polarization light intro
TRANSCRIPT
Polarization of Light:from Basics to Instruments
(in less than 100 slides)
N. Manset
CFHT
N. Manset / CFHT Polarization of Light: Basics to Instruments 2
Introduction
• Part I: Different polarization states of light
• Part II: Stokes parameters, Mueller matrices
• Part III: Optical components for polarimetry
• Part IV: Polarimeters
• Part V: ESPaDOnS
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Part I: Different polarization states of light
• Light as an electromagnetic wave
• Mathematical and graphical descriptions of polarization
• Linear, circular, elliptical light
• Polarized, unpolarized light
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Light as an electromagnetic wave
Light is a transverse wave,
an electromagnetic wave
Part I: Polarization states
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Mathematical description of the EM wave
Light wave that propagates in the z direction:
y)t-kzcos(E)tz,(E
xt)-kzcos(E)tz,(E
0yy
0xx
Part I: Polarization states
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Graphical representation of the EM wave (I)
One can go from:
to the equation of an ellipse (using trigonometric
identities, squaring, adding):
2
0y
y
0x
x
2
0y
y
2
0x
x sincosE
E
E
E2
E
E
E
E
y)t-kzcos(E)tz,(E
xt)-kzcos(E)tz,(E
0yy
0xx
Part I: Polarization states
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Graphical representation of the EM wave (II)
An ellipse can be represented by 4 quantities:
1. size of minor axis
2. size of major axis
3. orientation (angle)
4. sense (CW, CCW)
Light can be represented by 4 quantities...
Part I: Polarization states
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Vertically polarized light
If there is no amplitude in x (E0x = 0), there is only one component, in y (vertical).
y)t-kzcos(E)tz,(E
xt)-kzcos(E)tz,(E
0yy
0xx
Part I: Polarization states, linear polarization
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Polarization at 45º (I)
If there is no phase difference (=0) and
E0x = E0y, then Ex = Ey
y)t-kzcos(E)tz,(E
xt)-kzcos(E)tz,(E
0yy
0xx
Part I: Polarization states, linear polarization
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Polarization at 45º (II)
Part I: Polarization states, linear polarization
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Circular polarization (I)
If the phase difference is = 90º and E0x = E0y
then: Ex / E0x = cos , Ey / E0y = sin
and we get the equation of a circle:
1sin cosE
E
E
E 22
2
0y
y
2
0x
x
y)t-kzcos(E)tz,(E
xt)-kzcos(E)tz,(E
0yy
0xx
Part I: Polarization states, circular polarization
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Circular polarization (II)
Part I: Polarization states, circular polarization
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Circular polarization (III)
Part I: Polarization states, circular polarization
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Circular polarization (IV)
Part I: Polarization states, circular polarization... see it now?
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Elliptical polarization
Part I: Polarization states, elliptical polarization
• Linear + circular polarization = elliptical polarization
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Unpolarized light(natural light)
Part I: Polarization states, unpolarized light
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A cool Applet
Electromagnetic Wave
Location: http://www.uno.edu/~jsulliva/java/EMWave.html
Part I: Polarization states
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Part II: Stokes parameters and Mueller matrices
• Stokes parameters, Stokes vector
• Stokes parameters for linear and circular polarization
• Stokes parameters and polarization P
• Mueller matrices, Mueller calculus
• Jones formalism
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Stokes parametersA tiny itsy-bitsy little bit of history...
• 1669: Bartholinus discovers double refraction in calcite
• 17th – 19th centuries: Huygens, Malus, Brewster, Biot, Fresnel and Arago, Nicol...
• 19th century: unsuccessful attempts to describe unpolarized light in terms of amplitudes
• 1852: Sir George Gabriel Stokes took a very different approach and discovered that polarization can be described in terms of observables using an experimental definition
Part II: Stokes parameters
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Stokes parameters (I)
The polarization ellipse is only valid at a given instant of time (function of time):
εsinεcos(t)E
(t)E
(t)E
(t)E2
(t)E
(t)E
(t)E
(t)E 2
0y
y
0x
x
2
0y
y
2
0x
x
To get the Stokes parameters, do a time average (integral over time) and a little bit of algebra...
