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Polarization
Polarization:
Consider the superposition of two plane
polarized waves:
 x  Ex sin kz  t 
 y  E y sin kz  t   
If =2m, m being an integer,
   x2  y2  E sin kz  t 
E E E
2
x
2
y
tan  
Ey
Ex
If =(2m+1), m being an integer,
   x2  y2  E sin kz  t 
E  E x2  E y2
Optics II----by Dr.H.Huang, Department of Applied Physics
tan   
Ey
Ex
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Polarization
Right Circularly Polarized Light:
Left Circularly Polarized Light:
If =(2m-1/2), m being an integer,
If =(2m+1/2), m being an integer,
E x  E y  E1
E x  E y  E1
   x2   y2  E x2  E y2  E1
   x2   y2  E x2  E y2  E1
y
cos t
tan  

x
sin t
   2  t at z  0
 y cos t
tan  

 x sin t
    2  t at z  0
Right and left circular light can be
written as,
ER z, t   E1 isin kz  t   jcoskz  t 
EL z, t   E1 isin kz  t   jcoskz  t 
Their superposition becomes
Ez, t   2iE1 sin kz  t 
A plane polarized wave can be
synthesized from two oppositely
polarized circular waves.
Optics II----by Dr.H.Huang, Department of Applied Physics
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Polarization
Elliptically Polarized Light:
Suppose E x  E y and =(2m+1/2), m being an integer.
 x  Ex sin kz  t 
 x

 Ex



2
 y

E
 y
 y  E y sin kz  t   2  E y coskz  t 
2

 1


State of Polarization:
P-state
R-state and L-state
E-state
Natural Light:
Natural light is randomly polarized. We can mathematically represent natural light
in terms of two arbitrary, incoherent, orthogonal, linearly polarized waves of equal
amplitude (i.e., waves for which the relative phase difference varies rapidly and
randomly).
Optics II----by Dr.H.Huang, Department of Applied Physics
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The Hong Kong Polytechnic University
Polarization
Example:
A wave  has the components x=E1cos(kz-t) and y=-E1cos(kz-t). What is its state of
polarization?
   x2  y2  2 E1 coskz  t 
tan  
y
 1
x
  135
Polarizers:
a device to generate polarized light out of
natural one. It can also be used as an
analyzer to allow all E-vibrations parallel to
the transmission axis to pass.
Malus’ Law:
I   I 0 cos 2 
When a natural light passes though an ideal
polarizer, its intensity is reduced by half.
Optics II----by Dr.H.Huang, Department of Applied Physics
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Polarization
Dichorism:
selective absorption of one of the two orthogonal P-state in incident natural light.
wire-grid
polarizer
dichroic
crystal
Commercial Polaroid H-Sheet:
• It’s a dichroic sheet polarizer.
• An ideal H-sheet would transmit 50% of the incident natural light and is
designated HN-50.
• In practice, due to loss, the H-sheet might be labeled HN-46, HN-38, HN-32, and
HN-22 with the number indicating the percentage of natural light transmitted
through the H-sheet.
Optics II----by Dr.H.Huang, Department of Applied Physics
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Polarization
Example:
Natural light of intensity Ii is incident on three HN-32 sheets of Polaroid with their
transmission axes parallel. What is the intensity of the emergent light?
I1 
1
I i  64%;
2
I 2  I1  64%;
I 3  I 2  64% 
1
3
I i  0.64  0.131I i
2
Example:
Suppose the third Polaroid in the last question is rotated through 45. What is now the
intensity transmitted?
I 3  I 2 cos 2 45  64% 
1
3
I i  0.64  0.5  0.066 I i
2
Example:
Two sheets of HN-38 Polaroid are held in contact in the familiar crossed position. Let I1 be
the intensity emerging from sheet 1 natural light of intensity Ii is incident upon it. The
intensity emerging from sheet 2 must be I2=0. Now insert a third sheet of HN-32 between
them with its transmission axis at 45 to the other sheets’ transmission axes. What is I2 now?
1
I i  76%  0.38I i ;
I 3  I1 cos 2 45  64%  0.32 I1;
2
I 2  I 3 cos 2 45  76%  0.38I 3  0.046 I i
I1 
Optics II----by Dr.H.Huang, Department of Applied Physics
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Polarization
Birefringence:
A material which displays two different speeds of propagation in fixed and orthogonal
directions, and therefore displays two refractive indices, is known as birefringent.
Distinction: A dichroic material absorbs one of the orthogonal P-states is dichroic
while in birefringent material we usually neglect the absorption.
Rhombohedron
of
calcite. The optic axis
passes symmetrically
through
a
blunt
corner where the
three face angles
equal 102.
Atomic structure of a
CaCO3 tetrahedron.
ne 
c
v//
no 
c
v
n  ne  no is called birefringence.
If n  0 uniaxial positive
If
n  0
uniaxial negative
Optics II----by Dr.H.Huang, Department of Applied Physics
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Polarization
Double Refraction:
A narrow beam of natural light incident
normally to a cleavage plane of a calcite
crystal emerges as two parallel beams
displaced laterally.
Creation of an elliptical
Huygens’ wavelet by
extraordinary ray. The
material
is
uniaxial
negative. Ray direction
S for the extraordinary
ray in birefringence
material.
Optics II----by Dr.H.Huang, Department of Applied Physics
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The Hong Kong Polytechnic University
Polarization
Example:
A calcite plate is cut as shown in figure with the optic axis perpendicular to the plane of the
paper. A ray of natural light, =589.3 nm, is incident at 30 to the normal. The plane of the
paper is the plane of incidence. Find the angle between the rays inside the plate.
ne  1.4866;
no  1.6584
n sin i  ne sin ie
ie  19.65
n sin i  no sin io
 io  17.55
  ie  io  2.10
Optics II----by Dr.H.Huang, Department of Applied Physics
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The Hong Kong Polytechnic University
Polarization
Various Types of Plane Polarizers:
Glan-Air prism
Nicol prism
Optics II----by Dr.H.Huang, Department of Applied Physics
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Polarization
Example:
A 50 calcite prism is cut with its optic axis as shown in the figure. Sodium light is used in a
spectrometer experiment to find no and ne. Two images of the slit are seen and minimum
deviation is measured for each. Find no and ne if the angles of minimum deviation are 27.83
and 38.99. Explain how you would decide which image was formed by the o-rays and which
was formed by the e-rays.
When d min  27.83
n
sin


