Transcript Introduction to the techniques of ultrahigh resolution

Modern Optics II – Polarization of light
Special topics course in IAMS
Lecture speaker: Wang-Yau Cheng
2006/4
Outline
•
•
•
•
Wave properties of light
Polarization of light
Coherence of light
Special issues on quantum optics
• Polarization of light
 Polarization vs. dipole oscillation
– Ways to change the polarization status
– Ways to purify the polarization
– Polarization in light scattering
Concepts of induced dipole moment
Neutral atoms


P  0
Interact with
EM wave



P   eri



P  0


P  E

er
))))))


F  kx  r x 2  ...




(1)
( 2) 2
(3) 3
P   E   E   E 
Higher harmonic radiation
Scattering by molecules is not spherically
symmetric. It has a "dipole pattern."
The field emitted by an oscillating dipole excited by a vertically
polarized light wave:
Direction of light excitation
E-field and electron oscillation
Emitted intensity pattern
Directions of scattered light E-field
No light is emitted along
direction of oscillation!
Directions of scattered light E-field
Dipole Emission Pattern from an Antenna
Analogous to a molecule emitting light, an antenna emits a dipole
pattern at much lower frequency and longer wavelength:
The pattern is somewhat distorted by the earth and nearby objects.
Linear polarization
yˆ

P  Pxcoswtxˆ  Pycoswtyˆ
Left hand circular polarization

P  Pxcoswtxˆ  Pysin wtyˆ
Right hand circular polarization

P  Pxcoswtxˆ  Pysin wtyˆ
For stimulation emission process, all the dipoles will
be coherently induced and the radiation will go with
certain polarization
xˆ
Note that  could be a tensor
Interact with
EM wave

P  0


P  E

E

P
))))))

P

E
Optical Activity
Unlike birefringence, optical activity maintains a linear polarization
throughout. The rotation angle is proportional to the distance.
Right-handed quartz
• Polarization of light
– Polarization vs. dipole oscillation
Ways to change the polarization status
– Ways to purify the polarization
– Polarization in light scattering
Wave plates
When a beam propagates through a birefringent medium, one
polarization experiences more phase delay than the other.
If both polarizations are present, this has the effect of changing
relative phase of the x and y fields, and hence rotating the polarization.


E ( z , t )  Re E exp i  kz  t  
Input: Ex ( z , t )  Re E exp i  kz  t  
~0
y
Polarization state:
}
~0


}
E ( z , t )  Re E exp i  kz  kn d  t  
Output: Ex ( z , t )  Re E exp i  kz  kno d  t  
~0
y
1
1
 
~0
e
exp(ikno d ) 
 exp(ikn d )  
e


1




2



exp  i
no  ne  d  


 
 
Wave plates (continued)
Wave plate output polarization state:
(45-degree input polarization)
2

“Quarter-wave
plate”
“Half-wave
plate”
 no  ne  d
0
 /2

3 /2
2
1




2



exp  i
no  ne  d 


 

 2
exp  i
 no  ne  d 
 

Output
Polarization State
1
i
1
i
1
45 Linear
Left Circular
45 Linear
Right Circular
45 Linear
A quarter-wave plate creates circular polarization, and a half-wave plate
rotates linear polarization to its orthogonal state.
We can add an additional 2m without changing the polarization, so the
polarization cycles through this evolution as d increases further.
Half-Wave Plate
When a beam propagates through a half-wave plate, one polarization
experiences half of a wavelength more phase delay than the other.
If the incident polarization is 45° to the principal axes, then the
output polarization is rotated by 90°.
If the incident polarization is parallel to one of the principal axes of
the plate, then no polarization rotation occurs.
Jones Matrices for standard components
x
Multiplying Jones Matrices
y
Crossed polarizers:
E1  A y Ax E0
E0
x-pol
E1
y-pol
 0 0  1 0   0 0 
Ay Ax  





0 1  0 0  0 0 
Uncrossed polarizers
(slightly):
so no light leaks through.
rotated
x-pol
E0
0 0   1    0 0
Ay Ax    





0 1  0   0
 Ex   0 0   Ex   0 
A y A x      
E   


E

E

0
 y  x
 y 
E1
y-pol
So Iout ≈ 2 Iin,x
z
Steering mirrors
up
down
CRYSTAL QUARTZ WAVEPLATES
Phase shifts in reflection (glass to air)
n t < ni

Interesting phase
above the critical
angle
┴
0
0°
30°
60°
90°
Incidence angle

180° phase shift
for angles below
Brewster's angle;
0° for larger angles
||
0
0°
30°
60°
Incidence angle
90°
Total Internal Reflection occurs just as the
transmitted beam grazes the surface.
Note that the irradiance of the transmitted beam goes to zero as it
grazes the surface.
Total internal reflection is 100% efficient.
quarter wave Fresnel rhomb retarder
Winding fiber
The Faraday Effect
A magnetic field can induce optical activity.
The Faraday effect allows control over the polarization rotation.
• Polarization of light
– Polarization vs. dipole oscillation
– Ways to change the polarization status
 Ways to purify the polarization
– Polarization in light scattering
Phase shifts in reflection (air to glass)
n i < nt

