arX

iv:1

401.

1911

v1 [

astr

o-ph

.IM

] 9

Jan

201

4

Journal of The Korean Astronomical Society (Preprint - no DOI assigned)

00: 1 ∼ 25, 2013 December ISSN:1225-4614

Preprint prepared by the author http://jkas.kas.org

POLARIZATION AND POLARIMETRY: A REVIEW

Sascha Trippe

Department of Physics and Astronomy, Seoul National University, Seoul 151-742, South KoreaE-mail: trippe@astro.snu.ac.kr

(Received 30 August 2013; Revised 17 December 2013; Accepted 28 December 2013)

ABSTRACT

Polarization is a basic property of light and is fundamentally linked to the internal geometry of a sourceof radiation. Polarimetry complements photometric, spectroscopic, and imaging analyses of sources ofradiation and has made possible multiple astrophysical discoveries. In this article I review (i) the phys-ical basics of polarization: electromagnetic waves, photons, and parameterizations; (ii) astrophysicalsources of polarization: scattering, synchrotron radiation, active media, and the Zeeman, Goldreich-Kylafis, and Hanle effects, as well as interactions between polarization and matter (like birefringence,Faraday rotation, or the Chandrasekhar-Fermi effect); (iii) observational methodology: on-sky geom-etry, influence of atmosphere and instrumental polarization, polarization statistics, and observationaltechniques for radio, optical, and X/γ wavelengths; and (iv) science cases for astronomical polarime-try: solar and stellar physics, planetary system bodies, interstellar matter, astrobiology, astronomicalmasers, pulsars, galactic magnetic fields, gamma-ray bursts, active galactic nuclei, and cosmic microwavebackground radiation.

Key words : Polarization — Methods: polarimetric — Radiation mechanisms: general

Contents

1. INTRODUCTION . . . . . . . . . . . . . . . . . 2

2. PHYSICAL BASICS . . . . . . . . . . . . . . . . 2

2.1. Electromagnetic Waves . . . . . . . . . . . 2

2.1.1. Electric Field Vectors . . . . . . 2

2.1.2. Elliptical Polarization . . . . . . 2

2.1.3. Linear Polarization . . . . . . . . 2

2.1.4. Circular Polarization . . . . . . . 3

2.1.5. Macroscopic Polarization . . . . 3

2.2. Photons . . . . . . . . . . . . . . . . . . . 3

2.3. Parameterizations . . . . . . . . . . . . . . 4

2.3.1. Jones Calculus . . . . . . . . . . 4

2.3.2. Stokes Parameters . . . . . . . . 5

2.3.3. Muller Formalism . . . . . . . . . 6

3. POLARIGENESIS . . . . . . . . . . . . . . . . . 6

3.1. Scattering Polarization . . . . . . . . . . . 6

3.1.1. Microscopic Scattering . . . . . . 6

3.1.2. Scattering by Dust . . . . . . . . 7

3.2. Dichroic Media . . . . . . . . . . . . . . . 7

3.3. Optically Active Media . . . . . . . . . . . 8

3.4. Synchrotron Radiation . . . . . . . . . . . 8

3.5. Zeeman Effect . . . . . . . . . . . . . . . . 9

3.6. Goldreich–Kylafis Effect . . . . . . . . . . 10

3.7. Hanle Effect . . . . . . . . . . . . . . . . . 10

3.8. Interactions of Polarization and Matter . . 11

3.8.1. Partial Reflection . . . . . . . . . 11

3.8.2. Birefringence . . . . . . . . . . . 11

3.8.3. Faraday Rotation . . . . . . . . . 12

3.8.4. Faraday Depolarization . . . . . 12

3.8.5. Polarization Conversion . . . . . 13

3.8.6. Chandrasekhar-Fermi Effect . . . 13

4. OBSERVATIONS . . . . . . . . . . . . . . . . . 13

4.1. Sky Projection . . . . . . . . . . . . . . . 13

4.2. Terrestrial Atmosphere . . . . . . . . . . . 13

4.3. Instrumental Polarization . . . . . . . . . 14

4.4. Polarization Statistics . . . . . . . . . . . 14

4.5. Radio Observations . . . . . . . . . . . . . 14

4.6. Optical Observations . . . . . . . . . . . . 15

4.7. X and γ Ray Observations . . . . . . . . . 16

5. SCIENCE CASES . . . . . . . . . . . . . . . . . 17

5.1. Solar and Stellar Physics . . . . . . . . . . 17

5.2. Planetary System Bodies . . . . . . . . . . 17

5.2.1. Solid Surfaces . . . . . . . . . . . 17

5.2.2. Atmospheres . . . . . . . . . . . . 18

5.3. Interstellar Matter . . . . . . . . . . . . . 18

5.4. Astrobiology . . . . . . . . . . . . . . . . 19

5.5. Astronomical Masers . . . . . . . . . . . . 19

5.6. Pulsars . . . . . . . . . . . . . . . . . . . . 19

5.7. Active Galactic Nuclei . . . . . . . . . . . 20

5.8. Galactic Magnetic Fields . . . . . . . . . . 20

5.9. Gamma Ray Bursts . . . . . . . . . . . . 21

5.10. Cosmic Background Radiation . . . . . . 21

6. CONCLUSIONS . . . . . . . . . . . . . . . . . . 22

Acknowledgments . . . . . . . . . . . . . . . . . . . . 22

References . . . . . . . . . . . . . . . . . . . . . . . . . 22

– 1 –

2 S. TRIPPE

1. INTRODUCTION

“To many astrophysicists, stellar polarimetry is aCinderella subject considered as being so insignif-icant and, at the same time, being so esoteric asto be ignored and left alone. [...] There can beno doubt, however, that the study of polarizationwithin astronomy has a strong role to play either inits own right, or in combination with other obser-vational tools, as a diagnostic for understanding thebehaviour of celestial sources.”

— Clarke (2010), p.XIII

Historically, the study of polarized light began withthe discovery of birefringence in crystals by Eras-mus Bartholinus and its subsequent interpretation byChristian Huygens around the year 1670 (Brosseau1998). Astronomical observations of polarized lightcommenced in the middle of the 19th century; someof the earliest publications treat the linear polariza-tion of sun light reflected by the moon (Secchi 1860)and the linear polarization of the light from the solarcorona (Edlund 1860). Subsequently, the field of po-larimetry evolved closely with the technical progress ofobservational techniques in general: from optical po-larimetry to radio polarimetry in the 1940s (Wilson,Rohlfs & Huttemeister 2010) and eventually to space-based X-ray polarimetry in the 1970s (Weisskopf et al.1978).

Polarization is a fundamental property of electro-magnetic radiation. It is a rich source of information onthe physical properties – magnetic fields, internal con-ditions, particle densities, et cetera – of astronomicalobjects. Polarimetric observations complement analy-sis methods based on photometry as well as on spectral(spectroscopy) or angular (mapping, imaging) resolu-tion. Accordingly, polarimetry has contributed sub-stantially to the progress of astronomy. Milestoneshave been, for example:

• the mapping of solar and stellar magnetic fields(e.g., Schrijver & Zwaan 2000);

• the characterization of the surface composition ofsolar system bodies (e.g., Bowell & Zellner 1974);

• the discovery of synchrotron radiation from astro-nomical objects (Oort & Walraven 1956);

• the discovery and characterization of large-scale(many kpc) galactic magnetic fields (e.g., Kulsrud& Zweibel 2008);

• the analysis of the polarization modes of thecosmic microwave background (e.g., Kovac et al.2002).

This review provides a broad overview on the the-ory and phenomenology of polarization in astronomy.It covers (i) the physical basics of polarized radiation,(ii) sources of astrophysical polarization, (iii) conceptsand methods of astronomical polarimetry, and (iv) as-trophysical science cases.

2. PHYSICAL BASICS

2.1 Electromagnetic Waves

2.1.1 Electric Field Vectors

The concept of electromagnetic waves derives from non-zero solutions of Maxwell’s equations in vacuum, mean-ing here specifically the absence of electric charges (e.g.,Landau & Lifschitz 1997; Jackson 1999). Followingcommon conventions, we regard the electric field∗ Eof an electromagnetic wave traveling in z direction inEuclidean coordinates (x, y, z) with speed of light c.Accordingly, we have – in trigonometric notation –

E(t, z) = E(0, 0) cos(ωt− kz − φ) (1)

where t is the time, ω denotes the angular frequency,k = ω/c is the absolute value of the wave vector, and φdenotes an arbitrary phase. As the electric field vectoris perpendicular to z, we can decompose E(t, z) intoits x and y components. For simplicity, we regard thelocation z = 0 only, i.e., we regard the location of E(t)in the xy plane. The x and y components are then

Ex(t) = Ex(0) cos(ωt− φ1) (2)

Ey(t) = Ey(0) cos(ωt− φ2) .

Here, φ1,2 denote two – a priori arbitrary – phases.In addition, we denote the angle between E(t) andthe positive x axis – the polarization angle, countedin counterclockwise direction – with χ. The polar-ization of the wave is given by the relative values ofEx(0), Ey(0), φ1, and φ2 (Rybicki & Lightman 1979;Huard 1997; Landau & Lifschitz 1997; Jackson 1999;Born & Wolf 1999; Goldstein 2003).

2.1.2 Elliptical Polarization

In general, the tip of the electric field vector follows anelliptical trajectory in the xy plane; accordingly, thelight is denoted as elliptically polarized. The orientationof the ellipse in the xy plane is constant in time; thepolarization angle χ corresponds to the angle betweenthe positive x axis and the semi-major axis of the ellipse(counted in counterclockwise direction).

Elliptical polarization is the most general state ofpolarization of an electromagnetic wave. Linear po-larization occurs if the polarization ellipse degeneratesinto a line. Circular polarization corresponds to the –opposite – special case of the ellipse degenerating intoa circle.

2.1.3 Linear Polarization

For the case φ1 = φ2, using here specifically φ1 = φ2 =0 without loss of generality, we have

∗The magnetic field is perpendicular to the direction of travel andto the electric field. The amplitude of the magnetic field, B, isrelated to the amplitude of the electric field, E, like B = E/c.

POLARIZATION AND POLARIMETRY 3

Ex(t) = Ex(0) cos(ωt) (3)

Ey(t) = Ey(0) cos(ωt) .

The orientation of E then depends only on the magni-tudes of Ex(0) and Ey(0) and is independent of time;the angle χ is constant. The radiation is linearly polar-ized with polarization angle χ ∈ [0, π]. The orientationof the plane wherein the wave is located – the planeof polarization – as given by χ has an orientation butno direction; accordingly, the location of linear polar-ization in the xy plane is not a vector. In physics andastronomy, the x and y components of linearly polar-ized light are commonly identified with horizontal (H)and vertical (V) polarizations, respectively.

2.1.4 Circular Polarization

In case of a relative phase shift φ2 = φ1 ± π/2, usinghere specifically φ1 = 0 without loss of generality, andEx(0) = Ey(0), we have

Ex(t) = Ex(0) cos(ωt) (4)

Ey(t) = ±Ey(0) sin(ωt) .

