80
4 Mathematical Models of the Interaction of Laser Radiation with Turbid Media
a 33 = cos(θ ), a 23 = − sin(θ ) cos(ψ), a 31 = sin(ϕ) sin(θ ),
(4.59)
a 32 = cos(ϕ) sin(θ ).
(4.60)
In the coordinate system (x, y, z), the incident field is written as
E inc =
1
(2π) 2
∞
−∞
∞
−∞
dk
∧
1x dk
∧
1y ×
× exp[i(xk 1x + yk 1y − zk 1z )
∞
−∞
∞
−∞
dξ
∧
1 dξ
∧
2 exp[−ik
∧
1x ξ
∧
1 − ik
∧
1y ξ
∧
2 ]Φ(ξ
∧
1 , ξ
∧
2 ),
where
k 1x = −
k 31
1 −
ε
2 y
2 k
2∧
1y
2k 2 n
2
1
−
ε
2 x
2 k
2∧
1x
2k 2 n
2
1
+ ε
k
∧
1x
kn 1
k 11 − ε
k
∧
1y
kn 1
k 21
+
+ O(ε
4
),
(4.61)
k 1y = −
k 32
1 −
ε
2 y
2 k
2∧
1y
2k 2 n
2
1
−
ε
2 x
2 k
2∧
1x
2k 2 n
2
1
+ ε
k
∧
1x
kn 1
k 12 − ε
k
∧
1y
kn 1
k 22
+
+ O(ε
4
),
(4.62)
k 1z =
k 33
1 −
ε
2 y
2 k
2∧
1y
2k 2 n
2
1
−
ε
2 x
2 k
2∧
1x
2k 2 n
2
1
+ ε
k
∧
1x
kn 1
k 13 − ε
k
∧
1y
kn 1
k 23
+
+ O(ε
4
).
(4.63)
4.5 The Reflected Field
Upon reflection, each spectral component exp[i xk 1x + iyk 1y − i zk 1z ] gives rise to
a reflected wave A(ξ 1 , ξ 2 , ξ 3 , k 1x , k 1y ) exp[i xk 1x + iyk 1y + i zk 1z ], where A is the
amplitude determined by formula (4.47) and k 1x , k 1y and k 1z are given by formulas
(4.61)–(4.63). The reflected field will be written as
E re f =
1
(2π) 2
∞
−∞
∞
−∞
dk
∧
1x dk
∧
1y exp(i xk 1x + iyk 1y + i zk 1z )
A(ξ 1 , ξ 2 , ξ 3 , k 1x , k 1y )
∞
−∞
∞
−∞
dξ
∧
1 dξ
∧
2 exp(−ik
∧
1x ξ
∧
1 − ik
∧
1y ξ
∧
2 )Φ(ξ
∧
1 , ξ
∧
2 ).