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![Page 1: Physics 42200 Waves & Oscillationsjones105/phys42200_Spring2013/notes/Phys... · Physics 42200 Waves & Oscillations Spring 2013 Semester Matthew Jones Lecture 27 –Polarization of](https://reader030.fdocuments.in/reader030/viewer/2022021607/5c66556109d3f2f91c8c19b5/html5/thumbnails/1.jpg)
Physics 42200
Waves & Oscillations
Spring 2013 SemesterMatthew Jones
Lecture 27 – Polarization of Light
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Types of Polarization
• Light propagating through different materials:
– One polarization component can be absorbed
more than the other
– One polarization component can propagate with a
different speed
• What are the properties of the light that
emerges?
• Principle of superposition:
� = �∥ + ��
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Linear Polarization
• Electric field component along one axis is completely absorbed
• Components of the transmitted �-field are in phase,
� is aligned with the polarizing axis:
� �, � = (��̂ + � �̂) cos �� − ��
• Phases that differ by ±�, ±3�, … are still linearly polarized:
eg., � �, � = (��̂ − � �̂) cos �� − ��
�� ������ !��"#
![Page 4: Physics 42200 Waves & Oscillationsjones105/phys42200_Spring2013/notes/Phys... · Physics 42200 Waves & Oscillations Spring 2013 Semester Matthew Jones Lecture 27 –Polarization of](https://reader030.fdocuments.in/reader030/viewer/2022021607/5c66556109d3f2f91c8c19b5/html5/thumbnails/4.jpg)
Linear Polarization
• Both components are still in phase– Nodes occur at common points on the z-axis
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Circular Polarization
• What if the two components had the same
amplitude but a different phase?
�(�, �) = ���̂ cos �� − ��
� (�, �) = ���̂ cos �� − �� + $
• In particular, what if $ = ±%
&
� (�, �) = ���̂ sin �� − �
• Resultant electric field:
� �, � = ��(�̂ cos �� − �� + �̂ sin �� − �� )
![Page 6: Physics 42200 Waves & Oscillationsjones105/phys42200_Spring2013/notes/Phys... · Physics 42200 Waves & Oscillations Spring 2013 Semester Matthew Jones Lecture 27 –Polarization of](https://reader030.fdocuments.in/reader030/viewer/2022021607/5c66556109d3f2f91c8c19b5/html5/thumbnails/6.jpg)
Circular Polarization
x and y components oscillate: ( )tkzEEx ω−= cos0
( )tkzEEy ω−= sin0
Angle α = kz0-ωt
Vector E rotates in time
with angular frequency -ω
Vector E rotates in space
with angular spatial speed k
� �, � = ��(�̂ cos �� − �� + �̂ sin �� − �� )
![Page 7: Physics 42200 Waves & Oscillationsjones105/phys42200_Spring2013/notes/Phys... · Physics 42200 Waves & Oscillations Spring 2013 Semester Matthew Jones Lecture 27 –Polarization of](https://reader030.fdocuments.in/reader030/viewer/2022021607/5c66556109d3f2f91c8c19b5/html5/thumbnails/7.jpg)
Circular Polarization
( ) ( ) ( )[ ]tkztkzEtzE ωω −+−= sinˆcosˆ, 0 jir
Right circularly polarized light:
E rotates clockwise as seen by observer
Left circularly polarized light:
E rotates counter clockwise
( ) ( ) ( )[ ]tkztkzEtzE ωω −−−= sinˆcosˆ, 0 jir
What if we have a superposition of left and right circularly polarized light of
equal amplitude?
( ) ( )tkzEtzE ω−= cosˆ2, 0ir
- linearly polarized light
Vector makes full turn as wave
advances one wavelength
![Page 8: Physics 42200 Waves & Oscillationsjones105/phys42200_Spring2013/notes/Phys... · Physics 42200 Waves & Oscillations Spring 2013 Semester Matthew Jones Lecture 27 –Polarization of](https://reader030.fdocuments.in/reader030/viewer/2022021607/5c66556109d3f2f91c8c19b5/html5/thumbnails/8.jpg)
Circular Polarization and Angular Momentum
What would happen with an electron under
circularly polarized light?
