4.3 Reflection of a Plane Wave from a Layer with a Slowly Varying Thickness
73
E 3 = exp
i
ε
τ 3elap (ξ 1 , ξ 2 , ξ 3 )
C
+
(ξ 1 , ξ 2 , ξ 3 , ε x , ε y )+
+ exp
i
ε
τ 4re f (ξ 1 , ξ 2 , ξ 3 )
C
−
(ξ 1 , ξ 2 , ξ 3 , ε x , ε y ),
(4.14)
E 4 = exp
i
ε
τ 4elap (ξ 1 , ξ 2 , ξ 3 )
D
+
(ξ 1 , ξ 2 , ξ 3 , ε x , ε y )+
exp
i
ε
τ 5re f (ξ 1 , ξ 2 , ξ 3 )
D
−
(ξ 1 , ξ 2 , ξ 3 , ε x , ε y ),
(4.15)
E 5 = exp
i
ε
τ 5elap (ξ 1 , ξ 2 , ξ 3 )
E(ξ 1 , ξ 2 , ξ 3 , ε x , ε y ).
(4.16)
Here, A, B
±
, C
±
, D
± and E are the amplitudes, τ 1re f , τ 2elap , τ 3re f , τ 3elap ,
τ 4re f , τ 5re f and τ 5elap are unknown functions. By substituting the fields E 1 , E 2 , E 3 , E 4
into (4.4), we obtain the equations for the amplitudes and eikonals:
ε
2
A + iε(2∇ A∇τ 1re f + Aτ 1re f ) + A(k
2 n
2
2 − ∇τ 1re f ) = 0,
(4.17)
ε
2
B
+
+ iε(2∇ B
+
∇τ 2elap + B
+
τ 2elap ) + B
+
(k
2 n
2
2 − ∇τ 2elap )+
ε
2
B
−
+ iε(2∇ B
−
∇τ 3re f + B
−
τ 3re f )+
+ B
−
(k
2 n
2
3 − ∇τ 3re f ) = 0,
(4.18)
ε
2
C
+
+ iε(2∇C
+
∇τ 3elap + C
+
τ 3elap ) + C
+
(k
2 n
2
3 − ∇τ 3elap )+
+ε
2
C
−
+ iε(2∇C
−
∇τ 4re f + C
−
τ 4re f )+
+ C
−
(k
2 n
2
4 − ∇τ 4re f ) = 0,
(4.19)
ε
2
D
+
+ iε(2∇ D
+
∇τ 4elap + D
+
τ 4elap ) + D
+
(k
2 n
2
4 − ∇τ 4elap )+
+ε
2
D
−
+ iε(2∇ D
−
∇τ 5re f + D
−
τ 5re f )+
+ D
−
(k
2 n
2
5 − ∇τ 5re f ) = 0,
(4.20)
ε
2
E + iε(2∇ E∇τ 5elap + Eτ 5elap )+
+ E(k
2 n
2
5 − ∇τ 5elap ) = 0,
(4.21)