The tip of the electric field vector moves circularly inthe xy plane with angular frequency ω: the radiationis circularly polarized. The sign of Ey(t) – which de-rives from the relative phase – determines the senseof the motion of E. A positive sign, corresponding tocounterclockwise motion, is commonly referred to asright-hand circular (RHC) polarization. Accordingly,a negative sign, corresponding to clockwise motion, isdenoted as left-hand circular (LHC) polarization.

2.1.5 Macroscopic Polarization

An individual electromagnetic wave is necessarily po-larized as described above (microscopic polarization).Astrophysical observations do not deal with individualwaves but with radiation that is a (almost always inco-herent) superposition of a very large number of elemen-tary electromagnetic waves. Accordingly, astronomicalobservations are sensitive to the macroscopic polariza-tion of light. For most physical systems, all orientationsof the electric field vectors from the elementary emit-ters are equally probable – there is, a priori, no reasonto expect a macroscopic polarization of light. Confus-ingly, unpolarized light is therefore sometimes referredto as “natural” light.

A (macroscopic) polarization signal (Fowles 1975;Rybicki & Lightman 1979; Mandel & Wolf 1995; Born& Wolf 1999) can occur whenever the internal geome-try of the source of radiation or the properties of theinterstellar medium prefer a certain orientation of elec-tric field vectors. In those cases – discussed in detail in§ 3 – the light becomes partially polarized with a degreeof polarization

mP =IPI

∈ [0, 1] (5)

where I denotes the total intensity of the light andIP denotes the intensity of polarized light.† Inten-sity I and amplitude of electric field E are related likeI ∝ E2. The total polarization state of radiation canbe described as a superposition of linear and circularpolarization (cf. § 2.1.2–2.1.4); accordingly, we can de-fine separate degrees of linear and circular polarization.The degree of linear polarization is given by

mL =ILI

∈ [0, 1] (6)

where IL denotes the intensity of linearly polarizedlight. The degree of circular polarization is given by

mC =ICI

∈ [−1, 1] (7)

where IC denotes the intensity of circularly polar-ized light. The sign of IC and thus mC depends onthe orientation of the polarization. By convention,the positive (negative) sign is assigned to light withIRHC−ILHC > 0 (< 0); here, IRHC and ILHC denote theintensities of right-hand circularly and left-hand circu-larly polarized light, respectively (Hamaker & Bregman1996).

2.2 Photons

The wave-particle dualism of light (Einstein 1905) im-plies that polarization is a property of individual pho-tons; each photon can be assigned an individual stateof polarization (Dirac 1958).

Using the standard bra–ket notation for quantumstates,‡ we may write two arbitrary photon stateslike |X〉 and |Y 〉. These two states are orthogonal if〈X |Y 〉 = 0. We may further assume that all states arenormalized, meaning 〈X |X〉 = 1 for arbitrary X . Thebracket product 〈X |Y 〉 is the probability amplitude ofthe event “The system in state X is also in state Y ”,and |〈X |Y 〉|2 ∈ [0, 1] is the corresponding probability;|...| denotes the absolute value of the enclosed function.

For the specific case of photon polarization (Bachor& Ralph 2004), we have to consider photon states corre-sponding to horizontal linear polarization |H〉, verticallinear polarization |V 〉, right-hand circular polarization

†In the astronomical literature, degrees of polarization are com-monly quoted in units of per cent (%).

‡One may picture bras 〈X| as complex row vectors and kets |X〉as complex column vectors of equal – potentially infinite – di-mension. A bra transforms into a ket by transposition plus com-plex conjugation (Hermitian conjugate). Accordingly, the prod-uct |X〉〈Y | corresponds to a complex matrix, the product 〈X|Y 〉corresponds to a complex scalar (e.g. Dirac 1958).

4 S. TRIPPE

|R〉, and left-hand circular polarization |L〉.§ Evidently,|H〉 and |V 〉 on the one hand and |R〉 and |L〉 on theother hand are mutually exclusive, meaning

〈H |V 〉 = 〈V |H〉 = 0 (8)

〈R|L〉 = 〈L|R〉 = 0 .

Due to normalization, we further have

〈H |H〉 = 〈V |V 〉 = 1 (9)

〈R|R〉 = 〈L|L〉 = 1 .

In addition, we have to note the relations betweenlinear and circular polarization states. A standardtool employed in quantum optics is a polarizing beam-splitter that is sensitive to linear polarization: in a(thought) laboratory experiment, incident photons instate |H〉 are sent into one direction, photons in state|V 〉 into another. The beam-splitter is insensitive tocircular polarization; an incident photon in state |R〉or |L〉 is sent into either direction with equal probabil-ity of 50%. Accordingly, one finds

|〈R|V 〉|2 = |〈R|H〉|2 = 0.5 (10)

|〈L|V 〉|2 = |〈L|H〉|2 = 0.5 .

Combining the information from Eqs. 8, 9, and 10,it turns out (Bachor & Ralph 2004) that circular po-larization states can be expressed as superpositions oflinear polarization states like

|R〉 =1√2(|H〉+ i|V 〉) (11)

|L〉 =1√2(|H〉 − i|V 〉) .

with i being the imaginary unit. An arbitrary polar-ization state |P 〉 can be expressed as

|P 〉 = a |H〉+ eiφ b |V 〉 (12)

with φ, a, and b being real numbers, and a2 + b2 = 1.The representation given by Eq. 12 is not unique; anypair of orthogonal states can be used as base vectors.

In analogy to the case of electromagnetic waves, themacroscopic polarization of light is given by the super-position of a large number of photons with individualmicroscopic polarization states. The discussion pro-vided in § 2.1.5 is equally valid for waves and particles.

§Only up to this sub-section V denotes vertical linear polarization.In the remainder of this paper, V denotes the correspondingStokes parameter (§ 2.3.2).

2.3 Parameterizations

2.3.1 Jones Calculus

In Eq. 2, I introduced the components of the electricfield in trigonometric notation for convenience. Like-wise, the electric field can be given in complex expo-nential notation as

Ex(t) = Ex(0)ei(ωt−φ1) (13)

Ey(t) = Ey(0)ei(ωt−φ2) .

One may now define (Jones 1941; Fowles 1975; Huard1997; Goldstein 2003) the Jones vector

e ≡[

Ex(0) eiφ1

Ey(0) eiφ2

]

(14)

that expresses amplitudes and phases of the electricfield in vector form. A convenient – usually not normal-ized – form of the Jones vector is achieved by express-ing the components in units of the amplitude of one ofthem. Linearly polarized waves may be expressed like

ex =

[

10

]

; ey =

[

01

]

(15)

where ex and ey denote waves polarized in x and y di-rection, respectively. Likewise, by exploiting the iden-tity ±i = e±iπ/2, circularly polarized waves can be ex-pressed like

eL =

[

1i

]

; eR =

[

1−i

]

(16)

where eL and eR denote left-hand and right-hand cir-cular polarization, respectively. The result of a super-position of electric fields is given by the sum of theappropriate Jones vectors. A noteworthy example is

[

1i

]

+

[

1−i

]

=

[

20

]

= 2

[

10

]

(17)

which demonstrates that a linearly polarized wave canbe expressed as the sum of a left-hand and a right-handcircularly polarized wave with equal amplitudes – asdemanded by equivalence with Eqs. 11, 12. In general,any polarization state can be expressed as combinationof two Jones vectors e1, e2 that represent orthogonalpolarizations, meaning

e1e∗2 = 0 (18)

where the operator ∗ denotes complex conjugation.

A (linear) modification of the polarization state of awave is expressed by a 2×2 Jones matrix J that relatesinput wave e and output wave e′ like

e′ = Je . (19)

POLARIZATION AND POLARIMETRY 5

Fig. 1.— Fundamental coordinates and geometries of theStokes parameters Q and U ; ψ denotes the parallactic angle.

In optics, Jones matrices are commonly employedto characterize polarizing optical elements or trainsthereof. Successive modifications 1, 2, ..., n of the polar-ization state can be written in terms of a single Jonesmatrix. This matrix is given by the product of the ma-trices corresponding to the individual optical elementslike

J = Jn Jn−1 ...J2 J1 . (20)

Simple examples for Jones matrices of polarizing opti-cal elements are

Jx =

[

1 00 0

]

; JR =1

2

[

1 i−i 1

]

(21)

where Jx describes a linear polarizer with the x axisbeing the transmission axis, and JR corresponds to aright-hand circular polarizer.

2.3.2 Stokes Parameters

We understand from §§ 2.1.2–2.1.4 that the polariza-tion state of an electromagnetic wave can be charac-terized by means of three independent parameters: theamplitudes Ex(0) and Ey(0) and the phase differenceδ = φ2−φ1. In general, astronomical observations dealwith light intensities rather than with field amplitudes;accordingly, it is convenient to quantify polarizationvia characteristic intensities. For reasons that are go-ing to be evident soon, those characteristic intensities¶

are the Stokes parameters‖

¶In the convention we adopt here, intensity I and field amplitudeE are related like I = E2, whereas in SI units I = ε0cE2 withvacuum permittivity ε0. Accordingly, our convention implies arescaling of electric fields like E −→ E′ =

√ε0cE

‖Occasionally, the parameters I, Q, U , V are also denoted as S0,S1, S2, S3, respectively.

I = 〈E2x〉+ 〈E2

y〉 (22)

Q = 〈E2x〉 − 〈E2

y〉U = 2 〈ExEy cos δ〉V = 2 〈ExEy sin δ〉

(Stokes 1852) where Ex,y ≡ Ex,y(t) for simplicity,and 〈...〉 denotes the time average of the enclosedparameters taken over times much larger than 2π/ω(Huard 1997; Born & Wolf 1999; Goldstein 2003;Thompson, Moran & Swenson 2004; Wilson, Rohlfs &Huttemeister 2010). Notably, the parameters Q, U ,and V can take negative values.

By construction, I is the intensity of the wave. Theparameter Q quantifies a difference in the intensitiesin x and y, thus providing information on linear po-larization. The parameter U quantifies the differencebetween the two field components diagonal – at anglesof 45 and 135 counted from the positive x axis – tothe x and y coordinates, thus likewise probing linearpolarization. Finally, the parameter V corresponds tothe circularly polarized intensity. An illustration of thefundamental geometries is provided in Fig. 1.

For individual waves – microscopic polarization –the Stokes parameters are related via

I2 = Q2 + U2 + V 2, (23)

thus reducing the number of free parameters to three– as expected. As I is a constant, each polarizationstate of a wave corresponds to a point on a sphere, thePoincare sphere (Poincare 1892). In case of macro-scopic polarization, radiation with intensity I is formedfrom superposition of many elementary emitters; polar-ization is averaged out at least partially. Accordingly,Eq. 23 breaks down to

I2P = Q2 + U2 + V 2 (24)

with IP ≤ I being the polarized intensity; the numberof free parameters increases to four, the fourth param-eter being I. Using the definitions provided by Eqs. 5,6, and 7 as well as the definition of the polarization an-gle χ, the Stokes parameters relate to the parametersof linear polarization like

mL =

√

Q2 + U2

I∈ [0, 1] (25)

χ =1

2atan2

(

U

Q

)

∈ [0, π]

where atan2 denotes the quadrant-preserving arc tan-gent; mL and χ correspond to the length and the ori-entation of a vector centered at the origin of a planespanned by Q and U . The degree of circular polariza-tion relates to Stokes V like

6 S. TRIPPE

mC =V

I∈ [−1, 1] . (26)

When using complex exponential notation for theelectric field (Eq. 13), we find an alternative – thoughequivalent – definition of the Stokes parameters:

I = 〈ExE∗x〉+ 〈EyE

∗y〉 (27)

Q = 〈ExE∗x〉 − 〈EyE

∗y〉

U = 〈ExE∗y〉+ 〈EyE

∗x〉

V = −i[

〈ExE∗y〉 − 〈EyE

∗x〉]

.

