Download presentation

Presentation is loading. Please wait.

Published byArianna Nore Modified over 2 years ago

1
**Polarization of Light: from Basics to Instruments (in less than 100 slides)**

N. Manset CFHT

2
**Polarization of Light: Basics to Instruments**

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 N. Manset / CFHT Polarization of Light: Basics to Instruments

3
**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 N. Manset / CFHT Polarization of Light: Basics to Instruments

4
**Light as an electromagnetic wave**

Part I: Polarization states Light as an electromagnetic wave Light is a transverse wave, an electromagnetic wave N. Manset / CFHT Polarization of Light: Basics to Instruments

5
**Mathematical description of the EM wave**

Part I: Polarization states Mathematical description of the EM wave Light wave that propagates in the z direction: N. Manset / CFHT Polarization of Light: Basics to Instruments

6
**Graphical representation of the EM wave (I)**

Part I: Polarization states Graphical representation of the EM wave (I) One can go from: to the equation of an ellipse (using trigonometric identities, squaring, adding): N. Manset / CFHT Polarization of Light: Basics to Instruments

7
**Graphical representation of the EM wave (II)**

Part I: Polarization states Graphical representation of the EM wave (II) An ellipse can be represented by 4 quantities: size of minor axis size of major axis orientation (angle) sense (CW, CCW) Light can be represented by 4 quantities... N. Manset / CFHT Polarization of Light: Basics to Instruments

8
**Vertically polarized light**

Part I: Polarization states, linear polarization Vertically polarized light If there is no amplitude in x (E0x = 0), there is only one component, in y (vertical). N. Manset / CFHT Polarization of Light: Basics to Instruments

9
**Polarization of Light: Basics to Instruments**

Part I: Polarization states, linear polarization Polarization at 45º (I) If there is no phase difference (=0) and E0x = E0y, then Ex = Ey N. Manset / CFHT Polarization of Light: Basics to Instruments

10
**Polarization of Light: Basics to Instruments**

Part I: Polarization states, linear polarization Polarization at 45º (II) N. Manset / CFHT Polarization of Light: Basics to Instruments

11
**Circular polarization (I)**

Part I: Polarization states, circular polarization 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: N. Manset / CFHT Polarization of Light: Basics to Instruments

12
**Circular polarization (II)**

Part I: Polarization states, circular polarization Circular polarization (II) N. Manset / CFHT Polarization of Light: Basics to Instruments

13
**Circular polarization (III)**

Part I: Polarization states, circular polarization Circular polarization (III) N. Manset / CFHT Polarization of Light: Basics to Instruments

14
**Circular polarization (IV)**

Part I: Polarization states, circular polarization... see it now? Circular polarization (IV) N. Manset / CFHT Polarization of Light: Basics to Instruments

15
**Elliptical polarization**

Part I: Polarization states, elliptical polarization Elliptical polarization Linear + circular polarization = elliptical polarization N. Manset / CFHT Polarization of Light: Basics to Instruments

16
**Unpolarized light (natural light)**

Part I: Polarization states, unpolarized light Unpolarized light (natural light) N. Manset / CFHT Polarization of Light: Basics to Instruments

17
**Polarization of Light: Basics to Instruments**

Part I: Polarization states A cool Applet Electromagnetic Wave Location: N. Manset / CFHT Polarization of Light: Basics to Instruments

18
**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 N. Manset / CFHT Polarization of Light: Basics to Instruments

19
**Stokes parameters A tiny itsy-bitsy little bit of history...**

Part II: Stokes parameters Stokes parameters A 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 N. Manset / CFHT Polarization of Light: Basics to Instruments

20
**Polarization of Light: Basics to Instruments**

Part II: Stokes parameters Stokes parameters (I) The polarization ellipse is only valid at a given instant of time (function of time): To get the Stokes parameters, do a time average (integral over time) and a little bit of algebra... N. Manset / CFHT Polarization of Light: Basics to Instruments

21
**Stokes parameters (II) described in terms of the electric field**

Part II: Stokes parameters Stokes parameters (II) described in terms of the electric field The 4 Stokes parameters are: N. Manset / CFHT Polarization of Light: Basics to Instruments

22
**Stokes parameters (III) described in geometrical terms**

Part II: Stokes parameters Stokes parameters (III) described in geometrical terms N. Manset / CFHT Polarization of Light: Basics to Instruments

23
**Polarization of Light: Basics to Instruments**

Part II: Stokes parameters, Stokes vectors Stokes vector The Stokes parameters can be arranged in a Stokes vector: Linear polarization Circular polarization Fully polarized light Partially polarized light Unpolarized light N. Manset / CFHT Polarization of Light: Basics to Instruments

