Optical Polarisation
Parag Bhattacharya
Department of Basic Sciences
School of Engineering and Technology
Assam Don Bosco University
Description of Polarised Light
Optical Polarisation refers to the situations where the plane,
containing the electric field vector E and the propagation vector k, is
well-defined.
The electric field vector E for a given lightwave can be resolved into
two mutually perpendicular components Ex and Ey.
Based on their amplitudes and the relative phase difference between
them, we can obtain any of the following states of polarisation:



Linearly-polarised or plane-polarised light, i.e., P-state
Circularly-polarised light
 Right-circular light, i.e., R-state
 Left-circular light, i.e., L-state
Elliptically-polarised light, i.e., E-state



Linearly-polarised or plane-polarised light, i.e., P-state
Circularly-polarised light
 Right-circular light, i.e., R-state
 Left-circular light, i.e., L-state
Elliptically-polarised light, i.e., E-state
All the above states of polarization can be described in terms of two
orthogonal optical disturbances as given below:
(1)
(2)
where both the waves are travelling along the z-axis,
and ε is the relative phase between them
Note:
If ε > 0, Ey lags Ex
If ε < 0, Ey leads Ex
Linear Polarisation
Light polarised along the x-axis is described as:
(3)
Light polarised along the y-axis is described as:
(4)
Light polarised along any arbitrary direction is described as:
(5)
Each of the above states is known as a P-state.
Circular Polarisation
In circular polarisation, the tip of the electric field vector, traces out a
circle (either clockwise or anti-clockwise) as the wave propagates.
For circular polarisation, the constituent waves have the same
amplitude, i.e., E0x = E0y = E0

If ε = – π/2 + 2mπ, where m = 0, ±1, ±2, ±3, ...
(6)



To an observer looking back at the source, the vector
E rotates clockwise at an angular frequency ω.
This wave is said to be right-circularly polarised.
This state of polarisation is known as the R-state.
Circular Polarisation
In circular polarisation, the tip of the electric field vector, traces out a
circle (either clockwise or anti-clockwise) as the wave propagates.
For circular polarisation, the constituent waves have the same
amplitude, i.e., E0x = E0y = E0

If ε = + π/2 + 2mπ, where m = 0, ±1, ±2, ±3, ...
(7)



To an observer looking back at the source, the vector
E rotates anti-clockwise at an angular frequency ω.
This wave is said to be left-circularly polarised.
This state of polarisation is known as the L-state.
Circular Polarisation
A wave in P-state can be obtained from the superposition of a wave
in L-state and another wave in R-state, both of the same amplitude.
That is,
[P-state] = [L-state] + [R-state]
This can be easily proved as follows:
(8)
(9)
Adding (8) and (9), we obtain the resultant wave:
(10)
(10) represents a wave in P-state, polarised along the x-axis and
having a constant amplitude of 2E0.
Elliptical Polarisation
The tip of E traces out a ellipse as the wave propagates.
This means that the resultant electric field E will rotate, as well as
change its magnitude.
The relationship between the electric field components is:
(11)
The angle α made by the ellipse with the
(Ex, Ey) coordinate system is:
(12)
The state of polarisation
described by (11) is known as
the E-state.
E0y
Ey
E
α
E0x
Ex
Elliptical Polarisation
When the principal axes of the ellipse are aligned with the coordinate
axes, i.e., α = 0, or equivalently, ε = ±π/2, ±3π/2, ...
[E-state]
Further, if E0y = E0x = E0,
[L-state or R-state]
If ε = 0, ±2π, ±4π, ... (even multiples of π)
If ε = ±π, ±3π, ±5π, ... (odd multiples of π)
[P-states]
Natural Light







A natural light source consists of a very large number of randomly
oriented atomic emitters.
Each excited atom radiates a polarized wavetrain for
approximately 10–8 s.
All emissions with the same frequency combine to form a single
resultant polarised wave that does not persist for more than 10–8 s
on an average.
Overall polarisation changes in a completely random fashion.
If, due to these random changes in the state of polarisation, it
becomes impossible to assign any specific polarisation state, the
emitted waves are known as natural light.
Another term for natural light is randomly polarised light.
Mathematically, we can represent natural light in terms to two
arbitrary, incoherent, orthogonal, linearly polarised waves of equal
amplitude.
Production of Polarised Light
Polarisers

