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Schawlow–Townes Linewidth

Definition: linewidth of a single-frequency laser with quantum noise only

More general term: linewidth

German: Schawlow-Townes-Linienbreite

Categories: lasers, quantum optics, fluctuations and noise

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Even before the first laser was experimentally demonstrated, Schawlow and Townes calculated the fundamental (quantum) limit for the linewidth of a laser [1]. This lead to the famous Schawlow–Townes equation:

Schawlow--Townes linewidth

with the photon energy h ν, the resonator bandwidth Δν2.9979e8 (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 [2]. 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), we obtain the modified linewidth equation

modified Schawlow--Townes linewidth

A more general form of the equation [6, 7] is

Schawlow--Townes linewidth, modified equation

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.

Calculator for the Schawlow--Townes Linewidth

Emission wavelength:
Spontaneous emission factor:
Resonator round-trip time:
Output coupler transmission: (has to be ≪ 1)
Parasitic round-trip losses: (has to be ≪ 1)
Output power:
Linewidth: calc

Enter input values with units, where appropriate. After you have modified some inputs, click the "calc" button to recalculate the output.

The corresponding two-sided power spectral density of the phase noise is

Schawlow--Townes phase noise

This corresponds to white frequency noise with

Schawlow--Townes frequency noise

Essentially, the Schawlow–Townes linewidth is determined by the strength of fluctuations of the optical phase which are caused by quantum noise. 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 [3].

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):

Application to Mode-locked Lasers

Interestingly, the Schawlow–Townes formula can be applied even to actively mode-locked lasers [4], 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 [5]. 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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Bibliography

[1]A. L. Schawlow and C. H. Townes, “Infrared and optical masers”, Phys. Rev. 112 (6), 1940 (1958), doi:10.1103/PhysRev.112.1940
[2]M. Lax, “Classical Noise. V. Noise in Self-Sustained Oscillators”, Phys. Rev. 160 (2), 290 (1967), doi:10.1103/PhysRev.160.290
[3]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
[4]P.-T. Ho, “Phase an amplitude fluctuations in a mode-locked laser”, IEEE J. Quantum Electron. 21 (11), 1806 (1985), doi:10.1109/JQE.1985.1072594
[5]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
[6]R. Paschotta, "Derivation of the Schawlow–Townes linewidth"
[7]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

(Suggest additional literature!)

See also: linewidth, linewidth enhancement factor, laser noise
and other articles in the categories lasers, quantum optics, fluctuations and noise

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