Nonlinear Fiber Optics - 4 ed. Agrawal

10.4. Parametric Amplification 395
where κ = Δk+γ(P 1 +P 2 ) and P 1 and P 2 are the input pump powers, assumed to remain
undepleted. The amplification factor is related to g as indicated in Eq. (10.4.7) and is
given by
G s = P 3 (L)/P 3 (0)=1 +(2γ/g) 2 P 1 P 2 sinh 2 (gL). (10.4.15)
Similar to the single-pump case, the gain spectrum is affected by the frequency
dependence of the linear phase mismatch Δk. If we introduce ω c =(ω 1 + ω 2 )/2asthe
center frequency of the two pumps with ω d =(ω 1 − ω 2 )/2, and expand Δk around ω c ,
we obtain [89]:
Δk(ω 3 )=2
∞
∑
m=1
β c 2m
(2m)!
[
(ω3 − ω c ) 2m − ωd
2m ]
, (10.4.16)
where the superscript c indicates that dispersion parameters are evaluated at the frequency
ω c . This equation differs from Eq. (10.4.12) by the ω d term, which contributes
only when two pumps are used. The main advantage of dual-pump FOPAs over singlepump
FOPAs is that the ω d term can be used to control the phase mismatch. By properly
choosing the pump wavelengths, it is possible to use this term for compensating
the nonlinear phase mismatch γ(P 1 + P 2 ) such that the total phase mismatch κ is maintained
close to zero over a wide spectral range.
The most commonly used configuration for dual-pump FOPAs employs a relatively
large wavelength difference between the two pumps. At the same time, the center
frequency ω c is set close to the zero-dispersion frequency ω 0 of the fiber so that the
linear phase mismatch in Eq. (10.4.16) is constant over a broad range of ω 3 . To achieve
a fairly wide phase-matching range, the two pump wavelengths should be located on
the opposite sides of the zero-dispersion wavelength in a symmetrical fashion [90].
With this arrangement, κ can be reduced to nearly zero over a wide wavelength range,
resulting in a gain spectrum that is nearly flat over this entire range.
The preceding discussion assumes that only the nondegenerate FWM process, ω 1 +
ω 2 → ω 3 +ω 4 , contributes to the parametric gain. In reality, the situation is much more
complicated for dual-pump FOPAs because the degenerate FWM process associated
with each pump occurs simultaneously. In fact, it turns out that the combination of
degenerate and nondegenerate FWM processes can create up to eight other idler fields,
in addition to the one at the frequency ω 4 [66]. Only four among these idlers, say,
at frequencies ω 5 , ω 6 , ω 7 , and ω 8 , are important for describing the gain spectrum
of a dual-pump FOPA. These four idlers are generated through the following FWM
processes:
ω 1 + ω 1 → ω 3 + ω 5 , ω 2 + ω 2 → ω 3 + ω 6 , (10.4.17)
ω 1 + ω 3 → ω 2 + ω 7 , ω 2 + ω 3 → ω 1 + ω 8 . (10.4.18)
In addition to these, the energy conservation required for FWM is also maintained
during the following FWM processes:
ω 1 + ω 1 → ω 4 + ω 7 , ω 2 + ω 2 → ω 4 + ω 8 , (10.4.19)
ω 1 + ω 2 → ω 5 + ω 8 , ω 1 + ω 2 → ω 6 + ω 7 , (10.4.20)
ω 1 + ω 4 → ω 2 + ω 5 , ω 1 + ω 6 → ω 2 + ω 4 . (10.4.21)