24
**Pictorial representation of the Stokes parameters**

Part II: Stokes parameters Pictorial representation of the Stokes parameters N. Manset / CFHT Polarization of Light: Basics to Instruments

25
**Stokes vectors for linearly polarized light**

Part II: Stokes parameters, examples Stokes vectors for linearly polarized light LHP light LVP light +45º light -45º light N. Manset / CFHT Polarization of Light: Basics to Instruments

26
**Stokes vectors for circularly polarized light**

Part II: Stokes parameters, examples Stokes vectors for circularly polarized light RCP light LCP light N. Manset / CFHT Polarization of Light: Basics to Instruments

27
**Polarization of Light: Basics to Instruments**

Part II: Stokes parameters (Q,U) to (P,) In the case of linear polarization (V=0): N. Manset / CFHT Polarization of Light: Basics to Instruments

28
**Polarization of Light: Basics to Instruments**

Part II: Stokes parameters, Mueller matrices Mueller matrices If light is represented by Stokes vectors, optical components are then described with Mueller matrices: [output light] = [Muller matrix] [input light] N. Manset / CFHT Polarization of Light: Basics to Instruments

29
**Polarization of Light: Basics to Instruments**

Part II: Stokes parameters, Mueller matrices Mueller calculus (I) Element Element Element 3 I’ = M3 M2 M1 I N. Manset / CFHT Polarization of Light: Basics to Instruments

30
**Polarization of Light: Basics to Instruments**

Part II: Stokes parameters, Mueller matrices Mueller calculus (II) Mueller matrix M’ of an optical component with Mueller matrix M rotated by an angle : M’ = R(- ) M R() with: N. Manset / CFHT Polarization of Light: Basics to Instruments

31
**Polarization of Light: Basics to Instruments**

Part II: Stokes parameters, Jones formalism, not that important here... 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: BUT: Jones formalism can only deal with 100% polarization... N. Manset / CFHT Polarization of Light: Basics to Instruments

32
**Part III: Optical components for polarimetry**

Complex index of refraction Polarizers Retarders N. Manset / CFHT Polarization of Light: Basics to Instruments

33
**Complex index of refraction**

Part III: Optical components Complex index of refraction The index of refraction is actually a complex quantity: 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 N. Manset / CFHT Polarization of Light: Basics to Instruments

34
**Polarization of Light: Basics to Instruments**

Part III: Optical components, polarizers 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. N. Manset / CFHT Polarization of Light: Basics to Instruments

35
**Wire-grid polarizers (I) [dichroism]**

Part III: Optical components, polarizers 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 N. Manset / CFHT Polarization of Light: Basics to Instruments

36
**Wide-grid polarizers (II) [dichroism]**

Part III: Optical components, polarizers Wide-grid polarizers (II) [dichroism] N. Manset / CFHT Polarization of Light: Basics to Instruments

37
**Dichroic crystals [dichroism]**

Part III: Optical components, polarizers Dichroic crystals [dichroism] Dichroic crystals absorb one polarization state over the other one. Example: tourmaline. N. Manset / CFHT Polarization of Light: Basics to Instruments

38
**Polaroids [dichroism]**

Part III: Optical components, polarizers – Polaroids, like in sunglasses! 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. N. Manset / CFHT Polarization of Light: Basics to Instruments

39
**Crystal polarizers (I) [birefringence]**

Part III: Optical components, polarizers 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 N. Manset / CFHT Polarization of Light: Basics to Instruments

40
**Crystal polarizers (II) [birefringence]**

Part III: Optical components, polarizers Crystal polarizers (II) [birefringence] isotropic crystal (sodium chloride) anisotropic (calcite) The 2 output beams are polarized (orthogonally). N. Manset / CFHT Polarization of Light: Basics to Instruments

41
**Crystal polarizers (IV) [birefringence]**

Part III: Optical components, polarizers 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, ... N. Manset / CFHT Polarization of Light: Basics to Instruments

42
**Mueller matrices of polarizers (I)**

Part III: Optical components, polarizers Mueller matrices of polarizers (I) (Ideal) linear polarizer at angle : N. Manset / CFHT Polarization of Light: Basics to Instruments

43
**Mueller matrices of polarizers (II)**

Part III: Optical components, polarizers Mueller matrices of polarizers (II) Linear (±Q) polarizer at 0º: Linear (±U) polarizer at 0º : Circular (±V) polarizer at 0º : N. Manset / CFHT Polarization of Light: Basics to Instruments

44
**Mueller calculus with a polarizer**

Part III: Optical components, polarizers Mueller calculus with a polarizer Input light: unpolarized --- output light: polarized Total output intensity: 0.5 I N. Manset / CFHT Polarization of Light: Basics to Instruments

45
**Polarization of Light: Basics to Instruments**

Part III: Optical components, retarders 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. N. Manset / CFHT Polarization of Light: Basics to Instruments

