128
L. Parameswar
with
I =
n lin ω 0 c
2π
|A(z, t)|
2
(40)
and
k 1 =
dk
dω
=
1
c
n lin (ω) + ω
dn lin
dω
ω=ω 0
=
1
v g ω 0
(41)
and
k 2 =
d
2 k
dω 2 =
d
dω
1
v g (ω)
=
−
1
v 2
g
dv g
dω
ω=ω 0
(42)
In the above expression, k 1 is the reciprocal of the group velocity, and k 2 is a measure
of the dispersion of the group velocity. Equation (37) for is next introduced into the
reduced wave Eq. (36) and the equation becomes
∂ A
∂z
− ik N L A − ik 1 (ω − ω 0 )A −
1
2
ik 2 (ω − ω 0 )
2 A = 0
(43)
which is transformed from the frequency domain to the time domain. For this multiply
each term by the factor e
−i(ω−ω 0 )t and integrate the resulting equation over all values
of (ω − ω 0 ). The resulting integrals are evaluated as follows.
+∞
−∞
A(z, ω − ω 0 )e
−i(ω−ω 0 )t d(ω − ω 0 )
2π
= ˜
A(z, t)
(44)
+∞
−∞
(ω − ω 0 )A(z, ω − ω 0 )e
−i(ω−ω 0 )t d(ω − ω 0 )
2π
=
1
−i
∂
∂t
+∞
−∞
A(z, ω − ω 0 )e
−i(ω−ω 0 )t d(ω − ω 0 )
2π
= i
∂
∂t
˜
A(z, t)
(45)
+∞
−∞
(ω − ω 0 )
2 A(z, ω − ω 0 )e
−i(ω−ω 0 )t d(ω − ω 0 )
2π
= −
∂
2
∂t 2
˜
A(z, t)
(46)
Equation (43) then becomes
∂ ˜
A
∂z
+ k 1
∂ ˜
A
∂t
+
1
2
ik 2
∂
2 ˜
A
∂t 2 − ik N L ˜
A = 0
(47)
Introducing the retarded time τ as