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Laboratory Manual for determination of Brewster's Angle

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The laws of reflection and refraction
The full treatment of reflection and refraction is dervived from the Maxwell’s equations of electrodynamics. It is known as Fresnel equations. First of all, we suppose that a plane monochromatic wave is incident on the planar surface separating two isotropic media with nearly equal permittivity (
μ
1
=
μ
2
=
μ
0
). Whatever the polarization of the wave we shall resolve its electric
E
and magnetic
B
fields into components parallel and perpendiculat to the plane of incidence and treat these constituents seperately. Case 1.
E
perpendicular to the plane of incidence (s component) The most common forms of the
amplitude
reflection coefficient
r
s
and the amplitude transmission coefficient
t
s
are simply:
t t ii
iist t ii
t t ii
s
nnnt nnnnr
θ θ θ θ θ θ θ
coscoscos2coscoscoscos
+=+−=
Case 2.
E
parallel to the plane of incidence (p component) The most common forms of the
amplitude
reflection coefficient
r
p
and the amplitude transmission coefficient
t
p
are simply:
it t i
t i pit t i
t iit
p
nnnt nnnnr
θ θ θ θ θ θ θ
coscoscos2coscoscoscos
+=+−=
One further notational simplification can be made by using Snell’s law, where upon the Fresnel equations for dielectric media become
( )( )(( ))
t it i pt it is
r r
θ θ θ θ θ θ θ θ
+−=+−−=
tantansinsin
)cos()sin(
cossin2)sin(cossin2
t it i
it pt it is
t t
θ θ θ θ
θ θ θ θ θ θ
−+=+=
The measured reflectance R and transmittance T however are the ratio of flux (not amplitude), such that
( )
1coscos
22
=+==
T Rt nnT r R
iit t
θ θ
Hence
222222
coscoscoscos
siit t psiit t s p pss
t nnT t nnT r Rr R
⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ =⎟⎟ ⎠ ⎞⎜⎜⎝ ⎛ ===
θ θ θ θ
The reflectance and transmittance versus incident angle is shown below for refractive index of 1.5. Note the existence of Brewster’s angle in one case Note that the law is applicable at the point of incidence on the surface. Absorption, if any, is after penetration into the media and therefore does not enter into the Fresnel equations.
Brewster’s Angle and Polarization
I. Introduction
Light is a wave phenomenon with its electric field (or its magnetic field) vibrating in time and in space. If the electric field of a light beam is vibrating in a fixed planein space, such a light beam is referred to as
plane-polarized
light or
linearly polarized
light. This fixed plane is named as the
plane of vibration.
Light from most lasers is plane-polarized.On the other hand, light form the Sun is said to be un-polarized.However, sunlightcan be decomposed into two plane-polarized light beams with their planes of vibration perpendicular to each other.When un-polarized light is incident on the surface of a dielectric (such as a glass), it can be decomposed into two components, denoted as “
s
” and “
p
” component. The plane of vibration of the “
s
” or “
p
” component is perpendicular or parallel to the plane of incidence, respectively.Their intensities depend on the angle of incidence. At a specific angle of incidence, the intensity of “
p
” component of reflected light is zero; and the reflected light becomes plane-polarized with only the “
s
” component. This phenomenon was discovered by Sir David Brewster and, thus, the specific angle is called
Brewster’s angle
or
polarization angle.
From his experiments, Brewster confirmed that the reflected ray and the refracted ray are 90 degrees apart when the incident angle is set at Brewster’s angle.
Figure 1. Reflection and Refraction of Un-polarized Light
Shell’s law states that:
1122
sinsin
n n
(1) where
n
1
and
n
2
are the refractive index of the air and the dielectric, respectively;
1
and
2
are the angle of incidence and refraction, respectively. Because of
22
90,90,
p p
and
2
sinsin(90)cos
p p
, we have the

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