The rotating electric field would push it in a
circle…
Angular velocity ω - angular momentum L
Light is absorbed, and if it was circularly polarized:
ωE=L
energy of light
frequency of lightangular momentum
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Photons and Angular Momentum
ωE=L Photon has energy: ωω
πν h===
2
hhE
Angular momentum of a photon is
independent of its energy:h±=L
Photon has a spin, h+ - L-state
h− - R-state
Whenever a photon is absorbed or
emitted by a charged particle, along with
the change in its energy it will undergo a
change in its angular momentum
First measured in 1935 by Richard Beth
Linearly polarized light: photons exist in
either spin state with equal probability
![Page 10: Physics 42200 Waves & Oscillationsjones105/phys42200_Spring2013/notes/Phys... · Physics 42200 Waves & Oscillations Spring 2013 Semester Matthew Jones Lecture 27 –Polarization of](https://reader030.fdocuments.in/reader030/viewer/2022021607/5c66556109d3f2f91c8c19b5/html5/thumbnails/10.jpg)
Elliptic Polarization
�(�, �) = ���̂ cos �� − ��
� (�, �) = ���̂ cos �� − �� + $
• $ = 0,±� corresponds to linear polarization
• $ = ±%
&corresponds to circular polarization
• What about other values of $?
Elliptic polarizationLinear Circular
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Geometry Lesson
• Equation for a circle:
*& + +& = ,&
*
,
&
++
,
&
= 1
• Equation for an ellipse:
*
.
&
++
/
&
= 1
,
.
/
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Geometry Lesson
• Equation for an ellipse oriented at an angle 0with respect to the x-axis:
*
.
&
++
/
&
− 2*
.
+
/cos $ = sin& $
tan 20 =2./ cos $
.& − /&
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Most General Case
�(�, �) = ���̂ cos �� − ��
� (�, �) = �� �̂ cos �� − �� + $
• Magnitudes of *- and +-components are different
• Phase difference between *- and +-components
• Algebra:
� = �� cos �* − ��
� = �� cos �� − �� + $
�
�� = cos �* − �� cos $ − sin �* − �� sin $
�
�� −�
��cos $ = −sin �* − �� sin $
�
��
&
= cos& �* − �� = 1 − sin& �* − ��
sin& �* − �� = 1 −�
��
&
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Most General Case
�
�� −�
��cos $
&
= sin& �* − �� sin& $
= 1 −�
��
&
sin& $
�
��
&
− 2�
��
�
��cos $ +
�
��
&
sin& $ + cos& $ = sin& $
45
465
7
− 745
465
48
4689:; < +
48
468
7
= ;=>7 <
This is the equation for an ellipse.
![Page 15: Physics 42200 Waves & Oscillationsjones105/phys42200_Spring2013/notes/Phys... · Physics 42200 Waves & Oscillations Spring 2013 Semester Matthew Jones Lecture 27 –Polarization of](https://reader030.fdocuments.in/reader030/viewer/2022021607/5c66556109d3f2f91c8c19b5/html5/thumbnails/15.jpg)
Elliptic Polarization
ξξ 2
2
00
2
0
2
0
sincos2 =
−
+
x
x
y
y
x
x
y
y
E
E
E
E
E
E
E
E
ξ=
When � ? 2�:
![Page 16: Physics 42200 Waves & Oscillationsjones105/phys42200_Spring2013/notes/Phys... · Physics 42200 Waves & Oscillations Spring 2013 Semester Matthew Jones Lecture 27 –Polarization of](https://reader030.fdocuments.in/reader030/viewer/2022021607/5c66556109d3f2f91c8c19b5/html5/thumbnails/16.jpg)
Changing Polarization
• How can we turn unpolarized light into linearly
polarized light?
– Pass it through a polarizer
• How do we get equal magnitudes of the *- and +-
components?
– Rotate the polarizer so that it is 45° with respect to the *-
axis.
�� �C
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Changing Polarization
• How do we turn linearly polarized light into light with
circular polarization?
� �, � = ��(�̂ cos �� − �� + �̂ sin �� − �� )
– Shift the phase of one component by ±�/2
– But how?
• Remember the birefringent crystal?
– Light polarized along the optic axis travels at a different speed
compared with light polarized perpendicular to the optic axis
– Make the thickness such that light emerges with the desired
phase shift.
– This device is called a “quarter-wave plate” because it shifts one
component by one quarter of a wavelength.
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Quarter Wave Plates
• Consider a birefringent crystal with ordinary and extraordinary indices of refraction EF and E"
• How thick should the crystal be to retard one component by ∆H/H = 2I� + �/2 = (4I + 1)�/2?
