10.2 Scattering of a Plane Wave from a Rough Surface
165
(we omit factor exp(−iωt for brevity), where E inc (x, y, z) is the primary monochromatic field incident on the rough surface, E
0
01 (x, y, z) is the amplitude of the reflected
wave, and E
0
02 (x, y, z) is the amplitude of the transmitted wave. The remaining terms
of series (10.9) and (10.10) are propagating and attenuating scattered modes in the
upper and lower media. Substituting relations (10.9) and (10.10) into (10.7) and
(10.8), we obtain the boundary conditions for successive approximations of the field:
E inc | z=0 + E
0
01 | z=0 = E
0
02 | z=0 ,
(10.11)
E
1
01 | z=0 + H
∂ E inc
∂z
z=0
+ H
∂ E
0
01
∂z
z=0
=
= H
∂ E
0
02
∂z
z=0
+ E
1
02 | z=0 ,
(10.12)
E
2
01 | z=0 + H
∂ E
1
01
∂z
| z=0 +
H
2
2
∂ E
2
inc
∂z 2 +
∂
2 E
0
01
∂z 2
z=0
=
= E
0
02 + H
∂ E
1
02
∂z
z=0
+ E
1
02 | z=0 +
H
2
2
∂
2 E
0
02
∂z 2
z=0
,
(10.13)
1
n
2
1
∂ E inc
∂z
+
∂ E
0
01
∂z
z=0
=
1
n
2
2
∂ E
0
02
∂z
z=0
,
(10.14)
1
n
2
1
∂ E
1
01
∂z
z=0
−
∂ E inc
∂ x
+
∂ E
0
01
∂ x
z=0
∂ H
∂ x
−
∂ E inc
∂ y
+
∂ E
0
01
∂ y
z=0
∂ H
∂ y
+
+
1
n
2
1
H
∂
2 E
0
01
∂z 2
z=0
=
=
1
n
2
2
∂ E
1
02
∂z
z=0
−
∂ E
0
02
∂ x
z=0
∂ H
∂ x
−
∂ E
0
02
∂ y
z=0
∂ H
∂ y
+
+
1
n
2
2
H
∂
2 E
0
02
∂z 2
z=0
,
(10.15)
1
n
2
1
∂ E
2
01
∂z
z=0
−
∂ E inc
∂ x
+
∂ E
1
01
∂ x
z=0
∂ H
∂ x
−
∂ E inc
∂ y
+
∂ E
1
01
∂ y
z=0
∂ H
∂ y
+
+
1
n
2
1
H
∂
2 E inc
∂z 2 +
∂
2 E
1
01
∂z 2
z=0
+