Mar 23, 2001

Notes for Lecture Notes for Lecture #26

Reflectivity for anisotropic media - Extension of Section 7.3 in Jackson's text

Figure Consider the problem of determining the reflectance from an isotropic medium as shown above. For simplicity, we will assume that the dielectric tensor for the medium is diagonal and is given by:

k º æ
ç
ç
ç
è
kxx
0
0
0
kyy
0
0
0
kzz
ö
÷
÷
÷
ø 		.
(1)
We will assume also that the wave vector in the medium is confined to the x-y plane and will be denoted by
kt º w
c
 (nx ^
x
 
 + ny ^
y
 
 ).
(2)
We will assume that the electric field inside the medium is given by
E = (Ex ^
x
 
 + Ey ^
y
 
 + Ez ^
z
 
 ) ei[(w)/ c] ( nx x + ny y).
(3)
In terms of this electric field and the magnetic field H = B/m0, where H is assumed to have the same complex spatial and temporal form as (3), the four Maxwell's equations are given by:
Ñ·H = 0                           Ñ·k ·E = 0
(4)
Ñ×E - i wm0 H = 0                 Ñ×H + i we0 k·E = 0
Using these equations, we obtain the following equations for electric field amplitudes within the medium:
æ
ç
ç
ç
è
kxx - ny2
nx ny
0
nx ny
kyy - nx2
0
0
0
kzz - (nx2+ny2)
ö
÷
÷
÷
ø 		æ
ç
ç
ç
è
Ex
Ey
Ez
ö
÷
÷
÷
ø 		= 0.
(5)
Once the electric field amplitudes are determined, the magnetic field can be determined according to:
H = 1
m0 c
 ì
í
î 		Ez ( ny ^
x
 
 - nx ^
y
 
 ) + (Ey nx - Ex ny) ^
z
 
 ü
ý
þ 		ei[(w)/ c] ( nx x + ny y).
(6)
The incident and reflected electro-magnetic fields are given in your textbook. In the notation of the figure the wavevector for the incident wave is given by:
ki = w
c
 ( sini ^
x
 
 + cosi ^
y
 
 ),
(7)
and the wavevector for reflected wave is given by:
kR = w
c
 ( sini ^
x
 
 - cosi ^
y
 
 ).
(8)
In this notation, Snell's law requires that nx = sini. The continuity conditions at the y = 0 plane involve continuity requirements on the following fields:
H(x,0,z),     Ex(x,0,z),     Ez(x,0,z),     and     Dy(x,0,z).
(9)

Below we consider two different polarizations for the electric field.

Solution for s-polarization

In this case, Ex = Ey = 0, and ny2 = kzz - n2x. The fields in the medium are given by:
E = Ez ^
z
 
 ei[(w)/ c] ( nx x + ny y)                H = 1
m0 c
 ì
í
î 		Ez ( ny ^
x
 
 - nx ^
y
 
 ) ü
ý
þ 		ei[(w)/ c] ( nx x + ny y).
(10)
The amplitude Ez can be determined from the matching conditions:
E0 + E0¢¢ = Ez
(11)
(E0 - E0¢¢) cosi = Ez ny
(E0 + E0¢¢) sini = Ez nx.
In this case, the last equation is redundant. The other two equations can be solved for the reflected amplitude:
E0¢¢
E0
 = cosi - ny
cosi + ny
 .
(12)
This is very similar to the result given in Eq. 7.39 of Jackson for the isotropic media.

Solution for p-polarization

In this case, Ez = 0 and
ny2 = kxx
kyy
 (kyy - nx2).
(13)
In terms of the unknown amplitude Ex, the electric field in the medium is given by:
E = Ex æ
ç
è 		^
x
 
 - kxx nx
kyyny
 ^
y
 
 ö
÷
ø 		ei[(w)/ c] ( nx x +ny y).
(14)
The corresponding magnetic field is given by:
H = - Ex
m0 c
 kxx
ny
 ^
z
 
 ei[(w)/ c] ( nx x +ny y).
(15)
The amplitude Ex can be determined from the matching conditions:
(E0 - E0¢¢) cosi = Ex
(16)
(E0 + E0¢¢) = kxx
ny
 Ex
(E0 + E0¢¢) sini = kxx nx
ny
 Ex.
Again, the last equation is redundant, and the solution for the reflected amplitude is given by:
E0¢¢
E0
 = cosi kxx -ny
cosi kxx + ny
 .
(17)

This result reduces to Eq. 7.41 in Jackson for the isotropic case.

File translated from TEX by TTH, version 2.20.
On 23 Mar 2001, 18:12.