Nonlinear Fiber Optics - 4 ed. Agrawal

10.4. Parametric Amplification 389
10.4.2 Gain Spectrum and Its Bandwidth
The most important property of any optical amplifier—and FOPAs are no exception—is
the bandwidth over which the amplifier can provide a relatively uniform gain. Typically,
FOPAs are pumped using one or two CW lasers acting as pumps. In the case of
dual pumps, the FWM process is governed by Eqs. (10.2.1)–(10.2.4), and a complete
description of parametric amplification requires a numerical solution.
Considerable physical insight is gained by employing the approximate analytic solution
given in Eqs. (10.2.17) and (10.2.18) and obtained under the assumption that
the pumps are not depleted much. The constants a 3 , b 3 , a 4 , and b 4 in these equations
are determined from the boundary conditions. If we assume that both signal and idler
waves are launched at z = 0, we find that the constants a 3 and b 3 satisfy
a 3 + b 3 = B 3 (0), g(a 3 − b 3 )=(iκ/2)(a 3 + b 3 )+2iγ √ P 1 P 2 B ∗ 4(0). (10.4.1)
Solving these equations, we obtain
a 3 = 1 2 (1 + iκ/2g)B 3(0)+iC 0 B ∗ 4(0), b 3 = 1 2 (1 − iκ/2g)B 3(0) − iC 0 B ∗ 4(0),
(10.4.2)
where C 0 =(γ/g) √ P 1 P 2 . A similar method can be used to find the constants a 4 and
b 4 . Using these values in Eqs. (10.2.17) and (10.2.18), the signal and idler fields at a
distance z are given by
B 3 (z) ={B 3 (0)[cosh(gz)+(iκ/2g)sinh(gz)] + iC 0 B ∗ 4(0)sinh(gz)}e −iκz/2 , (10.4.3)
B ∗ 4(z) ={B ∗ 4(0)[cosh(gz) − (iκ/2g)sinh(gz)] − iC 0 B 3 (0)sinh(gz)}e iκz/2 . (10.4.4)
The preceding general solution simplifies considerably when only the signal is
launched at z = 0 (together with the pumps), a practical situation for most FOPAs.
Setting B ∗ 4 (0)=0 in Eq. (10.4.3), the signal power P 3 = |B 3 | 2 grows with z as [12]
P 3 (z)=P 3 (0)[1 +(1 + κ 2 /4g 2 )sinh 2 (gz)], (10.4.5)
where the parametric gain g is given in Eq. (10.2.19). The idler power P 4 = |B 4 | 2 can
be found from Eq. (10.4.4) in the same way. It can also be obtained by noting from
Eqs. (10.2.13) and (10.2.14) that d(P 3 − P 4 )/dz = 0, or P 4 (z) =P 3 (z) − P 3 (0). Using
this relation, we obtain
P 4 (z)=P 3 (0)(1 + κ 2 /4g 2 )sinh 2 (gz). (10.4.6)
Equation (10.4.6) shows that the idler wave is generated almost immediately after
the input signal is launched into the fiber. Its power increases as z 2 initially, but both
the signal and idler grow exponentially after a distance such that gz > 1. As the idler is
amplified together with the signal all along the fiber, it can build up to nearly the same
level as the signal at the FOPA output. In practical terms, the same FWM process can
be used to amplify a weak signal and to generate simultaneously a new wave at the
idler frequency. The idler wave mimics all features of the input signal except that its
phase is reversed (or conjugated). Among other things, such phase conjugation can be
used for dispersion compensation and wavelength conversion in a WDM system [53].