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# Round-trip gain

Round-trip gain refers to the laser physics, and laser cavitys (or laser resonators). It is gain, integrated along a ray, which makes a round-trip in the cavity.

## Round-trip gain in geometric optics

Generally, the Round-trip gain may depend on the frequency, on the position and tilt of the ray, and even on the polarization of light. Usually, we may assume that at some moment of time, at reasonable frequency of operation, the gain $~G(x,y,z)~$ is function of the Cartesian coordinates $~x~$, $~y~$, and $~z~$. Then, assuming that the geometrical optics is applyable the round-trip gain $~g~$ can be expressed as follows: $~g=\int G(x(a),y(a),z(a))~{\rm d}a~$,


where $~a~$ is path along the ray, parametrized with functions $~x(a)~$, $~y(a)~$, $~z(a)~$; 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 $~G~$ is assumed to be constant. In the case of simlest cavity, the round-trip gain $~g=2Gh~$, where $~h~$ 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 $~\theta~$ 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 deermined by the transmitivity of the output coupler, or the magnification coefficient in the case of unstable cavity .

## Round-trip loss

The round-trip loss $~\beta~$ 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 lates 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  .

At the steady-state operation, the round-trip gain $~g~$ exactly compensate both, the output coupling and losses: $~\exp(g)~(1-\beta-\theta)=1~$. Assuming, that the gain is small ( $~g~\ll 1~$), this relation can be written as follows: $~g=\beta+\theta~$


Such as relation is used in analytic estimates of the performance of lasers . In particular, the round-trip loss $~\beta~$ may be one of important parameters which limit the output power of a disk laser; at the power scaling, the gain $~G~$ should be decreased (in order to avoid the exponential growth of the amplified spontaneous emission), and the round-trip gain $~g~$ should remain larger than the background loss $~\beta~$; this requires to increase of the thickness of the slab of the gain medium; at certain thickness, the overheating prevents the efficient operation .

For the analysis of processes in active medium, the sum $~\beta+\theta~$ can be also called "loss" ; however, such a notation leads to a confusion as soon as we are interested, which part of the energy is absorbed and scattered, and which part of such a "loss" is actually useful output of the laser.

## References

1. ^ a b A.E.Siegman (1986). Lasers. University Science Books. ISBN 0-935702-11-3.
2. ^ G.L.Oppo; A.Politi (1985). "Toda potential in laser equations". Zeitschrift fur Physik B 59: 111-115.
3. ^ D.Kouznetsov; J.-F.Bisson, J.Li, K.Ueda (2007). "Self-pulsing laser as oscillator Toda: Approximation through elementary functions". Journal of Physics A 40: 1-18.
4. ^ D.Kouznetsov; J.-F.Bisson, K.Takaichi, K.Ueda (2005). "Single-mode solid-state laser with short wide unstable cavity". JOSAB 22 (8): 1605-1619.
5. ^ D. Kouznetsov; J.-F. Bisson, J. Dong, and K. Ueda (2006). "Surface loss limit of the power scaling of a thin-disk laser". JOSAB 23 (6): 1074-1082. Retrieved on 2007-01-26.;