38
2 Electrodynamics at Interface
For S polarization, the electric field amplitudes are
E
0
ˆ
e i = E
0
⎛
⎝
0
1
0
⎞
⎠ ,
E
t
ˆ
e j = E
t
⎛
⎝
0
1
0
⎞
⎠ = t S E
0
⎛
⎝
0
1
0
⎞
⎠ ,
and accordingly
F
i→j
yy
= t S .
(2.61)
F
i→j
xx , F
i→j
yy , and F
i→j
zz
in Eqs. (2.60) and (2.61) lead to Eq. (2.59), by using the
following notations,
q
i
=
n i ω
c
cos θ i , q
j
=
n j ω
c
cos θ j , ε
i
= n
2
i , ε
j
= n
2
j .
q i (j) denotes the z component of the wavevector in the medium i (j ), and ε i (j) is
the dielectric constant in the medium i (j ).
In the following, we examine the Fresnel coefficients at glass-air interface for
example, where we suppose n i = 1.49 for the glass (i) and n j = 1 for the air (j ).
The critical angle of total reflection is derived from sin θ c = n j /n i = 1/1.49 and
hence θ c = 42.2 ◦ . The Fresnel coefficients r P , r S , t P , t S in Eqs. (2.57) and (2.58)
are plotted as a function of θ i in Fig. 2.7. We notice that these coefficients become
30
60
90
-1.0
-0.5
0.5
1.0
0
0.0
30
60
90
-1.0
-0.5
0.5
1.0
0
0.0
30
60
90
-1
1
2
3
0
0
30
60
90
-1.0
-0.5
0.5
1.0
1.5
2.0
0
0.0
(a) r P
(b) t P
(c) r S
(d) t S
Fig. 2.7 The Fresnel coefficients (a) r P , (b) t P , (c) r S , (d) t S for the glass (n i = 1.49) – air (n j = 1)
interface in Eqs. (2.57) and (2.58). The abscissas denote the incident angle θ i in degrees. Red lines
stand for the real part while the blue lines for the imaginary part