Mar 21, 2001

Notes for Lecture 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 [^(z)] and using the text's notation the reflectivity is given by,

 = Re ì
ï
ï
í
ï
ï
î
Sr · ^
z
 

Si· ^
z
 
ü
ï
ï
ý
ï
ï
þ
= ê
ê
ê
E0¢¢
E0
ê
ê
ê
2

 
.
(1)
Similarly, the transmittance is given by
T = Re ì
ï
ï
í
ï
ï
î
St · ^
z
 

Si· ^
z
 
ü
ï
ï
ý
ï
ï
þ
= ê
ê
ê
E0¢
E0
ê
ê
ê
2

 
  Re ì
í
î
(n¢/m¢)* cost
(n/m)*   cosi
ü
ý
þ
.
(2)
It is apparent that these equations are consistent with energy conservation at the interface: Â + T = 1.

For the case that m¢ = m 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:

Âp
Âs
= ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
tan2 i -   æ
 ú
Ö

æ
ç
è
n¢
n
ö
÷
ø
2

 
+ tan2 i [( [(n¢)/ n] )2 -1 ]
 

tan2 i +   æ
 ú
Ö

æ
ç
è
n¢
n
ö
÷
ø
2

 
+ tan2 i [( [(n¢)/ n] )2 -1 ]
 
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
ê
2







 
.
(3)

This ratio vanishes at the Brewster angle - tani = [(n¢)/ n], and goes to 1 at normal incidence - tani = 0.


File translated from TEX by TTH, version 2.20.
On 21 Mar 2001, 10:29.