Polarization of Light

Polarization
Model the vector nature of light (Linear, Circular, Mueller/Stokes/Jones
Matrix) and optical components that manipulate polarization (e.g., polarizing
filters and quarter-wave plates).
© Haris Hassan

OPTICS KIT
2
© 𝐻𝑎𝑟𝑖𝑠 𝐻𝑎𝑠𝑠𝑎𝑛

OPTICS KIT
3

WHAT IS A POLARIZED LIGHT?
• When the direction of the electric field of light oscillates in a regular, predictable fashion, we say that
the light is polarized.
• Polarization describes the direction of the oscillating electric field, a distinct concept from dipoles per
volume in a material P – also called polarization.
• Polarization of a plane wave is the orientation of the oscillations of the E field; perpendicular to the
direction of propagation
• For a simple harmonic wave, the electric vector in orthogonal directions may have:
• Different amplitude
• Different phase
• The resulting wave is
• Linearly, elliptically or circularly polarized
• There is a convenient way for keeping track of polarization using a two-dimensional Jones vector.
4

MATHEMATICS
• Light is a propagating oscillation of the electromagnetic field. The general principles that govern
electromagnetic waves are Maxwell's equations. From these general relations, a vector wave equation can
be derived.
∇
2
Ԧ
𝑬 = 𝜇0𝜖0
𝜕
2
Ԧ
𝑬
𝜕𝑡2
• One of the simplest solutions is that of a plane wave propagating in the ො
𝒛 direction is
Ԧ
𝑬 𝑥, 𝑦, 𝑧, 𝑡 = 𝐸𝑥 Ƹ
𝒙 cos 𝜔𝑡 − 𝑘𝑧 + 𝜙𝑥 + 𝐸𝑦 Ƹ
𝒚 cos 𝜔𝑡 − 𝑘𝑧 + 𝜙𝑦
• Where 𝐸𝑥 and 𝐸𝑦 are the electric field magnitudes of the 𝑥-polarization and 𝑦-polarization, 𝜔 = 2𝜋𝑓 is the
angular frequency of the oscillating light wave, 𝑘 = Τ
2𝜋 𝜆 is the wave-number, and 𝜙𝑥 and 𝜙𝑦 are phase
shifts.
5

ELLIPTICAL POLARIZATION
• This results from the combination of
two linear components with different
amplitudes and/or a phase difference
that is not π/2. This is the most general
description of polarized light, and
circular and linear polarized light can
be viewed as special cases of
elliptically polarized light
• When the orthogonal components have
different phase and amplitude, resulting
wave is Elliptically Polarized
(General Case)
6

CIRCULAR POLARIZATION
• The electric field of the light consists of two
linear components that are perpendicular to
each other, equal in amplitude, but have a
phase difference of π/2. The resulting
electric field rotates in a circle around the
direction of propagation and, depending on
the rotation direction, is called left- or right-
hand circularly polarized
• When the orthogonal components have 90o
phase shift and equal amplitude, the
resulting wave is Circularly Polarized
(Special Case)
7

LINEAR POLARIZATION
• The electric field of light is confined to
a single plane along the direction of
propagation
• When the orthogonal components
have zero phase shift, resulting
wave is Linearly Polarized
• More useful
• Emitted by lasers
• Polarization control is possible
• Horizontal and vertical polarizations
8

LINEAR POLARIZATION
9

JONES MATRIX
• Many devices can affect polarization such as polarizers and
wave plates. Their effects on a light field can be represented by
2×2 Jones matrices that operate on the Jones vector
representing the light.
• A Jones matrix can describe, for example, a polarizer oriented
at an arbitrary angle or it can characterize the influence of a
wave plate, which is a device that introduces a relative phase
between two components of the electric field.
10