1
a  d min  sin 1 50  27.83
2
2

 1.4864
1
1 
sin a
sin 50
2
2
 
When d min  38.99
n
sin


1
a  d min  sin 1 50  38.99
2
2

 1.6584
1
1 
sin a
sin 50
2
2
 
Since calcite is uniaxial
negative, therefore
ne=1.4864 and no=1.6584
Optics II----by Dr.H.Huang, Department of Applied Physics
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The Hong Kong Polytechnic University
Polarization
Example:
A quartz Wollaston beam-splitting polarizer is used with a normally incident parallel beam of
sodium light for which ne=1.5534 and no=1.5443. If the wedge angle is 45, find the angular
separation of the emergent e- and o-rays.
i2  45
no sin i2  ne sin ie2
ie2  44.67 
ne sin i2  no sin io 2
io 2  45.34
ie3  45  ie2  0.33
io 3  io 2  45  0.34
ne sin ie3  nair sin ie3
ie3  0.51
no sin io 3  nair sin io 3
io 3  0.53
io 3  ie3  1.04
Optics II----by Dr.H.Huang, Department of Applied Physics
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Polarization
Scattering: The displacement of an electron oscillating harmonically under an
q m
E sin t
2
2
0  
where 0 is the natural frequency (resonant frequency) of the oscillation of the bound
electron. At resonant frequency, strong absorption occurs. At non-resonant
frequencies the absorption of the wave packet and its subsequent emission is known
as scattering.
external field is
x
Rayleigh Scattering:
Scattering centers have dimensions smaller than the wavelength. The radiated
power is inversely proportional to the fourth power of the wavelength of the incident
radiation.
Polarization due to scattering:
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Polarization by Reflection:
At Brewster’s angle ip,
r// 
Polarization
tan i p  i
0
tan i p  i
the reflected light becomes plane
polarized perpendicular to the plane of incidence.
pile-of-plates polarizer
Degree of polarization:
Brewster window
P
Ip
I p  Iu