As usual, the operator ∗ denotes complex conjugation.It is straightforward to see that this definition is equiv-alent to Eq. 22 (e.g. Hamaker & Bregman 1996).

In astronomy, the Euclidean coordinates we use hereare conventionally defined such that the x axis points tothe north, the y axis points to the east, and the z axispoints toward the observer. Accordingly, the polariza-tion angle χ is counted from north to east. The orien-tations of circular polarizations as defined in § 2.1.4 –RHC and LHC – are preserved.

As I indicated before in § 2.2 and § 2.3.1, the po-larization state of photons as well as electromagneticwaves can be described using left- and right-hand cir-cular polarization components with amplitudes EL,R

and phases φL,R as base. Using δ′ = φR − φL, we find(e.g. Cenacchi et al. 2009) for the Stokes parametersin trigonometric notation

I = 〈E2R〉+ 〈E2

L〉 (28)

Q = 2 〈EREL cos δ′〉U = 2 〈EREL sin δ′〉V = 〈E2

R〉 − 〈E2L〉

and in complex exponential notation

I = 〈ERE∗R〉+ 〈ELE

∗L〉 (29)

Q = 〈ERE∗L〉+ 〈ELE

∗R〉

U = −i [〈ERE∗L〉 − 〈ELE

∗R〉]

V = 〈ERE∗R〉 − 〈ELE

∗L〉 .

It is important to note that – in general – the treat-ment of combinations of polarized signals requires theuse of the Stokes parameters: only intensities can beadded or subtracted in a straightforward manner – de-grees of polarization or polarization angles cannot (cf.,e.g., Heiles 2002).

2.3.3 Muller Formalism

The Muller formalism (Muller 1948; Hamaker, Breg-man & Sault 1996; Huard 1997; Goldstein 2003) ex-

tends and combines Jones calculus and Stokes formal-ism. The Stokes parameters can be expressed as a four-dimensional vector, the Stokes vector

S =

IQUV

≡ TC (30)

with

C =

〈ExE∗x〉

〈ExE∗y〉

〈EyE∗x〉

〈EyE∗y〉

; T =

1 0 0 11 0 0 −10 1 1 00 −i i 0

. (31)

The vectorC is commonly referred to as coherency vec-tor ; evidently, this notation is equivalent to Eq. 27.

Modifications of the polarization state can be ex-pressed as modifications of the Stokes vector like

S′ = MS (32)

where M is a 4×4 Muller matrix. Simple examples are

Mref =

1 0 0 00 1 0 00 0 −1 00 0 0 −1

(33)

and

Mrot =

1 0 0 00 cos 2β sin 2β 00 − sin 2β cos 2β 00 0 0 1

(34)

that correspond to a reflection at a mirror and rotationby an angle β, respectively.

Else than the Jones calculus, the Muller formalismcan describe unpolarized light as well as depolarization,i.e. a reduction of the degree of (total) polarizationmP .

3. POLARIGENESIS

As I discussed briefly in § 2.1.5, the occurrence of amacroscopic polarization of light – the polarigenesis –is intimately linked to the internal symmetry of thephysical system under consideration. Macroscopic po-larization requires that the internal structure of thesource of light is anisotropic. We can identify a varietyof astrophysical sources of polarized light that I discussin the following.

3.1 Scattering Polarization

3.1.1 Microscopic Scattering

Microscopic scattering processes – meaning the scatter-ing of a photon at a free electric charge like an electron,

POLARIZATION AND POLARIMETRY 7

Fig. 2.— The geometry of scattering polarization. Anunpolarized light ray with linear polarization componentsEx,y propagates in positive z direction. The ray is scatteredby an angle θ and continues to propagate in the yz plane(shaded). From the point of view of an observer of thescattered light, the polarization components perpendicularto the direction of propagation are E′

x = Ex and E′y =

Ey cos θ, respectively: the light is linearly polarized.

an atom, or a molecule – lead to a characteristic linearpolarization of the scattered light. Notably, the follow-ing geometry argument holds for a variety of scatteringprocesses like Thomson scattering, Compton scattering,Rayleigh scattering, fluorescence, or Raman scatteringregardless of the different underlying physical mecha-nisms. From now on, I assume incident unpolarizedlight propagating in z direction with electric field com-ponents Ex,y in x and y directions. Upon interactionwith a charge, atom, or molecule, the light is scatteredand continues to propagate in the yz plane at an angleθ to the z axis.

From the point of view of an observer of the scat-tered light, the light intensities in x direction are thesame before and after the scattering, i.e. E′2

x = E2x,

with the prime denoting the scattered light. The ycomponent transforms like E′2

y = E2y cos

2 θ, meaning

the intensity is reduced by a factor cos2 θ. This im-plies that the scattered light is linearly polarized witha degree of polarization

mL =I ′x − I ′yI ′x + I ′y

=1− cos2 θ

1 + cos2 θ∈ [0, 1] (35)

with I ′x,y denoting the observed intensities (after thescattering); I provide an illustration in Fig. 2.

3.1.2 Scattering by Dust

Dust grains are an important ingredient of interstellarmatter (Dyson & Williams 1997; Kwok 2007). Grainsizes are typically on the order of a micrometer, approx-imately corresponding to the wavelengths of optical to

infrared light. A quantitative description of scatter-ing of light by dust is provided by Mie’s theory thatassumes scattering by small spherical particles. Fromgeometrical arguments equivalent to those presented in§ 3.1.1 one finds that initially unpolarized incident lightbecomes linearly polarized by dust scattering; the de-gree of polarization is given by an expression equivalentto Eq. 35 (Born & Wolf 1999).

The derivation of Eq. 35 assumes that the incidentlight is propagating along a single well-defined direc-tion. In clouds of interstellar matter, this is usuallynot the case: dust clouds tend to be optically thick,meaning that incident light experiences multiple scat-tering, absorption, and re-emission events, making theradiation field within the cloud isotropic. In this case,linear polarization can occur if (i) the dust grains haveelongated (cylindrical, ellipsoidal) shapes, and (ii) thegrains are oriented collectively along a preferred direc-tion by magnetic fields. The absorption of radiationby the dust becomes a function of orientation relativeto the magnetic field – resulting in linear polarization.Empirically, it has been found (Serkowski, Mathewson& Ford 1975; Draine 2003) that at optical to near-infrared wavelengths the linear polarization is

mL

mmaxL

≈ exp

[

−1.15 ln2(

λmax

λ

)]

(36)

scaled by the degree of linear polarization at a referencewavelength,

mmaxL ∼< 0.03A(λmax) (37)

where λ is the wavelength, λmax ≈ 550nm, A is theextinction in units of photometric magnitudes, and lndenotes the logarithm to base e; this relation is com-monly referred to as Serkowski’s law.

3.2 Dichroic Media

Dichroism is a further effect of the electric anisotropyof certain materials. Here the attenuation of light dueto absorption by the material is anisotropic. Assum-ing initially unpolarized light, one component of thewave experiences stronger attenuation than the otherone; the light becomes polarized.∗∗ Depending on ifthe difference in absorption affects the linear or circularwave components, the medium is referred to as lineardichroic or circular dichroic, respectively. Accordingly,the light becomes either linearly or circularly polar-ized. The most efficient linear dichroic polarizers arepolaroids, sheets of organic polymers with long-chainmolecules which are aligned by stretching (Huard 1997;Born & Wolf 1999).

∗∗The term “dichroism” is actually misleading. Historically, thefirst dichroic crystals studied showed a strong dependence of theeffect on the wavelength of the light, leading to rays with differentpolarization having different colors.

8 S. TRIPPE

In the presence of a magnetic field, a plasma be-comes dichroic with respect to circular polarization.Assuming a propagation of light along the magneticfield lines, the ratio of the absorption coefficients forLHC and RHC polarized light, κL and κR, respectively,is

κLκR

=

(

ω + ωB

ω − ωB

)2

(38)

where ω denotes the (angular) frequency of the lightand ωB denotes the (angular) gyration frequency ofcharged particles (usually electrons). This relation as-sumes ωB ≪ ω; the resulting circular polarization ismC = 2ωB/ω (Angel 1974).

3.3 Optically Active Media

Materials composed of helically shaped molecules af-fect the polarization state of reflected or transmittedlight very similar to birefringent and/or dichroic me-dia (§§ 3.2, 3.8.2). Macroscopic polarization arises ifone of the two possible helix orientations is preferred;this is the case in a variety of biological materials. Ini-tially unpolarized light reflected from helically layeredsurface structures on some insects can reach circularpolarizations up to mC ≈ 100% (Wolstencroft 1974).

3.4 Synchrotron Radiation

Synchrotron radiation is arguably the most importanttype of non-thermal continuum radiation from astro-nomical sources. It is emitted by electric charges –usually electrons – gyrating around magnetic field linesat relativistic velocities. Assuming a magnetic field di-rected in z direction, the magnetic Lorentz force en-forces a circular motion in the xy plane. In addition,the electron will usually have a non-zero velocity in zdirection, meaning that the overall trajectory of theelectron has a helical shape. From the point of viewof an external, not co-moving, observer the radiationis emitted from the electron in forward direction into anarrow cone with half opening angle θ ≈ 1/γ, with γbeing the relativistic Lorentz factor†† (Ginzburg & Sy-rovatskii 1965; Rybicki & Lightman 1979; Bradt 2008).Due to geometry, the observer sees the orbit of the elec-tron as an ellipse in projection – the radiation emittedby a single oscillating charge is thus elliptically polar-ized.

Collimated electron beam. Macroscopically, theamounts of linear and circular polarization observed byan external observer are functions of γ and the viewingangle φ between the line of sight and the plane of par-ticle motion. For a collimated beam of mono-energeticelectrons – i.e., all electrons have the same Lorentz fac-tor γ – moving in the xy plane the circularly polarizedflux (Stokes V ) is

†† γ =(

1− v2/c2)−1/2

, with v being the electron speed and c

being the speed of light.

Fig. 3.— Polarization of synchrotron radiation from acollimated electron beam as function of re-scaled viewingangle η = γ sinφ. Given here are the Stokes parameters

I and V and the linearly polarized flux IL =√

Q2 + U2,normalized to I(η = 0) ≡ 1.

V = − 64η

7π√3(1 + η2)3

(39)

with η = γ sinφ. In this notation, the total intensity(Stokes I) of the radiation is given by

I =7 + 12η2

7(1 + η2)7/2(40)

(Michel 1991). The linearly polarized flux follows fromEqs. 39 and 40 via Eq. 23 in a straightforward manner.Notably, I normalized the expressions for V and I suchthat I = 1 at φ = 0.