A polariser is an optical device whose input is natural light and
whose output is some form of polarised light.
Polariser
Natural Light
Polarised Light

All polarisers are based on 4 fundamental physical mechanisms:
a) Reflection (and refraction)
b) Scattering
c) Birefringence, or double refraction
d) Dichroism, or selective absorption

All 4 mechanisms share 1 underlying property: there must be
some form of asymmetry associated with the process.
Polarisers


Any polariser has an associated transmission axis.
Only the component parallel to the transmission axis will pass.
Y
Transmission Axis
Y'
Y''
X
θ
X'
θ
X''
I0
Z
I0/2
Malus' Law:

This law gives the output intensity when natural light is incident on
a set of 2 polarisers.

When linearly polarised light is incident on a polariser (known as
an analyser), the output intensity is propotional to the square of
the cosine of the angle between the transmission axis and the
plane of polarisation of the incident light.

If I0 is intensity of the light incident on an analyser then the output
intensity I is:
where θ is the angle between the plane of polarisation of the
incident light, and the transmission axis of the analyser.
Malus' Law:
I0
I0 cos2 θ
θ
θ
Polariser
Polarisation due to Reflection




Light reflected off an interface is partially polarised.
The degree of polarisation can change depending on the angle of
incidence.
At a particular angle of incidence, called the Brewster's Angle, the
reflected light attains the P-state.
This effect was discovered empirically by Sir David Brewster (also
the inventor of the kaleidoscope)
Natural Light
Partially Polarised
Light
n1
n2
Partially Polarised
Light
Polarisation due to Reflection


When the angle of incidence is at Brewster's Angle θB the
reflected and the refracted rays become normal to each other.
By Snell's Law, we have,
(1)
From the figure, it is clear that,
(2)
Natural Light
n1
n2
θB
θB
P-state Light
θT
Partially Polarised
Light
Polarisation by Reflection
(1)
(2)
Using (2) in (1) gives,
Therefore,
(3)
(3) gives the angle of incident required for the reflected light to be in
the P-state.
For instance, in case of air-water interface, n1 = 1 and n2 = 1.33.
This gives the Brewster's angle to be
θB = tan–1 1.33 = 53.06º
This means that one needs to be at an angle of approx 37º above
the water-surface in order to obtain light in P-state.
Polarisation due to Refraction


Using Brewster's angle to make a polariser is not practical
because of 2 reasons:
1) Reflected wave is plane polarised but is of low intensity.
2) Refracted wave is of high intensity but is partially polarised.
Solution: construct a pile-of-plates polariser.
θB
Polarisation due to Scattering




When natural light passes through the atmosphere, it sets the
molecular dipoles into oscillations.
Now an oscillating dipole emits EM waves.
The electromagnetic field is maximum in a direction perpendicular
to the dipole axis.
The electromagnetic field is zero in a direction along the dipole
axis.
Zero Field
+
Maximum Field
-
Polarisation due to Scattering
Atmospheric Dipoles
Natural
Light
Partially
Polarised
Light
Natural
Light
Linearly
Polarised
Light
Partially
Polarised
Light
Polarisation due to Dichroism (Selective Absorption)
Dichroism refers to the selective absorption of one of the two
orthogonal P-state components of incident natural light.
 A dichroic polariser is physically anisotropic.
 It has a strong asymmetric or preferential absorption of one field
component while being almost transparent to the other.
 Example of a dichroic crystal is tourmaline
Chemically, NaFe3B3Al6Si6O27(OH)4
 For any dichroic crystal, there is a specific direction within the
crystal known as the Principal Axis or Optic Axis (OA).
(this direction is determined by its atomic configuration)
 The E-field that is perpendicular to the OA is strongly absorbed
by the crystal.
 The E-field along the OA is weakly absorped (almost
unabsorbed) by the crystal.
 The thicker the crystal, the stronger would be the absorption.