Part II: Stokes parameters
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Stokes parameters (II)described in terms of the electric field
20y0x2
0y0x
220y
20x
220y
20x εsinEE2εcosEE2EEEE
The 4 Stokes parameters are:
εsinEE2V
εcosEE2U
E E Q
EEI
0y0x3
0y0x2
20y
20x1
20y
20x0
S
S
S
S
Part II: Stokes parameters
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Stokes parameters (III)described in geometrical terms
2sin
2sin2cos
2cos2cos
V
U
Q
I
2
2
2
2
a
a
a
a
Part II: Stokes parameters
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Stokes vectorThe Stokes parameters can be arranged in a Stokes vector:
LCPIRCPI
135I45I
90I0I
intensity
εsinEE2
εcosEE2
EE
EE
V
U
Q
I
0y0x
0y0x
20y
20x
20y
20x
• Linear polarization• Circular polarization• Fully polarized light• Partially polarized light• Unpolarized light 0VUQ
VUQI
VUQI
0V 0, U0,Q
0V 0, U0,Q
2222
2222
Part II: Stokes parameters, Stokes vectors
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Pictorial representation of the Stokes parameters
Part II: Stokes parameters
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Stokes vectors for linearly polarized light
LHP light
0
0
1
1
I0
LVP light +45º light -45º light
0
0
1
1
I0
0
1
0
1
I0
0
1
0
1
I0
Part II: Stokes parameters, examples
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Stokes vectors for circularly polarized light
RCP light
1
0
0
1
I0
LCP light
1
0
0
1
I0
Part II: Stokes parameters, examples
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(Q,U) to (P,)In the case of linear polarization (V=0):
I
UQP
22
Q
Uarctan
2
1
2 cos PQ 2 sin PU
Part II: Stokes parameters
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Mueller matrices
If light is represented by Stokes vectors, optical components are then described with Mueller matrices:
[output light] = [Muller matrix] [input light]
V
U
Q
I
V'
U'
Q'
I'
44434241
34333231
24232221
14131211
mmmm
mmmm
mmmm
mmmm
Part II: Stokes parameters, Mueller matrices
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Mueller calculus (I)
Element 1 Element 2 Element 3
1M 2M 3M
I’ = M3 M2 M1 I
Part II: Stokes parameters, Mueller matrices
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Mueller calculus (II)
Mueller matrix M’ of an optical component with Mueller matrix M rotated by an angle :
M’ = R(- ) M R() with:
1000
02cos2sin0
02sin2cos0
0001
)R(
Part II: Stokes parameters, Mueller matrices
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Jones formalism
Stokes vectors and Mueller matrices cannot describe interference effects. If the phase information is important (radio-astronomy, masers...), one has to use the Jones formalism, with complex vectors and Jones matrices: • Jones vectors to describe the polarization of light:
• Jones matrices to represent optical components:
(t)E
(t)E(t)J
y
x
2221
1211
jj
jjJ
BUT: Jones formalism can only deal with 100% polarization...
Part II: Stokes parameters, Jones formalism, not that important here...
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Part III: Optical components for polarimetry
• Complex index of refraction
• Polarizers
• Retarders
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Complex index of refraction
The index of refraction is actually a complex quantity:
iknm • real part
• optical path length, refraction: speed of light depends on media
• birefringence: speed of light also depends on P
• imaginary part
• absorption, attenuation, extinction: depends on media
• dichroism/diattenuation: also depends on P
Part III: Optical components
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Polarizers
Polarizers absorb one component of the polarization but not the other. The input is natural light, the output is polarized light (linear, circular, elliptical). They work by dichroism, birefringence, reflection, or scattering.
Part III: Optical components, polarizers
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Wire-grid polarizers (I)[dichroism]
• Mainly used in the IR and longer wavelengths
• Grid of parallel conducting wires with a spacing comparable to the wavelength of observation
• Electric field vector parallel to the wires is attenuated because of currents induced in the wires
Part III: Optical components, polarizers
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Wide-grid polarizers (II) [dichroism]
Part III: Optical components, polarizers
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Dichroic crystals [dichroism]
Dichroic crystals absorb one polarization state over the other one.
Example: tourmaline.
Part III: Optical components, polarizers
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Polaroids [dichroism]
Made by heating and stretching a sheet of PVA laminated to a supporting sheet of cellulose acetate treated with iodine solution (H-type polaroid). Invented in 1928.
Part III: Optical components, polarizers – Polaroids, like in sunglasses!
