Birefringence
Outline
• Polarized Light (Linear & Circular)
• Birefringent Materials
• Quarter-Wave Plate & Half-Wave Plate
Reading: Ch 8.5 in Kong and Shen
1
True / False
1. The plasma frequency ωp is the frequency above
which a material becomes a plasma.
-field of this wave is 1
2. The magnitude of the E
= (x̂ + ŷ)ej(ωt−kz)
E
3. The wave above is polarized 45o with respect
to the x-axis.
2
Microscopic Lorentz Oscillator Model
ωo2
ωo
ωp
3
kspring
=
m
Sinusoidal Uniform Plane Waves
Ey = A1 cos(ωt − kz)
EEyx==AA12cos(ωt
cos(ωt−−kz)
kz)
4
45° Polarization
Ey
Ex (z, t) = x̂Re Ẽo ej(ωt−kz)
Ey (z, t) = ŷRe Ẽo ej(ωt−kz)
E
E
Ex
ŷ
E
Ey
Ex
E
Ey
Ex
E
x̂
The complex
amplitude,
Ẽo ,is the
same for
both
components.
Ex
E
Ey
E
Ex
Ex
E
E
ẑ
Ey
Ey
Where is the magnetic field?
5
Therefore Ex
and Ey are
always in
phase.
Superposition of Sinusoidal Uniform Plane Waves
Can it only be at 45o ?
6
Arbitrary-Angle Linear Polarization
y
Here, the y-component is in phase
with the x-component,
but has different magnitude.
E-field variation
over time
(and space)
x
7
Arbitrary-Angle Linear Polarization
Specifically:
0° linear (x) polarization: Ey /Ex = 0
90° linear (y) polarization: Ey /Ex = ∞
45° linear polarization: Ey /Ex = 1
Arbitrary linear polarization: E /E = constant
y
8
x
Circular (or Helical) Polarization
Ex (z, t) = x̂E˜o sin(ωt − kz)
Ey (z, t) = ŷ E˜o cos(ωt − kz)
E
… or, more generally,
ŷ
Ey
x̂
Ex
Ex (z, t) = x̂Re{−j E˜o ej(ωt−kz) }
Ey (z, t) = ŷRe{j E˜o ej(ωt−kz) }
E
The complex amplitude of the xcomponent is -j times the complex
amplitude of the
y-component.
ẑ
The resulting E-field rotates
counterclockwise around the
propagation-vector
(looking along z-axis).
Ex and Ey are always
90 If projected on a constant z plane the
E-field vector would rotate clockwise !!!
9
Right vs. Left Circular (or Helical) Polarization
Ex (z, t) = −x̂E˜o sin(ωt − kz)
Ey (z, t) = ŷ E˜o cos(ωt − kz)
E-field variation
over time
(at z = 0)
ŷ
… or, more generally,
Ex (z, t) = x̂Re{+j E˜o ej(ωt−kz }
Ey (z, t) = ŷRe{j E˜o ej(ωt−kz) }
Here, the complex amplitude
of the x-component is +j times
the complex amplitude of the
y-component.
x̂
The resulting E-field rotates
clockwise around the
propagation-vector
(looking along z-axis).
So the components are always
90 If projected on a constant z plane the E-field
vector would rotate counterclockwise !!!
10
Unequal arbitrary-relative-phase components
yield elliptical polarization
Ex (z, t) = x̂Eox cos(ωt − kz)
Ey (z, t) = ŷEoy cos(ωt − kz − θ)
E-field variation
over time
(and space)
ŷ
where
x̂
… or, more generally,
Ex (z, t) = x̂Re{Eox ej(ωt−kz) }
Ey (z, t) = ŷRe{Eoy ej(ωt−kz−θ) }
… where
complex amplitudes
are arbitrary
11
The resulting E-field can
rotate clockwise or counterclockwise around the k-vector
(looking along k).
Sinusoidal Uniform Plane Waves
Left
IEEE Definitions:
12
Right
A linearly polarized wave can be represented
as a sum of two circularly polarized waves
13
A linearly polarized wave can be represented
as a sum of two circularly polarized waves
Ex (z, t) = −x̂E˜o sin(ωt − kz)
Ey (z, t) = ŷ E˜o cos(ωt − kz)
Ex (z, t) = x̂E˜o sin(ωt − kz)
Ey (z, t) = ŷ E˜o cos(ωt − kz)
CIRCULAR
LINEAR
14
CIRCULAR
Polarizers for Linear and Circular Polarizations
CASE 1:
CASE 2:
Linearly polarized light with
magnitude Eo oriented 45o
with respect to the x-axis.
Circularly polarized light with
magnitude Eo .
Sin = Eo2 /(2η)
Sin = Eo2 /η
Wire grid
polarizer
Sout =
Sout = Eo2 /(2η)
Eo2 /(4η)
What is the average power at the input and output?
15
Todays Culture Moment
Image is in the public domain.
vision
16
Anisotropic Material
The molecular
"spring constant" can
be different for
different directions
If
,
then the material has
a single optics axis
and is called
uniaxial crystal
17
Microscopic Lorentz Oscillator Model
In the transparent regime
…
yi
xi
yr
18
xr
Uniaxial Crystal
Optic
axis
Uniaxial crystals have one
refractive index for light
polarized along the optic axis (ne)
Extraordinary
polarizations
and another for light polarized in
either of the two directions
perpendicular to it (no).
Ordinary
polarizations
Light polarized along the optic axis
is called the extraordinary ray,
and light polarized perpendicular to
it is called the ordinary ray.
Ordinary…
These polarization directions are
the crystal principal axes.
Extraordinary…
19
Birefringent Materials
o-ray
no
e-ray
ne
Image by Arenamotanus http://www.flickr.com/photos/
arenamontanus/2756010517/ on flickr
All transparent crystals with non-cubic lattice structure are birefringent.
20
Polarization Conversion
Linear to Circular
inside
Polarization of output wave is determined by…
21
Quarter-Wave Plate
left circular
output
45o
Circularly polarized output
ŷ
E
x̂
Ey
λ/4
Ex
fast axis
linearly polarized
input
Example:
If we are to make quarter-wave plate using calcite (no = 1.6584, ne = 1.4864),
for incident light wavelength of
= 590 nm, how thick would the plate be ?
dcalcite QWP = (590nm/4) / (no-ne) = 858 nm
22
Half-Wave Plate
The phase difference between the waves linearly polarized
parallel and perpendicular to the optic axis is a half cycle
Optic
axis
45o
LINEAR IN LINEAR OUT
23
ŷ
Key Takeaways
Ey
EM Waves can be linearly, circularly, or
elliptically polarized.
E
Ex
A circularly polarized wave can be represented
as a sum of two linearly polarized waves having
π/2 phase shift.
A linearly polarized wave can be represented as
a sum of two circularly polarized waves.
E
In the general case, waves are elliptically
polarized.
Circular
Polarization
Waveplates can be made from birefringent materials:
Quarter wave plate:
Half wave plate:
λ/4 = (no − ne )d
λ/2 = (no − ne )d
24
x̂
(gives π/2 phase shift)
(gives π phase shift)
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6.007 Electromagnetic Energy: From Motors to Lasers
Spring 2011
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