Round-trip gain explained
Round-trip gain refers to the laser physics, and laser cavities (or laser resonators). It is gain, integrated along a ray, which makes a round-trip in the cavity.
At the continuous-wave operation, the round-trip gain exactly compensates both the output coupling of the cavity and its background loss.
Round-trip gain in geometric optics
is function of the
Cartesian coordinates
,
, and
. Then, assuming that the
geometrical optics is applicable the round-trip gain
can be expressed as follows:
~g=\intG(x(a),y(a),z(a))~{\rmd}a~
,where
is path along the ray, parametrized with functions
,
,
; the integration is performed along the whole ray, which is supposed to form the closed loop.
In simple models, the flat-top distribution of pump and gain
is assumed to be constant. In the case of simplest cavity, the round-trip gain
, where
is length of the cavity; the laser light is supposed to go forward and back, this leads to the coefficient 2 in the estimate.
In the steady-state continuous wave operation of a laser, the round-trip gain is determined by the reflectivity of the mirrors (in the case of stable cavity) and the magnification coefficient in the case of unstable resonator (unstable cavity).
Coupling parameter
The coupling parameter
of a laser resonator determines, what part of the energy of the laser field in the cavity goes out at each round-trip. This output can be determined by the transmitivity of the
output coupler, or the magnification coefficient in the case of unstable cavity.
[1] Round-trip loss (background loss)
The background loss, of the round-trip loss
determines, what part of the energy of the laser field becomes unusable at each round-trip; it can be absorbed or scattered.
At the self-pulsation, the gain is late to respond the variation of number of photons in the cavity. Within the simple model, the round-trip loss and the output coupling determine the damping parameters of the equivalent oscillator Toda.[2] [3]
At the steady-state operation, the round-trip gain
exactly compensate both, the output coupling and losses:
~\exp(g)~(1-\beta-\theta)=1~
.Assuming, that the gain is small (
), this relation can be written as follows:
Such as relation is used in analytic estimates of the performance of lasers.[4] In particular, the round-trip loss
may be one of important parameters which limit the output power of a
disk laser; at the power scaling, the gain
should be decreased (in order to avoid the
exponential growth of the
amplified spontaneous emission), and the round-trip gain
should remain larger than the background loss
; this requires to increase of the thickness of the slab of the
gain medium; at certain thickness, the
overheating prevents the efficient operation.
[5] For the analysis of processes in active medium, the sum
can be also called "loss".
[1] This notation leads to confusions as soon as one is interested, which part of the energy is absorbed and scattered, and which part of such a "loss" is actually wanted and useful output of the laser.
References
- Book: Lasers. A.E.Siegman. 1986. University Science Books. 978-0-935702-11-8. 2007-05-24. 2016-12-06. https://web.archive.org/web/20161206073700/http://www.uscibooks.com/siegman.htm. dead.
- Toda potential in laser equations. G.L.Oppo. A.Politi . . 59. 1. 111–115. 1985. 10.1007/BF01325388. 1985ZPhyB..59..111O . 119657810. 0722-3277.
- Self-pulsing laser as oscillator Toda: Approximation through elementary functions. D. Kouznetsov. J.-F. Bisson. J. Li. K. Ueda. Journal of Physics A. 40. 1–18. 2007. 10.1088/1751-8113/40/9/016. 2007JPhA...40.2107K. 9 . 10.1.1.535.5379. 53330023 .
- D. Kouznetsov. J.-F. Bisson. K. Takaichi. K. Ueda. Single-mode solid-state laser with short wide unstable cavity. Journal of the Optical Society of America B. 22. 8. 1605–1619. 2005. 10.1364/JOSAB.22.001605. 2005JOSAB..22.1605K.
- D. Kouznetsov. J.-F. Bisson. J. Dong. K. Ueda. Surface loss limit of the power scaling of a thin-disk laser. Journal of the Optical Society of America B. 23. 6. 1074–1082. 2006. 2007-01-26. 10.1364/JOSAB.23.001074. 2006JOSAB..23.1074K.
- http://www.ils.uec.ac.jp/~dima/disk.pdf