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Crystal polarizers (I) [birefringence]
• Optically anisotropic crystals
• Mechanical model:• the crystal is anisotropic, which means that the electrons are bound with different ‘springs’ depending on the orientation
• different ‘spring constants’ gives different propagation speeds, therefore different indices of refraction, therefore 2 output beams
Part III: Optical components, polarizers
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Crystal polarizers (II)[birefringence]
The 2 output beams are polarized (orthogonally).
isotropiccrystal (sodiumchloride)
anisotropiccrystal(calcite)
Part III: Optical components, polarizers
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Crystal polarizers (IV)[birefringence]
• Crystal polarizers used as:• Beam displacers,• Beam splitters,• Polarizers,• Analyzers, ...
• Examples: Nicol prism, Glan-Thomson polarizer, Glan or Glan-Foucault prism, Wollaston prism, Thin-film polarizer, ...
Part III: Optical components, polarizers
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Mueller matrices of polarizers (I)
• (Ideal) linear polarizer at angle :
0000
0χ2sinχ2cosχ2sinχ2sin
0χ2cosχ2sinχ2cosχ2cos
0χ2sinχ2cos1
2
12
2
Part III: Optical components, polarizers
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Mueller matrices of polarizers (II)
Linear (±Q) polarizer at 0º:
0000
0000
0011
0011
5.0
Linear (±U) polarizer at 0º :
0000
0101
0000
0101
5.0
Part III: Optical components, polarizers
Circular (±V) polarizer at 0º :
1001
0000
0000
1001
5.0
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Mueller calculus with a polarizer
Input light: unpolarized --- output light: polarized
0
I-
0
I
5.0
0
0
0
I
0000
0101
0000
0101
5.0
V'
U'
Q'
I'
Total output intensity: 0.5 I
Part III: Optical components, polarizers
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Retarders
• In retarders, one polarization gets ‘retarded’, or delayed, with respect to the other one. There is a final phase difference between the 2 components of the polarization. Therefore, the polarization is changed.
• Most retarders are based on birefringent materials (quartz, mica, polymers) that have different indices of refraction depending on the polarization of the incoming light.
Part III: Optical components, retarders
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Half-Wave plate (I)
• Retardation of ½ wave or 180º for one of the polarizations.
• Used to flip the linear polarization or change the handedness of circular polarization.
Part III: Optical components, retarders
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Half-Wave plate (II)
Part III: Optical components, retarders
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Quarter-Wave plate (I)
• Retardation of ¼ wave or 90º for one of the polarizations
• Used to convert linear polarization to elliptical.
Part III: Optical components, retarders
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• Special case: incoming light polarized at 45º with respect to the retarder’s axis
• Conversion from linear to circular polarization (vice versa)
Quarter-Wave plate (II)
Part III: Optical components, retarders
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Mueller matrix of retarders (I)
• Retarder of retardance and position angle :
cosτ12
1Handcosτ1
2
1G :with
cosτcos2ψsinτsin2ψsinτ0
cos2ψsinτcos4ψHGsin4ψH0
sin2ψsinτsin4ψHcos4ψHG0
0001
Part III: Optical components, retarders
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Mueller matrix of retarders (II)
• Half-wave oriented at 0º or 90º
• Half-wave oriented at ±45º
1000
0100
0010
0001
k
1000
0100
0010
0001
k
Part III: Optical components, retarders
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Mueller matrix of retarders (III)
• Quarter-wave oriented at 0º
• Quarter-wave oriented at ±45º
0100
1000
0010
0001
k
0010
0100
1000
0001
k
Part III: Optical components, retarders
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Mueller calculus with a retarder
1
0
0
1
0
0
1
1
0010
0100
1000
0001
V'
U'
Q'
I'
kk
• Input light linear polarized (Q=1)• Quarter-wave at +45º• Output light circularly polarized (V=1)
Part III: Optical components, retarders
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(Back to polarizers, briefly)
Circular polarizers• Input light: unpolarized --- Output light: circularly polarized
• Made of a linear polarizer glued to a quarter-wave plate oriented at 45º with respect to one another.
Part III: Optical components, polarizers
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Achromatic retarders (I)
• Retardation depends on wavelength• Achromatic retarders: made of 2 different materials with
opposite variations of index of refraction as a function of wavelength
• Pancharatnam achromatic retarders: made of 3 identical plates rotated w/r one another
• Superachromatic retarders: 3 pairs of quartz and MgF2
plates
Part III: Optical components, retarders
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Achromatic retarders (II)
Part III: Optical components, retarders
=140-220º
not very achromatic!
