Physics 228 Today: Polarization, Scattering
Website: Sakai 01:750:228 or www.physics.rutgers.edu/ugrad/228
First exam (Thursday, Feb 28, 9:40 PM) room assignments TBD,
but rooms Physics Lecture Hall (here) and ARC 103.
Please bring: 1 formula sheet (both sides may be used, handwritten or
computer generated or photocopied all are OK), pencils, erasers, a calculator,
and a working brain.
There will be: 16 questions, multiple choice, about 4 each from chapters
33-36. About 1/2 direct calculation, 1/4 harder calculation, 1/4 conceptual
Monday, February 18, 2013
Linear Polarization
Polarization refers to the orientation of the electric (and magnetic)
fields of a light wave. For a wave moving to +z, we have written:
y B
�
E(z, t) = E0 cos(kz − ωt)x̂
E
� t) = B0 cos(kz − ωt)ŷ
B(z,
x
But the orientation of the fields could be different. For example:
� t) = E0 cos(kz − ωt)ŷ
E(z,
or
� t) = −B0 cos(kz − ωt)x̂
B(z,
(x̂
+
ŷ)
� t) = E0 cos(kz − ωt) √
E(z,
2
(ŷ
−
x̂)
� t) = B0 cos(kz − ωt) √
B(z,
2
y
E
B
x
B y
E and B must be perpendicular, but can be in any direction.
(Or rotate the coordinate system.)
Monday, February 18, 2013
E
x
Circular Polarization
The polarization direction does not have to be fixed - it can
rotate. Consider ``right handed’’ circularly polarized light:
B
� t) = E0 cos(kz − ωt)(cos(ωt)x̂ + sin(ωt)ŷ) y
E(z,
� t) = B0 cos(kz − ωt)(cos(ωt)ŷ − sin(ωt)x̂)
B(z,
As the light heads towards us, we see the fields rotating CCW.
There is also ``left handed’’ circularly polarized light.
B
y
�
E(z, t) = E0 cos(kz − ωt)(cos(ωt)x̂ − sin(ωt)ŷ)
� t) = B0 cos(kz − ωt)(cos(ωt)ŷ + sin(ωt)x̂)
B(z,
As the light heads towards us, we see the fields rotating CW.
You can also see that if you add the RH to the LH light, the ``sin’’
terms have opposite signs and cancel, we we get linearly polarized
light with E in the x direction, B in the y direction.
Any direction of linearly polarized light can be represented as a
sum of RH + LH circularly polarized light, and vice versa.
Monday, February 18, 2013
E
x
E
x
Circular Polarization
The picture from the text...
Monday, February 18, 2013
Elliptical Polarization
If the two linearly-polarized waves we add have
different amplitudes, the sum is an elliptically
polarized wave. The amplitude will, for example,
be bigger when it is in the ±x direction than in
the ±y direction.
Monday, February 18, 2013
Pick a Direction?
With a rope, you can make a wave traveling to the right with the
displacement in the vertical or in the horizontal direction.
You can also ``polarize’’ the wave build a filter that only allows
waves with displacement in a
particular direction to pass. By
waving the end of rope around in
a circle we generate a
``corkscrew’’ (or circularly
polarized) wave, which is linearly
polarized by a vertical slit.
Monday, February 18, 2013
Polarization of Light Sources
As we discussed earlier, if, as with the broadcast antenna shown
to the left, you make the electrons oscillate vertically, then you
get light with a vertically polarized E field. But if, as in the bulb,
there is no preferred direction, the light is unpolarized - the E
field of the light at any point varies randomly with time.
PhET simulation
Monday, February 18, 2013
Polarizing Light iClicker 1 of 3
We can polarize light, similar to how we polarize the transverse
wave on a rope. We use materials that have slots smaller than the
wavelength of light.
Consider a metal slab with
slots in it. Can we polarize
microwaves, which have a few
cm wavelength, with it?
a) No.
b) Yes. E is parallel to the slot.
c) Yes. E ⊥ slot.
d) Yes. But E is not oriented
any particular way.
e) Yes, but you really need
small holes rather than slots.
Microwave Demo
Monday, February 18, 2013
Polarizing Visible Light
Although the wavelength of visible
light is < 1 μm, we can linearly
polarize it using arrays of
molecules, as in a polaroid filter.
We can think of this similarly to
microwaves and the metal plate: if
the electric field orientation can
accelerate electrons in the
material - E is parallel to the long
molecules - the electrons will
accelerate, absorbing the energy
from the field and screening out
the field components in that
direction. If the electrons cannot
be accelerated, no energy is
absorbed and the wave passes
through.
