Nonlinear Fiber Optics - 4 ed. Agrawal

9.2. Quasi-CW SBS 333
[20]. A part of the increase is due to the guided nature of acoustic modes in optical
fibers [15]. Most of the increase, however, can be attributed to inhomogeneities in
the fiber-core cross section along the fiber length. The numerical aperture of the fiber
also plays a role in broadening the SBS gain spectrum [25]. Because such factors are
specific to each fiber, Δν B is generally different for different fibers and can exceed
100 MHz in the 1.55-μm spectral region.
Equation (9.1.4) for the Brillouin gain is obtained under steady-state conditions and
is valid for a CW or quasi-CW pump (pulse width T 0 ≫ T B ), whose spectral width Δν p
is much smaller than Δν B . For pump pulses of width T 0 < T B , the Brillouin gain is substantially
reduced [5] compared with that obtained from Eq. (9.1.5). Indeed, if the pulse
width becomes much smaller than the phonon lifetime (T 0 < 1 ns), the Brillouin gain
is reduced below the Raman gain; such a pump pulse generates a forward-propagating
Raman pulse through SRS, as discussed in Section 8.3.
Even for a CW pump, the Brillouin gain is reduced considerably if the spectral
width Δν p of the pump exceeds Δν B . This can happen when a multimode laser is used
for pumping. It can also happen for a single-mode pump laser whose phase varies
rapidly on a time scale shorter than the phonon lifetime T B . Detailed calculations show
that the Brillouin gain, under broadband pumping conditions, depends on the relative
magnitudes of the pump-coherence length [26]–[28], defined by L coh = c/(n p Δν p ), and
the SBS-interaction length L int , defined as the distance over which the Stokes amplitude
varies appreciably. If L coh ≫ L int , the SBS process is independent of the mode
structure of the pump laser provided the longitudinal-mode spacing exceeds Δν B , and
the Brillouin gain is nearly the same as for a single-mode laser after a few interaction
lengths [26]. In contrast, the Brillouin gain is reduced significantly if L coh ≪ L int .
The latter situation is generally applicable to optical fibers, where interaction length is
comparable to the fiber length L whenever fiber losses are not too large. In the case
of a pump laser with a Lorentzian spectral profile of width Δν p , the gain spectrum is
still given by Eq. (9.1.4) but the peak value of Brillouin gain is reduced by a factor
1 + Δν p /Δν B [28]. As a result, the SBS threshold increases by a large factor when
Δν p ≫ Δν B .
9.2 Quasi-CW SBS
Similar to the SRS case, the development of SBS in optical fibers requires consideration
of mutual interaction between the pump and Stokes waves. In this section we develop a
simple theory valid under CW or quasi-CW conditions and use it to discuss the concept
of Brillouin threshold.
9.2.1 Brillouin Threshold
Under steady-state conditions, applicable for a CW or quasi-CW pump, SBS is governed
by the two coupled equations similar to Eqs. (8.1.2) and (8.1.3). The only difference
is that the sign of dI s /dz should be changed to account for the counterpropagating
nature of the Stokes wave with respect to the pump wave. Two simplifications can be
made; the first ω p ≈ ω s owing to a relatively small value of the Brillouin shift and the