15.4 Non-linear dispersion formulation and instability threshold 447and also
ω−ω 3 =x+iy−ω 3 ≃−ω 1. (15.105)
These assumptions and definitions have been made so thatε 1 (ω−ω 3 )is close to zero,
ε 1 (ω+ω 3 )is not close to zero, and positiveycorresponds to instability. The term 1 /ε 1 (ω+
ω 3 )can therefore be discarded as being non-resonant and the nonlinear dispersion relation
simplifies to
ε 2 (ω)ε 1 (ω−ω 3 )=(
λZ 3
2) 2
. (15.106)
The linear dispersion relationsε 2 andε 1 on the left hand side are each Taylor-expanded to
give
ε 2 (ω) = ε 2 (ω 2 +x−ω 2 +iy)≃ ε 2 (ω 2 )+(x−ω 2 +iy)dε 2
dω∣
∣
∣∣
ω=ω 2
= −2iω 2 Γ 2 − 2 ω 2 (x−ω 2 +iy) (15.107)and
ε 1 (ω−ω 3 ) = ε 1 (−ω 1 +x+iy+ω 1 −ω 3 )≃ ε 2 (−ω 1 )+(x+iy+ω 1 −ω 3 )dε 1
dω∣
∣∣
∣
ω=−ω 1
= 2iω 1 Γ 1 +2ω 1 (x+iy+ω 1 −ω 3 ). (15.108)Using these expansions, the nonlinear dispersion relation becomes
{−i (Γ 2 +y)−(x−ω 2 )}{i (Γ 1 +y) + (x+ω 1 −ω 3 )}=1
ω 1 ω 2(
λZ 3
4) 2
(15.109)
or, in more compact form,
( ̄x+i (Γ 2 +y))( ̄x−∆+i (Γ 1 +y)) =−1
ω 1 ω 2(
λZ 3
4) 2
(15.110)
where
x ̄=x−ω 2 (15.111)
and
∆=ω 3 −(ω 1 +ω 2 ) (15.112)
is the frequency mismatch.
The real and imaginary parts of Eq.(15.110) are
( ̄x−∆) ̄x−(Γ 2 +y)(Γ 1 +y) = −1
ω 1 ω 2(
λZ 3
4) 2
(15.113)
x ̄(Γ 1 +y)+( ̄x−∆)(Γ 2 +y) = 0. (15.114)Solving Eq.(15.114) gives
x ̄=(Γ 2 +y)
(2y+Γ 1 + Γ 2 )∆, ̄x−∆=−(Γ 1 +y)
(2y+Γ 1 + Γ 2 )