Isotropic electron motion. In most astrophysicalplasmas, the electron velocities are distributed ran-domly and (more or less) isotropically. This impliesthat both right-handed and left-handed electron orbitscontribute with equal probability, meaning the circularcomponent of the radiation averages out: macroscopi-cally, the synchrotron radiation becomes linearly polar-ized (though not perfectly; see the discussion below).

Taking into account the various projection effects,the degree of linear polarization is given by

mL =I⊥ − I||

I⊥ + I||(41)

with I⊥ and I|| denoting the intensities perpendicularand parallel to the magnetic field lines as projectedonto the plane of the sky. Notably, the direction of po-larization is perpendicular to the (projected) directionof the magnetic field. The actual value of mL dependson the spectrum of the synchrotron radiation which inturn is a function of the distribution of the electronenergies. For an ensemble of mono-energetic electrons

POLARIZATION AND POLARIMETRY 9

the result is mL = 75% (Rybicki & Lightman 1979).For a power law distribution of electron energies, thenumber of electrons N varies with γ like N ∝ γ−Γ,with Γ being the energy index. In this case, the fluxdensity of the synchrotron radiation Sν varies with fre-quency ν like Sν ∝ ν−α, where Γ = 2α+1. The degreeof linear polarization is given by

mL =Γ+ 1

Γ + 7/3(42)

(Ginzburg & Syrovatskii 1965). For astrophysically re-alistic plasmas with α ≈ 0...1, mL ≈ 60...80%.

The relation given by Eq. 42 corresponds to a highlyidealized situation, assuming optically thin plasmas,isotropic distributions of electrons, perfectly orderedhomogeneous magnetic fields, and the absence of sub-stantial perturbations. A major modification occurs foroptically thick plasmas where each light ray experiencesmultiple scattering events. In those cases,

mL =1

2Γ + 13/3(43)

(Pacholczyk 1970); for α ≈ 0, mL ≈ 16%. Likewise,any disordering of the magnetic field leads to the po-larization signal partially being averaged out, thus re-ducing mL well below the idealized theoretical values.In optically thick plasmas, the polarization is orientedparallel to the magnetic field component projected onthe sky.

Yet another characteristic modulation of polariza-tion is caused by shocks propagating through theplasma. The degree of linear polarization in a partiallycompressed plasma, compared to the case without com-pression, is reduced by a factor

µ =δ

2− δ∈ [0, 1] with δ = (1− k2) cos2 ǫ (44)

where 0 ≤ k ≤ 1 is the factor by which the lengthof the shocked region is reduced by compression, andǫ is the angle between the line of sight and the planeof compression in the frame of reference of the emit-ter (Hughes, Aller & Aller 1985; Cawthorne & Wardle1988).

The amount of circular polarization follows roughlythe relationmC ∼ (ω/ωB)

−1/2 where ω is the (angular)frequency of the light and ωB is the (angular) gyrationfrequency of the charged particle. As before, this rela-tion assumes a uniform magnetic field and an isotropicdistribution of particle velocities. In the extreme caseωB ∼> ω, the relation changes to mC ∼ ω/ωB. In real-

istic astrophysical plasmas, mC ∼< 1% (Angel 1974).

3.5 Zeeman Effect

Spectral emission or absorption lines experience modi-fications if the emitting or absorbing material is perme-

ated by a magnetic field. For atomic or molecular tran-sitions involving the orbital angular momentum only(i.e., no spin–orbit coupling), the quantum mechanicalselection rules demand that the change of the mag-netic quantum number m has to obey ∆m ∈ [−1, 0, 1]– meaning there are three different transitions possible.As long as the atom is not exposed to external electricor magnetic fields, the three transitions are energeti-cally degenerate; all three transitions have the sameenergy and cause spectral lines at the same frequencyν0. This changes in the presence of an external mag-netic field as reported first by Zeeman (1897), henceZeeman effect. As shown by a simple classical analy-sis (Rybicki & Lightman 1979; Haken & Wolf 1990),the magnetic field causes a splitting of the three ini-tially degenerate energy levels. The spectral line splitsinto a set of three lines located at frequencies ν0 andν± = ν0±∆νz for ∆m = 0 and ∆m = ±1, respectively.The frequency offset is given by

∆νz =1

4π

e

meB = 14GHz×B (45)

where e is the electric charge of the electron, me is theelectron mass, and B is the strength of the magneticfield in units of Tesla. The lines at ν−, ν0, and ν+are denoted as σ−, π (“parallel”), and σ+ components,respectively.

The three spectral lines have distinct polarizationproperties that depend on the viewing geometry.

Transversal view. If the line of sight is perpendicu-lar to the magnetic field lines, the observer notes threespectral lines corresponding to the π and σ± compo-nents. All lines are linearly polarized. The polarizationof the π component is parallel to the magnetic field(hence the name), the polarizations of the σ± compo-nents are perpendicular to the field lines.

Longitudinal view. If the line of sight is parallel tothe magnetic field lines, only the σ± components arevisible. Both lines are circularly polarized, with σ+

and σ− being RHC and LHC, respectively. Notably,the orientation of polarization is defined relative to thedirection of the magnetic field lines, not the directionof propagation of the light.

Assuming an angle θ between the magnetic field andthe line of sight, the Stokes parameters resulting fromthe viewing geometry (e.g., Elitzur 2000) are

I = I0 sin2 θ (46)

Q = I0 sin2 θ

U = 0

V = 0

for the ∆m = 0 transition and

I =1

2I± (1 + cos2 θ) (47)

10 S. TRIPPE

Q = −1

2I± sin2 θ

U = 0

V = ±I± cos θ

for the ∆m = ±1 transitions, respectively. The I0, I±

denote the maximum intensities of the respective lines,with I+(ν+) = I−(ν−) = I0(ν0).

As yet, I assumed strong Zeeman splitting that leadsto distinct spectral lines, meaning intrinsic line widths∆ν ≪ ∆νz. In astronomy, this is not always the case;the situation ∆ν ≫ ∆νz is common. In the latter case,the total intensity as function of frequency, I(ν), ex-hibits a single line only. The Zeeman components areunveiled by analysis of the Stokes parameters as func-tion of frequency, resulting in

I = I0 + I+ + I− = 2 I0 (48)

Q = −d2I(ν)

dν2(∆νz sin θ)

2

U = 0

V =dI(ν)

dν∆νz cos θ

Accordingly, Q(ν) and V (ν) are the scaled derivativesof I(ν). Observations especially of V (ν) are importantexamples of spectro-polarimetry.

As yet, I discussed the normal Zeeman effect occur-ring in atomic transitions without spin–orbit coupling.If spin–orbit coupling has to be taken into account,more complex patterns with multiple line componentsspaced non-equally occur. Details for this anomalousZeeman effect depend on the total spins S, orbital an-gular momentum quantum numbers L, and total an-gular quantum numbers J . The π and σ± componentsare given by ∆M = 0 and ∆M = ±1 respectively, withM denoting the total magnetic quantum number.

3.6 Goldreich–Kylafis Effect

We now revisit the Zeeman effect for the case of weakZeeman splitting (∆ν ≫ ∆νz), meaning that the threeZeeman line components are not resolved. As indicatedby Eq. 48, the linear polarization is rather weak in gen-eral (and zero at ν = ν0). However, this result is basedon the assumption that each transition ∆m = −1, 0, 1is equally likely.

In case that the transitions ∆m occur at differentrates, increased linear polarization can occur (Goldre-ich & Kylafis 1981, 1982). This is the case if the radi-ation field within the source is anisotropic: dependingon the relative orientation of magnetic field and inci-dent radiation, the amount of anisotropy, and the ratioof collisional and radiative excitation rates, the σ± andπ line components are excited with different probabili-ties. The radiative transition rates T for the σ± and πcomponents, which are proportional to the line inten-sities, are given by

T± ∝ 1

2

∫

dΩ[

I⊥ + I|| cos2 α

]

(49)

T0 ∝∫

dΩ I|| sin2 α

respectively, where I⊥ is the incident light intensitypolarized perpendicular to the magnetic field and thedirection of propagation, I|| is the intensity of the inci-dent light polarized perpendicular to I⊥ and the direc-tion of propagation, Ω is the solid angle, and α is theangle between the travel path of the incident light andthe magnetic field.

Depending on which transitions are excited prefer-entially, the resulting linear polarization is oriented ei-ther parallel or perpendicular to the (sky-projected)magnetic field lines. Under realistic conditions, onemay expect degrees of linear polarization up to ≈10%in molecular lines from cool (temperatures ∼<100K) in-terstellar matter at millimeter-radio wavelengths.

3.7 Hanle Effect

The Hanle effect (Hanle 1924) is a phenomenon thatappears in fluorescent light. In the following, I assumea fluorescent gas permeated by a weak magnetic fieldB. The primary source of radiation is located in xdirection from the fluorescent gas, the line of sight aswell as the magnetic field are directed along the z axis.The primary light source emits radiation in x directionwith electric field components Ey,z. By geometry, thelinear polarization component Ey causes fluorescenceobservable in z direction.

When regarding normal Zeeman splitting (§ 3.5) inthe limit of a vanishing magnetic field B → 0, onereaches the regime of coherent resonances. As longas the Zeeman splitting is on the order of the nat-ural line width, i.e., ∆νz ∼> ∆ν, the σ± transitionsare excited independently. The observer notes two cir-cularly polarized waves with amplitudes E+ and E−

and a combined intensity I = E2+ + E2

−. For van-ishing magnetic fields, ∆νz ≪ ∆ν and both transi-tions σ± can be excited by the same photon (whichis linearly polarized and thus can be decomposed inone RHC and one LHC component for exciting σ+ andσ−, respectively). The observer notes fluorescent lightwith intensity I = (E+ + E−)

2 and linear polarization(mL = 100% in ideal situations) directed along y.∗

Ideal coherence with maximum light intensity I andmaximum linear polarization occurs at B = 0. For anincreasing field strength (|B| > 0), the degree of coher-ence decreases, causing several effects: (i) the intensityof the fluorescent light decreases; (ii) the “de-phasing”

∗Similar situations can also occur at non-zero magnetic fieldswhere magnetic term levels belonging to different angular mo-mentum quantum numbers can cross. This is the base of level-crossing spectroscopy.

POLARIZATION AND POLARIMETRY 11

of σ+ and σ− causes (a) a depolarization, and (b) aturning of the plane of linear polarization by an angleβ such that tanβ ≈ ωL/Γline, with ωL being the Lamorfrequency and Γline being the effective (i.e., natural pluscollisional) line width (e.g., Stenflo 1982).

