82
4 Mathematical Models of the Interaction of Laser Radiation with Turbid Media
A
ξ
1
k
1
x
kn 1
+ ξ
2
k
1
y
kn 1
−
ξ
1
k 13
kn 1
k 1x k 1y
k 1z
+ ξ
2
k 23
kn 1
k 1x k 1y
k 1z
, k 1y , k 1x
=
= A
oo (ξ
∼
1 + ξ
∼
2 , k 1y , k 1x )−
−ε x
A
10 (ξ
∼
1 + ξ
∼
2 , k 1y , k 1x ) +
k 13
kn 1
ξ
1 A 0000 (ξ
∼
1 + ξ
∼
2 , k 1y , k 1x )
−
−ε y
A
01 (ξ
∼
1 + ξ
∼
2 , k 1y , k 1x ) +
k 23
kn 1
ξ
2 A 0000 (ξ
∼
1 + ξ
∼
2 , k 1y , k 1x )
−
−ε x ε y
A
11 (ξ
∼
1 + ξ
∼
2 , k 1y , k 1x ) +
k 13
kn 1
ξ
1 A 0000 (ξ
∼
1 + ξ
∼
2 , k 1y , k 1x )
−
−ε x ε y
k 23
kn 1
ξ
2 A 0000 (ξ
∼
1 + ξ
∼
2 , k 1y , k 1x )
−
−
∂ A 0000 (ξ
∼
1 + ξ
∼
2 , k 1y , k 1x )
∂k 1x
(k
∧
1x k 11 − k
∧
1y k 21 )
kn 1
−
−
∂ A 0000 (ξ
∼
1 + ξ
∼
2 , k 1y , k 1x )
∂k 1y
(k
∧
1x k 12 − k
∧
1y k 22 )
kn 1
+ O(ε
2
).
(4.66)
The substitution of (4.66) into (4.65) and integration yield the following expression for the reflected field along the beam axis z
= 0:
E re f =
A
oo (ξ
∼
1 + ξ
∼
2 , k 1y , k 1x )Φ(ξ
1 , ξ
2 )
α
−
(4.67)
−
ε x
α
A
10 (ξ
∼
1 + ξ
∼
2 , k 1y , k 1x ) +
k 13
kn 1
ξ
1 A 0000 (ξ
∼
1 + ξ
∼
2 , k 1y , k 1x )
Φ(ξ
1 , ξ
2 )−
−
ε y
α
A
01 (ξ
∼
1 + ξ
∼
2 , k 1y , k 1x ) +
k 23
kn 1
ξ
2 A 0000 (ξ
∼
1 + ξ
∼
2 , k 1y , k 1x )
Φ(ξ
1 , ξ
2 )−
−
ε x ε y
α
A
11 (ξ ∼
1 + ξ ∼
2 , k 1y , k 1x ) +
k 13
kn 1
ξ
1 A 0000 (ξ ∼
1 + ξ ∼
2 , k 1y , k 1x )
Φ(ξ
1 , ξ
2 )−
−
ε x ε y
α
k 23
kn 1
ξ
2 A 0000 (ξ ∼
1 + ξ ∼
2 , k 1y , k 1x )
Φ(ξ
1 , ξ
2 )−
−
ε x k o
x
ikn 1 α
∂ A
oo (ξ ∼
1 + ξ ∼
2 , k 1y , k 1x )
∂k 1x
+
∂ A
oo (ξ ∼
1 + ξ ∼
2 , k 1y , k 1x )
∂k 1y
∂Φ(ξ
1 , ξ
2 )
∂ξ
1
−