Download Types of polarization

Types of polarization
Plane of polarization
Light is an electromagnetic wave, i.e. a transverse sinusoidally varying electric field
whose direction, represented by vector E, is perpendicular to the direction of propagation of
the wave. If the orientation of E (q in Fig 1) remains fixed as the wave moves forward, the
light is plane-polarized. The plane of polarization is the plane containing E and the direction
of propagation.
Fig.1.
Natural or unpolarized light from an ordinary source such as a discharge lamp consists
of the total output from a very large number of randomly oriented atomic emitters. Each
excited atom radiates a linearly polarized wave train for ~ 10 -8 s and new wave trains are
constantly emitted so that the overall polarization changes in a completely random fashion,
equivalent to q in Fig 1 varying randomly on a 10 -8 s timescale.
Circular polarization.
The plane polarized light represented in Fig 1 can be described by the equation
E = E o sin ωt
where
= angular frequency, t = time, Eo = amplitude. E can be resolved into perpendicular
components along y and x:
E x = E 0 sin ωt ⋅ cos θ ; E y = E 0 sin ωt ⋅ sin θ
Conversely, if we originally had two separate beams polarized along x and y, we could add
them to produce a resultant:
[
E = ( E x2 + E y2 ) 2 = ( E 0 cos θ ) 2 + ( E 0 sin θ ) 2
1
]
1
2
sin ωt = E 0 sin ωt
Suppose we have two plane polarized beams of equal amplitudes, with electric vectors along
y and x, and differing in phase by
π
2
:
E x = E 0 sin ωt.....
E x = E 0 sin(ωt +
π
2
) = E 0 cos ωt.....
[
Resultant E = ( E x2 + E y2 ) 2 = ( E 0 sin 2 ωt + E 2 0 cos 2 ωt
1
2
(3)
]
1
2
= E0
independent of t, i.e. the magnitude of E is constant.
If we plot components Ex and Ey (Fig 2) using Eq.2 and Eq.3 for various times starting at t
=0, we see that E = E0 starts off oriented along OY and rotates in a clockwise direction with
increasing t, pointing along OX after a quarter period ( ωt =
π
2
) and completing the circle in
one full period.
Fig. 2 Right Handed circular polarization
The resultant light is said to be circularly polarized. Note that two plane polarized beams
represented by E x = E 0 sin ωt and E y = E 0 cos ωt would add to produce a circularly polarized
beam with the E vector rotating in an anticlockwise direction. By convention, the two senses
of circular polarization are called; Right handed if E0 rotates clockwise when viewed against
the direction of propagation, and left-handed if E0 rotates anti-clockwise.
Elliptical polarization
Two beams plane polarized in perpendicular directions are
π
2
out of phase but unequal in
am1plitude (i.e. E x ≠ E y ) can be represented E x = E a sin ωt and E x = Eb cos ωt . When these
E
are combined , x
Ea
2
+
Ey
Eb
2
= sin 2 ωt + cos 2 ωt = 1
which is the equation of an ellipse whose major and minor axes coincide with OX and OY
(Fig.3). In this case, both the magnitude and direction vary with time.
Fig.3 Right Hand elliptical polarization
Two perpendicularly plane polarized beams out of phase by an arbitrary angle
can be
written as E x = E a sin ωt and E x = Eb sin(ωt + δ ) Eliminating t from these equations shows
that the resultant vibration traces out an ellipse whose major and minor axes don't coincide
with the co-ordinate axes.
The elliptic polarization becomes circular when Ea = Eb and δ =
π
2
, and plane when
= 0.
Elliptical polarization can, like circular polarization, be right or left-handed. (Fig. 5 below)
Fig.5 Various polarization configurations corresponding to different values of d. Ex leads Ey by
polarization will be circular when
δ=
π
2
or
3
π , if Ea = Eb .
2
. The
Birefringence
An optical device whose input is natural light and whose output is some form of
polarized light is called a polarizer.
The physical mechanisms used for this purpose are dichroism or selective absorption,
scattering, reflection and birefringence or double refraction.
Birefringence is due to strains and anisotropy within the crystal. It implies that light
propagates differently in the material according to the direction taken. When light passes
through a medium it is slowed down to a velocity v with respect to its velocity in a vacuum c
and it is usual to denote this by the refractive index
n=
c
v
This variation of the refractive index n in any direction of space is represented by the
indicatrix. In the most general case it can be described as an ellipsoid.
The indicatrix depends on the structure of the crystal so that at the phase transition, a
change will occur in its shape (translating a change in the optical properties, see Fig.6)
Orientation is a tool as the angle between the indicatrix in two different domains reproduces
the angle existing in the structure. This means that a right angle can be observed in the
tetragonal crystals, a nearly right angle will be observed in rhombohedral crystals. The
difference is so weak that it will not give any indication of what the phase are. However, the
change is noticeable and will provide information on Tc. Moreover, the transition to the cubic
phase is clearer: temperature can be known and phase designed as cubic when the
birefringence completely disappears in isotropic cubic structure.
As suggests Fig.6 the birefringence of the material is related to its structure
(symmetry) and thus a good indicator of the possible changes and transmission.
Optic axis OA
Tetragonal
Trigonal
Hexagonal
n3
Extraordinary n0 or
n
n
Ordinary n0 or
Optical isotropy
n1 = n2 = n3
uniaxial indicatrix
n1 = n2 n3
O
A
2V
O
A
Orthorhombi
c
Monoclinic
biaxial indicatrix
n1 n2 n3
Fig.6 Main types of the indicatrix shapes and structures related
Polarized light beam passing a linearly birefringent crystal is divided into two beams
linearly polarized in perpendicular planes, propagates, with different velocities.
Wherefore the light beam passing a crystal, the phase shift between the fast and slow
light wave is related to the linear birefringence by:
Γ=
2π
λ0
∆n ⋅ L ,
where λ0 - is the wavelength of the light in the vacuum,
L - is the length of the light path through the sample,
n- is the linear birefringence.
(1)
When the light exits the crystal, both light beam interfering giving in effect a linearly,
circularly or elliptically beam depending on the phase shift.
For parallel arrangement of polarizer and analyzer the intensity of light exiting the system is
given by
I = I 0 sin 2 (2ϕ ) ⋅ sin 2
Γ
,
2
(2)
where I0 – initial light intensity,
ϕ - angle between plate of light polarization and the largest indicatrix axis
Using the formula (2) we can calculate that the change in lateral positions of maximum
intensity for two different beams in crossed-polars achieve when ϕ = 45° then formula (2) is
given by
I = I 0 sin 2
Γ
.
2
(3)
In centosymmetric in paraelectic phase ferroelectrics crystals the spontaneous
birefringence has nature of spontaneous Kerr effect:
δΓ ~ rPs2
(4)
where: r - electrooptical coefficient,
Ps – spontaneous polarization.
It is known, that the temperature dependence of spontaneous polarization Ps at
ferroelectrics exhibiting the 2nd order phase transition in a crystal can be described as.
Ps = ± −
α
(T − Tc )
β
where: α, β – free energy expansion coefficients, Tc - the Curie-Weiss temperature.
(5)