Polarisation due to Dichroism
Optic Axis
Natural
Light
Dichroic
Crystal
Light in
P-state
Polarisation due to Birefringence (Double Refraction)







Some crystalline substances are optically anisotropic.
The optical properties are different along different directions within
the crystal.
The E-field of the wave travelling through such a crystal interacts
with the lattice atoms in the crystal, setting them into oscillations.
The speed of the wave, and hence the refractive index, will
depend on the difference between the frequency of the E-field and
the natural frequency of the oscillating atoms.
An anisotropy in the binding force will be manifest in an anisotropy
in the refractive index.
Along 2 different directions, different binding of the atoms leads to
different speeds of the wave, and hence 2 different refractive
indices.
Such materials, that display 2 different refractive indices, cause
double refraction of light, and are said to be birefringent.
Polarisation due to Birefringence







Most common type of birefringence is described to be uniaxial.
This refers to the single direction governing the anisotropy.
This direction is referred to as the optic axis (OA).
Optical behaviour along directions normal to the OA are
equivalent.
Hence rotation of the crystal about the OA does not change the
optical behaviour of the crystal.
Based on the orientation of the direction of polarisation with
respect to the OA, we can distinguish between 2 types of waves:
1) Ordinary: when the polarisation is perpendicular to the OA.
Such waves experience a refractive index nO.
2) Extraordinary: when at least some component of the
polarisation is parallel to the OA.
Such waves experience a refractive index nE.
Birefringence is defined as Δn = nE – nO
Polarisation due to Birefringence


Birefringence is defined as Δn = nE – nO
 Δn > 0 (i.e., nE > nO): positive uniaxial crystals
 Δn < 0 (i.e., nE < nO): negative uniaxial crystals
Examples
nE
nO
Δn
Ice
1.313
1.309
0.004
Quartz (SiO2)
1.553
1.544
0.009
Zircon (ZrSiO4)
1.968
1.923
0.045
Calcite (CaCO3)
1.486
1.658
–0.172
Tourmaline
1.638
1.669
–0.031
Sodium Nitrate
1.337
1.585
–0.248
Crystal
Type
Positive
Uniaxial
Crystal
Negative
Uniaxial
Crystal
Polarisation due to Birefringence

In a positive uniaxial crystal
Δn > 0, i.e., nE > nO
Hence, vE < vO
Thus, the O-ray travels faster
than the E-ray.

The wavelets for O-rays are spherical in shape.
 The bending of O-rays due to refraction takes place as per
Snell's Law.
The wavelets for E-rays are ellipsoidal in shape.
 The bending of E-rays due to refraction does not take place as
per Snell's Law.



In a negative uniaxial crystal
Δn < 0, i.e., nE < nO
Hence, vE > vO
Thus, the E-ray travels faster
than the O-ray.
This idea can be better understood by using the index ellipsoid.
Refractive Index Ellipsoid for Positive Uniaxial Crystal
vE <
vO
O-Ray
E-Ray
vO
v
v
E-wavelet
(ellipsoidal)
E
v
E
E
vO
vO
v
E
vO
O-wavelet
(spherical)
OA
Huygen's Principle in a Positive Uniaxial Crystal
E-wavefront
O-wavefront
OA
O- and E-rays (travel together with path difference)
O- and E-wavefronts
OA
O- and E-rays (travel together without path difference)
Huygen's Principle in a Positive Uniaxial Crystal
E-wavefront
O-wavefront
OA
OA
O-ray
E-ray
O-ray
E-ray
Note: In a positive uniaxial crystal, the E-ray lies
between the O-ray and the OA.
Refractive Index Ellipsoid for Negative Uniaxial Crystal
v
vE >
vO
O-Ray
E-Ray
E
vO
v
O-wavelet
(spherical)
v
E
E
vO
vO
vO
v
E
E-wavelet
(ellipsoidal)
OA
Huygen's Principle in a Negative Uniaxial Crystal
O-wavefront
E-wavefront
OA
O- and E-rays (travel together with path difference)
O- and E-wavefronts
OA
O- and E-rays (travel together without path difference)
Huygen's Principle in a Negative Uniaxial Crystal
OA
OA
O-wavefront
E-wavefront
O-ray
E-ray
O-ray
E-ray
Note: In a negative uniaxial crystal, the O-ray lies
between the E-ray and the OA.
Polarising Prisms