3.8 Interactions of Polarization and Matter

3.8.1 Partial Reflection

A plane wave that falls onto the boundary between twohomogeneous media “1” and “2” with indices of refrac-tion n1,2 – like, e.g., the boundary between two plane-parallel plates made of different types of glass – is splitinto a reflected and a transmitted wave according to thelaw of reflection and Snell’s law of diffraction, respec-tively. Except of the special case of normal incidence,reflectivity R and transmissivity T – the fractions ofintensity being reflected and transmitted – are func-tions of linear polarization (Born & Wolf 1999). Bothparameters have to be re-written like

R = R|| cos2 α+R⊥ sin2 α (50)

T = T|| cos2 α+ T⊥ sin2 α

with || and ⊥ denoting linear polarizations parallel andperpendicular to the plane of incidence, respectively,and α being the angle between the electric field vec-tor and the plane of incidence. Conservation of energydemands

R+ T = R|| + T|| = R⊥ + T⊥ = 1 . (51)

All reflectivities and transmittivities depend on theratio n = n2/n1 and the angle between the normal ofthe boundary and the incident light ray, θ, via Fres-nel’s formulae. In general, R|| 6= R⊥ and T|| 6= T⊥;both, reflected and transmitted light become linearlypolarized even if the incident light is unpolarized. Inthe specific case

tan θB = n (52)

the component R|| vanishes; θB is the Brewster angle.For the case of a transition from air to glass, n ≈ 1.5and θB ≈ 57. If the incident light is unpolarized, thereflected light is completely polarized; the transmittedlight is polarized with a degree of linear polarizationmL ≈ 8%.

3.8.2 Birefringence

Birefringence is a consequence of the electric anisotropyof crystals, meaning that the response of the mediumto incident radiation (usually) depends on the directionof the electric field. This anisotropy is described by asymmetric dielectric tensor ε that defines a system ofprincipal dielectric axes with permittivities εx, εy, εz(Fowles 1975; Huard 1997; Born & Wolf 1999). In case

of isotropic crystals, εx = εy = εz; this is the casefor cubic crystals. In uniaxial crystals, εx = εy whileεy 6= εz; in biaxial crystals, εx 6= εy 6= εz. The two lin-ear polarization components of a light ray – which areperpendicular to the direction of propagation – pass-ing through an anisotropic crystal experience differentpermittivities and thus different indices of refractionn ∝ √

ε. The optic axes of a crystal correspond tolight travel paths for which the two linear polarizationsexperience equal refraction. For the case of uniaxialcrystals, the optic axis is the z axis.

Birefringence is employed in various optical elementsthat modify the polarization state of light.

Polarizers. For a uniaxial crystal, the dielectric ten-sor is an ellipsoid. The index of refraction describesa circle in the xy plane and ellipses in the xz and yzplanes,with the optic axis corresponding to either themajor or the minor axis of the ellipse depending on thematerial. For a wave polarized perpendicular to the op-tic axis, E1, the propagation in the crystal is isotropic,the surfaces of equal phase in the plane of incidence arecircular (regardless of the relative orientation of opticaxis and plane of incidence). E1 propagates throughthe crystal according to Snell’s law; therefore it is re-ferred to as ordinary wave. For a wave E2 polarizedperpendicular to E1 (i.e., E1 ⊥ E2), the componentsparallel and perpendicular to the optic axis experiencedifferent refractive indices and thus different phase ve-locities. The surfaces of equal phase in the plane ofincidence are elliptical, rendering Snell’s law invalid;accordingly, E2 is referred to as extraordinary wave.Anisotropic crystals can be used to separate the ordi-nary and extraordinary waves, and thus the two linearpolarizations, of unpolarized light. Various crystal ge-ometries and combinations of crystals with different op-tic axis orientations – adding internal reflections – areused. The most common types of polarizers are polar-izing beam splitters, Nicol prisms, and Glan-Thompsonprisms.

Linear-to-circular converters. Let us consider athin, plane-parallel plate cut from a uniaxial crystallocated in the xy plane, with the optic axis being they axis. In this case, the linear polarization componentsof a light wave propagating in z direction experiencetwo different refractive indices nx,y. The correspondingphase velocities – the speeds of light within the crystal– are cx = c/nx and cy = c/ny, respectively, with cdenoting the speed of light in vacuum. For light withwavelength λ in vacuum, traveling a distance d withinthe crystal, the two polarizations experience a phaseshift

δ =2π

λd (ny − nx) . (53)

When choosing d such that d(ny − nx) = λ/4, we findδ = π/2. Light with 〈E2

x〉 = 〈E2y〉, i.e. Stokes param-

eter Q = 0 (cf. Eq. 22), becomes circularly polarized(cf. Eq. 4); for Q 6= 0, it becomes elliptically polar-

12 S. TRIPPE

ized. Optical elements with this property are referredto as quarter wave plates. For realistic uniaxial crystalswith |ny−nx| ≈ 0.01...0.1 (Fowles 1975), d ≈ (3...25)λ,meaning that quarter wave plates are fragile devices.

Polarization plane turners. The relation providedby Eq. 53 has an additional consequence for linearlypolarized light with components Ex,y. When choosingd such that d(ny − nx) = λ/2, the phase shift becomesδ = π; the plate is a half wave plate. The componentperpendicular to the optic axis, here Ex, is mirrored atthe optic axis, i.e. Ex −→ −Ex (cf. Eq. 3). If planeof polarization and optic axis are tilted by an angle β,the crystal turns the polarization plane of the light byan angle of 2β.

In a variety of materials, birefringence can be in-duced by external electric – Kerr effect – or magnetic– Cotton-Mouton effect – fields (Fowles 1975). This isemployed in light modulators that need to be switchedbetween different states at high speed. In case of theKerr effect, the difference between the indices of refrac-tion parallel – n|| – and perpendicular – n⊥ – to theorientation of an external electric field with amplitudeE is

n|| − n⊥ = K E2 λ (54)

where λ is the wavelength of the light in vacuum andK is Kerr’s constant which is a function of the mate-rial. The Cotton-Mouton effect is the magnetic ana-logue of the (electric) Kerr effect; here the differencebetween the two indices of refraction is proportional tothe squared strength of the external magnetic field.

A further variety is introduced by the Pockels ef-fect observed in certain kinds of birefringent crystalsupon application of external electric fields. Here thedifference between the two indices of refraction is pro-portional to the electric field strength. This effect isused in Pockels cells that permit a rapid modulation oflight. A common setup comprises a Pockels cell locatedbetween two static linear polarizers with perpendiculartransmission axes. Via appropriate switching of thePockels cell, it turns the plane of polarization of theinfalling linearly polarized light, making the setup actas a very fast shutter (Fowles 1975).

3.8.3 Faraday Rotation

An external, static magnetic field B permeating amedium introduces an electric anisotropy. The impacton a light wave propagating through the medium isfound by solving the equation of motion for an elec-tron influenced by B and the oscillating electric fieldof the light wave E(t) (Fowles 1975; Rybicki & Light-man 1979). From this, one finds a circular electricanisotropy with permittivities εR 6= εL, with R andL denoting right-hand and left-hand circular polariza-tion, respectively. As any linearly polarized wave canbe expressed as a superposition of one left-hand andone right-hand circularly polarized wave (Eq. 17), the

electric anisotropy implies a characteristic rotation ofthe plane of polarization of linearly polarized light. Thechange of polarization angle can be expressed like

∆χ = V B|| l (55)

with l denoting the length of the light travel pathwithin the medium, B|| being the magnetic field strengthparallel to the light travel path, and V denotingVerdet’s constant which is a function of wavelength andmaterial (Fowles 1975).

In astrophysical situations, Faraday rotation oc-curs when light passes through magnetized interstellarplasma. This effect is quantified like

∆χ = RM× λ2 (56)

where λ is the wavelength of the radiation (in the rest-frame of the medium) and RM is the rotation measure(in units of radm−2)

RM = 8.1× 105∫ l

0

B|| ne dz (57)

with B|| being the strength of the magnetic field (inunits of Gauss) parallel to the line of sight (l.o.s.),ne being the electron number density (in cm−3), andz being the coordinate (in parsec) directed along thel.o.s. (Rybicki & Lightman 1979; Wilson, Rohlfs &Huttemeister 2010).

3.8.4 Faraday Depolarization

In case of spatially inhomogeneous media, especiallyastrophysical plasmas, the Faraday effect can lead to aloss of linearly polarized intensity. If the rotation mea-sure RM shows modulations with amplitudes ∆RM onspatial scales smaller than the source, the source radi-ation experiences different Faraday rotation dependingon the position. Observations that do not resolve theRM structure spatially superimpose waves with differ-ent orientations of their planes of linear polarization.This partially averages out the polarization signal, re-ducing the degree of linear polarization observed. Acomplete depolarization occurs when the medium is“Faraday thick”; from Eq. 56 one can estimate thatthis is the case if

∆RM× λ2 ≫ 1 . (58)

A more sophisticated calculation is possible when as-suming that the RM fluctuations follow a Gaussian dis-tribution with dispersion ζ ≈ ∆RM. For a source thatis not resolved spatially by observations, one finds adepolarization law

ξ = exp(

−2ζ2λ4)

(59)

(Burn 1966; Tribble 1991). The parameter ξ ∈ [0, 1]is the ratio of observed and intrinsic degree of linearpolarization.

POLARIZATION AND POLARIMETRY 13

3.8.5 Polarization Conversion

Under certain conditions, effects corresponding to thoseof birefringence in crystals can be observed also in as-trophysical plasmas. In the following, I assume an elec-tromagnetic wave with components Ex,y propagatingthrough a plasma in z direction. The plasma is per-meated by an ordered, static magnetic field directedalong the x axis. Plasma electrons accelerated by Ex

can move freely, whereas those accelerated by Ey ex-perience an additional magnetic Lorentz force – the re-sponse of the plasma to light becomes anisotropic, theplasma effectively becomes birefringent. In analogy tothe relation given by Eq. 53, this effective birefringenceintroduces a phase shift between Ex and Ey that con-verts linear into circular polarization and vice versa;this effect is also referred to as Faraday conversion orFaraday pulsation (Pacholczyk & Swihart 1970).

In relativistic astrophysical plasmas and at radio fre-quencies, one may expect to observe a certain level ofcircularly polarized light generated from initially lin-early polarized radiation. Details depend strongly onthe physical conditions within the plasma. Pacholczyk(1973) provides an estimate for the relation betweenthe degrees of linear (mL) and circular (mC) polariza-tion,

mC

mL∝ neB

2⊥ ν

−3 (60)

where ne denotes the electron density, B⊥ is thestrength of the magnetic field component perpendicu-lar to the line of sight, and ν is the observing frequency.In general, one may expect mC ∼< 1%.

3.8.6 Chandrasekhar-Fermi Effect

Linear polarization generated within a magnetized tur-bulent plasma – via, e.g., dust scattering (§ 3.1.2)or synchrotron radiation (§ 3.4) – is sensitive to thestrengths of turbulence and magnetic field. The mag-netic field is assumed to be “frozen” in the plasma. Incase of weak fields, the field lines are dragged aroundby the turbulence, leading to a large r.m.s. disper-sion in polarization angles. In case of strong fields,the field lines remain rather unimpressed by the tur-bulence, the dispersion in polarization angles is small.Magnetic field, turbulence, and polarization angle arerelated (Chandrasekhar & Fermi 1953) like

B⊥ =

(

4

3πρ

)1/2σvσχ

(61)

where B⊥ is the strength of the magnetic field perpen-dicular to the line of sight (in Gauss), ρ is the massdensity of the gas (in g cm−3), σv is the r.m.s. velocitydispersion of the gas (in cms−1), and σχ is the disper-sion of polarization angles (in radians).

