3.4 Solutions to Problems
73
=
x∼z
q,r,u
F
α→β
qq
(() ˆ
e
α
S,q (()
χ
(2)
qru ((, ω 1 , ω 2 )
F
α→β
rr
(ω 1 ) ˆ
e
α
S,r (ω 1 )
F
α→β
uu (ω 2 ) ˆ
e
α
P,u (ω 2 )
=
F
α→β
yy
(()
χ
(2)
yyx (Ω, ω 1 , ω 2 )
F
α→β
yy
(ω 1 )
F
α→β
xx
(ω 2 ) cos θ
α (ω 2 )
+
F
α→β
yy
(()
χ
(2)
yyz ((, ω 1 , ω 2 )
F
α→β
yy
(ω 1 )
F
α→β
zz
(ω 2 ) sin θ
α (ω 2 )
= F
α→β
yy
(()F
α→β
yy
(ω 1 )F
α→β
zz
(ω 2 ) sin θ
α (ω 2 )χ
(2)
yyz ((, ω 1 , ω 2 ),
(3.49)
where the term in a shade box vanishes for the selection rule. The last expression
takes account of χ
(2)
yyx = 0 for the C ∞v interface.
Similar expressions can be derived in the SPS and PSS cases.
χ
(2)
eff,SPS =
x∼z
q,r,u
e S,q (()χ
(2)
qru ((, ω 1 , ω 2 )e P,r (ω 1 )e S,u (ω 2 )
= F
α→β
yy
(()F
α→β
zz
(ω 1 )F
α→β
yy
(ω 2 ) sin θ
α (ω 1 )χ
(2)
yzy ((, ω 1 , ω 2 ),
(3.50)
χ
(2)
eff,PSS =
x∼z
q,r,u
e P,q (()χ
(2)
qru ((, ω 1 , ω 2 )e S,r (ω 1 )e S,u (ω 2 )
= F
α→β
zz
(()F
α→β
yy
(ω 1 )F
α→β
yy
(ω 2 ) sin θ
α (()χ
(2)
zyy ((, ω 1 , ω 2 ).
(3.51)
In the PPP case,
χ
(2)
eff,PPP =
x∼z
q,r,u
e P,q (()χ
(2)
qru ((, ω 1 , ω 2 )e P,r (ω 1 )e P,u (ω 2 )
=
x∼z
q,r,u
F
α→β
qq
(() ˆ
e
α
P,q (()
χ
(2)
qru ((, ω 1 , ω 2 )
F
α→β
rr
(ω 1 ) ˆ
e
α
P,r (ω 1 )
F
α→β
uu (ω 2 ) ˆ
e
α
P,u (ω 2 )
=
−F
α→β
xx
(() cos 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)
xxx
+
−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
xx
(() cos 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)
xzz
+
F
α→β
zz
(() sin 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)
zxz