Nonlinear Optical Probes and Processes in Polymers and Liquid ...

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−2ik2∇E20 + k 2 0F ∗ E10 = 0
Expressing the wave vectors k1 and k2 in coordinates as k1 = k(ˆz cos θ + ˆx sin θ),
√ √
k2 = k(ˆz cos θ − ˆx sin θ) where k = k0 ε0 (= 2π ε0 /λ), we rewrite the Eqs. 2.14 in
the form
∂E10
∂z
F k0
+ i
2 √ ε0 cos θ E20 = 0 (2.15)
∂E20
∂z + i F ∗ k0
2 √ ε 0 cos θ E10 = 0
Here we recall that F ∼ Esc,max. The amplitude of space charge field Esc,max depends
on the depth of the intensity modulation
as [5]
m =
2E10E ∗ 20
|E10| 2 + |E20| 2
Esc,max = mEw
(2.16)
(2.17)
where Ew is the part of the space charge field that does not depend on the intensity
of the incident beams. Substitution of Eqs. 2.16, 2.17 and Eq. 2.10 into Eqs. 2.15
yields
∂E10
∂z − in3 reEwk0
2 cos θ
∂E20
∂z − in3 reEwk0
2 cos θ
|E20| 2E10 |E10| 2 = 0
+ |E20|
2
|E10| 2E20 |E10| 2 = 0
+ |E20|
2
(2.18)