Feb. 9, 2011

Fourier TransformsPolarizations

Fourier Transforms

dxexFf ix 2)()(

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)

dxefxF ix 2)()(

see Bracewell’s book:FT and Its Applications

The inverse transform is

note change in sign

Some examples

22

for 1

2 and

2for 0)(

x

xxxB(1) F.T. of box function

€

b(σ ) = B(x)e2πixσ dx−∞

+∞

∫

= e2πixσ dx−ω 2

ω2

∫

=e2πixσ

2πiσ

⎡

⎣ ⎢

⎤

⎦ ⎥−ω 2

ω2

=1

2πiσe2πi(ω 2)σ − e−2πi(ω 2 )σ

[ ]

=1

2πiσcos πωσ( ) + isin(πωσ ) − cos −πωσ( ) + isin(−πωσ )[ ][ ]

=ωsin(πωσ )

πωσ=ω sinc(πωσ )

“Ringing” -- sharp discontinuity ripples in spectrum

When ω is large, the F.T. is narrow: first zero at

1

other zeros at

1

(2) Gaussian

F.T. of gaussian is a gaussian with narrower width

22 /1)(

xexG

222

)( egFT

Dispersion of G(x) β

Dispersion of g(σ)

1

(3) delta- function

11 for 0)( xxxx

Note:

1)( dxx

)(

)()()()(

1

111

xF

dxxxxFdxxFxx

x

)( 1xx

x1

1

1

2

12

21

)(

)()(

:function delta theof

ix

ix

ix

e

dxxxe

dxexxf

FT

Amplitude of F.T. of delta function = 1 (constant with sigma)Phase = 2πxiσ linear function of sigma

(4) )()()( 11 xxxxxF

)2cos(2

)(

1

22 11

x

eefFT ixix

So, cosine with wavelength1

1

x

transforms to delta functions at +/ x1

x

0-x1 +x1

)()()( 11 xxxxxF (5)

)2sin(2)( 1 xifFT

x-x1

x10

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

This describes a monochromatic beam which is linearly polarized –

)(01ˆ

trkieEaE

1ˆ direction, same in the always is aE

More generally, consider a wave propagating in direction z

The electric vector is the real part of

tieEyExE 21 ˆˆ

where E1 and E2 are complex numbers. They can be written in the form

212211 ii eEeE

)cos(

)cos(

22

11

tE

tE

y

x

E

The real part of

The tip of the vector traces an ellipse with time (1) describes anELLIPTICALLY POLARIZED wave

is

E

φ1 and φ2 are phases

The tips of the Ex and Ey trace out an ellipsewhose major axis is tilted with respect to the x- and y- axes, by angle χ

or, in the x’, y’ coordinate system

tE

tE

y

x

sinsin

coscos

0

0

'

'

22 where

yx EE and between difference phase is

20For

Ellipse traced clockwise as viewed by an observer toward whom the wave propagates

Called RIGHT-HAND polarization, or negative helicity

02

-For

Ellipse trace counter-clockwise as viewed by an observer LEFT-HAND polarization, or positive helicity

Negative and Positive Helicity

ellipsean generally is )cos(

)cos(

that Show

22

11

tE

tE

y

x

11

11

sin)sin(cos)cos(

)cos(

tt

tEx

222

sin)sin(cos)cos(

ttEy

bababa sinsincoscos)cos(

So...

)sin()cos(

sincos)cos(sincos)cos(

sinsin

21

1221

12

21

t

tt

EE yx

)sin()sin(

sincos

21

22

21

t

EE yx

bababa sincoscossin)sin( since

(1)

(2)

Square (1) and (2) and add

212

2

221

21

2

1

sin

cos2

yyxxEEEE

Equation for an Ellipse

ellipse. s' respect to with 90 rotated is ellipse

its and polarizedly elliptical also is

, lar toperpendicu always is Since

polarizedly elliptical is

0 E

B

EB

E

What does E look like for special cases?

21 )1(

212

2

221

21

2

1

sincos2

yyxxEEEE

022

221

2

1

yyxxEEEEbecomes

02

21

yxEE

21 yxEE

LINEARLY POLARIZED

Ex and Ey are in phase reach maxima together = 0 together

x

y

ε1

ε2

-ε2

-ε1

21 )2(

212

2

221

21

2

1

sincos2

yyxxEEEE

becomes

02

21

yxEE

21 yxEE

LINEARLY POLARIZED

x

y

ε1

ε2

-ε2

-ε1

LINEAR POLARIZATION

2 )3( 21

212

2

221

21

2

1

sincos2

yyxxEEEE

becomes

12

2

2

1

yxEE

ELLIPTICALLY POLARIZED

x

y

ε1

ε2

-ε2

-ε1

circleE a describes then If 21

x

y

CIRCULARLY POLARIZED

Circular Polarization

Note phase shift

SUMMARY

Stokes Parameters

)cos( and )cos( 2211 tEtE yx

, of in terms axes) (rotatingwritten -re becan

ttE

ttE

y

x

sincossincossincos

sinsinsincoscoscos

0

0

cossinsin

sincoscos

sinsinsin

coscoscos

22

22

11

11

o

o

o

o

where

,,for solve ,,,Given 02211

Define STOKES PARAMETERS

2sin

sin2

2sin2cos

cos2

2cos2cos

20

2121

20

2121

20

22

21

20

22

21

V

U

Q

I

Q

UI

V

I

2tan

2sin

20

21210 ,,,or ,, of instead

U)V,Q,(I, PARAMETERS STOKES theof in terms

wave theofon polarizati thedescribe tocustomary isIt

Q

UI

V

I

2tan

2sin

20

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:

For pure elliptical polarization

,, .. 0ge

2222 VUQI

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 theε1, ε2, χ

(note – in one nanosecond, a visible wave has ~106 oscillations)

2121

2121

22

21

22

21

sin2

cos2

V

U

Q

I

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:

22

21

So...

0

0222

VUQ

VUQ

The intensity will consist of a polarized part (for which I2 = U2 + V2 + Q2) and an unpolarized part.Thus,

2222 VUQI

Degree of Polarization

I

VUQ 222

waveofintensity total

waveofpart polarized ofintensity

Special case: V=0 no circular polarization, but can have “partial linear polarization”

i.e. Some of I is unpolarized

Some of I is polarized

22 UQII dunpolarize

22 UQI polarized

Angle of Polarization

onpolarizati maximum of angle

Q

U2tan

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

mc

eB

20

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 paramagneticdust grains – they tend to orient their long axes perpendicular to theB-field.

E-field parallelto the long axisis blocked morethan 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

Cleary, Heiles & Haslam 1979

The direction of polarization is shown below as short lines superimposed on amap of the hydrogen gas distribution in Galactic latitude and longitude.Note that the hydrogen gas filaments lie mostly parallel to the polarizationdirections of starlight, indicating that the gas concentrations are elongatedparallel to the local magnetic field. This indicates that the gas filamentscannot be strongly self-gravitating.

PolarimetersMost 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 onewith different Stokes Parameters I’, Q’, U’, V’

Quarter Wave Plate2

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

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 theraindrop, 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 anti-crepuscular rays created by storm clouds along the western horizon.

David Lien, PSIEPOD03/02/06

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.