72
3 Microscopic Expressions of Nonlinear Polarization
Then ρ (2) in Eq. (3.28) is derived,
ρ
(2)
mn (t) =
i
¯
h 2
l
j,k
x−z
p,q
m|μ p |l
(ρ
(0)
l − ρ
(0)
n )
l|μ q |n
1
ω k − ω ln + ii ln
−(ρ
(0)
m − ρ
(0)
l )
m|μ q |l
1
ω k − ω ml + ii ml
l|μ p |n
E p (ω j )E q (ω k )
e
−i(ω j +ω k )t
−i(ω j + ω k ) + iω mn + mn
= −
1
¯
h 2
l
j,k
x−z
p,q
(ρ
(0)
l − ρ
(0)
n )
m|μ p |l
l|μ q |n
ω k − ω ln + ii ln
− (ρ
(0)
m − ρ
(0)
l )
m|μ q |l
l|μ p |n
ω k − ω ml + ii ml
·
1
(ω j + ω k ) − ω mn + ii mn
E p (ω j )E q (ω k )e
−i(ω j +ω k )t .
(3.28)
3.4.4 Effective χ (2) Formula
[Problem 3.4] Derive Eqs. (3.49), (3.50), (3.51), and (3.52). Also explain that other
combinations of light polarization vanish for the C ∞v interface.
Equations (3.49), (3.50), (3.51), and (3.52) are derived from the definition of χ
(2)
eff
in Eq. (2.23) and the unit vectors of electric fields below.
• S polarization:
ˆ
e
α
S (() = ˆ
e
α
S (ω 1 ) = ˆ
e
α
S (ω 2 ) =
⎛
⎝
0
1
0
⎞
⎠
• P polarization:
ˆ
e
α
P (()=
⎛
⎝
− cos θ α (()
0
sin θ α (()
⎞
⎠ , ˆ
e
α
P (ω 1 )=
⎛
⎝
cos θ α (ω 1 )
0
sin θ α (ω 1 )
⎞
⎠ , ˆ
e
α
P (ω 2 )=
⎛
⎝
cos θ α (ω 2 )
0
sin θ α (ω 2 )
⎞
⎠ .
In the SSP case,
χ
(2)
eff,SSP =
x∼z
q,r,u
e S,q (()χ
(2)
qru ((, ω 1 , ω 2 )e S,r (ω 1 )e P,u (ω 2 )