= 177-183º
much better!
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Retardation on total internal reflection
• Total internal reflection produces retardation (phase shift)
• In this case, retardation is very achromatic since it only depends on the refractive index
• Application: Fresnel rhombs
Part III: Optical components, retarders
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Fresnel rhombs
• Quarter-wave and half-wave rhombs are achieved with 2 or 4 reflections
Part III: Optical components, retarders
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Other retarders
• Soleil-Babinet: variable retardation to better than 0.01 waves
• Nematic liquid crystals... Liquid crystal variable retarders... Ferroelectric liquid crystals... Piezo-elastic modulators... Pockels and Kerr cells...
Part III: Optical components, retarders
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Part IV: Polarimeters
• Polaroid-type polarimeters
• Dual-beam polarimeters
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Polaroid-type polarimeterfor linear polarimetry (I)
• Use a linear polarizer (polaroid) to measure linear polarization ... [another cool applet] Location: http://www.colorado.edu/physics/2000/applets/lens.html
• Polarization percentage and position angle:
)II(
II
IIP
max
minmax
minmax
Part IV: Polarimeters, polaroid-type
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Polaroid-type polarimeterfor linear polarimetry (II)
• Advantage: very simple to make
• Disadvantage: half of the light is cut out
• Other disadvantages: non-simultaneous measurements, cross-talk...
• Move the polaroid to 2 positions, 0º and 45º (to measure Q, then U)
Part IV: Polarimeters, polaroid-type
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Polaroid-type polarimeterfor circular polarimetry
• Polaroids are not sensitive to circular polarization, so convert circular polarization to linear first, by using a quarter-wave plate
• Polarimeter now uses a quarter-wave plate and a polaroid
• Same disadvantages as before
Part IV: Polarimeters, polaroid-type
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Dual-beam polarimetersPrinciple
• Instead of cutting out one polarization and keeping the other one (polaroid), split the 2 polarization states and keep them both
• Use a Wollaston prism as an analyzer• Disadvantages: need 2 detectors (PMTs, APDs) or
an array; end up with 2 ‘pixels’ with different gain• Solution: rotate the Wollaston or keep it fixed and
use a half-wave plate to switch the 2 beams
Part IV: Polarimeters, dual-beam type
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Dual-beam polarimetersSwitching beams
Part IV: Polarimeters, dual-beam type
• Unpolarized light: two beams have identical intensities whatever the prism’s position if the 2 pixels have the same gain
• To compensate different gains, switch the 2 beams and average the 2 measurements
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Dual-beam polarimetersSwitching beams by rotating the prism
rotate by 180º
Part IV: Polarimeters, dual-beam type
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Dual-beam polarimetersSwitching beams using a ½ wave plate
Rotated by 45º
Part IV: Polarimeters, dual-beam type
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A real circular polarimeterSemel, Donati, Rees (1993)
Quarter-wave plate, rotated at -45º and +45º
Analyser: double calcite crystal
Part IV: Polarimeters, example of circular polarimeter
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A real circular polarimeterfree from gain (g) and atmospheric transmission () variation effects
• First measurement with quarter-wave plate at -45º, signal in the (r)ight and (l)eft beams:
• Second measurement with quarter-wave plate at +45º, signal in the (r)ight and (l)eft beams:
• Measurements of the signals:
rl SS 11 ,
rl SS 22 ,
)()(
)()(
22222222
11111111
VIgSVIgS
VIgSVIgSrrll
rrll
Part IV: Polarimeters, example of circular polarimeter
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A real circular polarimeterfree from gain and atmospheric transmission variation effects
• Build a ratio of measured signals which is free of gain and variable atmospheric transmission effects:
1for 2
1
2
11
4
1
2
2
1
1
21211221
2112
1
2
2
1
VI
V
I
VF
VVVIVIII
VIVI
S
S
S
SF
r
r
l
l
average of the 2 measurements
Part IV: Polarimeters, example of circular polarimeter
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Polarimeters - Summary• 2 types:
– polaroid-type: easy to make but ½ light is lost, and affected by variable atmospheric transmission
– dual-beam type: no light lost but affected by gain differences and variable transmission problems
• Linear polarimetry: – analyzer, rotatable– analyzer + half-wave plate
• Circular polarimetry:– analyzer + quarter-wave plate
2 positions minimum
1 position minimum
Part IV: Polarimeters, summary
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Part V: ESPaDOnS
Optical components of the polarimeter part :
• Wollaston prism: analyses the polarization and separates the 2 (linear!) orthogonal polarization states
• Retarders, 3 Fresnel rhombs:– Two half-wave plates to switch the
beams around– Quarter-wave plate to do circular
polarimetry
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ESPaDOnS: circular polarimetry
• Fixed quarter-wave rhomb
• Rotating bottom half-wave, at 22.5º increments
• Top half-wave rotates continuously at about 1Hz to average out linear polarization when measuring circular polarization
Part V: ESPaDOnS, circular polarimetry mode
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ESPaDOnS: circular polarimetry of circular polarization
• half-wave
• 22.5º positions
• flips polarization
• gain, transmission
• quarter-wave
• fixed
• circular to linear
• analyzer
Part V: ESPaDOnS, circular polarimetry mode
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ESPaDOnS: circular polarimetry of (unwanted) linear polarization
• half-wave
• 22.5º positions
• gain, transmission
• quarter-wave
• fixed
• linear to elliptical
• analyzer• circular part goes through not analyzed and adds same intensities to both beams
• linear part is analyzed!