JONES VECTORS FOR SEVERAL
COMMON POLARIZATION STATES
Linearly polarized along x
1
0
Linearly polarized along y
0
1
Linearly polarized at angle 𝜃 (measured from the x-axis)
𝑐𝑜𝑠 𝜃
𝑠𝑖𝑛 𝜃
Right circularly polarized
1
2
1
−𝑖
Left circularly polarized
1
2
1
𝑖
11

POLARIZERS
Polarizers can be broadly divided into reflective, dichroic, and
birefringent polarizers.
• Reflective polarizers transmit the desired polarization while reflecting the
rest.
• Dichroic polarizers absorb a specific polarization of light, transmitting the
rest; modern nanoparticle polarizers are dichroic polarizers.
• Birefringent polarizers rely on the dependence of the refractive index on
the polarization of light. Different polarizations will refract at different
angles and this can be used to select certain polarizations of light.
12

LINEAR POLARIZERS
The function of a polarizer is to pass only the component of electric field
that is oriented along the polarizer transmission axis. If a polarizer is
oriented with its transmission axis along the x-dimension, only the x-
component of polarization transmits; the y-component is killed.
A polarizer can be represented as a 2×2 matrix that operates on Jones
vectors.
Linearly polarizer along x
1
0
0
0
Linearly polarizer along y
0
0
0
1
13

CIRCULAR POLARIZERS
• To produce polarized light, the light must go through the linear polarizer first,
circularly
• then the 1/4-wave plate. If you send it through the other way, you get linear
polarized
• light.
• 2. To analyze circularly polarized light, the light must go through the 1/4-wave
plate first,
• then the linear polarizer.
• 3. A right-handed analyzer will not pass left-hand polarized light and vice-versa.
• 4. If circularly polarized light is reflected, the handedness reverses.
14

APPLICATIONS OF POLARIZERS
Polarizing the ring light output and the lens separately can greatly
reduce glare effect to reveal important surface details.
15

A polarizer is placed in front of the lens of a DSLR camera reducing the
glare coming from the partially reflective surface of the water.
16

A polarizer is placed in front of the lens of a machine vision camera, reducing the
stray light coming from a reflective surface between the lens and electronic chip.
17

A pair of glasses appears clear without polarization; however, the use of polarizers
makes visible the material stress variations and they appear as color variations.
18

EXAMPLE
Use the Jones matrix to calculate the effect of a horizontal polarizer on light that is initially horizontally polarized,
vertically polarized, and arbitrarily polarized.
First we consider a horizontally polarized plane wave traversing a polarizer with its transmission axis oriented also
horizontally (x-dimension):
1 0
0 0
1
0
=
1
0
As expected, the polarization state is unaffected by the polarizer. (We have ignored possible attenuation from surface
reflections.) Now consider vertically polarized light traversing the same horizontal polarizer. In this case, we have:
1 0
0 0
0
1
=
0
0
As expected, the polarizer extinguishes the light.
Finally, when a horizontally oriented polarizer operates on light with an arbitrary Jones vector (6.11), we have
1 0
0 0
𝐴
𝐵𝑒𝑖𝛿 =
𝐴
0
19

BIREFRINGENCE PHENOMENA
• Birefringence is the optical phenomena which happens when a material
having a refractive index that changes with respect to the polarization and
propagation direction of light wave. These optically anisotropic materials
are said to be birefringent (or birefractive). The birefringence is often
quantified as the maximum difference between refractive indices exhibited
by the material.
• There is also a slow time-varying birefringence component that is random
and unpredictable. The result is that the original state of polarization at
launch is transformed as it propagates through the fiber [11]. The good
news is that if two mutually orthogonal light signals are launched into a
fiber, they emerge at the receiver with their orthogonality preserved [15].
20

WAVE PLATES
• Wave plates (or retarders), which are usually made from birefringent
crystals. The index of refraction in the crystal depends on the
orientation of the electric field polarization.
• A wave plate has the appearance of a thin window through which the
light passes. The crystal is cut such that the wave plate has a fast
and a slow axis, which are 90◦ apart in the plane of the window.
• If the light is polarized along the fast axis, it experiences index 𝑛𝑓𝑎𝑠𝑡.
The orthogonal polarization component experiences higher index
𝑛𝑠𝑙𝑜𝑤.
21

