We assume constant input power for all channels. We will now consider dynamic simulations, where all input powers may be time-dependent. After all, fiber amplifiers and lasers typically operate with optical pulses and/or pulsed pump sources.
As long as we do not enter the realm of ultrashort pulses, as we discuss in the next section, a simple generalization of the equations and numerical methods used can often be made:
For example, fiber amplifiers can first be pumped for a while and then injected with short signal pulses (e.g. Gaussian or super-Gaussian temporal shape). During the pumping phase, the (not yet correlated) signal gain rises steadily. When a signal pulse is injected, it initially benefits from high gain, but then quickly saturates that gain. As a result, other parts of the pulse will experience reduced gain and correspondingly lower output power. This can lead to significant distortion of the temporal pulse shape. Figure 1 shows a numerical example, taken from the detailed case study.
Figure 1: Output power and ytterbium excitation versus time for an ytterbium-doped fiber amplifier with pulsed pump and signal.
In pulse amplifier systems, propagation time is usually negligible.
Note that in this case we can safely ignore the propagation time, the time delay caused by the light traveling through the amplifier fiber. Even if the pulse duration is shorter than the propagation time, it usually doesn’t matter that different parts of the fiber “see” the pulse at slightly different times. After all, the usual level has a much longer lifetime. The time step used in the numerical algorithm can be larger or smaller than the propagation time; it only needs to be small enough to properly sample all temporal features, including saturation effects. Note that if the amplifier gain is saturated by very high signal strengths, it will drop rapidly. Therefore, the numerical step size may have to be much lower than the lifetime of the metastable energy level.
See also Part VII of our fiber amplifier tutorial, which discusses the behavior of fiber amplifiers in amplifying nanosecond pulses.
When simulating Q-switched fiber lasers, something new comes into play. We now need to consider the propagation time, since the round-trip time of the laser resonator now plays a crucial role. Therefore, we need to substantially expand the algorithm:
For modeling Q-switched lasers, the propagation time of light in the resonator must be considered.
In principle, neither the required equations nor the algorithms are very complex. However, the implementation is a bit cumbersome due to various “bookkeeping” requirements.
As an example, we can use the case study of a Q-switched fiber laser made with the software RP Fiber Power.
Figure 2: Output Power and Ytterbium Excitation vs. Time.
Figure 2 shows the evolution of the output power and excitation level during the first two pulse periods after switching on the laser. The resulting pulse shape may look very surprising. There are large variations in power within a single round-trip time, which typically do not occur in Q-switched bulk lasers. An important factor behind this phenomenon is the high gain of active optical fibers. ASE results in a very non-uniform distribution of optical power (at fairly low levels) within the active fiber until the Q-switch is turned on (transparent). If the Q-switch is turned on very quickly, then an uneven power distribution starts to propagate around the resonator, creating a spiked structure at the output. Think about it and the details will become apparent, but it’s definitely more complicated at these high gains – certainly more work on this number is needed. One can also study aspects such as the finite switching time of Q-switches.