• Add a rotating half-wave to “spread out” the unwanted signal
Part V: ESPaDOnS, circular polarimetry mode
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ESPaDOnS: linear polarimetry
• Half-Wave rhombs positioned at 22.5º increments
• Quarter-Wave fixed
Part V: ESPaDOnS, linear polarimetry
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ESPaDOnS: linear polarimetry
• Half-Wave rhombs positioned as 22.5º increments– First position gives Q– Second position gives U– Switch beams for gain and atmosphere effects
• Quarter-Wave fixed
Part V: ESPaDOnS, linear polarimetry
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ESPaDOnS - Summary
• ESPaDOnS can do linear and circular polarimetry (quarter-wave plate)
• Beams are switched around to do the measurements, compensate for gain and atmospheric effects
• Fesnel rhombs are very achromatic
Part V: ESPaDOnS, summary
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Credits for pictures and movies• Christoph Keller’s home page – his 5 lectures
http://www.noao.edu/noao/staff/keller/
• “Basic Polarisation techniques and devices”, Meadowlark Optics Inc. http://www.meadowlark.com/
• Optics, E. Hecht and Astronomical Polarimetry, J. Tinbergen • Planets, Stars and Nebulae Studied With Photopolarimetry, T.
Gehrels
• Circular polarization movie http://www.optics.arizona.edu/jcwyant/JoseDiaz/Polarization-Circular.htm
• Unpolarized light movie http://www.colorado.edu/physics/2000/polarization/polarizationII.html
• Reflection of wave http://www.physicsclassroom.com/mmedia/waves/fix.html
• ESPaDOnS web page and documents
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References/Further reading On the Web
• Very short and quick introduction, no equation http://www.cfht.hawaii.edu/~manset/PolarIntro_eng.html
• Easy fun page with Applets, on polarizing filters http://www.colorado.edu/physics/2000/polarization/polarizationI.html
• Polarization short course http://www.glenbrook.k12.il.us/gbssci/phys/Class/light/u12l1e.html
• “Instrumentation for Astrophysical Spectropolarimetry”, a series of 5 lectures given at the IAC Winter School on Astrophysical Spectropolarimetry, November 2000 –http://www.noao.edu/noao/staff/keller/lectures/index.html
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References/Further reading Polarization basics
• Polarized Light, D. Goldstein – excellent book, easy read, gives a lot of insight, highly recommended
• Undergraduate textbooks, either will do:– Optics, E. Hecht– Waves, F. S. Crawford, Berkeley Physics Course vol. 3
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References/Further readingAstronomy, easy/intermediate
• Astronomical Polarimetry, J. Tinbergen – instrumentation-oriented
• La polarisation de la lumière et l'observation astronomique, J.-L. Leroy – astronomy-oriented
• Planets, Stars and Nebulae Studied With Photopolarimetry, T. Gehrels – old but classic
• 3 papers by K. Serkowski – instrumentation-oriented
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References/Further readingAstronomy, advanced
• Introduction to Spectropolarimetry, J.C. del Toro Iniesta – radiative transfer – ouch!
• Astrophysical Spectropolarimetry, Trujillo-Bueno et al. (eds) – applications to astronomy