Nonlinear Fiber Optics - 4 ed. Agrawal

10.2. Theory of Four-Wave Mixing 371
the fiber together with the pump, the signal is amplified while a new idler wave at ω 4
is generated simultaneously. The gain responsible for such amplification is called the
parametric gain. In this section, we consider the FWM mixing process in detail and
derive an expression for the parametric gain. The nondegenerate case (ω 1 ≠ ω 2 )is
considered for generality.
10.2.1 Coupled Amplitude Equations
The starting point is, as usual, the wave equation (2.3.1) for the total electric field
E(r,t) with P NL given in Eq. (10.1.1). We substitute Eqs. (10.1.2) and (10.1.3) in the
wave equation, together with a similar expression for the linear part of the polarization,
and neglect the time dependence of the field components E j ( j = 1 to 4) assuming
quasi-CW conditions. Their spatial dependence is, however, included using E j (r) =
F j (x,y)A j (z), where F j (x,y) is the spatial distribution of the fiber mode in which the
jth field propagates inside the fiber [12]. Integrating over the spatial mode profiles, the
evolution of the amplitude A j (z) inside an optical fiber is governed by the following set
of four coupled equations:
dA 1
dz = in [(
2ω 1
f 11 |A 1 | 2 + 2
c
∑ f 1k |A k | 2) A 1 + 2 f 1234 A ∗ 2A 3 A 4 e iΔkz] , (10.2.1)
k≠1
dA 2
dz = in [(
2ω 2
f 22 |A 2 | 2 + 2
c
∑ f 2k |A k | 2) A 2 + 2 f 2134 A ∗ 1A 3 A 4 e iΔkz] , (10.2.2)
k≠2
dA 3
dz = in [(
2ω 3
f 33 |A 3 | 2 + 2
c
∑ f 3k |A k | 2) A 3 + 2 f 3412 A 1 A 2 A ∗ 4e −iΔkz] , (10.2.3)
k≠3
dA 4
dz = in [(
2ω 4
f 44 |A 4 | 2 + 2
c
∑ f 4k |A k | 2) A 4 + 2 f 4312 A 1 A 2 A ∗ 3e −iΔkz] , (10.2.4)
k≠4
where the wave-vector mismatch Δk is given in Eq. (10.1.8). The overlap integral f jk
is defined in Eq. (7.1.14) of Section 7.1. The new overlap integral f ijkl is given by [12]
f ijkl =
〈Fi ∗F∗
j F kF l 〉
[〈|F i | 2 〉〈|F j | 2 〉〈|F k | 2 〉〈|F l | 2 , (10.2.5)
〉] 1/2
where angle brackets denote integration over the transverse coordinates x and y. In
deriving Eqs. (10.2.1) through (10.2.4), we kept only nearly phase-matched terms and
neglected frequency dependence of χ (3) . The parameter n 2 is the nonlinear-index coefficient
defined earlier in Eq. (2.3.13).
10.2.2 Approximate Solution
Equations (10.2.1) through (10.2.4) are quite general in the sense that they include the
effects of SPM, XPM, and pump depletion on the FWM process; a numerical approach
is necessary to solve them exactly. Considerable physical insight is gained if the pump
waves are assumed to be much more intense than the other waves and to remain undepleted
during the FWM process. As a further simplification, we assume that all overlap