• Ordinary component: �F �, � = � cos �F� − ��
• Extraordinary component: �" �, � = � cos(�"� − ��)
�F =�
JF=�EF
K= 2�
EF
H�
�" =�
J"=�E"
K= 2�
E"
H�• After propagating a distance, L, we want
(�F − �")L = (4I + 1)�/2
EF − E" L = H�(4I + 1)/4
M =N6
>: − >O(PQ + R)/P
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Quarter Wave Plates
• Consider calcite: (ne - no) = -0.172
• The thickness at normal incidence depends on the wavelength
of light.
– Suppose H� = 560EI (green) and pick I = 1
M =N6
>: − >O(PQ + R)/P
=560EI
0.172
1
4= 814EI
• This is very thin… if the thickness were to be 500XI then
I ? 150
L =560EI
0.172
1
24 × 150 + 1 = 489.186XI
– Less fragile but the path length would be significantly different if light
was not at precisely normal incidence.
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Quarter Wave Plate
• Additional quarter-wave plates introduce additional
phase shifts:
– Two quarter-wave plates produces linear polarization
rotated by 90° with respect to the polarization axis of the
incident light.
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Reflective Retarders
Total internal reflection: phase shift between the two components.
Glass - n=1.51, and 45o shift occurs at incidence angle 54.6o.
Fresnel rhomb
Unlike devices that use
birefringent materials this
is achromatic.
(works for all wavelengths)
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Optical Activity
Discovered in 1811, Dominique F. J. Arago
Optically active material: any
substance that cause a plane of
polarization to appear to rotate
Dextrorotatory (d-rotatory) - polarization rotates right (CW)
Levorotatory (l-rotatory) - polarization rotates left (CCW)
Latin: dextro - right
levo - left
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Optical Activity
Many materials have
molecules or crystal
structures that are non
mirror-symmetric.
These usually interact with
left- and right-circular
polarized light in different
ways.
Circular birefringence:
different indices of refraction
for L- or R-polarized light.
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Optical Activity
• Fresnel described linearly polarized light as the
superposition of L- and R-polarization:
�[ =��
2�̂ cos �[� − �� + �̂ sin �[� − ��
�\ =��
2�̂ cos �\� − �� − �̂ sin �\� − ��
• Add them together:
� = �� cos�[ + �\
2� − �� �̂ cos
�[ − �\
2� + �̂ sin
�[ − �\
2�
• Linear polarization: direction at a fixed point in � is
constant.
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Optical Activity
�[ =��
2�̂ cos �[� − �� + �̂ sin �[� − ��
�\ =��
2�̂ cos �\� − �� − �̂ sin �\� − ��
� = �� cos�[ + �\
2� − �� �̂ cos
�[ − �\
2� + �̂ sin
�[ − �\
2�
Time dependence of the � vector at � = 0: � = ���̂ cos(��)
![Page 26: Physics 42200 Waves & Oscillationsjones105/phys42200_Spring2013/notes/Phys... · Physics 42200 Waves & Oscillations Spring 2013 Semester Matthew Jones Lecture 27 –Polarization of](https://reader030.fdocuments.in/reader030/viewer/2022021607/5c66556109d3f2f91c8c19b5/html5/thumbnails/26.jpg)
Optical Activity
� = �� cos�[ + �\
2� − �� �̂ cos
�[ − �\
2� + �̂ sin
�[ − �\
2�
• After propagating a distance � = L through an optically active
material, � = �� �̂ cos ] + �̂ sin ] cos ��
(�[ ^ �\)
] =�L
H�(E\ − E[)
_
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Optical Activity
• Application in organic chemistry:
• Experimental setup:
– Sample concentration: 1 gram per millileter
– Path length: 10 cm
– Wavelength is 589 nm (Sodium “D-line”)
– The “specific rotation” is the angle 0 that linearly polarized
light is rotated
• Examples: Substance Specific rotation
Sucrose +66.47°
Lactose +52.3°
Levulose (D-Fructose) −92.4°
Dextrose (D-Glucose) +52.5°
L-Glucose −52°
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Stereoisomers
• Same chemical formula but mirror-symmetric structure:
• Measurements of optical rotation provide information about sample purity and composition.
• Useful for analytic chemistry, regardless of the underlying physics.