4. OBSERVATIONS

Similar to the cases of photometry and spectroscopy,the techniques used for polarimetry of radiation fromastronomical sources depend strongly on the energy ofthe light. In general, we can distinguish three differentwavelength regimes. At radio wavelengths, we are ableto record electromagnetic waves with their amplitudesand phases. At optical wavelengths, we are usuallydealing with light intensity information. At X/γ-rayenergies, a combination of high frequency and low fluxusually implies that we observe and analyze individualphotons. This being said, I note that the distinctioncan be blurred depending on the physical situation, andvarious techniques find application over a wide rangeof radiation energies.

4.1 Sky Projection

Our usual use of polarization parameters, especially ofthe Stokes parameters (§ 2.3.2), implicitly assumes thatemitter and receiver of radiation are placed in a com-mon, stationary system of coordinates. In astronom-ical observations, this is usually not the case: Earthrotation leads to a rotation of the field of view withrespect to the observer. Assuming electric fields EV,H

measured vertical and horizontal, respectively, with re-spect to the telescope, these are related to the Stokesparameters in the frame of reference of the source onsky like

2 〈EVE∗V〉 = I +Q cos 2ψ + U sin 2ψ (62)

2 〈EHE∗H〉 = I −Q cos 2ψ − U sin 2ψ

2 〈EVE∗H〉 = −Q sin 2ψ + U cos 2ψ + iV

2 〈EHE∗V〉 = −Q sin 2ψ + U cos 2ψ − iV

where ψ denotes the parallactic angle counted fromnorth to east and i is the imaginary unit. Compar-ison to Eq. 27 shows that Earth rotation leads to aconversion from Q to U and vice versa from the pointof view of the observer; V remains unaffected (Thomp-son, Moran & Swenson 2004). The same result followsfrom Eq. 34 in a straightforward manner.

4.2 Terrestrial Atmosphere

In most situations, the influence of Earth’s atmosphereon polarization can be neglected; the atmosphere isneither birefringent nor dichroic. An important excep-tion occurs at radio frequencies where the interactionof ionosphere and terrestrial magnetic field causes sub-stantial Faraday rotation (§ 3.8.3). At an observing fre-quency ν = 100MHz, the angle of polarization is ro-tated by ∆χ ≈ 300 at night to ∆χ ≈ 3 000 at daytimeunder typical atmospheric conditions; a reliable deriva-tion of the true polarization angle is very difficult. As∆χ ∝ ν−2, this effect can be circumvented by select-ing a sufficiently high observing frequency (Thompson,Moran & Swenson 2004; Clarke 2010).

14 S. TRIPPE

An additional effect relevant mostly at optical wave-lengths is the polarization of scattered sun or moonlight. From the geometry argument presented in § 3.1.1it is straightforward to see that scattered light is lin-early polarized. The degree of polarization reaches itsmaximum at an angular distance of 90 from the lightsource, the polarization is oriented perpendicular tothe line on sky connecting the source and the pointobserved. This behavior can be exploited for the cali-bration of polarimetric observations via dedicated ob-servations of scattered light.

4.3 Instrumental Polarization

The design and geometry of a telescope inevitably in-fluence the polarization of the collected light. Ex-cept of highly symmetric situations – most notably inCassegrain focus telescopes – the (usually) multiple re-flections within an optical system alter the polarizationstate of the light. The resulting instrumental polariza-tion is given by the product of the Muller matricesof the individual telescope components as described in§ 2.3.3. These geometric effects need to be corrected inthe course of data analysis and/or by dedicated correc-tive optics in the telescope (see, e.g., Thum et al. 2008for a discussion of Nasmyth optics).

Even though one may correct for the influence ofthe telescope geometry, realistic instruments are notperfect. In the most general case, the observed Stokesparameter values deviate from the actual ones by

∆S = −

1

2

γ++ γ+− δ+− −iδ−+

γ+− γ++ δ++ −iδ−−

δ+− −δ++ γ++ iγ−−

−iδ−+ iδ−− −iγ−− γ++

S (63)

where S is the Stokes vector of the infalling light, theγxx ≪ 1 are error terms related to the gains, or efficien-cies, of the optical paths for the Ex,y components, andthe δxx ≪ 1 are error terms related to the leakage, alsoknown as cross-talk, meaning the mutual influence ofthe optical paths for separate polarizations;† the (max-imum) number of error terms is seven (Sault, Hamaker& Bregman 1996). The expression given by Eq. 63 as-sumes that Stokes parameters are derived from linearpolarization components via Eq. 27. If Stokes parame-ters are derived from circular polarization components(Eq. 29), Eq. 63 can be applied to a Stokes vector withQ, U , and V being interchanged with V , Q, and U ,respectively (cf. Eq. 27 vs. Eq. 29).

The actual calibration procedure depends stronglyon the telescope(s) used. In general, calibration in-volves observations of one or more unpolarized astro-nomical reference sources – probing interactions be-tween I on the one hand and Q, U , and V on the

†Even though the δ and γ terms are small, they can easily be ofthe order of few per cent – i.e. the same order as the actualpolarization signal in many cases.

other hand – and observations of one or more polarizedcalibration sources, possibly several times at differentparallactic angles – probing interactions between Q, U ,and V by comparison of observed and expected values(e.g., Clarke 2010).

4.4 Polarization Statistics

Whereas the effects discussed in §§ 4.1–4.3 introducesystematic errors into polarization data, we now dis-cuss the statistical uncertainties and limits to be takeninto account. First of all, it is important to note that, ingeneral, polarimetric observations require much bettersignal-to-noise ratios (S/N) than photometric ones. Asthe degrees of linear or circular polarization mL,C ≤ 1are usually much smaller than unity, the signal-to-noiseratio (S/N)I of the total intensity (Stokes I) signal canbe related to the S/N of the polarized intensity (S/N)Plike (S/N)P ≈ mL,C(S/N)I .

‡ This implies that a de-tection of a weak polarization signal may require veryhigh (S/N)I .

In case of linear polarization, statistical measure-ment uncertainties lead to a bias in the measured val-ues for mL. This is due to mL being positive definiteby construction (Eq. 25): even if Q and U are symmet-ric random variables centered at zero, the sum of theirsquares is not; the values of mL follow a Rice distri-bution. A de-biasing can be attempted by subtractingfrom each of Q and U the corresponding statistical un-certainty in squares before the calculation of mL. Forhigh (S/N)P , the statistical errors of mL, σm, and po-larization angle χ, σχ, are related like σχ = σm/(2mL)(in units of radians); the values of χ follow a normaldistribution. For low (S/N)P , the values for Q and Uscatter around the origin of the QU plane, the distribu-tion of the χ values becomes more and more platykurticfor lower and lower (S/N)P (Clarke 2010).

4.5 Radio Observations

At radio wavelengths, the infalling radiation can berecorded and analyzed as waves with full amplitude andphase information; due to fundamental quantum lim-its, this is possible at frequencies up to about one THz(Thompson, Moran & Swenson 2004; Wilson, Rohlfs& Huttemeister 2010). Regardless of the actual de-sign details, a radio telescope can be modeled as across of two dipoles aligned along the x and y axes,respectively. We may assume, as usual, light propa-gating along the z direction with linear polarizationcomponents Ex,y. Each of the two dipoles receives thecorresponding polarization component and converts itinto an electric voltage that can be recorded and pro-cessed electronically – radio receivers are polarimeters

‡This relation is strictly valid only when the polarization is de-rived from sums or differences of intensities, especially in opti-cal polarimetry. In cases where the polarization is derived frommultiplications of fields or from correlations, the process of mul-tiplication leads to non-Gaussian error distributions, modifyingthe noise estimates by factors of several.

POLARIZATION AND POLARIMETRY 15

by construction§ (see also Hamaker, Bregman & Sault1996; Hamaker & Bregman 1996; Sault, Hamaker &Bregman 1996; Hamaker 2000, 2006 for an exhaustivediscussion). The signals received by the dipoles can beautocorrelated – resulting in the time-averaged prod-ucts 〈ExE

∗x〉 and 〈EyE

∗y 〉 – as well as crosscorrelated –

resulting in 〈ExE∗y〉 and 〈EyE

∗x〉 (using complex expo-

nential notation). The Stokes parameters I,Q, U, V arederived from these products via Eq. 27 in a straight-forward manner. A receiving system sensitive to bothpolarizations Ex,y (or ER,L for circular polarization) isreferred to as a dual-polarization receiver.

The cross of dipoles also serves as a model for radioreceivers sensitive to circular polarization. For this weassume that (i) the signals from the two dipoles aresent to a common electronic processor and summedup coherently and (ii) a phase shift of ±π/2 is ap-plied to the signal from the y dipole. Accordingly, thetwo voltages will be in phase and trigger a signal ifthe infalling light is either RHC or LHC polarized, de-pending on the sign of the phase shift. The receiveris a single-polarization receiver sensitive to either RHCor LHC; it can be extended to a dual-polarization re-ceiver by adding a second cross of dipoles with oppo-site phase shift. By symmetry, these arguments holdalso for the case of sending waves for radar astron-omy; here, usually circularly polarized radio light isused (Ostro 1993). Due to the technical simplicity ofradio polarimetry, recent efforts have been directed to-ward simultaneous multi-wavelength polarimetry (cf.,e.g., K.-T. Kim et al. 2011; Lee et al. 2011) aimed atthe measurement of differential parameters like disper-sion measures.

The choice of polarization is important when com-bining the signals from two antennas located at largedistance, e.g., in Very Long Baseline Interferometry(VLBI). The discussion provided in § 4.1 also impliesthat the raw observed values for Q and U are functionsof the geographic positions when using linear polariza-tion receivers. This problem is circumvented by usingcircular polarization receivers which are insensitive toEarth rotation.

4.6 Optical Observations

At optical wavelengths, polarimetry is limited to lightintensities rather than electric waves. Polarimetricmeasurements require the use of polarizers plus aux-iliary optical elements placed in the optical path be-fore the detector (usually a CCD array; e.g., Tinbergen1996).

Linear polarization can be probed by measuring theintensity of the received light, I(ψ), polarized at aparallactic angle ψ to the x (north-south) axis of theusual xy coordinate system of the Stokes parameters

§This excludes radio techniques sensitive to total intensities only,notably bolometers. These have to be treated like optical tele-scopes (§ 4.6).

Fig. 4.— Polarimetric imaging of Sagittarius A* (Sgr A*),the supermassive black hole at the center of the Milky Way,at 2.2µm. Sgr A* is indicated by an arrow, the surround-ing sources are stars; the angular resolution is ≈50mas.The difference between the synchrotron source Sgr A* andthe stars emitting thermal radiation becomes evident in theStokes Q image; at the time of observation, mL ≈ 20%(Trippe et al. 2007).