Monday, February 18, 2013
Single Polarizer
Algebra: light has components relative to polarizer direction of:
E|| = (E.n)n cos(kz-ωt) = Ecos(φ)cos(kz-ωt)n|| which passes through
E⊥ = [E-(E.n)n] cos(kz-ωt) = Esin(φ)cos(kz-ωt)n⊥ which is blocked
Note: The notation n|| and n⊥ to indicates the unit vectors
parallel and perpendicular to the polarizing axis.
Monday, February 18, 2013
What intensity of light makes it though a polarizer?
Since I
∝ E , and E
2
||
/ Eincident = cos(φ), the intensity after the
polarizer is Iout = Iincident cos2(φ). If the incident light is
unpolarized, the average cos2(φ) is 1/2, so Iout = Iincident/2.
One Polarizer Demo
Monday, February 18, 2013
Polarizer + Analyzer
-or- Polarizers Crossed at Arbitrary Angles
``Malus’s Law’’
Monday, February 18, 2013
Crossed Polarizers
Two Polarizer Demo
Monday, February 18, 2013
iClicker 2 of 3
I have 3 polarizers in a row, with the 1st in the x direction, the 2nd
rotated 45o, and the 3rd in the y direction. What fraction of the light
that makes it through the 1st polarizer also makes it through the 3rd
polarizer?
a) 0 = cos(φ1-φ3) = cos(90).
b) 1/4 = cos2(45) x cos2(45).
c) 1/2 = cos(45) x cos(45).
d) It depends on whether
the light is in the +x or -x
direction initially.
e) It depends on whether
the light is linearly
polarized or circularly
polarized.
Monday, February 18, 2013
Polarization in Reflection
When light is incident upon a surface, it generally partially
refracts and partially reflects. The ``plane of incidence’’ is the
plane that contains the incident and reflected light rays. The
electric field can be split into a component in the plane of
incidence, and a component perpendicular to the plane of incidence
(also parallel to the surface). Normally each component is partially
reflected and partially transmitted, but not to the same degree, so
the reflected / transmitted light is partially polarized.
Monday, February 18, 2013
Brewster’s angle
Brewster’s angle is the angle for
which θp + θb = 90o: the
reflected and refracted rays are
90o apart. From Snell’s Law,
na sin(θp) = nb sin(θb)
na sin(θp) = nb sin(90-θp)
na sin(θp) = nb cos(θp)
tan(θp) = nb/na
For this angle, the reflected ray
is polarized completely
perpendicular to the plane of
incidence, and the component of
the E field in the plane of
incidence is completely refracted.
Monday, February 18, 2013
Brewster’s angle
For this angle, the reflected ray
is polarized completely
perpendicular to the plane of
incidence, and the component of
the E field in the plane of
incidence is completely refracted.
Practical application: reflected
sunlight has large horizontal
polarization, the glare of which
can be largely eliminated with
vertical polarizing sunglasses.
http://en.wikipedia.org/wiki/
Brewster%27s_angle
Monday, February 18, 2013
Brewster’s angle
Practical application: Spying on your neighbors?
http://en.wikipedia.org/wiki/Brewster's_angle
Left: reflection of light from a window prevents you
from seeing in.
Right: a polarizer eliminates most of the reflections, so
you can see light form inside the room
Monday, February 18, 2013
Scattering of Light by Air
Why are clouds white?
Why is the sky blue?
Thin clouds are white because
they scatter all wavelengths of
light. Thick clouds turn dark /
gray because too much of the
light is absorbed.
The sky is blue because the scattering of light is proportional to f4 or 1/
λ4. Blue light scatters into your eye. Similarly, in the evening the sun
appears more red because the blue light scatters out. Blue scatters more
by a factor of ≈ (750/450)4 ≈ 8.
The scattered blue light is
polarized. You can see
that the sunlight can
accelerate atoms in the yz
plane, but the sunbather
sees acceleration in the xz
plane. The sunbather sees
linearly (z) polarized light.
Monday, February 18, 2013
Calcite crystal
from Furrfu,
Wikimedia
Commons
Calcite is a material where the index of refraction depends
on the polarization of the light (birefringence). Thus the
light is refracted into two images, each polarized
differently. We can see this with a polarizer.
Monday, February 18, 2013
iClicker 3 of 3
Light is normally incident on two polarizers that are
crossed at an angle of 60 degrees.
(It might help to know that cos(60) = 1/2.)
What fraction of incident unpolarized light intensity is
transmitted through both polarizers?
a) 1/2.
b) 1/4.
c) 1/8.
d) 1/16.
e) It depends on
whether the incident
light is unpolarized
linearly or unpolarized
circularly.
Monday, February 18, 2013