Soliton Propagation Through Photorefractive Media
129
τ = t −
z
v g
= t − ik 1 z
(48)
and we introduce the optical pulse defined by a function
˜
A(z, τ ) = ˜
A(z, t)
(49)
Next the chain rule of differentiation is used to show that
∂ ˜
A
∂z
=
∂ ˜
A s
∂z
+
∂ ˜
A s
∂τ
∂τ
∂z
=
∂ ˜
A s
∂z
− k 1
∂ ˜
A s
∂τ
(50)
∂ ˜
A
∂t
=
∂ ˜
A s
∂z
∂z
∂t
+
∂ ˜
A s
∂τ
∂τ
∂t
=
∂ ˜
A s
∂τ
(51)
and analogously that
∂
2 ˜
A s
∂t 2 =
∂
2 ˜
A s
∂τ 2 These expressions are now introduced into Eq.
(37), which becomes
∂ ˜
A s
∂z
+
1
2
ik 2
∂
2 ˜
A s
∂t 2 − ik N L ˜
A s = 0
( 5 2 )
The two equations obtained for the signal and pump beams in the case of two-beam
coupling in a photorefractive material are, therefore,
∂ A 1
∂z
− ik N L ˜
A 1 +
iβ 1
2
∂
2 ˜
A 1
∂z 2 = 0
∂ A 2
∂z
− ik N L ˜
A 2 +
iβ 2
2
∂
2 ˜
A 2
∂z 2 = 0
(53)
where
β i =
d
2 k i
dω
2
i
, ω i = ω 0
(54)
with i = 1, 2 and
k N L =
n 2 ωn 0
2π
|A 1 |
2
+ |A 2 |
2
+
A
∗
1 A 2
1 + iδτ
(55)
Substituting the value of, the Eq. (53) reduces to
∂ A 1
∂z
+
iβ 1
2
∂
2 A 1
∂z 2 −
in 2 ωn 0
2π
|A 1 |
2
+ |A 2 |
2
A 2 −
in 2 ωn 0
2π(1 + iδτ )
|A 1 |
2 A 2 = 0
∂ A 2
∂z
+
iβ 2
2
∂
2 A 2
∂z 2 −
in 2 ωn 0
2π
|A 1 |
2
+ |A 2 |
2
A 1 −
in 2 ωn 0
2π(1 + iδτ )
|A 2 |
2 A 1 = 0 (56)