(§ 2.3.2). The Stokes parameters Q and U are relatedto these intensities like

Q

I=

I(0)− I(90)

I(0) + I(90)(64)

U

I=

I(45)− I(135)

I(45) + I(135)

with I ≡ I(0)+I(90) ≡ I(45)+I(135) being StokesI as usual (Kitchin 2009; Witzel et al. 2011); see Fig. 4for an example. Alternatively, one may measure I(ψ)at multiple – at least four – values of ψ and model themeasurement values with the function

q(ψ) =I(ψ)− I(ψ + 90)

I(ψ) + I(ψ + 90)= mL cos [2(ψ − χ)] (65)

with mL denoting the degree of linear polarization andχ denoting the polarization angle as defined in § 2.1.5(e.g., Ott, Eckart & Genzel 1999; Trippe et al. 2010).Using Eq. 65 with a sufficiently large number of mea-surement values (≥8) with a good sampling of ψ valueshelps to recognize instrumental polarization effects inthe data.¶

¶This is straightforward to see in the special case of Cassegrainfocus observations. In this case, the target polarization is fixedwith respect to the sky whereas the instrumental polarization isfixed with respect to the telescope. We may obtain observationsat two (or more) different hour angles and model each data setas a superposition of two cosine profiles as defined in Eq. 65: onecorresponding to the target polarization and one correspondingto the instrumental polarization. A polarization signal whichremains unchanged – in sky coordinates – at different hour anglesis intrinsic to the target.

16 S. TRIPPE

In order to filter I(ψ) out of the infalling radiation,various types of polarizers can be used. One possi-bility are wire-grid polarizers (Huard 1997) that passlight polarized perpendicular to the grid and reflect theother component. For each value of ψ, the grid is ro-tated into the required position and an image of thetarget is taken (e.g., Ott, Eckart & Genzel 1999). Amore efficient approach is provided by using a combi-nation of (i) a Wollaston prism that splits the infallinglight into ordinary and extraordinary linearly polarizedrays, and (ii) a half wave plate (HWP) that permitsturning the plane of linear polarization (cf. § 3.8.2).A complete measurement cycle involves taking two im-ages, each showing the ordinary and extraordinary rayimages of the target: one with the HWP turned to a po-sition corresponding to ψ = 0/90, one with the HWPturned such that ψ = 45/135 is observed. The linearpolarization of the target is then derived via Eq. 64 ina straightforward manner (e.g., Witzel et al. 2011).

An analysis of circular polarization requires the useof a quarter wave plate (QWP). The QWP convertscircular into linear polarization; the linearly polarizedlight can be analyzed as discussed above. For a QWPwith its axis of minimum index of refraction – its fastaxis – being the x axis in our usual (§ 2.1) coordinatesystem, its impact on circularly polarized light can bewritten in Jones calculus like

[

1 00 i

] [

1∓i

]

=

[

1±1

]

(66)

which denotes the application of the Jones matrix of theQWP to a circularly (RHC or LHC) polarized wave, re-sulting in a linearly polarized wave with diagonal planeof polarization (Fowles 1975). Comparison of the re-sult to the definition of the Stokes parameters (§ 2.3.2)shows that the QWP converts V to U . Accordingly, wecan now derive V from an analysis of linear polarizationaccording to Eq. 64, resulting in

V

I=I(45)− I(135)

I(45) + I(135)(67)

(cf., e.g., Goodrich, Cohen & Putney 1995). I note thatthe choice of QWP orientation – here along the x axis– is arbitrary; for example, a diagonal orientation ofthe fast axis leads to a conversion from V to Q (e.g.,Fowles 1975).

4.7 X and γ Ray Observations

Due to the high photon energies and short wavelengthsinvolved, optical elements used at optical wavelengthsbecome transparent at X/γ ray energies. Polarimetryat these wavelengths can be based on any of three dis-tinct physical effects.

Bragg diffraction. Light with sufficiently short (lessthan a few nanometers) wavelength λ falling onto acrystal is reflected by the crystal according to Bragg’slaw

Fig. 5.— Illustration of Bragg diffraction polarimetry.

nλ = 2 d sin θ ; n = 1, 2, 3, ... (68)

with d denoting the distance between two consecutiveatomic layers measured perpendicular to the surfaceof the crystal, θ denoting the angle of incidence mea-sured between the infalling light ray and the surface ofthe crystal, and n being the order of diffraction (Born& Wolf 1999). Using the relation derived in § 3.1.1it becomes evident that the reflected light is linearlypolarized, with the component parallel to the surfaceprevailing. If incident and reflected rays are perpendic-ular – meaning θ = 45 – only the polarization com-ponent parallel to the surface of the crystal remainsin the reflected light. The linear polarization state ofa science target can be derived by rotating the fieldof view of the instrument and observing the resultingcosinusoidal profile equivalent to Eq. 65; I illustratethe measurement geometry in Fig. 5. As Bragg’s lawis strictly valid only for one specific wavelength, thismethod is, a priori, limited to very narrow energy bandsat each order of diffraction. This condition can be re-laxed by using “mosaicked” crystals composed of manysmall crystalets, thus providing a range of d values fordiffraction (Silver & Schnopper 2010).

Bragg diffraction polarimetry was the method usedfor the only X-ray polarimeter ever implemented in aspace telescope, the OSO-8 satellite. It was used tomeasure the polarization of the Crab nebula (Weis-skopf et al. 1978), resulting in the “first and only high-precision X-ray polarization measurement obtained forany cosmic source” (Silver & Schnopper 2010).

Scattering polarimetry. As discussed in § 3.1.1,Thomson or Compton scattering of photons by elec-trons is sensitive to the linear polarization of the inci-dent photons. This is exploited in scattering polarime-ters. We may assume, as usual, partially polarized lightcomposed of photons propagating in z direction. Thesephotons arrive at a scattering detector that providesmaterial for scattering the incident light; the scatter-

POLARIZATION AND POLARIMETRY 17

ing detector is located at the origin of the xy plane.A certain fraction of the arriving photons will be scat-tered at right angles into the xy plane where they arerecorded by a calorimeter. In case of no polarization,the distribution of scattered photons in the xy planewill be isotropic. If the light is partially linearly polar-ized, photons will be scattered preferentially perpen-dicular to the direction of polarization projected ontothe xy plane; the resulting distribution is given by ex-pressions equivalent to Eq. 65 when taking into accountthe instrument and scattering geometries (McConnell2010).

Photoelectron tracking. Irradiation of X rays on amedium can cause the release of photoelectrons. Thedirection of photoelectron emission is a function of thepolarization of the incident light. Assuming a linearlypolarized X ray photon propagating in z direction thatcauses the emission of a photoelectron at the origin ofthe xy plane, the differential cross section of photoelec-tron emission is given by

dσ

dΩ∝ sin2 θ cos2 φ

[1− β cos θ]4(69)

where σ is the cross section, Ω denotes the solid an-gle, β is the electron speed in units of speed of light,θ is the angle between the path of the incident photonand the path of the emitted photoelectron, and φ is theangle between the path of the photoelectron and the di-rection of polarization of the photon projected onto thexy plane (Bellazzini & Spandre 2010). Accordingly, thedistribution of photoelectrons is a function of photonpolarization, resulting in a characteristic cos2 φ patternin the xy plane.

Compared to classical Thomson scattering, the pho-toelectric effect is more efficient in analyzing the photonpolarization: whereas the differential cross section ofThomson scattering decreases with θ increasing from 0

to 90 (cf., e.g., Rybicki & Lightman 1979), it stronglyincreases in case of photoelectron emission (Eq. 69).At energies of few keV (i.e., β ∼< 0.1), the differen-tial cross section peaks at θ ≈ 90, meaning most ofthe photoelectrons are emitted within or close to thexy plane. The electron paths can be traced by semi-conductor (CCD), gas, or scintillation photo-detectorslocated in the xy plane; again, the degree and orien-tation of macroscopic polarization can be derived fromthe photoelectron distribution via Eq. 65 (or equivalentexpressions).

The use of a certain method for X/γ ray polarime-try is largely dictated by the photon energies. Soft Xrays can be analyzed with either method; the analysisof hard X and γ rays is usually limited to Comptonscattering polarimetry (cf., e.g., Bloser et al. 2010).

5. SCIENCE CASES

5.1 Solar and Stellar Physics

Across the Hertzsprung-Russell diagram, stars, includ-ing the sun, are known to posess magnetic fields withfield strengths ranging from a few to several ten thou-sand Gauss (e.g., Schrijver & Zwaan 2000; Berdyug-ina 2009). In case of hot stars with radiative outerlayers (roughly, spectral classes O–A), magnetic fieldsare supposed to be “fossil”, i.e., inherited from the in-tergalactic medium the stars formed from; in case ofstars with convective outer layers (approximately spec-tral classes F–M), magnetic fields are generated by dy-namo processes (e.g., Berdyugina 2009). Accordingly,analyses of stellar magnetic fields are able to constrainsolar and stellar dynamo models.

Stellar magnetic fields – including here also mag-netic white dwarfs with B ∼< 109G (Jordan 2009) –can be analyzed via spectropolarimetry of absorptionlines that are affected by Zeeman splitting (§ 3.5). Ac-cording to Eqs. 46, 47 and 48, the orientation of thefield lines can be assessed from the relative strengthof linear and circular polarization (see, e.g., Donati &Landstreet 2009 for a review). In case of weak Zeemansplitting (Eq. 48) – a common case in stellar spectrallines – the magnetic field strength B enters (via Eq. 45)linearly into V and quadratically into Q. On the onehand, this makes it possible to estimate B directly fromthe V (ν) profile; on the other hand, this complicatesthe analysis of the field orientation. Spatially resolvedmaps of magnetic fields of the sun (e.g., Stenflo 2013) orstars (e.g., Arzoumanian et al. 2011), usually based oncircular polarization, are referred to as magnetograms.

Complementary to Zeeman effect measurements, theHanle effect (§ 3.7) can be used to probe the magneticfield of the sun (Berdyugina 2004; Milic & Faurobert2012). This is achieved by simultaneous spectropo-larimetric observations of several molecular fluores-cent lines; important diagnostic molecules are C2 andMgH. In addition, scattering polarization by Rayleighand Raman scattering probes the physical conditionsin stellar atmospheres (e.g., Sampoorna, Nagendra &Stenflo 2013).

5.2 Planetary System Bodies

5.2.1 Solid Surfaces

Sunlight reflected at a solid surface – like the ones ofrocky planets or asteroids – becomes partially linearlypolarized due to scattering polarization (§ 3.1). Un-surprisingly, the observed degree of polarization is afunction of the relative position of observer, reflectingbody, and the star (the phase angle in case of the so-lar system). The maximum degree of linear polariza-tion, in the following denoted with mL, is a function ofwavelength and of the structure of the reflecting mate-rial. The interplay between absorption and scatteringof light causes the Umov effect, a characteristic anti-

18 S. TRIPPE

correlation between mL and geometric albedo A of asolid surface (Bowell & Zellner 1974). For a given ma-terial – like lunar regolith, sand, basalt, or granite –polarization and albedo are related like

log(A) = −c1 log(mL) + c2 (70)

at visible wavelengths, with constants c1 ≈ 1 andc2 ≈ −2 (for A,mL ∈ [0.01, 1]). Notably, this im-plies degrees of polarization close to 100% for very lowalbedos. Observed deviations from this relation indi-cate a change in the structure of the surface material;accordingly, A − mL diagrams can be used to assessthe surface composition of a planet or any other solidbody. In addition, characteristic variations of mL withtime can be used to estimate the rotation period and/orsurface profile of a small body (e.g., an asteroid) thatis not resolved spatially by observations (e.g., Ishiguroet al. 1997; Cellino et al. 2005).