46
**Polarization of Light: Basics to Instruments**

Part III: Optical components, retarders 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. N. Manset / CFHT Polarization of Light: Basics to Instruments

47
**Polarization of Light: Basics to Instruments**

Part III: Optical components, retarders Half-Wave plate (II) N. Manset / CFHT Polarization of Light: Basics to Instruments

48
**Quarter-Wave plate (I)**

Part III: Optical components, retarders Quarter-Wave plate (I) Retardation of ¼ wave or 90º for one of the polarizations Used to convert linear polarization to elliptical. N. Manset / CFHT Polarization of Light: Basics to Instruments

49
**Quarter-Wave plate (II)**

Part III: Optical components, retarders Quarter-Wave plate (II) Special case: incoming light polarized at 45º with respect to the retarder’s axis Conversion from linear to circular polarization (vice versa) N. Manset / CFHT Polarization of Light: Basics to Instruments

50
**Mueller matrix of retarders (I)**

Part III: Optical components, retarders Mueller matrix of retarders (I) Retarder of retardance and position angle : N. Manset / CFHT Polarization of Light: Basics to Instruments

51
**Mueller matrix of retarders (II)**

Part III: Optical components, retarders Mueller matrix of retarders (II) Half-wave oriented at ±45º Half-wave oriented at 0º or 90º N. Manset / CFHT Polarization of Light: Basics to Instruments

52
**Mueller matrix of retarders (III)**

Part III: Optical components, retarders Mueller matrix of retarders (III) Quarter-wave oriented at ±45º Quarter-wave oriented at 0º N. Manset / CFHT Polarization of Light: Basics to Instruments

53
**Mueller calculus with a retarder**

Part III: Optical components, retarders Mueller calculus with a retarder Input light linear polarized (Q=1) Quarter-wave at +45º Output light circularly polarized (V=1) N. Manset / CFHT Polarization of Light: Basics to Instruments

54
**(Back to polarizers, briefly) Circular polarizers**

Part III: Optical components, polarizers (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. N. Manset / CFHT Polarization of Light: Basics to Instruments

55
**Achromatic retarders (I)**

Part III: Optical components, retarders 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 N. Manset / CFHT Polarization of Light: Basics to Instruments

56
**Achromatic retarders (II)**

Part III: Optical components, retarders Achromatic retarders (II) = º not very achromatic! = º much better! N. Manset / CFHT Polarization of Light: Basics to Instruments

57
**Retardation on total internal reflection**

Part III: Optical components, retarders 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 N. Manset / CFHT Polarization of Light: Basics to Instruments

58
**Polarization of Light: Basics to Instruments**

Part III: Optical components, retarders Fresnel rhombs Quarter-wave and half-wave rhombs are achieved with 2 or 4 reflections N. Manset / CFHT Polarization of Light: Basics to Instruments

59
**Polarization of Light: Basics to Instruments**

Part III: Optical components, retarders 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... N. Manset / CFHT Polarization of Light: Basics to Instruments

60
**Polarization of Light: Basics to Instruments**

Part IV: Polarimeters Polaroid-type polarimeters Dual-beam polarimeters N. Manset / CFHT Polarization of Light: Basics to Instruments

61
**Polaroid-type polarimeter for linear polarimetry (I)**

Part IV: Polarimeters, polaroid-type Polaroid-type polarimeter for linear polarimetry (I) Use a linear polarizer (polaroid) to measure linear polarization ... [another cool applet] Location: Polarization percentage and position angle: N. Manset / CFHT Polarization of Light: Basics to Instruments

62
**Polaroid-type polarimeter for linear polarimetry (II)**

Part IV: Polarimeters, polaroid-type Polaroid-type polarimeter for linear polarimetry (II) Move the polaroid to 2 positions, 0º and 45º (to measure Q, then U) Advantage: very simple to make Disadvantage: half of the light is cut out Other disadvantages: non-simultaneous measurements, cross-talk... N. Manset / CFHT Polarization of Light: Basics to Instruments

63
**Polaroid-type polarimeter for circular polarimetry**

Part IV: Polarimeters, polaroid-type Polaroid-type polarimeter for 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 N. Manset / CFHT Polarization of Light: Basics to Instruments

64
**Dual-beam polarimeters Principle**

Part IV: Polarimeters, dual-beam type Dual-beam polarimeters Principle 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 N. Manset / CFHT Polarization of Light: Basics to Instruments

65
**Dual-beam polarimeters Switching beams**

Part IV: Polarimeters, dual-beam type Dual-beam polarimeters Switching beams 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 N. Manset / CFHT Polarization of Light: Basics to Instruments

