76
4 Mathematical Models of the Interaction of Laser Radiation with Turbid Media
+ ε x ε y
A
11 (t 0 ) + ξ 3 A 0000 (t 0 )
+ O(ε
2
),
(4.47)
t 0 = ξ 1 + ξ 2
k 2y
k 2z
− ξ 3
k 2x
k 2z
, A 0000 (t 0 ) = −
1
2i
∂
2 A
00 (t 0 )
∂t
2
0
k
2 n
2
2
k
3
2z
,
B(ξ 1 , ξ 2 , ξ 3 , ε x , ε y ) = B
+
00 (t 1 ) + ε x
B
+
10 (t 1 ) + B
+
0000 (t 1 )ξ 3
+
+ε y
B
+
01 (t 1 ) + B
+
0000 (t 1 )ξ 3
+ ε x ε y
B
+
11 (t 1 ) + B
+
0000 (t 1 )ξ 3
+ B
−
00 (t 2 )+
+εx
B
−
01 (t 1 ) + B
−
0000 (t 1 )ξ 3
+ ε y
B
−
01 (t 2 ) + B
−
0000 (t 2 )ξ 3
+ ε x ε y
B
−
11 (t 2 ) + B
−
0000 (t 2 )ξ 3
+ O(ε
2
),
(4.48)
B
+
0000 (t 1 ) =
1
2i
∂
2
∂t
2
1
B
+
00 (t 1 )
k
2 n
2
2
k
3
2z
, B
−
0000 (t 2 ) = −
1
2i
∂
2
∂t
2
2
B
−
00 (t 2 )
k
2 n
2
2
k
3
3z
,
t 1 = ξ 1 + ξ 2
k 2y
k
2z
+ ξ 3
k 2x
k
2z
, t 2 = ξ 1 + ξ 2
k 3y
k
3z
− ξ 3
k 3x
k
3z
C(ξ 1 , ξ 2 , ξ 3 , ε x , ε y ) = C
+
00 (t 3 ) + ε x
C
+
10 (t 3 ) + C
+
0000 (t 3 )ξ 3
+
ε y
C
+
01 (t 3 ) + C
+
0000 (t 1 )ξ 3
+ ε x ε y
C
+
11 (t 3 ) + C
+
0000 (t 1 )ξ 3
+ C
−
00 (t 4 )+
+ε x
C
−
10 (t 4 ) + C
−
0000 (t 4 )ξ 3
+ ε y
C
−
01 (t 2 ) + C
−
0000 (t 4 )ξ 3
+
+ ε x ε y
C
−
11 (t 4 ) + C
−
0000 (t 4 )ξ 3
+ O(ε
2
),
(4.49)
C
+
0000 (t 3 ) =
1
2i
∂
2
∂t
2
3
C
+
00 (t 3 )
k
2 n
2
2
k
3
3z
, C
−
0000 (t 4 ) = −
1
2i
∂
2
∂t
2
4
C
−
00 (t 4 )
k
2 n
2
2
k
3
4z
t 3 = ξ 1 + ξ 2
k 3y
k
3z
+ ξ 3
k 3x
k
3z
, t 4 = ξ 1 + ξ 2
k 4y
k
4z
− ξ 3
k 4x
k
4z
D(ξ 1 , ξ 2 , ξ 3 , ε x , ε y ) = D
+
00 (t 5 ) + εx
D
+
10 (t 5 ) + D
+
0000 (t 5 )ξ 3
+
+εy
D
+
01 (t 5 ) + C
+
0000 (t 5 )ξ 3
+ ε x ε y
D
+
11 (t 5 ) + D
+
0000 (t 5 )ξ 3
+ D
−
00 (t 6 )+
+ε x
D
−
10 (t 6 ) + D
−
0000 (t 6 )ξ 3
+ ε y
D
−
01 (t 6 ) + D
−
0000 (t 6 )ξ 3
+
+ ε x ε y
D
−
11 (t 6 ) + D
−
0000 (t 6 )ξ 3
+ O(ε
2
),
(4.50)
D
+
0000 (t 5 ) =
1
2i
∂
2
∂t
2
5
D
+
00 (t 5 )
k
2 n
2
2
k
3
4z
, D
−
0000 (t 6 ) = −
1
2i
∂
2
∂t
2
6
D
−
00 (t 6 )
k
2 n
2
2
k
3
5z
,