72
4 Mathematical Models of the Interaction of Laser Radiation with Turbid Media
The conditions that the tangential components of E and H should be continuous at
the interfaces between media lead to the following boundary conditions
E 1 | ξ 3 =0 = E 2 | ξ 3 =0 , E 2 | ξ 3 =εh 1 (ξ 1 ,ξ 2 ) = E 3 | ξ 3 =εh 1 (ξ 1 ,ξ 2 ) ,
(4.6)
E 3 | ξ 3 =εh 2 (ξ 1 ,ξ 2 ) = E 4 | ξ 3 =εh 2 (ξ 1 ,ξ 2 ) ,
E 4 | ξ 3 =εh 3 (ξ 1 ,ξ 2 ) = E 5 | ξ 3 =εh 3 (ξ 1 ,ξ 2 ) ,
(4.7)
1
n
2
1
∂ E 1
∂ξ 3
| ξ 3 =0 =
1
n
2
2
∂ E 2
∂ξ 3
| ξ 3 =0 ,
(4.8)
1
n
2
2
∂
∂ξ 3
− ε
∂h 1
∂ξ 1
∂
∂ξ 1
− ε
∂h 1
∂ξ 2
∂
∂ξ 2
E 2 | ξ 3 =εh 1 (ξ 1 ,ξ 2 ) =
=
1
n
2
3
∂
∂ξ 3
− ε
∂h 1
∂ξ 1
∂
∂ξ 1
− ε
∂h 1
∂ξ 2
∂
∂ξ 2
E 3 | ξ 3 =εh 1 (ξ 1 ,ξ 2 ) ,
(4.9)
1
n
2
3
∂
∂ξ 3
− ε
∂h 2
∂ξ 1
∂
∂ξ 1
− ε
∂h 2
∂ξ 2
∂
∂ξ 2
E 3 | ξ 3 =εh 2 (ξ 1 ,ξ 2 ) =
=
1
n
2
4
∂
∂ξ 3
− ε
∂h 2
∂ξ 1
∂
∂ξ 1
− ε
∂h 2
∂ξ 2
∂
∂ξ 2
E 4 | ξ 3 =εh 2 (ξ 1 ,ξ 2 ) ,
(4.10)
1
n
2
4
∂
∂ξ 3
− ε
∂h 3
∂ξ 1
∂
∂ξ 1
− ε
∂h 3
∂ξ 2
∂
∂ξ 2
E 4 | ξ 3 =εh 3 (ξ 1 ,ξ 2 ) =
=
1
n
2
5
∂
∂ξ 3
− ε
∂h 3
∂ξ 1
∂
∂ξ 1
− ε
∂h 3
∂ξ 2
∂
∂ξ 2
E 5 | ξ 3 =εh 3 (ξ 1 ,ξ 2 ) .
(4.11)
Since H 1 , H 2 , and H 3 are slowly varying functions of x and y, it is natural to seek
the reflected field in the form of waves with slowly varying amplitudes and quickly
oscillating phases:
E 1 = exp
i
ε
τ inc (ξ 1 , ξ 2 , ξ 3 )
+ exp
i
ε
τ 1re f (ξ 1 , ξ 2 , ξ 3 )
×
× A(ξ 1 , ξ 2 , ξ 3 , ε x , ε y ),
(4.12)
E 2 = exp
i
ε
τ 2elap (ξ 1 , ξ 2 , ξ 3 )
B
+
(ξ 1 , ξ 2 , ξ 3 , ε x , ε y )+
+ exp
i
ε
τ 3re f (ξ 1 , ξ 2 , ξ 3 )
B
−
(ξ 1 , ξ 2 , ξ 3 , ε x , ε y ),
(4.13)