The refractive index of the O-ray is constant, independent of the
direction.
The refractive index of the E-ray is variable, changing with
direction.
A polarising prism utilises this property of two different refractions
for the O- and E-rays (or double refraction) in order to physically
separate the two refracted rays which are in mutually
perpendicular P-states.
A polarising prism can be constructed using either positive or
negative uniaxial crystals.
Some examples of polarising prisms used are:
1) Nicol prism
2) Glan-Air prism
3) Wollaston prism
4) Rochon prism
5) Sernamont prism
Glan-Air Prism


Two calcite prisms with apex
angle θ, are combined with
their long faces opposed and
separated by an air space.
Critical angle for total internal
reflection is given as
OA
θ
E-ray
θ
θ
OA
O-ray

For calcite,
nE = 1.486 & θC(E) = 42.3º
nO = 1.658 & θC(O) = 37.1º


θ is also the angle of incidence on
the long face for both P-states.
Using prisms where 37.1º < θ < 42.3º, ensures only the O-ray
undergoes TIR while the E-ray refracts out.
The 2nd prism is used to re-orient the E-ray along the original
beam direction.
Wollaston Prism
O-Ray
E-Ray
Rochon Prism
O-Ray
E-Ray
Sernamont Prism
O-Ray
E-Ray
Positive Uniaxial
E-wavefront
O-wavefront
OA
Path difference between the O- and E-rays, Δ
Negative Uniaxial
O-wavefront
E-wavefront
OA
If t is the thickness of the
plate, then the total path
difference between the Oand E-rays is:
Positive Uniaxial
And the corresponding
phase difference between
the O- and E-rays is:
OA
t
Δ
Therefore, by changing the thickness of the
plate, we can obtain any desired phase
difference between the 2 P-states.
Quarter-Wave Plate (QWP)
 A QWP is used to produce light in R-, L- or E-states.
 In a QWP, the thickness of the plate is adjusted so that between
the O- and E-rays, the path difference is quarter of a wavelength,
i.e., Δ = λ/4.

And the phase difference is δ = π/2.
Then the required thickness of a QWP found as:

Since λ is extremely small, it is more practical to make QWPs with




Δ = mλ + λ/4
If input on a QWP is either natural light or P-state at 45º to the OA,
the output is in R- or L-state.
If input is P-state at 0º to the OA, the output is also in P-state.
If input is P-state neither at 0º nor at 45º to the OA, the output is in
E-state.
Half-Wave Plate (HWP)
 An HWP is used to rotate the plane of polarisation by 90º.
 In an HWP, the thickness of the plate is adjusted so that between
the O- and E-rays, the path difference is half of a wavelength, i.e.,
Δ = λ/2.

And the phase difference is δ = π.
Then the required thickness of an HWP found as:

Since λ is extremely small, it is more practical to make HWPs with



Δ = mλ + λ/2
If input on an HWP is in P-state at any angle to the OA, the output
is in the orthogonal P-state.
If input is natural light, the output is also natural light.
Half-Wave Plate (HWP)
HWP
HWP
HWP
References:
1)
2)
3)
“Optics”, 4th Ed., Eugene Hecht & A. R. Ganesan,
Pearson
“Introduction to Optics”, 3rd Ed., Frank L. Pedrotti,
Leno M. Pedrotti & Leno S. Pedrotti, Pearson
“Physics for Engineers and Technologists”, Samrat
Dey, EBH Publishers
Thank You!
Assam Don Bosco University