Nonlinear Fiber Optics - 4 ed. Agrawal

9.4. SBS Dynamics 345
9.4.1 Coupled Amplitude Equations
The coupled intensity equations (9.2.1) and (9.2.2) are valid only under steady-state
conditions. To include the transient effects, we need to solve the Maxwell wave equation
(2.3.1) with the following material equation [10]:
∂ 2 ρ ′
∂t 2
− Γ A∇ 2 ∂ρ′
∂t
− v 2 A∇ 2 ρ ′ = −ε 0 γ e ∇ 2 (E · E), (9.4.1)
where ρ ′ = ρ − ρ 0 is a change in the local density from its average value ρ 0 , Γ A is
the damping coefficient, and γ e = ρ 0 (dε/dρ) ρ=ρ0 is the electrostrictive constant introduced
earlier in Section 9.1. The nonlinear polarization P NL appearing in Eq. (2.3.1)
also has an additional term that depends on ρ ′ as
P NL = ε 0 [χ (3) . . . EEE +(γe /ρ 0 )ρ ′ E], (9.4.2)
where we have neglected the Raman contribution.
To simplify the following analysis, we assume that all fields remain linearly polarized
along the x axis and introduce the slowly varying fields, A p and A s ,as
E(r,t)= ˆxRe[F p (x,y)A p (z,t)exp(ik p z − iω p t)+F s (x,y)A s (z,t)exp(−ik s z − iω s t)],
(9.4.3)
where F j (x,y) is the mode profile for the pump ( j = p) orStokeswave(j = s). The
density ρ ′ is expanded in a similar fashion as
ρ ′ (r,t)=Re[F A (x,y)Q(z,t)exp(ik A z − iΩt)], (9.4.4)
where Ω = ω p −ω s and F A (x,y) is the spatial distribution of the acoustic mode with the
amplitude Q(z,t). If several acoustic modes participate in the SBS process, a sum over
all modes should be used in Eq. (9.4.4). In the following we consider a single acoustic
mode responsible for the dominant peak in the Brillouin-gain spectrum.
By using Eqs. (9.4.1)–(9.4.4) and Eq. (2.3.1) with the slowly varying envelope
approximation, we obtain the following set of three coupled amplitude equations:
∂A p
∂z + 1 ∂A p
v g ∂t
= − α 2 A p + iγ(|A p | 2 + 2|A s | 2 )A p + iκ 1 A s Q, (9.4.5)
− ∂A s
∂z + 1 ∂A s
= − α v g ∂t 2 A s + iγ(|A s | 2 + 2|A p | 2 )A s + iκ 1 A p Q ∗ , (9.4.6)
∂Q
∂t + v ∂Q
A
∂z = −[ 1 2 Γ B + i(Ω B − Ω)]Q + iκ 2
A p A ∗ s ,
A eff
(9.4.7)
where Γ B = k 2 A Γ A is the acoustic damping rate, A p is normalized such that |A p | 2 represents
power, and the coupling coefficients are defined as
κ 1 =
γ eω p
2n p cρ 0
,
κ 2 = γ eω p
c 2 v A
. (9.4.8)
Equations (9.4.5) through (9.4.7) govern the SBS dynamics under quite general
conditions [86]–[94]. They include the nonlinear phenomena of SPM and XPM, but