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Feb. 9, 2011
Fourier Transforms
Polarizations
see Bracewell’s book:
FT and Its Applications
Fourier Transforms
A function’s Fourier Transform is a specification of the
amplitudes and phases of sinusoidals, which, when
added together, reproduce the function
Given a function F(x)
The Fourier Transform of F(x) is f(σ)

f(

)
F
(
x
)
e
dx

2

i
x


The inverse transform is

F
(
x
)

f
(

)
e




2

ix

dx
note change in sign
Some examples
(1) F.T. of box function
b() 

 B(x)e
2ix





1 for

x
dx
2



B
(x
)
0for
xand
x

2
2
2
 e   dx
2 ix
 2

e 2ix  2
 

2

i




2

1
2i( 2 )
2i( 2 )

e
e
2i
1

cos  isin( )  cos  isin( )
2i
sin( )


  sinc( )


2
“Ringing” -- sharp discontinuity  ripples in spectrum
When ω is large, the F.T. is narrow:
first zero at

other zeros at
1

 
1

(2)
Gaussian
F.T. of gaussian is a gaussian with narrower width
G
(
x
)
1
2 2

x
/
e





FT

g
(
)

e
2
2
2

Dispersion of G(x)  β
Dispersion of g(σ) 
1

(3)
delta- function

(
x

x
)

0
for
x

x
1
1
(x
x
)
1
x
x1

Note:
(x)dx1




(
x

x
)
F
(
x
)
d

x
F
(
x
)

(
x

x
)
dx


1



F
(
x
)
1
1


1
FT of thedeltafunction:

f ( )    (x  x1)e
2ix
dx

2ix1
e

 (x  x )dx
1

2ix1
e
Amplitude of F.T. of delta function = 1 (constant with sigma)
Phase = 2πxiσ  linear function of sigma
(4)

F
(
x
)

(
x

x
)

(
x

x
)
1
1
x
-x1
+x1
0


2
cos(
2

x

)



2

ix

1
FT

f
()

e 
e
2
ix
1
1
So, cosine with wavelength
1
x1
transforms to delta functions at +/ x1
(5)

F
(
x
)

(
x

x
)

(
x

x
)
1
1
-x1
x
0
x1


FT

f
(
)

2
i
sin
2
x
)
1
Summary of Fourier
Transforms
Fourier Transforms:
• Sharp features in the time domain  ringing in frequency domain
• Narrow feature in time domain  broad radiation spectrum
• Broad feature in the time domain  narrow radiation spectrum
POLARIZATION
The solution to the wave equation we considered was


i ( k r t )
E  aˆ1E0e
This describes a monochromatic beam which is
linearly polarized –

E is always in the same direction, aˆ1
More generally, consider a wave propagating in direction z
The electric vector is the real part of

 it
E  xˆE1  yˆ E2 e
where E1 and E2 are complex numbers. They can be written in the form
E1  1e
The real part of

E
i1
is
E2   2e
i2
φ1 and φ2 are phases
E x   1 cos(t  1 )
E y   2 cos(t  2 )
The tip of the

E
vector traces an ellipse with time  (1) describes an
ELLIPTICALLY POLARIZED wave
The tips of the Ex and Ey trace out an ellipse
whose major axis is tilted with respect to the x- and y- axes,
by angle χ
or, in the x ’, y ’ coordinate system
E x '   0 cos  cos t
E y '   0 sin  sin t
where 

2
 
 is phase difference between E x and E y

2
For 0   

2
Ellipse traced clockwise as viewed by an observer
toward whom the wave propagates
Called RIGHT-HAND polarization, or negative helicity
For -

