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= Eosinωt
where = angular frequency, t = time, Eo = amplitude. E can be resolved into perpendicular components along y and x:
θ ω cos
0sin ⋅
=E t
Ex ; Ey =E0sinωt⋅sinθ
Conversely, if we originally had two separate beams polarized along x and y, we could add them to produce a resultant:
[
E E]
t E tE E
E=( x2 + y2)12 = ( 0cosθ)2 +( 0sinθ)2 12sinω = 0sinω
Suppose we have two plane polarized beams of equal amplitudes, with electric vectors along y and x, and differing in phase by
2 π :
...
0sin t E
Ex = ω
...
cos 2)
sin( 0
0 t E t
E
Ex = ω +π = ω (3)
Resultant
[
2]
020 2 2
0 2
2 12 12
cos sin
( )
(E E E t E t E
E = x + y = ω + ω =
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 (
2
ωt=π ) 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 Ex = E0sinωtand Ey =E0cosωt 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 2
π out of phase but unequal in
am1plitude (i.e.Ex ≠ Ey) can be represented Ex =EasinωtandEx =Ebcosωt. When these
are combined , sin2 cos2 1
2 2
= +
=
+ t t
E E E
E
b y a
x ω ω
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 Ex =Easinωtand Ex =Ebsin(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 2
δ =π , 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 . The polarization will be circular when
2 δ =π or π
2
3 , if Ea = Eb .
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
v n= c
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.
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= ∆n⋅L
0
2
λπ , (1) 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.
Tetragonal Trigonal Hexagonal n3
n
n
Ordinary n0 or Extraordinary n0 or
Optic axis OA
Optical isotropy
n1 = n2 = n3 uniaxial indicatrix
n1 = n2 n3
OA O
2V A Orthorhombi
c Monoclinic
biaxial indicatrix n1 n2 n3
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
⋅ Γ
=I0sin2(2ϕ) sin2 2
I , (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
= Γ
sin2 2 I0
I . (3)
In centosymmetric in paraelectic phase ferroelectrics crystals the spontaneous birefringence has nature of spontaneous Kerr effect:
~rPs2
δΓ (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.
(
c)
s T T
P =± − − β
α (5)
where: α, β – free energy expansion coefficients, Tc - the Curie-Weiss temperature.
