Download Lecture 2. Electromagnetic waves at the interfaces

Lecture 2. Electromagnetic waves at the interfaces
In the previous lecture we summed up the properties of electromagnetic waves and its
representation.
In this lecture we will be looking into the modification which an EM wave under goes
when it strikes the interface of two dielectric medium
Questions addressed:
1. Electromagnetic waves at the interface of two dielectrics medium
.
2. Reflection and the transmission coefficient as a function of angle of incidence
Frensel’ equations.
3. Brewster’s angle
1. Electromagnetic waves at the interface of two dielectric mediums.
When an electromagnetic wave strikes at and air dielectric interface, from air to the
dielectric medium, part of it is reflected back and part of it is transmitted into the
dielectric medium. The amount of light reflected and transmitted depends on the angle of
incidence, refractive indices of the medium forming the interface and the angle of
incidence.
Let us consider an interface formed by two dielectric medium I and II of refractive index
n1 and n2 respectively as shown in fig 1, below.
A ray of light from the medium 1 incident on the interface making an angle of incidence
1 with normal partially gets reflected at the angle of reflection being equal to the angle
of incidence and partially refracted at an angle of incidence 2 fowling the Snell’s law
given by
n1 sin 1  n2 sin 2
2. Reflection and the transmission coefficient as a function of angle of incidence
The amount of light reflected and refracted or transmitted depends on the optical property
of the medium viz; refractive index, direction of electric field and the angle of incidence.
The direction of electric field is always perpendicular to the direction of propagation of
the wave. If a wave is traveling in a certain plane the electric field vector could be in two
mutually perpendicular planes which are normal to the direction of propagation. For
example if the wave is propagating along z direction the electric field can be in the z-x
(or z-y) plane or in x-y plane (or could be in both these planes; randomly polarized) but
always perpendicular to the direction of propagation.
Similarly in the present case of interface of two dielectric media, two mutually
perpendicular planes can be defined as
a. The plane of the incidence containing the incidence ray, the reflected ray and the
normal to the interface as shown in figure 2a and
b. The plane perpendicular to the plane of incidence as shown in fig 2b.
Fig 2.
If the electric field is only in the plane of incidence then the wave is called as p
polarized and if the electric field is perpendicular to the plane of incidence then the wave
is called as s polarized.
2.1 Fresnel’s equations
Fresnel’s equation relates the ratio of the amplitude of the electric field reflected and
transmitted to that of the incident field, angle of incidence and the refractive indices of
two media forming the interface. Let us denote the amplitude of incident electric field in
the plane of incidence as E oi and that of nomal to the plane of incidence as
Accordingly the reflected and transmitted amplitude of electric fields are:
Eor
Eoi .
Amplitude of the parallel component of electric field reflected.
Eot
Amplitude of the parallel component of electric field transmitted
Eot
Amplitude of the normal component of electric field reflected.
Eor
Amplitude of the normal component of electric field transmitted.
The ratio of the reflected amplitude to that of the incidence amplitude is called as
amplitude reflected coefficient r ( r or r , for corresponding to parallel or normal
component respectively) and that of for transmitted component is called as t ( t and t ).
The expressions for these coefficients are given by
r 
E0r n1 cos 1  n2 cos  2

E0i n1 cos 1  n2 cos  2
(1)
t 
E0t
2n1 cos 1


E0i n1 cos 1  n2 cos  2
(2)
r 
E0 r n2 cos 1  n1 cos  2

E0i n1 cos  2  n2 cos 1
(3)
t 
E0t
2n1 cos 1

E0i n1 cos  2  n2 cos 1
(4)
These four equations are called as Fresnel’s equations and can be obtained by solving the
wave propagation equations under the appropriate boundary conditions (ref).
2. Brewster’s angle
The Fresnel’s equations eqs 1-4 tell us the variation of amplitude coefficient fro reflected
and the transmitted ray for the given interface. The amplitude reflection Coefficient for
the air –glass interface as a function of angle of incidence is shown in fig 3 for both the s
polarized (perpendicular component) as well as p polarized (parallel component) wave.
Here air is taken as the first medium with refractive index n1 =1 and the glass as the
second medium having refractive index n=1.5 .
Fig 3. variation of amplitude coefficients for reflection as a function of angle of
incidence.
Equation 1 shows that in the present case where n1<n2 (air glass interface), the amplitude
coefficient of reflection for the s wave (perpendicular component) is negative, indicating
that the reflected and the incident wave are out of phase. The magnitude of reflection
coefficient for this particular components increases with the increasing angle of incidence

going to the value of 1 at 1  (fig 3)
2
Using the snall’s law, equation 3 can be re-written as
r 
E0 r sin 1 cos 1  sin  2 cos  2

E0i
sin 1 cos  2  in  2 cos 1
(5)
This will be positive and decrease with the angle of incidence 1 for
sin 1 cos 1  sin 2 cos 2  0
or equivalently
(6)
sin 1  2  cos 1  2   0
for the present case of air-glass interface, where n1<n2, the above inequality will hold for

1   2  
2
for
1   2  

2
,
(7)
the r  0 , indicating that the reflected light will only be only s polarized. Beyond this
1   2  

(8)
2
r will be negative and its magnitude will increase with the angle of incidence taking the
value of 1 at 1 

as shown in fig 3. the negative sign in the region of inequality (70
2
indicate that the electric field for the reflected wave is out of phase with the
corresponding incident wave.
For the angle of incidence satisfying equation (7), the reflected light is purely s polarized
. This particular angle of incidence denoted by
from equation (7) it can be easily shown
tan  B  n2
B
is termed as the Brewsetr angle and
(9)
In case if the first medium happen to be other than air than than equation (9) can be
shown to be modified as
tan  B 
n2
n1