66
**Dual-beam polarimeters Switching beams by rotating the prism**

Part IV: Polarimeters, dual-beam type Dual-beam polarimeters Switching beams by rotating the prism rotate by 180º N. Manset / CFHT Polarization of Light: Basics to Instruments

67
**Dual-beam polarimeters Switching beams using a ½ wave plate**

Part IV: Polarimeters, dual-beam type Dual-beam polarimeters Switching beams using a ½ wave plate Rotated by 45º N. Manset / CFHT Polarization of Light: Basics to Instruments

68
**Polarization of Light: Basics to Instruments**

Part IV: Polarimeters, dual-beam type Dual-beam polarimeter for circular polarization - Wollaston and quarter-wave plate The measurements V/I is: Switch the beams to compensate the gain effects N. Manset / CFHT Polarization of Light: Basics to Instruments

69
**A real circular polarimeter Semel, Donati, Rees (1993)**

Part IV: Polarimeters, example of circular polarimeter A real circular polarimeter Semel, Donati, Rees (1993) Quarter-wave plate, rotated at -45º and +45º Analyser: double calcite crystal N. Manset / CFHT Polarization of Light: Basics to Instruments

70
**Polarization of Light: Basics to Instruments**

Part IV: Polarimeters, example of circular polarimeter A real circular polarimeter free 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: N. Manset / CFHT Polarization of Light: Basics to Instruments

71
**Polarization of Light: Basics to Instruments**

Part IV: Polarimeters, example of circular polarimeter A real circular polarimeter free from gain and atmospheric transmission variation effects Build a ratio of measured signals which is free of gain and variable atmospheric transmission effects: average of the 2 measurements N. Manset / CFHT Polarization of Light: Basics to Instruments

72
**Polarimeters - Summary**

Part IV: Polarimeters, summary 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 N. Manset / CFHT Polarization of Light: Basics to Instruments

73
**Polarization of Light: Basics to Instruments**

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 N. Manset / CFHT Polarization of Light: Basics to Instruments

74
**ESPaDOnS: circular polarimetry**

Part V: ESPaDOnS, circular polarimetry mode 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 N. Manset / CFHT Polarization of Light: Basics to Instruments

75
**ESPaDOnS: circular polarimetry of circular polarization**

Part V: ESPaDOnS, circular polarimetry mode ESPaDOnS: circular polarimetry of circular polarization half-wave 22.5º positions flips polarization gain, transmission quarter-wave fixed circular to linear analyzer N. Manset / CFHT Polarization of Light: Basics to Instruments

76
**ESPaDOnS: circular polarimetry of (unwanted) linear polarization**

Part V: ESPaDOnS, circular polarimetry mode ESPaDOnS: circular polarimetry of (unwanted) linear polarization analyzer half-wave 22.5º positions gain, transmission quarter-wave fixed linear to elliptical 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 N. Manset / CFHT Polarization of Light: Basics to Instruments

77
**ESPaDOnS: linear polarimetry**

Part V: ESPaDOnS, linear polarimetry ESPaDOnS: linear polarimetry Half-Wave rhombs positioned at 22.5º increments Quarter-Wave fixed N. Manset / CFHT Polarization of Light: Basics to Instruments

78
**ESPaDOnS: linear polarimetry**

Part V: ESPaDOnS, linear polarimetry 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 N. Manset / CFHT Polarization of Light: Basics to Instruments

79
**Polarization of Light: Basics to Instruments**

Part V: ESPaDOnS, summary 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 N. Manset / CFHT Polarization of Light: Basics to Instruments

80
**Polarization of Light: Basics to Instruments**

N. Manset / CFHT Polarization of Light: Basics to Instruments

81
**Credits for pictures and movies**

Christoph Keller’s home page – his 5 lectures “Basic Polarisation techniques and devices”, Meadowlark Optics Inc. Optics, E. Hecht and Astronomical Polarimetry, J. Tinbergen Planets, Stars and Nebulae Studied With Photopolarimetry, T. Gehrels Circular polarization movie Unpolarized light movie Reflection of wave ESPaDOnS web page and documents N. Manset / CFHT Polarization of Light: Basics to Instruments

82
**References/Further reading On the Web**

Very short and quick introduction, no equation Easy fun page with Applets, on polarizing filters Polarization short course “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 N. Manset / CFHT Polarization of Light: Basics to Instruments

83
**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 N. Manset / CFHT Polarization of Light: Basics to Instruments

84
**References/Further reading Astronomy, 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 N. Manset / CFHT Polarization of Light: Basics to Instruments

85
**References/Further reading Astronomy, advanced**

Introduction to Spectropolarimetry, J.C. del Toro Iniesta – radiative transfer – ouch! Astrophysical Spectropolarimetry, Trujillo-Bueno et al. (eds) – applications to astronomy N. Manset / CFHT Polarization of Light: Basics to Instruments

Similar presentations

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google