Nonlinear Fiber Optics - 4 ed. Agrawal

10.2. Theory of Four-Wave Mixing 373
The same equation is obtained for B ∗ 4 . Thus, the general solution of the form [12]
B 3 (z) =(a 3 e gz + b 3 e −gz )exp(−iκz/2), (10.2.17)
B ∗ 4(z) =(a 4 e gz + b 4 e −gz )exp(iκz/2), (10.2.18)
where a 3 , b 3 , a 4 , and b 4 are determined from the boundary conditions. The parametric
gain g depends on the pump power and is defined as
√
g = (γP 0 r) 2 − (κ/2) 2 , (10.2.19)
where we have introduced the parameters r and P 0 as
r = 2(P 1 P 2 ) 1/2 /P 0 , P 0 = P 1 + P 2 . (10.2.20)
The solution given in Eqs. (10.2.17) and (10.2.18) is valid only when the pump
waves remain largely undepleted. Pump depletion is included by solving the complete
set of four equations, Eqs. (10.2.1) through (10.2.4). Such a solution can be written in
terms of elliptic functions [27] but is not discussed here because of its complexity.
10.2.3 Effect of Phase Matching
The preceding derivation of the parametric gain has assumed that the two pumps are
distinct. If the pump fields cannot be distinguished on the basis of their frequency,
polarization, or spatial mode, the entire procedure should be carried out with only three
terms in Eq. (10.1.2). The parametric gain is still given by Eq. (10.2.19) if we choose
P 1 = P 2 = P 0 /2(r = 1) and use
κ = Δk + 2γP 0 . (10.2.21)
Figure 10.1 shows variations of g with Δk in this specific case for several values of γP 0 .
The maximum gain (g max = γP 0 ) occurs at κ = 0, or at Δk = −2γP 0 . The range over
which the gain exists is given by 0 > Δk > −4γP 0 , as dictated by Eqs. (10.2.19) and
(10.2.21). The shift of the gain peak from Δk = 0 is due to the contribution of SPM and
XPM to the phase mismatch as apparent from Eq. (10.2.21).
It is useful to compare the peak value of the parametric gain with that of the Raman
gain [7]. From Eq. (10.2.19) the maximum gain is given by (using r = 1)
g max = γP 0 = g P (P 0 /A eff ), (10.2.22)
where γ is used from Eq. (10.2.7) and g P is defined as g P = 2πn 2 /λ 1 at the pump
wavelength λ 1 . Using λ 1 = 1 μm and n 2 ≈ 2.7 × 10 −20 m 2 /W, we obtain g P ≈ 1.7 ×
10 −13 m/W. This value should be compared with the peak value of the Raman gain
g R in Figure 8.1. The parametric gain is larger by about 70% compared with g R .Asa
result, the pump power for the FWM gain is expected to be lower than that required for
Raman amplification, if phase matching can be realized. In practice, SRS dominates
for long fibers because it is difficult to maintain phase matching over long fiber lengths.
We can define a length scale, known as the coherence length, using L coh = 2π/|Δκ|,
where Δκ is the maximum value of the wave-vector mismatch that can be tolerated.