15.4 Non-linear dispersion formulation and instability threshold 445Thus, if 1 /LC >> R/Lthe imaginary part of the frequency isωi=−R/ 2 Land the
real part of the frequency isωr=± 1 /
√
LC.This means that the resistor can be identified
with damping by the relationR→− 2 ωiL. InterpretingQjnow as the dependent variable
for oscillation modej, the appropriate differential equation for the damped mode can be
written as
d^2 Qj
dt^2
+2ΓjdQj
dt+ω^2 jQj=0 (15.88)whereΓj=|Imωj|is the linear damping rate for modej.Modejtherefore has the time
dependenceexp(±iωjt−Γjt)where bothωjandΓjare positive quantities.
Thus, when dissipation is included, the typical system given by Eq.(15.83) generalizes
to
(
∂^2
∂t^2
+2Γ 3
∂
∂t+ω^23)
̃E 3 = −λψ ̃E ̃ 2
(
∂^2
∂t^2+2Γ 2
∂
∂t
+ω^22)
̃E 2 = −λψ ̃E ̃ 3
(
∂^2
∂t^2+2Γ 1
∂
∂t+ω^21)
ψ ̃ = −λE ̃ 2 ·E ̃ 3. (15.89)Let us now consider again the situation where the pump waveE ̃ 3 is very large and so
may be considered as having approximately constant amplitude. The decay waves then do
not grow to sufficient amplitude to deplete the pump energy and so only the last two of the
three coupled equations have to be considered. We defineˆs 2 as a unit vector in the direction
ofE ̃ 2 so that
E ̃ 2 = ˆs 2 ·E ̃ 2 (15.90)
and so the last two of the three coupled equations become
(
∂^2
∂t^2
+2Γ 2
∂
∂t+ω^22)
E ̃ 2 = −λψ ̃sˆ 2 ·E ̃ 3
(
∂^2
∂t^2+2Γ 1
∂
∂t+ω^21)
ψ ̃ = −λE ̃ 2 ˆs 2 ·E ̃ 3. (15.91)The wavevector selection rules are assumed to be satisfied but a slight mismatching in the
frequency selection rules will be allowed. The pump wave is now writtenas
ˆs 2 · ̃E 3 =Z 3 cosω 3 t (15.92)whereZ 3 is the effective pump wave amplitude. The coupled daughter equations then
become
(
∂^2
∂t^2
+2Γ 2
∂
∂t+ω^22)
E ̃ 2 = −λZ^3
2(
eiω^3 t+e−iω^3 t) ̃
ψ
(
∂^2
∂t^2+2Γ 1
∂
∂t+ω^21)
̃ψ = −λZ^3
2(
eiω^3 t+e−iω^3 t)
E ̃ 2. (15.93)
The Fourier transform of a quantityf(t)is defined as
f(ω)=∫
dtf(t)eiωt (15.94)