March 21, 2001
Notes for Lecture #24
Reflectivity – Section 7.3 in Jackson’s text
In Section 7.3 of your text, the electric field amplitudes of the reflected and transmitted
electromagnetic waves are derived, for the cases of s-polarization (E perpendicular to the
plane of incidence) and p-polarization (E parallel to the plane of incidence). The reflectivity
can by determined from the ratio of the surface normal components of Poynting vectors.
Denoting the surface normal as ẑ and using the text’s notation the reflectivity is given by,
E 00 2
0
= .
E0 (1)
(
)
E 0 2
(n0 /µ0 )∗ cos t
0
= Re
.
E0
(n/µ)∗ cos i
(2)
(
Sr · ẑ
R = Re
Si · ẑ
)
Similarly, the transmittance is given by
(
St · ẑ
T = Re
Si · ẑ
)
It is apparent that these equations are consistent with energy conservation at the interface:
R + T = 1.
For the case that µ0 = µ and that the refractive indices are real, using the equations 7.39
and 7.41 and some algebra, we can show that the ratio of the reflectivity at p-polarization
to that at s-polarization is:
s
2
0 2
2
n0
n
2
2
+ tan i n − 1 tan i −
n
Rp .
s
=
0 2
0 2
Rs n
n
2
tan2 i +
+ tan i n − 1 n
This ratio vanishes at the Brewster angle – tan i =
tan i = 0.
n0
,
n
(3)
and goes to 1 at normal incidence –