![Page 1: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/1.jpg)
Feb 14, 2011Midterm: Mon., Feb. 21 Presentations on Friday:
Carleton (HST)Chromey (Chandra)Robertson (Spitzer)
![Page 2: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/2.jpg)
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
![Page 3: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/3.jpg)
,,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
![Page 4: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/4.jpg)
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
![Page 5: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/5.jpg)
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
![Page 6: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/6.jpg)
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
![Page 7: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/7.jpg)
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
![Page 8: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/8.jpg)
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
![Page 9: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/9.jpg)
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
![Page 10: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/10.jpg)
Angle of Polarization
onpolarizati maximum of angle
Q
U2tan
![Page 11: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/11.jpg)
Sources of Polarization of Light in Celestial Objects
(1) Refelection off solid surfaces e.g. moon; plane mirrors
![Page 12: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/12.jpg)
(2) Scattering of light by molecules: Rayleigh Scattering e.g. the blue sky
![Page 13: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/13.jpg)
(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
![Page 14: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/14.jpg)
(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
![Page 15: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/15.jpg)
(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
![Page 16: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/16.jpg)
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.
![Page 17: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/17.jpg)
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
![Page 18: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/18.jpg)
![Page 19: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/19.jpg)
(2) Birefringent crystal e.g. calcite
Has different index of refraction for waves oscillating in x-direction vs. the y-direction
![Page 20: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/20.jpg)
(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.
![Page 21: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/21.jpg)
(4) Pockels Cell
Single crystal emersed in a controllable E field. The external E field induces bi-refringence; can be varied.
![Page 22: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/22.jpg)
(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
![Page 23: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/23.jpg)
(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
![Page 24: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/24.jpg)
![Page 25: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/25.jpg)
![Page 26: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/26.jpg)
Single channel polarimeters combine these elements to measure either circular or linear polarization:
![Page 27: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/27.jpg)
![Page 28: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/28.jpg)
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
![Page 29: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/29.jpg)
(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
![Page 30: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/30.jpg)
(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
![Page 31: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/31.jpg)
![Page 32: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/32.jpg)
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
![Page 33: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/33.jpg)
Anti-glare windows
![Page 34: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/34.jpg)
LCD Displays
![Page 35: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/35.jpg)
Bees can see polarized light polarization of blue sky enables them to navigate
Humans: Haidinger’s Brush
Vikings: Iceland Spar?
![Page 36: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/36.jpg)
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
![Page 37: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/37.jpg)
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.
![Page 38: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/38.jpg)
![Page 39: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/39.jpg)
![Page 40: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/40.jpg)
CD cover seen in polarized light from monitor
![Page 41: Feb 14, 2011 Midterm: Mon., Feb. 21 Presentations on Friday: Carleton (HST) Chromey (Chandra) Robertson (Spitzer)](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d6a5503460f94a48035/html5/thumbnails/41.jpg)
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.