Page 466 - High Power Laser Handbook

434 Fi b er L a s er s Intr oduction to Optical Fiber Lasers 435

the higher extraction efficiency in fiber amplifiers is the much larger
available small signal gain enabled by the waveguide structure,
which can exceed 50 dB. In fact, in high-power amplifiers, the small
signal gain is limited by the onset of parasitic amplifier oscillations
due to Rayleigh scattering, backscattering from optical fiber end
faces, as well as external optical components. Therefore, it is often
advisable to reduce the amplifier gain to values of 20 to 40 dB by pro-
viding a sufficiently strong seed signal. A ytterbium optical fiber
amplifier operating at 100 W output power typically requires an
injection power of 0.1 to 1.0 W for reasonably safe operation.
For operation at repetition rates less than 10 kHz or at a pulse
fluence close to the saturation fluence, the time-dependent amplifier
saturation characteristics must be considered. In a timeframe mov-
ing with the pulse, the amplified pulse and the evolution of the
amplifier inversion during pulse propagation as a function of z can
be obtained using

(,
∂Iz t) = s ∆ σ N (, )( (15.29a)
z tI zt,)
∂t e eff s
z t)
∂∆N (, =− 1 σ s ∆N (, )( ,) (15.29b)
eff
z tI zt
∂t hν e eff s
σ
where ∆ eff =N N 2 −( e s / σ s a ) N is the effective population inversion at
1
position z as a function of time. Here we have neglected both ASE and the
influence of signal-dependent changes in pump absorption. For opera-
tion at repetition rates less than 10 kHz, it is also beneficial to use pulsed
pumping in order to minimize the buildup of ASE between pulses. Equa-
tions (15.29a) and (15.29b) can be readily extended to include pulsed
25
pumping. Analytic solutions to these equations for arbitrary pulse
shapes can also be readily found. 26
The total energy extracted from the amplifier is given by

E = F A ln( G G ) (15.30)
/
extr sat eff 0 f
where G are the amplifier gain values before and after the pulse
0,f
propagates through the amplifier and A is the mode area. E extr is
eff
maximized when the amplifier is totally saturated, such that G = 1.
f
An amplifier needs to be seeded at the saturation fluence to
achieve significant energy extraction from an inverted system. F sat =
3+
2
100 J/cm , or 1 µJ/µm , for Yb -doped optical fibers at the gain peak
2
2
s
of 1030 nm, where σ = 0.2 pm . Because peak pulse powers are lim-
e
ited by SRS, SPM, damage threshold, and nonlinear self-focusing, the
minimum pulse duration to reach saturation fluence while staying
below various nonlinear thresholds can be determined for known
saturation fluence and effective mode area A . Because the power
eff