Schawlow-Townes limit

In the physics of a laser, the Schawlow-Townes limit (rarely also Schawlow-Townes line width ) describes the minimum spectral line width of a laser beam that cannot be fallen below. It is named after the two Nobel laureates in physics Arthur L. Schawlow and Charles H. Townes , who predicted this limit value in 1958 and thus before the construction of the first laser in 1960 . A laser cannot have an “infinitely narrow” line width, i.e. a single and precisely defined light frequency. The reasons for this limit are quantum mechanical (especially the spontaneous emission , which takes place out of phase in contrast to the stimulated emission , and the Heisenberg uncertainty principle ) and optical effects (interaction of light with the components of the laser, "noise").

The line width is given by the formula

${\displaystyle \Delta \nu _{\text{Laser}}={\frac {2\pi h\nu (\Delta \nu _{\mathrm {c} })^{2}}{P_{\mathrm {aus} }}},}$

wherein the Planck's constant , the bandwidth of the laser resonator and the output power are of the laser. In some sources the factor 2 is missing in this formula, this depends on whether the derivation of the formula is carried out with the full or only half the width at half maximum of the normal distribution . However, this does not change the result, since the bandwidths and for the calculation must then also be halved. ${\displaystyle h}$ ${\displaystyle \Delta \nu _{\mathrm {c} }}$ ${\displaystyle P_{\mathrm {aus} }}$ ${\displaystyle \Delta \nu _{\text{Laser}}}$ ${\displaystyle \Delta \nu _{\mathrm {c} }}$

There are works that undercut this limit. However, this still does not mean that infinitely narrow line shapes are possible, but only that the minimal width of Schawlow and Townes was estimated to be too "pessimistic".

It was recently pointed out in a publication that in the original work by Gordon, Zeiger and Townes the maser line width was derived in a purely semi-classical manner, while quantum fluctuations were not taken into account, and that when Schawlow and Townes transferred to the optical regime, no quantum fluctuations were either were taken into account. Since the Schawlow-Townes line width was derived in a purely semi-classical manner, it is obvious that this line width must have a semi-classical physical cause. In this publication, the fundamental line width of a laser was derived semi-classically. It was shown that the original Schawlow-Townes line width represents a four-fold approximation of this fundamental line width of a laser, whereby the four approximations from the two original publications are already clear. Furthermore it was shown that the Schawlow-Townes line width does not represent a lower limit for the line width of a laser due to these approximations. It was already shown before that spontaneous emission in the laser mode cannot take place with any phase shift, as this would violate energy conservation.

Related Links

• Schawlow and Townes: Infrared and Optical Masers , original work
• Schawlow – Townes Linewidth in the Encyclopedia of Laser Physics and Technology (engl.)

Individual evidence

1. ↑ Derivation of the formula (PDF; 67 kB)
2. ↑ Y. Shevy, J. Iannelli, J. Kitching, and A. Yariv: Self-quenching of the semiconductor laser linewidth below the Schawlow-Townes limit by using optical feedback (PDF; 463 kB)
3. ↑ K. Yoshida, M. Kourogi, K. Nakagawa and M. Ohtsu: 1/8 Correction factor of Schawlow-Townes limit in FM noise of negative frequency feedback lasers (PDF; 283 kB)
4. ↑ ^{a b} M. Pollnau, M. Eichhorn: coherence Spectral Part I: Passive resonator linewidth, fundamental laser linewidth, and Schawlow-Townes approximation . In: Progress in Quantum Electronics . In press, 2020. doi : 10.1016 / j.pquantelec.2020.100255 .
5. ↑ ^{a b} J. P. Gordon, HJ Zeiger, CH Townes: The maser − New type of microwave amplifier, frequency standard, and spectrometer . In: Physical Review . 99, No. 4, 1955, pp. 1264-1274. doi : 10.1103 / PhysRev.99.1264 .
6. ↑ ^{a b} A. L. Schawlow, CH Townes: Infrared and optical masers . In: Physical Review . 112, No. 6, 1958, pp. 1940-1949. doi : 10.1103 / PhysRev.112.1940 .
7. ↑ M. Pollnau: Phase aspect in photon emission and absorption . In: Optica . 5, No. 4, 2018, pp. 465–474. doi : 10.1364 / OPTICA.5.000465 .