74
3 Microscopic Expressions of Nonlinear Polarization
+
F
α→β
zz
(() sin θ
α (()
F
α→β
zz
(ω 1 ) sin θ
α (ω 1 )
F
α→β
xx
(ω 2 ) cos θ
α (ω 2 )
χ
(2)
zzx
+
F
α→β
zz
(() sin θ
α (()
F
α→β
zz
(ω 1 ) sin θ
α (ω 1 )
F
α→β
zz
(ω 2 ) sin θ
α (ω 2 )
χ
(2)
zzz
=
−F
α→β
xx
(() cos θ
α (()
F
α→β
xx
(ω 1 ) cos θ
α (ω 1 )
F
α→β
zz
(ω 2 ) sin θ
α (ω 2 )
χ
(2)
xxz
+
−F
α→β
xx
(() cos θ
α (()
F
α→β
zz
(ω 1 ) sin θ
α (ω 1 )
F
α→β
xx
(ω 2 ) cos θ
α (ω 2 )
χ
(2)
xzx
+
F
α→β
zz
(() sin θ
α (()
F
α→β
xx
(ω 1 ) cos θ
α (ω 1 )
F
α→β
xx
(ω 2 ) cos θ
α (ω 2 )
χ
(2)
zxx
+
F
α→β
zz
(() sin θ
α (()
F
α→β
zz
(ω 1 ) sin θ
α (ω 1 )
F
α→β
zz
(ω 2 ) sin θ
α (ω 2 )
χ
(2)
zzz .
(3.52)
The shaded components of χ (2) also vanish for the C ∞v interface to derive the last
expression.
The other combinations of polarization result in null amplitude and yield no SFG
signal for the C ∞v interface. The relevant tensor components of χ (2) included in
χ
(2)
eff for the other polarization combinations are listed in the following.
• SPP — χ
(2)
yxx , χ
(2)
yxz , χ
(2)
yzx , χ
(2)
yzz .
• PSP — χ
(2)
xyx , χ
(2)
xyz , χ
(2)
zyx , χ
(2)
zyz
• PPS — χ
(2)
xxy , χ
(2)
xzy , χ
(2)
zxy , χ
(2)
zzy
• SSS — χ
(2)
yyy
We notice that all these components vanish.
Appendix
A.1 Off-Diagonal Elements of Density Matrix
We have learned in Sect. 3.1 that the density matrix can represent statistical
ensemble of states and a pure state in the common formulas. It is instructive to
illustrate the distinction between a superposition of quantum states and a statistical
ensemble of states. This example is useful to clarify the concept of coherence.
Let us consider two wavefunctions, φ 1 and φ 2 , for example. If the two states are
superposed in the quantum sense, the state is represented by a wavefunction,
ψ(t) = c 1 (t)φ 1 + c 2 (t)φ 2
where |c 1 (t)|
2
+ |c 2 (t)|
2
= 1
or equivalently by a density matrix
ρ
pure (t) =
c 1 c ∗
1 c 1 c ∗
2
c 2 c ∗
1 c 2 c ∗
2
.
(3.57)