WAVE PLATES
When a plane wave passes through a wave plate, the component
of the electric field oriented along the fast axis travels faster than
its orthogonal counterpart, which introduces a relative phase
between the two polarization components. As light passes
through a wave plate of thickness d, the phase difference that
accumulates between the fast and the slow polarization
components is
𝑘𝑠𝑙𝑜𝑤𝑑 − 𝑘𝑓𝑎𝑠𝑡𝑑 =
2𝜋𝑑
𝜆𝑣𝑎𝑐
(𝑛𝑠𝑙𝑜𝑤 − 𝑛𝑓𝑎𝑠𝑡)
22

QUARTER-WAVE PLATES
The most common types of wave plates are the quarter-wave plate and the half-wave plate.
The quarter-wave plate introduces a phase difference of
𝑘𝑠𝑙𝑜𝑤𝑑 − 𝑘𝑓𝑎𝑠𝑡𝑑 = 𝜋/2 + 2𝜋𝑚
between the two polarization components, where m is an integer. This means that the
polarization component along the slow axis is delayed spatially by a quarter wavelength (or
five quarters, etc.).
The half-wave plate introduces a phase difference of
𝑘𝑠𝑙𝑜𝑤𝑑 − 𝑘𝑓𝑎𝑠𝑡𝑑 = 𝜋 + 2𝜋𝑚
This means that the polarization component along the slow axis is delayed spatially by a
half wavelength (or three halves, etc.). When m = 0 in either (6.34) or (6.35), the wave plate
is said to be zero order.
23

1/4-WAVE PLATES
1/4-wave plates are true quarter wave retarders only for one wavelength
(usually chosen to be green light) because it is near the center of the visible
spectrum. For other wavelengths the plate will shift the two polarization
components by more or less than 90𝑜
. Since a circular polarizer contains a
1/4-wave plate, it will work best at the design wavelength of the 1/4-wave
plate while for other wavelengths the transmitted light will be elliptically
polarized. Thus if you put an L and an R polarizer back-to-back, instead of
complete extinction, you get some blue light transmitted. Linear polarizers
do not involve 1/4-wave plates and give nearly complete linear polarization
over the entire visible spectrum
24

JONES VECTORS FOR SEVERAL
COMMON POLARIZER
Linearly polarizer
𝑐𝑜𝑠2
𝜃 𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃
𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛 𝜃 𝑠𝑖𝑛2
𝜃
Half wave Plate
𝑐𝑜𝑠 2𝜃 𝑠𝑖𝑛 2𝜃
𝑠𝑖𝑛 2𝜃 − cos 2𝜃
Quarter wave Plate
𝑐𝑜𝑠2
𝜃 + 𝑖 𝑠𝑖𝑛2
𝜃 (1 − 𝑖) 𝑠𝑖𝑛 𝜃 𝑐𝑜𝑠 𝜃
(1 − 𝑖) 𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛 𝜃 𝑠𝑖𝑛2
𝜃 + 𝑖 𝑐𝑜𝑠2
𝜃
Right polarizer
1
2
1 𝑖
−𝑖 1
Left polarizer
1
2
1 −𝑖
𝑖 1
Reflection from an interface
−𝑟𝑝 0
0 𝑟𝑠
Reflection from an interface
𝑡𝑝 0
0 𝑡𝑠
25

CIRCULAR POLARIZER USING LP
AND QWP
A linear polarizing filter followed by a quarter-
wave plate whose slow and fast axes are at 45°
to the axis of the polarizer becomes a circular
polarizing filter, and incident unpolarized light
emerges as circularly polarized light. This will
not work if the order of the polarizer and wave
plate is reversed. A quarter-wave plate converts
circularly polarized light into linearly polarized
light.
26