180° phase shift
for all angles
┴
0
0°
30°
60°
90°
Incidence angle

180° phase shift
for angles below
Brewster's angle;
0° for larger angles
||
0
0°
30°
60°
Incidence angle
90°
Phase shifts in reflection (glass to air)
n t < ni

Interesting phase
above the critical
angle
┴
0
0°
30°
60°
90°
Incidence angle

180° phase shift
for angles below
Brewster's angle;
0° for larger angles
||
0
0°
30°
60°
Incidence angle
90°
Glare is horizontally polarized
Puddle reflection viewed
through polarizer that
transmits only horizontally
polarized light
Puddle reflection viewed
through polarizer that
transmits only vertically
polarized light
Light reflected
into our eyes
from the puddle
reflects at about
Brewster's Angle.
So parallel
(i.e., vertical)
polarization sees
zero reflection.
Polarizer sunglasses transmit only vertically polarized light.
Brewster's Angle
A complex trigonometric calculation reveals that the reflection
coefficient for parallel-polarized
light goes to zero for Brewster's
angle incidence, tan(qi) = nt / ni
When the reflected beam makes a
right angle with the transmitted beam,
and the polarization is parallel, then
no scattering can occur, due to the
scattered dipole emission pattern.
ni sin(qi )  nt sin(qt )
ni
But our right-angle assumption
implies that qi + qt = 90°. So:
qi qi
nt
qt
qi +qt = 90°
ni sin(qi )  nt sin(90  qi )
 nt cos(qi )
Thus,
nt
tan(q i ) 
ni
BREWSTER ANGLE
Extinction 5*10-5
Piles of plates
BROAD BAND POLARIZING CUBES
The extinction of transmitted p component is at least 1 part
in 500
GLAN THOMPSON
The Glan Thompson polarizer is made of two
calcite prisms cemented together. Two types of Glan
Thompsons are available. One is the standard form
and the other is the long form. Their length to
aperture ratios are 2.5 : 1 and 3.0 : 1 respectively.
Glan Thompsons tend to have higher extinction
ratio than air spaced polarizers. In the ultra violet
spectrum, their transmission is limited by
absorption in calcite as well as the cement layer.
These polarizers can be used from about 350 to
2300 nm.
Extinction 5*10-5
GLAN TAYLOR
The Glan Taylor prism polarizer is made of two
calcite prisms which are assembled with an air
space . It has a length to aperture ratio of
approximately 1.0 which makes it a relatively thin
polarizer. It is made of UV selected calcite. A 10
mm thick calcite plate having 50% or more
transmission at 250 nm is considered UV selected.
The spectral range of this polarizer is from 2502300 nm. Below 250 nm, transmission cutoff
wavelength varies from crystal to crystal.
Extinction 1*10-4
CALCITE WOLLASTON
Calcite Wollaston prism polarizer is made of two
prisms of calcite which are cemented together. The
two output beams in a Wollaston polarizer exit with
unequal beam deviation ( asymmetry ) which is
given in the table below. The beam separation angle
is dependent on wavelength. Useable range of this
polarizer is from 350 nm to 2300 nm.
Extinction 1*10-5
MAGNESIUM FLUORIDE ROCHON
extinction ratio of at least 10-3
The magnesium fluoride Rochon polarizer is made of two prisms
of single crystal magnesium fluoride which are optically
contacted. This polarizer can be used over the spectral range of
140 to 6000 nm and has an extinction ratio of at least 10-3.
CALCITE BEAM DISPLACER
A calcite Beam Displacer splits the input
unpolarized beam of light into two
orthogonally polarized components which exit
parallel to each other. The ordinary
polarization transmits straight through while
the extraordinary transmits through the
crystal making approximately 6 degree angle
with the straight through beam and emerges
parallel to it. The beam displacement varies
slightly with wavelength. Non standard beam
displacers are available by special order.
Polarizer used in camera
The extinction of transmitted p component is at least 1 part
in 200
• Polarization of light
– Polarization vs. dipole oscillation
– Ways to change the polarization status
– Ways to purify the polarization
 Polarization in light scattering
Scattering of polarized light
No light is scattered along the input field direction, i.e. with k
parallel to E.
Vertically polarized
input light
Horizontally polarized
input light
Scattering of unpolarized light
Again, no light is scattered along the input field direction,
i.e. with k parallel to Einput.
Scattering in the Earth's atmosphere leads to
interesting polarization properties of skylight.
Sun's rays
Skylight is polarized if the sun is to your
side.
Right-angle scattering
is polarized
This polarizer transmits
horizontal polarization
(of which there is very little).
Polarizer transmitting vertical polarization
Multiple scattering yields some light of the other polarization.
In clouds, much multiple scattering occurs, and light there is
unpolarized.
Polarization Spectroscopy
The 45°-polarized “Pump” pulse reorients molecules, which induces
some birefringence into the medium, which then acts like a wave plate
until the molecules re-orient back to their initial random distribution.