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112 Angermann and Yatsyk
+ 1 4
∑
χ (3)
⎧
n,m,p∈Z\{0}
⎪⎨ n≠−m, p=s
m≠−p, n=s
⎪⎩ n≠−p, m=s
n+m+p=s
1111 (sω; nω, mω, pω)E 1(r, nω)E 1 (r, mω)E 1 (r, pω), (5)
·
where the symbol = means that higher-order terms are neglected.
If we study nonlinear effects involving the waves at the first
three frequency components of E 1 only, it is possible to restrict the
system (4), (5) to three equations. Then the analysis of the scattering
problem for the plane wave packet
{
E1 inc (r, nκ) := E1 inc (nκ; y, z) := a inc
nκ exp
(i ( φ nκ y − Γ nκ (z − 2πδ) ))} 3
, (6)
z > 2πδ, δ > 0, with amplitudes a inc
nκ, angles of incidence ϕ nκ ,
|ϕ| < π/2
√
(cf. Figure 1) and κ := ω/c = 2π/λ, φ nκ := nκ sin ϕ nκ ,
Γ nκ := (nκ) 2 − φ 2 nκ, on the nonlinear structure can be simplified by
means of Kleinman’s rule (i.e., the equality of all the coefficients χ (3)
1111
at the multiple frequencies, [10, 12]) and reduces finally to the following
system of boundary-value problems ([4–6, 11, 17]):
[
∇ 2 +κ 2 ε κ (z, α(z), E 1 (r, κ), E 1 (r, 2κ), E 1 (r, 3κ)) ] × E 1 (r, κ)
= −α(z)κ 2 E 2 1(r, 2κ)Ē1(r, 3κ),
[
∇ 2 +(2κ) 2 ε 2κ (z, α(z),E 1 (r, κ),E 1 (r, 2κ),E 1 (r,3κ)) ] ×E 1 (r, 2κ)=0,
[
∇ 2 +(3κ) 2 ε 3κ (z, α(z), E 1 (r, κ), E 1 (r, 2κ), E 1 (r, 3κ)) ] × E 1 (r, 3κ)
= −α(z)(3κ) 2{ 1
}
3 E3 1(r, κ) + E1(r, 2 2κ)Ē1(r, κ) ,
where κ := ω/c = 2π/λ,
{
ε nκ := ε (L) + ε (NL)
nκ , |z| ≤ 2πδ,
and ε (L) := 1 + 4πχ (1)
11
1, |z| > 2πδ,
,
[
ε (NL)
nκ
:= α(z)
|E 1 (r, κ)| 2 + |E 1 (r, 2κ)| 2 + |E 1 (r, 3κ)| 2
[Ē1 (r, κ) ] ]
2
+δ n1
E 1 (r, κ) E Ē 1 (r, 2κ)
1(r, 3κ)+δ n2
E 1 (r, 2κ) E 1(r, κ)E 1 (r, 3κ)
with α(z) := 3πχ (3)
1111 (z) and δ nm — Kronecker’s symbol. In addition,
the following conditions are met (n = 1, 2, 3):
(C1) E 1 (nκ; y, z) = U(nκ; z) exp(iφ nκ y),
(the quasi-homogeneity condition w.r.t. y),
n=1
(7)