6.007 lecture 25: birefringence - mit opencourseware · birefringence outline • polarized light...
TRANSCRIPT
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Birefringence
Outline • Polarized Light (Linear & Circular) • Birefringent Materials • Quarter-Wave Plate & Half-Wave Plate
Reading: Ch 8.5 in Kong and Shen
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True / False
1. The plasma frequency is the frequency above which a material becomes a plasma.
3. The wave above is polarized 45o with respect to the x-axis.
2. The magnitude of the -field of this wave is 1
ωp
�E = (x+ y)ej(ωt−kz)
�E
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ω2o =
kspring
m
Microscopic Lorentz Oscillator Model
oω pω3
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Sinusoidal Uniform Plane Waves
Ey = A1cos(ωt− kz)Ex = A2cos(ωt− kz)
Ey = A1cos(ωt− kz)
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45° Polarization
The complex amplitude, ,is the same for both components. Therefore and are always in phase.
Where is the magnetic field?
x
y
z
�E
Ey
ExEy
Ex
Ex
Ex
Ex
Ex
Ey
Ey
Ey
Ey�E
�E
�E
�E
�E
�E
�E
�E
Eo
ExEy
Ex(z, t) = xRe(Eoe
j(ωt−kz))
Ey(z, t) = yRe(Eoe
j(ωt−kz))
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Superposition of Sinusoidal Uniform Plane Waves
Can it only be at 45o ?
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Arbitrary-Angle Linear Polarization
x
y E-field variation over time (and space)
Here, the y-component is in phase with the x-component, but has different magnitude.
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Arbitrary-Angle Linear Polarization
Specifically: 0° linear (x) polarization: 90° linear (y) polarization: 45° linear polarization: Arbitrary linear polarization:
Ey/Ex = 0Ey/Ex = ∞
Ey/Ex = 1
Ey/Ex = constant
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Circular (or Helical) Polarization
The resulting E-field rotates counterclockwise around the
propagation-vector (looking along z-axis).
If projected on a constant z plane the
E-field vector would rotate clockwise !!!
The complex amplitude of the x-component is -j times the complex amplitude of the y-component.
Ex and Ey are always 90
… or, more generally,
x
y
z�E
�E
Ex
EyEx(z, t) = xEosin(ωt− kz)
Ey(z, t) = yEocos(ωt− kz)
Ex(z, t) = xRe{−jEoej(ωt−kz)}
Ey(z, t) = yRe{jEoej(ωt−kz)}
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Right vs. Left Circular (or Helical) Polarization
Here, the complex amplitude of the x-component is +j times the complex amplitude of the y-component. So the components are always 90
E-field variation over time (at z = 0)
… or, more generally,
The resulting E-field rotates clockwise around the propagation-vector (looking along z-axis). If projected on a constant z plane the E-field vector would rotate counterclockwise !!!
Ey(z, t) = yRe{jEoej(ωt−kz)}
Ex(z, t) = xRe{+jEoej(ωt−kz}
Ey(z, t) = yEocos(ωt− kz)
Ex(z, t) = −xEosin(ωt− kz)
x
y
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Unequal arbitrary-relative-phase components yield elliptical polarization
The resulting E-field can rotate clockwise or counter-clockwise around the k-vector (looking along k).
where
E-field variation over time (and space)
… or, more generally,
… where are arbitrary complex amplitudes
Ey(z, t) = yEoycos(ωt− kz − θ)
Ex(z, t) = xEoxcos(ωt− kz)
Ex(z, t) = xRe{Eoxej(ωt−kz)}
Ey(z, t) = yRe{Eoyej(ωt−kz−θ)}
x
y
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Left Right
Sinusoidal Uniform Plane Waves
IEEE Definitions:
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A linearly polarized wave can be represented as a sum of two circularly polarized waves
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A linearly polarized wave can be represented as a sum of two circularly polarized waves
CIRCULAR LINEAR CIRCULAR
Ex(z, t) = xEosin(ωt− kz)
Ey(z, t) = yEocos(ωt− kz)Ey(z, t) = yEocos(ωt− kz)
Ex(z, t) = −xEosin(ωt− kz)
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Polarizers for Linear and Circular Polarizations
What is the average power at the input and output?
CASE 1: Linearly polarized light with magnitude oriented 45o with respect to the x-axis.
Wire grid polarizer
CASE 2: Circularly polarized light with magnitude .
EoEo
Sin = E2o/(2η)
Sout = E2o/(4η)
Sout = E2o/(2η)
Sin = E2o/η
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Today s Culture Moment
vision Image is in the public domain.
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The molecular "spring constant" can
be different for different directions
If , then the material has
a single optics axis and is called
uniaxial crystal
Anisotropic Material
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Microscopic Lorentz Oscillator Model
In the transparent regime …
εxr
εxiεyi
εyr
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Uniaxial Crystal
Uniaxial crystals have one refractive index for light
polarized along the optic axis (ne)
and another for light polarized in either of the two directions
perpendicular to it (no).
Light polarized along the optic axis is called the extraordinary ray,
and light polarized perpendicular to
it is called the ordinary ray.
These polarization directions are the crystal principal axes.
Ordinary…
Extraordinary…
Optic axis
Ordinary polarizations
Extraordinarypolarizations
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Birefringent Materials
All transparent crystals with non-cubic lattice structure are birefringent.
no
ne
o-ray
e-ray
Image by Arenamotanus http://www.flickr.com/photos/ arenamontanus/2756010517/ on flickr
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Polarization Conversion Linear to Circular
inside
Polarization of output wave is determined by…
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Quarter-Wave Plate
Example:
Circularly polarized output
linearly polarized input
fast axis
left circular output
45o
�E
Ex
Ey
x
y
λ/4
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
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Half-Wave Plate
The phase difference between the waves linearly polarized parallel and perpendicular to the optic axis is a half cycle
LINEAR IN LINEAR OUT
45o
Optic axis
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Key Takeaways
EM Waves can be linearly, circularly, or elliptically polarized. A circularly polarized wave can be represented as a sum of two linearly polarized waves having phase shift. A linearly polarized wave can be represented as a sum of two circularly polarized waves. In the general case, waves are elliptically polarized.
Circular Polarization
Waveplates can be made from birefringent materials: Quarter wave plate: (gives phase shift) Half wave plate: (gives phase shift)
�E
Ex
Ey
�E
x
y
λ/4 = (no − ne)dλ/2 = (no − ne)d
π/2
π/2
π
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