Even before the first laser was experimentally demonstrated, Schawlow and Townes calculated the fundamental (quantum) limit for the linewidth of a laser . This lead to the Schawlow–Townes equation, which became one of the most famous equations of laser physics:
with the photon energy hν, the resonator bandwidth Δνc (half width at half-maximum, HWHM), and the output power Pout. It has been assumed that there are no parasitic cavity losses. Note that the result is interpreted as a half width at half-maximum.
Later, Melvin Lax showed that the linewidth in lasing operation (above the threshold pump power) must be two times smaller than derived by Schawlow and Townes . If we take into account that factor 1 / 2 and also convert the linewidth values to full width at half maximum (FWHM) (for both the laser linewidth and the cavity linewidth), which is more common, we obtain the modified linewidth equation
A more general form of the equation [6, 8] is
where Toc denotes the output coupler transmission, ltot the total resonator losses (which may be larger than Toc), Trt the resonator round-trip time, and θ the spontaneous emission factor, which takes into account increased spontaneous emission in (quasi-)three-level gain media.
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. Although this picture is intuitive, it is not completely 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 decrease to half of the Schawlow–Townes value .
Carefully constructed solid-state lasers can have very small linewidths in the region of a few kilohertz, which is still far above their Schawlow–Townes limit: technical excess noise makes it very difficult to reach that limit.
The linewidth of semiconductor lasers is also normally much larger than according to the original equation (without the α factor):
- There is also a spontaneous emission factor, e.g. of the order of 2 or 3, due to reabsorption in the semiconductor.
- More importantly, there are strong amplitude-to-phase coupling effects, which can be quantified with the linewidth enhancement factor α and can increase the linewidth by a factor 1 + α2, which can contribute more than an order of magnitude.
- In addition, there can be excess noise with 1 / f-type shape of the power spectral density of the instantaneous frequency. This is related to fluctuations of the charge carriers in the semiconductor.
- Particularly for external-cavity diode lasers, there can also be significant excess noise due to mechanical vibrations.
Application to Mode-locked Lasers
Interestingly, the Schawlow–Townes formula can be applied even to actively mode-locked lasers , using the total average power (rather than the power in a particular line of the frequency comb) for Pout. The latter rule may be surprising, but it can be understood considering that the phases of the individual lines in the optical spectrum can not evolve independently; they are locked together by the modulator. (Otherwise, the circulating pulse would fall apart.)
A further generalization is possible for passively mode-locked lasers . Here, the Schawlow–Townes formula provides an estimate for the linewidth near the center of the spectrum, whereas the linewidth in the spectral wings is somewhat increased due to quantum-noise-induced timing jitter. Note, however, that the nonlinear dynamics occurring in many passively mode-locked lasers can lead to strong excess noise, invalidating the Schawlow–Townes results for such cases.
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|||A. L. Schawlow and C. H. Townes, “Infrared and optical masers”, Phys. Rev. 112 (6), 1940 (1958), doi:10.1103/PhysRev.112.1940 (contains the famous Schawlow–Townes equation)|
|||M. Lax, “Classical noise. V. Noise in self-sustained oscillators”, Phys. Rev. 160 (2), 290 (1967), doi:10.1103/PhysRev.160.290|
|||H. M. Wiseman, “Light amplification without stimulated emission: beyond the standard quantum limit to the laser linewidth”, Phys. Rev. A 60 (5), 4083 (1999), doi:10.1103/PhysRevA.60.4083|
|||P.-T. Ho, “Phase and amplitude fluctuations in a mode-locked laser”, IEEE J. Quantum Electron. 21 (11), 1806 (1985), doi:10.1109/JQE.1985.1072594|
|||R. Paschotta et al., “Optical phase noise and carrier–envelope offset noise of mode-locked lasers”, Appl. Phys. B 82 (2), 265 (2006), doi:10.1007/s00340-005-2041-9|
|||R. Paschotta, "Derivation of the Schawlow–Townes linewidth"|
|||C. J. McKinstrie, T. J. Stirling and A. S. Helmy, “Laser linewidths: tutorial”, J. Opt. Soc. Am. B 38 (12), 3837 (2021), doi:10.1364/JOSAB.439882|
|||R. Paschotta, H. R. Telle, and U. Keller, “Noise of Solid State Lasers”, in Solid-State Lasers and Applications (ed. A. Sennaroglu), CRC Press, Boca Raton, FL (2007), Chapter 12, pp. 473–510|