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# Laser Dynamics

Author: the photonics expert

Definition: the temporal evolution of quantities such as the optical power and gain in a laser

The dynamic behavior of a laser is determined by the interaction of the intracavity light field with the gain medium. Essentially, the intracavity laser power can grow or decay exponentially according to the difference between gain and resonator losses, whereas the rate of change in the gain is determined by stimulated and spontaneous emission (and possibly by other effects such as quenching and energy transfer).

With certain approximations, including a not too high laser gain, the dynamics of the intracavity optical power <$P$> and the gain coefficient <$g$> in a continuous-wave laser can be described with the following coupled differential equations:

$$\frac{\partial P}{\partial t} = \frac{g - l}{T_\rm{rt}}P$$ $$\frac{{\partial g}}{{\partial t}} = - \frac{{g - {g_{{\rm{ss}}}}}}{{{\tau _{\rm{g}}}}} - \frac{{g\;P}}{{{E_{{\rm{sat}}}}}}$$

where <$T_\textrm{rt}$> is the cavity round-trip time, <$l$> the cavity loss per resonator round trip (including both the output coupler transmission and parasitic losses), <$g_\rm{ss}$> the small-signal gain (for a given pump power), <$\tau_\textrm{g}$> the gain relaxation time (often close to the upper-state lifetime), and <$E_\rm{sat}$> the saturation energy of the gain medium. The equations can be generalized in various ways, for example allowing for high gain and loss and for a time-dependent pump power, or separately describing the optical powers in individual resonator modes, which involves mode competition phenomena. A wide range of phenomena can be simulated in such ways.

## Typical Phenomena of Laser Dynamics

Dynamic aspects of special interest in the case of a continuous-wave laser are the following:

• When the pump source of a laser is switched on, the laser undergoes some dynamics until steady-state operation is reached. This often includes spiking of the output power.
• One can study how a laser reacts to disturbances during operation, e.g. sudden changes of pump power. Usually, such disturbances lead to relaxation oscillations.

In these respects, different types of lasers exhibit very different behaviors. For example, doped-insulator lasers (exhibiting low transition cross-sections and a long upper-state lifetime) exhibit a strong tendency for spiking and relaxation oscillations, whereas for laser diodes this is not the case.

Dynamic aspects are particularly important in Q-switched lasers, where the energy stored in the gain medium undergoes a large change during pulse emission. Q-switched fiber lasers, typically having a high laser gain, can exhibit additional dynamical phenomena. This can lead to a temporal sub-structure of the pulses, which cannot be explained with equations such as those given above.

Similar equations can be used for passively mode-locked lasers [1]; an additional term appears in the first equation, which describes the losses of the saturable absorber. The effect of this is typically that the damping of the relaxation oscillations is reduced. The relaxation oscillations may even become undamped, so that the steady-state solution becomes unstable, and the laser exhibits Q-switched mode locking or other kinds of Q-switching instabilities.

In optical amplifiers based on laser amplification, there can be a similar interplay of signal power and gain. In the case of a regenerative amplifier, the similarity is strongest.

## More to Learn

Encyclopedia articles:

### Bibliography

 [1] 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 [2] O. Svelto, Principles of Lasers, Plenum Press, New York (1998) [3] A. E. Siegman, Lasers, University Science Books, Mill Valley, CA (1986)

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