• Calculate the Jones matrix for a quarter-wave plate at θ =
45◦ , and determine its effect on horizontally polarized light.
• Calculate the effect of a half wave plate at an arbitrary θ on
horizontally polarized light.
• Horizontally polarized light (α = 0) is sent through two
polarizers, the first oriented at θ1 = 45◦ and the second at θ2 =
90◦ . What fraction of the original intensity emerges? What is
the fraction if the ordering of the polarizers is reversed?
27

POLARIZATION EFFECTS OF
REFLECTION AND TRANSMISSION
• When light encounters a material interface, the amount of
reflected and transmitted light depends on the polarization.
• The Fresnel coefficients dictate how much of each polarization
is reflected and how much is transmitted.
• In addition, the Fresnel coefficients keep track of phases
intrinsic in the reflection phenomenon.
28

POLARIZATION EFFECTS OF
REFLECTION AND TRANSMISSION
A linearly-polarized field upon reflection
can become elliptically polarized. Even
when a wave reflects at normal
incidence so that the s and p
components are indistinguishable, right-
circular polarized light becomes left-
circular polarized. This is the same effect
that causes a right-handed person to
appear left-handed when viewed in a
mirror.
29

POLARIZATION EFFECT OF
REFLECTION
• If a beam of light encounters a series of mirrors, the final
polarization is determined by multiplying the sequence of
appropriate Jones matrices
• If the beam path deviates from this plane (due to vertical tilt on
the mirrors), then we must reorient our coordinate system before
each mirror to have a new ‘horizontal’ and the new ‘vertical’. The
rotation can be accomplished by multiplying the following matrix
onto the incident Jones vector:
𝑐𝑜𝑠 𝜃 𝑠𝑖𝑛 𝜃
−𝑠𝑖𝑛 𝜃 cos 𝜃
The angle of rotation θ is chosen such that the rotated x-axis lies
in the plane of incidence for the mirror. When such a reorientation
of coordinates is necessary, the two orthogonal field components
in the initial coordinate system are stirred together to form the field
components in the new system. This does not change the intrinsic
characteristics of the polarization, just its representation.
30

IS SUNLIGHT POLARISED?
• Most common sources of light (e.g. sunlight or a light bulb) have
an electric-field direction that varies rapidly and randomly. Such
sources are commonly referred to as unpolarized.
• It is also common to have a mixture of unpolarized and
polarized light, called partially polarized light.
• Jones vectors are designed for pure polarization states. How
to formalise the partially unpolarised light?
• Stokes vectors are used to keep track of the partial
polarization (and attenuation) of a light beam as the light
progresses through an optical system.5
31

PARTIALLY POLARIZED LIGHT
• We can consider any light beam as an intensity sum of
completely unpolarized light and perfectly polarized light:
𝐼 = 𝐼𝑝𝑜𝑙 + 𝐼𝑢𝑛
• The main characteristic of unpolarized light is that it cannot be
extinguished by a single polarizer (even in combination with a
wave plate).
• Moreover, the transmission of unpolarized light through an ideal
polarizer is always 50%.
32

DEGREE OF POLARIZATION
• The fraction of the intensity that is in a definite polarization
state:
𝜉𝑝𝑜𝑙 ≡
𝐼𝑝𝑜𝑙
𝐼𝑝𝑜𝑙 + 𝐼𝑢𝑛
• The degree of polarization takes on values between zero and
one. Thus, if the light is completely unpolarized (such that 𝐼𝑝𝑜𝑙 =
0), the degree of polarization is zero, and if the beam is fully
polarized (such that 𝐼𝑢𝑛 = 0), the degree of polarization is one.
33