2
 0
Ellipse trace counter-clockwise as viewed by an observer
 LEFT-HAND polarization, or positive helicity
Negative and Positive Helicity
Show that
E x  1 cos(t  1 )
E y   2 cos(t  2 ) is generally an ellipse
Ex
1
 cos(t  1 )
cos( a  b)  cos a cos b  sin a sin b
 cos(t ) cos 1  sin( t ) sin 1
Ey
2
 cos(t ) cos 2  sin( t ) sin 2
So...
Ex
1
sin 2 
Ey
sin 1
2
 cos(t ) cos 1 sin 2  cos(t ) cos 2 sin 1
(1)
 cos(t ) sin( 1  2 )
since sin( a  b)  sin a cos b  cos a sin b
Ex
1
cos 2 
Ey
sin 2
2
 sin( t ) sin( 1  2 )
(2)
Square (1) and (2) and add 
2
 Ex 
 E x  E y 
 Ey 
   2   cos1  2    
 1 
  1   2 
 2 
 sin 2 1  2 
2
 Equation for an Ellipse

E is elliptical ly polarized


Since B is always perpendicu lar to E ,

B is also elliptical ly polarized and its

0
ellipse is rotated 90 with respect to E ' s ellipse.
What does E look like for special cases?
(1) 1  2
2
2
 Ex 
 E x  E y 
 Ey 
   2   cos1  2      sin 2 1  2 
 1 
  1   2 
 2 
2
becomes
2
 Ex 
 E x  E y   E y 
   2       0
 1 
  1   2    2 
y
2
 Ex E y 

  0

 1  2 
ε2
-ε1
x
ε1
-ε2
Ex
1

Ey
2
 LINEARLY POLARIZED
Ex and Ey are in phase
reach maxima together
= 0 together
(2) 1  2  
 Ex 
 
 1 
2
2
 E x  E y 
 Ey 
 2   cos1  2      sin 2 1  2 
  1   2 
 2 
becomes
2
 Ex E y 

  0

 1  2 
y
Ex
ε2
ε1
-ε1
-ε2
x
1

Ey
2
 LINEARLY POLARIZED
LINEAR POLARIZATION
(3) 1  2 
 Ex 
 
 1 
2

2
2
 E x  E y 
 Ey 
 2   cos1  2      sin 2 1  2 
  1   2 
 2 
becomes
2
 Ex   E y 
      1
 1    2 
 ELLIPTICALLY POLARIZED

If 1   2 then E describes a circle
y
y
ε2
-ε1
2
x
-ε2
ε1
x
 CIRCULARLY
POLARIZED
Circular Polarization
Note phase shift
SUMMARY
Stokes Parameters
E x  1 cos(t  1 ) and E y   2 cos(t  2 )
can be re - written (rotating axes) in terms of  , 
Ex   0 cos  cos  cos t  sin  sin  sin t 
E y   0 cos  sin  cos t  sin  cos  sin t 
where
 1 cos 1   o cos  cos 
 1 sin 1   o sin  sin 
 2 cos 2   o cos  sin 
 2 sin 2   o sin  cos 
Given 1 , 1 ,  2 , 2  solve for  0 ,  , 
Define STOKES PARAMETERS
I     
2
1
2
2
2
0
Q       cos 2  cos 2 
2
1
2
2
2
0
U  2 1 2 cos1  2 
  cos 2  sin 2 
2
0
V  2 1 2 sin 1  2 
  sin 2 
2
0
 02  I
V
sin 2  
I
U
tan 2  
Q
It is customary to describe the polarizati on of the wave
in terms of the STOKES PARAMETERS (I, Q, V, U)
instead of  0 ,  ,   or 1 ,  2 , 1 , 2 
 02  I
V
sin 2  
I
U
tan 2  
Q
I: always positive
proportional to flux or intensity of wave
V: measures circular polarization
V=0 linear polarization
V>0 right hand ellipticity
V<0 left hand ellipticity
Q,U: measure orientation of ellipse relative to
x-axis
Q=U=0 for circular polarization
For a monochromatic wave, you only need 3 parameters
to describe it:
e.g.  0 ,  , 
For pure elliptical polarization
I  Q U V
2
2
2
2
The Stokes parameters are not independent:
you need only specify 3, then can compute the 4th
A more general situation will involve the superposition of many waves,
each with their own wavelength and polarization.
Then one defines the Stokes parameters as time averages of the
ε1, ε2, χ
(note – in one nanosecond, a visible wave has ~106 oscillations)
I   
2
1
2
2
Q   
2
1
2
2
U  2 1 2 cos1  2 
V  2 1 2 sin 1  2 
time average
Sometimes waves are completely unpolarized:
phase difference between Ex and Ey are random
No prefered direction in x-y plane, so Ex and Ey don’t trace
an ellipse, circle, line etc.
In this case:

