# Relaxation Oscillations

Definition: small mutually coupled oscillations of the laser power and laser gain around their steady-state values

German: Relaxationsoszillationen

Category: laser devices and laser physics

Author: Dr. Rüdiger Paschotta

Cite the article using its DOI: https://doi.org/10.61835/l1n

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When a laser is disturbed during operation, e.g. by fluctuations of the pump power, its output power does not immediately return to its steady state. Many lasers (e.g. solid-state lasers and most laser diodes) operate in the so-called *class B regime*, with the upper-state lifetime often being much longer than the laser resonator's damping time. In that regime, the laser dynamics are such that changes in pump power lead to so-called *relaxation oscillations*. These are usually damped, eventually leading back to the steady state. Particularly pronounced oscillatory behavior with relatively low oscillation frequencies (often in the kilohertz regime) occurs in doped insulator solid-state lasers, whereas semiconductor lasers normally exhibit strongly damped relaxation oscillations with very high frequencies in the gigahertz region. Other lasers, e.g. many gas lasers, operating in the *class A regime* with an upper-state lifetime below the cavity damping time, do not exhibit relaxation oscillations, but only an exponential relaxation to the steady state.

As Figure 1 shows, class B lasers can exhibit strong spiking e.g. when the pump power is suddenly turned on. After the emission of a few spikes (pulses), the laser power exhibits damped relaxation oscillations. The oscillation frequency is similar to the inverse period of the spikes.

Calculations of relaxation phenomena can be based on the dynamic equations as presented in the article on laser dynamics, which can (for small fluctuations, not for spiking) be linearized around the steady state. In the following, the main results of such an analysis for class-B lasers are given. The frequency of the relaxation oscillations is determined by the intracavity power <$P_\rm{int}$>, the resonator losses <$\rho$>, the round-trip time <$T_\textrm{rt}$> of the resonator, and the saturation energy <$E_\rm{sat}$> and the upper-state lifetime <$\tau_\textrm{g}$> of the laser gain medium:

$$f_{\rm ro} = \frac{1}{2\pi} \sqrt{\frac{\rho \; P_{\rm int}}{T_{\rm rt}\;E_{\rm sat}} - \frac{1}{4}{\left( \frac{1}{\tau_{\rm g}} + \frac{P_{\rm int}}{E_{\rm sat}} \right)}^2} $$The cavity damping time corresponds to <$T_\textrm{rt} / \rho$>, and the first term in the radicand is larger than the second one in the mentioned class B regime. For solid-state lasers (with <$\tau_\textrm{g} \gg T_\textrm{rt}$>), the second term of the radicand is negligible (except for operation close to the laser threshold), so that the equation simplifies to

$${f_\rm{ro}} \approx \frac{1}{2\pi} \sqrt{\frac{{\rho \; P_{\rm int}}}{{T_{\rm rt}\;E_{\rm sat}}}} $$The equations are valid for both four-level and three-level laser gain media. However, only for four-level gain media can the former equation be transformed into

$${f_\rm{ro}} = \frac{1}{{2\pi}} \sqrt{\frac{{\rho \; (r - 1)}}{{T_\rm{rt}\;{\tau _{\rm{g}}}}} - {{\left(\frac{r}{{2{\tau _{\rm{g}}}}} \right)}^2}} $$where <$r$> is the so-called pump parameter, which is the ratio of pump power to threshold pump power.

The damping time of the oscillations can be calculated from

$${\tau _{{\rm{ro}}}} = \frac{2}{{\frac{1}{{{\tau _{\rm{g}}}}} + \frac{P}{{{E_{{\rm{sat}}}}}}}}$$For operation just above the laser threshold, the relaxation oscillations are slow, and their damping time is about twice the upper-state lifetime of the gain medium. For higher powers, the oscillations can faster, and the damping time gets shorter. For four-level lasers, the damping time is inversely proportional to the pump parameter <$r$>.

Note that a saturable absorber in the laser resonator, which may be used for passive mode locking, can strongly reduce the damping [2]; the oscillations can even become undamped, so that the steady state becomes unstable. This leads to the phenomenon of Q-switching instabilities and Q-switched mode locking.

The characterization of the laser dynamics can deliver useful information on the laser parameters such as the resonator losses or the gain saturation energy, thus also the transition cross-sections of the laser gain medium.

### Bibliography

[1] | K. J. Weingarten et al., “In situ small-signal gain of solid-state lasers determined from relaxation oscillation frequency measurements”, Opt. Lett. 19 (15), 1140 (1994); https://doi.org/10.1364/OL.19.001140 |

[2] | A. Schlatter et al., “Pulse-energy dynamics of passively mode-locked solid-state lasers above the Q-switching threshold”, J. Opt. Soc. Am. B 21 (8), 1469 (2004); https://doi.org/10.1364/JOSAB.21.001469 |

[3] | A. E. Siegman, Lasers, University Science Books, Mill Valley, CA (1986) |

[4] | O. Svelto, Principles of Lasers, Plenum Press, New York (1998) |

See also: laser dynamics, spiking, Q-switching instabilities

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2023-09-20

Why does the frequency of relaxation oscillation increase with increasing intracavity power?

The author's answer:

To understand this, one needs to analyze the equations given in the article on laser dynamics. For low laser powers, one can see that the laser gain relaxes with a time constant which is the upper-state lifetime. For higher powers, the laser power speeds up the change of gain with time; the system gets “more stiff”.