STOKES VECTOR
A Stokes vector, which characterizes a partially polarized beam, is written as
𝑆0
𝑆1
𝑆2
𝑆3
=
𝐼0 + 𝐼90
𝐼0 − 𝐼90
𝐼45° − 𝐼−45°
𝐼𝑟𝑐𝑝 − 𝐼𝑙𝑐𝑝
Where 𝑆0 ≡
𝐼
𝐼𝑖𝑛
=
𝐼𝑖𝑛
is a comparison of the beam’s intensity (or power) to a benchmark or ’input’ intensity, Iin, measured before the beam
enters the optical system under consideration. I represents the intensity at the point of investigation, where one wishes to
characterize the beam. Thus, the value 𝑆0 = 1 represents the input intensity, and 𝑆0 can drop to values less than one, to
account for attenuation of light by polarizers in the system
𝐽𝐻𝐿𝑖𝑛𝑒𝑎𝑟 = 1
0
𝑀𝐻𝐿𝑖𝑛𝑒𝑎𝑟 =
1
1
0
0
𝐽𝑅𝐶𝑖𝑟𝑐𝑢𝑙𝑎𝑟 = 1
0
𝑀𝐻𝐿𝐶𝑖𝑟𝑐𝑢𝑙𝑎𝑟 =
1
0
0
1
34

POINCARÉ SPHERE
The Poincare sphere is a graphical tool in real, three-dimensional space that allows
convenient description of polarized signals and polarization transformations caused by
propagation through devices. The axes of the sphere are the Stokes parameters S₁, S₂,
S₃ and the radius corresponds to the total power of the light, S₀. Similar to the 2D
polarisation ellipse, but provides additional information about light polarisation such as
ellipticity angles and the orientation of light The induced polarisation, P, can be
expressed as a power series, where the quantities χ(1)
, χ(2)
, χ(3)
are the linear susceptibility and the second- and third-order nonlinear optical
susceptibilities, respectively
𝐏 = χ(1)
𝐄 + χ 2
𝐄𝐄 + χ 3
𝐄𝐄𝐄
The greater the susceptibility, the more the material will polarise in response to the
electric field Nonlinear susceptibilities can be determined by modulating the polarisation
of the laser and detecting the polarisation of the nonlinear signal
35
S1
S3
S2
X
S0
2γ
Y
2β
Z

POINCARÉ SPHERE
36
Poincaré sphere showing
right and left circular
polarisation(RCP/LCP),
horizontal and vertical linear
polarisation (HLP/VLP),
linear polarisation at +/-45°
and right and left elliptical
polarisation (REP/LEP).

STOKES VECTOR
The next parameter, 𝑺𝟏, describes how much the light looks either horizontally or
vertically polarized, and it is defined as
𝑆1 =
2𝐼ℎ𝑜𝑟
𝐼𝑖𝑛
− 𝑆0
• Here, 𝐼ℎ𝑜𝑟 represents the amount of light detected if an ideal linear polarizer is placed with
its axis aligned horizontally directly in front of the detector (inserted where the light is
characterized). 𝑆1 ranges between −1 and 1, taking on its extremes when the light is
linearly polarized either horizontally or vertically, respectively.
• If the light has been attenuated, it may still be perfectly horizontally polarized even if S1
has a magnitude less than one. (Alternatively, one might examine 𝑆1/𝑆0, which is
guaranteed to range between negative one and one.)
37

STOKES VECTOR
The parameter 𝑺𝟐 describes how much the light looks linearly
polarized along the diagonals. It is given by
𝑆2 ≡
2𝐼45°
𝐼𝑖𝑛
− 𝑆0
• Similar to the previous case, 𝐼45° represents the amount of light
detected if an ideal linear polarizer is placed with its axis at 45°
directly in front of the detector (inserted where the light is
characterized).
• As before, 𝑆2 ranges between negative one and one, taking on
extremes when the light is linearly polarized either at 45° or 135° .
38