So...
2
1
 
2
2
Q U V  0
Q U V  0
2
2
2
The intensity will consist of a polarized part
(for which I2 = U2 + V2 + Q2) and an unpolarized part.
Thus,
I  Q U V
2
2
2
2
Degree of Polarization
intensity of polarized part of wave

total intensity of wave
Q U V

I
2
Special case:
V=0
2
2
no circular polarization, but can have “partial
linear polarization”
i.e. Some of I is unpolarized 
Some of I is polarized 
I unpolarized  I  Q  U
2
I polarized  Q 2  U 2
2
Angle of Polarization
  angle of maximum polarizati on
U
tan 2 
Q
Sources of Polarization of Light in Celestial Objects
(1) Refelection off solid surfaces
e.g. moon; plane mirrors
(2) Scattering of light by molecules: Rayleigh Scattering
e.g. the blue sky
(3) Zeeman Effect
e.g. Sunspots
In the presence of a magnetic field of strength B, a line will split
into several components, each with different polarization
e.g. classical “Normal” Zeeman effect:
An oscillating charge of mass m radiates with frequency ω0
in the absence of a B field.
Apply B-field of strength B  splits into 3 lines
eB
   0 
2mc
0
circularly polarized
linearly polarized
(4) Scattering of light by free electrons (Thomson scattering)
e.g. solar corona
(5) Synchrotron emission (e.g. radio galaxies)
Radiation from relativistic electrons in B-field
(6) Scattering by dust grains
e.g. polarization of starlight by dust grains aligned in the Milky
Way’s B-field --- The Davis-Greeenstein Effect
The interstellar magnetic field in the Milky Way will align paramagnetic
dust grains – they tend to orient their long axes perpendicular to the
B-field.
E-field parallel
to the long axis
is blocked more
than E-field
perpendicular
to the dust grains
Light from stars is unpolarized, but becomes polarized as it traverses the
interstellar medium (ISM).
Light becomes polarized parallel to the magnetic field  map of B-field
The direction of polarization is shown below as short lines superimposed on a
map of the hydrogen gas distribution in Galactic latitude and longitude.
Note that the hydrogen gas filaments lie mostly parallel to the polarization
directions of starlight, indicating that the gas concentrations are elongated
parallel to the local magnetic field. This indicates that the gas filaments
cannot be strongly self-gravitating.
Cleary, Heiles & Haslam 1979
Polarimeters
Most polarimeters rely on linear polarizers, i.e. “analyzers”
which sub-divide the incident light into 2 beams:
one beam linearly polarized parallel to the “principal plane”
of the analyzer
other beam perpendicular to it.
TYPES OF ANALYZERS
(1) Polarizer, polarizer film
invented by Land in 1928, at age 19
Absorbs the component of the electric vector which oscillates in
a particular direction  usually not used in astronomy, since you
hate to throw out light
(2) Birefringent crystal
e.g. calcite
Has different index of refraction for waves oscillating in x-direction
vs. the y-direction
(3) Wollaston Prism; Nicol Prism
* To get equal intensities for the parallel and perpendicular beams
when the incident beam is unpolarized, you can cement 2 pieces
of birefringent crystal together, with the principle planes crossed.
* This
configuration also
results in the widest
separation of the 2
beams
* Nicol prism: one
beam reflected at
the interface.
(4) Pockels Cell
Single crystal emersed in a controllable E field.
The external E field induces bi-refringence; can be varied.
(5) Wire grid analyzers and Dipole Antennas
* Grid of parallel wires
* Dipole antenna -- radio receivers
Most sensitive to radiation which is linearly polarized with E parallel
to the dipole
 for optimum efficiency, need to build “dual-polarization” receivers
(6) Retarders e.g. sheet of mica
A material produces a phase difference between the beams
polarized parallel and perpendicular to the principle plane of the
crystal – called retardance, τ
Transforms a beam with Stokes Parameters I,Q,U,V to one
with different Stokes Parameters I’, Q’, U’, V’
Quarter Wave Plate
  2
If incident beam is circularly polarized: I=V, then the output
beam is linearly polarized I=sqrt(Q2 + U2)
Half Wave Plate
 
