the fundamental (quantum) limit for the linewidth of a laser [1]. This lead to the famous Schawlow-Townes formula:
with the photon energy hν, the resonator bandwidth Δνc (full width at half maximum), and the output power Pout. It has been assumed that there are no parasitic cavity losses. (Compared with the original formula, a factor 4 has been removed because of a different definition of the resonator bandwidth.)
A more general and perhaps more useful form of the equation is
where Toc denotes the output coupler transmission, ltot the total cavity losses (which may be larger than Toc), Trt the cavity round-trip time, and θ is the spontaneous emission factor which takes into account increased spontaneous emission in three-level gain media.
The corresponding two-sided power spectral density of the phase noise is
This corresponds to white frequency noise with
It is often claimed that the phase noise level corresponding to the Schawlow-Townes linewidth is a result of spontaneous emission into the laser mode. While this picture is intuitive, it is not quite correct. Both the laser gain and the linear losses of the laser resonator contribute equal amounts of quantum noise to the intracavity light field. This means that even when replacing laser gain with some noiseless amplification process, the phase noise would only go down to one half the Schawlow-Townes value [2].
Carefully constructed solid-state lasers can have very small linewidths in the region of a few kHz, which is still significantly above their Schawlow-Townes limit: technical excess noise makes it difficult to reach that limit. The linewidth of semiconductor lasers is also normally much larger than according to the original formula (without the α factor). This, however, is largely caused by amplitude-to-phase coupling effects, which can be quantified with the linewidth enhancement factor, and not by technical excess noise.
Bibliography
| [1] | A. L. Schawlow and C. H. Townes, "Infrared and optical masers", Phys. Rev. 112 (6), 1940 (1958) |
| [2] | H. M. Wiseman, "Light amplification without stimulated emission: Beyond the standard quantum limit to the laser linewidth", Phys. Rev. A 60 (5), 4083 (1999) |
See also: linewidth, linewidth enhancement factor, laser noise
Categories: fluctuations and noise, lasers, quantum optics