STOKES VECTOR
Finally, 𝑺𝟑 characterizes the extent to which the beam is either
right or left circularly polarized:
𝑆3 ≡
2𝐼𝑟𝑐𝑖𝑟
𝐼𝑖𝑛
− 𝑆0
• Here, 𝐼𝑟,𝑐𝑖𝑟 represents the amount of light detected if an ideal right-
circular polarizer is placed directly in front of the detector. A right-
circular polarizer is one that passes right-handed polarized light, but
blocks left handed polarized light.
• Again, this parameter ranges between negative one and one, taking
on the extremes for right and left circular polarization, respectively.
39

POLARIZED LIGHT IN STOKES
VECTOR
Importantly, if any of the parameters 𝑆1, 𝑆2, 𝑜𝑟 𝑆3 take on their
extreme values (i.e. a magnitude equal to 𝑆0), the other two
parameters necessarily equal zero.
As an example, if a beam is linearly polarized in the horizontal
direction with 𝐼 = 𝐼𝑖𝑛, then we have 𝐼ℎ𝑜𝑟 = 𝐼𝑖𝑛, 𝐼45° = 𝐼𝑖𝑛/
2, 𝑎𝑛𝑑 𝐼𝑟𝑐𝑙𝑟 = 𝐼𝑖𝑛/2. This yields
𝑆0 = 1, 𝑆1 = 1, 𝑆2 = 0, 𝑆3 = 0
40

POLARIZED LIGHT IN STOKES
VECTOR
As a second example, suppose that the light has been
attenuated to 𝐼 = 𝐼𝑖𝑛/3 but is purely left circularly polarized.
Then we have
𝐼ℎ𝑜𝑟 = 𝐼𝑖𝑛/6, 𝐼45° = 𝐼𝑖𝑛/6, 𝐼𝑟𝑐𝑖𝑟 = 0
Whereas the Stokes parameters are
𝑆0 = 1/3, 𝑆1 = 0, 𝑆2 = 0, 𝑆3 = −1/3.
41

UNPOLARIZED LIGHT IN STOKES
VECTOR
Another interesting case is completely unpolarized light, which transmits 50%
through all of the polarizers discussed above. In this case
𝐼ℎ𝑜𝑟 = 𝐼45° = 𝐼𝑟𝑐𝑙𝑟 =
𝐼
2
𝑆1 = 𝑆2 = 𝑆3 = 0
Find the Stokes parameters for perfectly polarized light, represented by an
arbitrary Jones vector
𝑨
𝑩
where A and B are complex. Depending on the
values A and B, the polarization can follow any ellipse.
42

STOKES VECTOR IN TERMS OF
POLARIZATION
General form of Stokes vector may then be written as
𝑆0
𝑆1
𝑆2
𝑆3
=
1
𝐼𝑖𝑛
𝐴 2 − 𝐵 2
𝐴∗𝐵 + 𝐴𝐵∗
𝑖(𝐴∗𝐵 − 𝐴𝐵∗)
where the Jones vector for the polarized portion of the light is
𝐴
𝐵
and the intensity of the polarized portion of the light is 𝐼𝑝𝑜𝑙 = 𝐴 2 + 𝐵 2
Degree of polarization can be written as 𝜉𝑝𝑜𝑙 ≡
𝐼𝑝𝑜𝑙
=
1
𝑆0
𝑆1
2
+ 𝑆2
2
+ 𝑆3
2
43

MUELLER MATRICES
• Since Stokes vectors are four-dimensional, the matrices used are
four-by-four. These are known as Mueller matrices
• Mueller matrix for a half wave plate
• Mueller matrix for a linear polarizer
44

FURTHER READING
• https://optics.byu.edu/docs/opticsbook.pdf
• http://www.sfu.ca/phys/141/1134/Lectures/SP%20Lecture%202
4%20-%20Polarization.pdf
• https://www.brown.edu/research/labs/mittleman/sites/brown.edu
.research.labs.mittleman/files/uploads/lecture17_0.pdf
• http://instructor.physics.lsa.umich.edu/int-labs/
• https://htwins.net/scale2/
45

Polarization of Light

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