Changes right-handed circular polarization to left-handed, or
Changes linear polarization at angle θ to linear pol. at angle -θ
Single channel polarimeters combine these elements to measure
either circular or linear polarization:
Some Practical Considerations for Polarimeters
(1) Birefringent materials are scarce, and difficult to obtain in large sizes
 limits sizes of instruments, resolution of spectrographs
(2) Refractive indices of material are a function of λ
 waveplates have retardance τ which is a function of λ
 need to achromatize
(3) The night sky is polarized.
Moonlight is VERY polarized.
 sky polarization must be monitored.
(4) Spectropolarimetry
* The wavelength dependence of the % polarization  origin of
polarization (e.g. dust vs. electron scattering vs. synchrotron)
* Diffraction gratings, however, polarize light very strongly,
like mirrors.
 design spectropolarimetry so that the polarizer is before
diffraction grating in light path, not after
(5) Telescope induced polarization
* light is polarized by reflection
* turns out, the induced linear polarization by each reflection is
exactly canceled on axis for
Cassegrain focus reflectors
Prime focus reflectors
refractors
* TERRIBLE at Newtonian, Coude, Naysmyth foci \
because of the plane mirror reflection
Other Applications of Polarization
Rainbows are
polarized
David Lien, PSI
EPOD
03/02/06
These two images of a late afternoon rainbow in Tucson, Arizona were taken within 20 seconds of each other (taken in July of
2005). The difference? Each picture was taken through a polarizing filter which was rotated 90 degrees between the two
photographs. The light waves that are ultimately redirected to create the rainbow are reflected at the back of the
raindrop, and the angle of redirection is very close to Brewster's angle for a water-air interface. Light reflecting off of a surface
at Brewster's angle is 100% polarized. Since the sunlight resulting in a rainbow reflects off the back of a drop over a small range
of angles, rainbows are not 100% polarized (~95% polarization is typical). Note that these images also show faint anticrepuscular rays created by storm clouds along the western horizon.
Anti-glare windows
LCD Displays
Bees can see polarized light 
polarization of blue sky enables them to navigate
Humans:
Haidinger’s Brush
Vikings:
Iceland Spar?
Many invertebrates can see polarization, e.g. the Octopus
Not to navigate (they don’t go far)
Perhaps they can see transparent jellyfish better?
unpolarized
polarized
Polarization and Stress Tests
In a transparent object, each wavelength of light is polarized by
a different angle. Passing unpolarized light through a polarizer,
then the object, then another polarizer results in a colorful pattern
which changes as one of the polarizers is turned.
CD cover seen in polarized light from monitor
3D movies
Polarization is also used in the entertainment industry to produce and show 3-D movies.
Three-dimensional movies are actually two movies being shown at the same time
through two projectors. The two movies are filmed from two slightly different camera
locations. Each individual movie is then projected from different sides of the audience
onto a metal screen. The movies are projected through a polarizing filter. The polarizing
filter used for the projector on the left may have its polarization axis aligned horizontally
while the polarizing filter used for the projector on the right would have its polarization
axis aligned vertically. Consequently, there are two slightly different movies being
projected onto a screen. Each movie is cast by light which is polarized with an
orientation perpendicular to the other movie. The audience then wears glasses which
have two Polaroid filters. Each filter has a different polarization axis - one is horizontal
and the other is vertical. The result of this arrangement of projectors and filters is that
the left eye sees the movie which is projected from the right projector while the right
eye sees the movie which is projected from the left projector. This gives the viewer a
perception of depth.