Many multipass amplifiers use laser crystals, which are either end-pumped or side-pumped, and then contain a number of laser mirrors to make the beam pass through the crystal multiple times. Since the beams of different channels need to be highly separated, they have different angular directions, although in principle their propagation directions can be made parallel by using a specific spatial offset. If the crystal is relatively thin, the different beams will overlap strongly inside the crystal, the limit case being thin-disk lasers.

Figure 1: Schematic of the multi-pass amplifier setup.

When the light of different channels overlaps strongly in the crystal, the overall gain is approximately equal to the product of the number of channels and the gain of a single channel. In addition, the effective saturation energy will also decrease. For small overall gain, it will decrease as the channel decreases.
When setting up amplifiers with many channels, the directions of the beams are usually not in the same plane. The design and arrangement of such amplifiers can be very complex issues.
In addition to being amplified, the signal beam also experiences other effects, such as thermal lensing or nonlinear effects in the crystal. Thermal lensing in particular can greatly affect the lateral beam shape; gain steering can also have this effect. This effect (natural beam divergence) can be counteracted by incorporating focusing optics (usually curved laser mirrors). Thermal lenses can also deflect light beams if they do not travel along the center of the thermal lens. The optimal arrangement depends on the pump power and signal light power.
Positive feedback amplifiers are considered a special kind of multipass amplifier. Here, instead of obtaining multiple channels by setting the geometric path of the beam, optical switches are used. This applies to ultrashort pulses, where the length of the pulse is much smaller than the round-trip time. So you can inject a pulse, let it cycle many times, and then output it. This is very convenient for obtaining very high overall gain, much more feasible than multipass amplifiers with geometrical setups.