I
I
p
p
 I u 2  I u 2
 I u 2  I u 2

I max  I min
I max  I min
Optics II----by Dr.H.Huang, Department of Applied Physics
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Polarization
Example:
The figure shows a ray of monochromatic light incident at an angle ip on extra-dense flint
glass for which ng=1.653. Show that the angle between the transmitted and reflected rays is
generally equal to 90, and find ip and i in this particular case.
 r// 
tan i p  i
0
tan i p  i
i p  arctan
 i p  i  90
n
 58.83
n
i  90  i p  31.17 
Example:
Natural light is incident on a plane air/glass boundary. Suppose the angle of incidence is
about 55 and the refractive index of glass is 1.5. Calculate the degree of polarization.
nair sin i  ng sin i
R// 
R 
2
r //
2
i //
I r // E

I i // E
2
r
2
i
Ir E

Ii E
 i  33.1
 r//2 
tan i  i
4

1
.
778

10
tan 2 i  i
 r2 
sin i  i
 0.1393
2

sin i  i 
2
2
P

Ip
I p  Iu
R  R//
 0.997
R  R//
Optics II----by Dr.H.Huang, Department of Applied Physics
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Polarization
Retarders:
Retarders are devices that cause one orthogonal P-state component to lag behind
the other on emerging from the retarders.


The path difference is OPD  ne  no t
Full wave plate:
OPD   ne  no t  m, m  1,2,3...
Half-wave plate:
1

OPD   ne  no t   m   , m  1,2,3...
2

Quart-wave plate:
1

OPD   ne  no t   m   , m  1,2,3...
4

The component E// and E
that travels faster defines the
fast axis of the plate.
Optics II----by Dr.H.Huang, Department of Applied Physics
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Polarization
Compensator:
―A device that allows a
continuous adjustment of
the relative phase shift, the
retardance.
Soleil-Babinet compensator. (Left) Zero retardation.
(Right) Maximum retardation.
A circular light can be changed into Pstate with a quarter-wave plate.
The handiness of circular light can be
checked by this method.
Optics II----by Dr.H.Huang, Department of Applied Physics
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Polarization
Example:
A beam of left circular light meets a quarter-wave plate with its fast axis vertical. What is the
state of the emergent light?
 x  Ex sin kz  t 


 y  E y sin  kz  t  
2

x leads y by /2.
On passing through the wave plate, the y component travels faster than the x one. It
emerges with a phase advanced by /4 and the two components are now in phase.
 x  Ex sin kz  t 
 y  E y sin kz  t 
The resultant wave is a P-state, making an angle =45 with the x-axis.
Optics II----by Dr.H.Huang, Department of Applied Physics
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Polarization
Example:
A quarter-wave plate made from calcite, ne=1.4864 and no=1.6584, is placed in a beam of
normally incident light, =656 nm, plane polarized at 45 to the x-axis. Right circular light
emerges. Find the minimum thickness of the plate and the orientation of the optic axis.
1

OPD   ne  no t   m   , m  0,1,2,3...
4

t
4m  1
4no  ne 
For minimum thickness, m=0, therefore, t=953.49 nm.
Example:
Show that the reflectance R// is equal to the amplitude reflection coefficient squared, r//2, and
also show that R=r2.
I r // Er2// r// Ei // 
2
 2 

r
By definition: R// 
//
I i // Ei //
Ei2//
2
I r  Er2 r Ei  
2
R 
 2 

r

I i  Ei 
Ei2
2
Optics II----by Dr.H.Huang, Department of Applied Physics
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Polarization
Optical Activity:
The property that causes the plane of polarization of P-state light to rotate when it
passes through certain material.
Viewing the beam head-on, if the rotation is clockwise, the material is referred to as
dextrorotatory or d-rotatory. If the rotation is anticlockwise, the material is referred
to as laevorotatory or l-rotatory.

z
nL  nR 
0
Phenomenologically, the optical activity can be
explained by viewing that the linearly polarized
light as to be the superposition of equal amounts
of left- and right-circularly polarized components
through an optically active materials with different
velocities, vL and vR, respectively.
Optics II----by Dr.H.Huang, Department of Applied Physics
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Polarization
Some Applications:
Photoelasticity
Kerr effect
n  0 KE 2
Homework: 12.1; 12.3; 12.4; 12.6;
12.7; 12.10
Optics II----by Dr.H.Huang, Department of Applied Physics
21