Radar astronomical observations (Ostro 1993; Camp-bell 2002) exploit the polarization state of the reflectedradio light. The transmitted radar signal has a well de-fined polarization state (usually 100% circular). In caseof a single reflection at an ideal dielectric surface, thecircular polarization state of the echo signal is invertedwith respect to the transmitted signal, the linear polar-ization state (expressed via Stokes Q by proper choiceof coordinates) remains unchanged (cf. Eq. 33). Multi-ple scattering and/or refraction at rough surfaces leadto some of the echo light being in the same circularpolarization state and/or inverted linear polarizationstate compared to the infalling light. Denoting the po-larization states as the “same” (S) and “opposite” (O)ones with respect to the transmitted radar signal, onecan define the polarization ratios

RC =ΣSC

ΣOCand RL =

ΣOL

ΣSL(71)

with “L” and “C” referring to linear and circular po-larization, respectively, and Σ denoting the radar crosssection of the target. Accordingly, both RL and RC

would be zero for an ideal smooth surface. For mostsolar system objects, RC ∼< 0.3, with the notable ex-ception of the icy moons of Jupiter for which RC ∼> 1

(Ostro 1993).

5.2.2 Atmospheres

Reflection of light at (sufficiently dense) planetary at-mospheres (e.g., Buenzli & Schmid 2009, and referencestherein) is mainly affected by two (linearly) polariz-ing processes: (i) Rayleigh scattering at molecules andaerosol haze particles, and (ii) refraction and reflec-tion at liquid droplets in clouds. Whereas individualinteractions can lead to degrees of linear polarizationup to 100%, the signal observed by a distant observeris the average over multiple light rays, which partiallyaverages out the polarization signal and reduces the ob-served degree of polarization. The actual polarization

levels depend strongly on the reflection geometry andthe chemical composition of the atmosphere. Withinand around the regime of visible wavelengths, observedlevels of polarization – integrated over the planetarydisks – are <5% for Venus, 5–10% for Jupiter and Sat-urn, and up to ≈50% for Titan (Saturn’s moon).

Quantitative investigations of the polarization prop-erties of planetary atmospheres require numerical mod-eling (e.g., Buenzli & Schmid 2009). The polarizationof – intrinsically unpolarized – starlight reflected fromplanets is used for direct imaging of exoplanets via po-larimetric differential imaging (e.g., Milli et al. 2013).

5.3 Interstellar Matter

Interstellar space is filled with diffuse matter occurringin a large variety of states, from cold dense molec-ular (main species being H2, CO, and H2O) cloudswith temperatures T of few Kelvin and (hydrogen)particle densities 103...5 cm−3 up to the hot ionized(coronal) medium (main species being H ii, C iv, Nv,and Ovi) with T ≈ 106K and hydrogen densities≈ 3 × 10−3 cm−3. In addition, interstellar dust is om-nipresent throughout galaxies (see, e.g., Kwok 2007 fora detailed overview).

The interplay of interstellar dust and galactic mag-netic fields (§§ 3.1.2, 5.8) is responsible for the inter-stellar polarization of scattered starlight (see, e.g., Das,Voshchinnikov & Il’in 2010; Matsumura et al. 2011 forrecent discussions); the degree of linear polarization isapproximately given by Serkowski’s law and, accord-ingly, ranges from a few to about ten per cent (Draine2003). In case of circumstellar material in the immedi-ate vicinity of a star, scattering polarization can arisefrom:

(i) The alignment of dust grains in the magnetic fieldof a circumstellar disk or star-forming nebula.

(ii) Scattering at spherical or randomly oriented dustgrains; in this case, polarization arises from ge-ometry because the incident light arrives from awell-defined direction – the star.

(iii) Scattering at magnetically aligned dust grainsplus dichroic absorption by foreground material,leading (also) to circular polarization with mC ∼<20% (Kwon et al. 2013).

In case (i), infrared polarimetric imaging has revealedthe magnetic field structures in disks around youngstars as well as characteristic “hour-glass” field geome-tries in star forming regions (e.g., Cho & Lazarian 2007;Sugitani et al. 2010). In case (ii), polarimetric imagingof circumstellar material shows a highly symmetric cir-cular pattern centered at the star, with the orientationof polarization being perpendicular to the direction ofthe incident radiation. This has been used to ana-lyze (proto)stars embedded in dense interstellar matter(e.g., Saito et al. 2009). Circular scattering/absorptionpolarization (case iii) has been observed only in a fewstar forming regions (Kwon et al. 2013).

POLARIZATION AND POLARIMETRY 19

5.4 Astrobiology

Complex helical organic molecules in terrestrial lifeforms – like amino acids – show homochirality: outof two helix orientations possible, only one is used ex-clusively. This phenomenon implies that one of thetwo orientations was preferred in pre-biotic chemistry.A possible cause is circularly polarized light in starforming regions (§ 5.3) leading to preferential photo-dissociation of organic molecules with one specific ori-entation. This causes an excess of molecules with agiven orientation and, eventually, on Earth to which or-ganic matter is transported via comets and meteoroids(De Marcellus et al. 2011; Kwon et al. 2013).

Homochirality causes light reflected from certain bi-ological surfaces to be circularly polarized (cf. § 3.3).This effect can – in principle – be exploited for de-tecting life on other planets via (spectro)polarimetry ofstarlight reflected from the surface (e.g., Sparks et al.2012).

5.5 Astronomical Masers

Stimulated emission of radiation at radio frequencies –maser radiation – can be observed from the interstellarmatter in star-forming regions and from the circumstel-lar envelopes of late-type (super)giant stars (e.g., Kwon& Suh 2012). Maser radiation is emitted as molecu-lar line emission with very high brightness tempera-tures up to roughly 1012 K. Species known to act asastrophysical maser media are the molecules OH, H2O,CH3OH, NH3, HC3N, H2CO, CH, SiO, SiS, and HCN,plus atomic hydrogen (H). Maser radiation has beenobserved at frequencies from 1.61GHz (from OH) to662.4GHz (from H), i.e., across the entire radio regime(Reid & Moran 1981; Elitzur 1982; Townes 1997).

Astronomical masers tend to show substantial Zee-man line splitting caused by magnetic fields permeatingthe maser medium. In the case of strong (∆νz ≫ ∆ν)Zeeman splitting, the resulting line polarizations aregiven by Eqs. 46 and 47. In case of weak Zeeman split-ting (∆νz ≪ ∆ν) however we have to take into accountthat maser radiation is caused by stimulated, coherentemission, meaning a coherent superposition of electricwaves. For a single electromagnetic wave, the resultingStokes parameters, in units of Stokes I, are

Q

I= −1 +

2

3 sin2 θ(72)

U

I= ± 2

3 sin2 θ

(

3 sin2 θ − 1)1/2

V

I= 0

with θ denoting the angle between the magnetic fieldand the line of sight (Elitzur 1991, 2000). For sin2 θ ≤1/3 (i.e., θ ∼< 35), Q/I = 1 and U/I = 0. As

(Q/I)2 + (U/I)2 = 1 for all θ, a wave emitted by an

ideal maser is always fully linearly polarized. Whenaveraging over multiple waves – as in any realistic as-tronomical observation – the sign ambiguity in U/Icauses Stokes U to average out; only Q remains, im-plying a partial linear polarization with mL = Q/I.However, more recent calculations based on numeri-cal simulations of realistic maser radiation fields findthat the analytical estimates quoted above suffer fromover-simplifications; the actual levels of linear polariza-tion should be substantially smaller than the ones pre-dicted by Eq. 72 (Dinh-V-Trung 2009). Furthermore,already moderate (∆νz < ∆ν) Zeeman splitting intro-duces circular polarization with amplitudes as high asmC ≈ 20%, with frequency-dependent profiles V (ν)similar to Eq. 48 for sufficiently small θ ∼< 30 (Elitzur

2000; Dinh-V-Trung 2009).

5.6 Pulsars

Pulsars are neutron stars with strong magnetospheres.Their radiation is composed of thermal radiation fromthe neutron star surface – at temperatures T ≈ 106K– and, predominantly, non-thermal synchrotron andcurvature radiation created within the stellar magne-tosphere. The observational pulsar phenomenology isgiven by geometry: the magnetic axis of the star istilted relative to its spin axis. If the magnetic axispoints to the observer during a rotation period, theradiation from the magnetosphere becomes visible as ashort pulse of light. Observed pulse periods are locatedroughly in the range from few milliseconds to tens ofseconds, with most pulsars having periods about fewhundred milliseconds. To date, approximately 2000pulsars are known which are distributed throughoutthe Milky Way (see, e.g., Lyne & Graham-Smith 2012for a review).

The magnetic field of the neutron star can be as-sumed to be a relic of the field of the progenitor star.Conservation of magnetic flux demands very high fieldstrengths nearby the star, with values in the rangeB ≈ 106−10T. The field geometry is bipolar (at leastwithin the light cylinder, i.e. the regime of co-rotationspeeds below the speed of light). The combination ofstrong magnetic field plus fast rotation leads to thecreation of a strong electric field at the stellar surface,with field strengths up to E ≈ 1012Vm−1. The elec-tric field extracts charged particle (electrons, ion) atand around the magnetic poles. The charges propa-gate along the magnetic field lines at highly relativis-tic (Lorentz factors γ ≈ 107) energies. Those primaryelectric charges, plus secondary charges with γ ≈ 1000originating from electron-positron pair creation, pro-duce synchrotron and (mostly) curvature radiation di-rected along the magnetic field lines. The emission ge-ometry provides the “lighthouse effect” necessary forthe observational pulsar phenomenology. The emittedradiation is partially coherent; the highest flux den-sities are usually observed at low – few GHz – radiofrequencies (Michel 1991; Beskin et al. 1993).

20 S. TRIPPE

By geometry, the radiation from pulsars can roughlybe approximated as synchrotron radiation from colli-mated beams of electrons (§ 3.4). Accordingly, one ob-serves (e.g., Rankin 1983) both linear and circular po-larization approximately following the pattern outlinedin Fig. 3, with details depending on the actual viewinggeometry. The angle of linear polarization “swings”through a range of values during a pulse because ofthe rotation of the star (Michel 1991). Historically, po-larimetric observations of the Crab nebula, the super-nova remnant surrounding the Crab pulsar, providedthe first evidence ever for synchrotron radiation fromastronomical objects (Oort & Walraven 1956).

5.7 Active Galactic Nuclei

With luminosities up to approximately 1015L⊙, activegalactic nuclei (AGN; see, e.g., Beckmann & Shrader2012 for a recent review) are the most luminous persis-tent objects in the universe. Their source of energy isthe accretion of interstellar matter onto supermassive– meaning M• ≈ 106−10M⊙ – black holes which arelocated in the centers of most, if not all, galaxies. Theenergy gained from accretion is (largely) radiated awayin the form of broad-band continuum emission thatis observed from low-frequency radio to high-energy γenergies. AGN emission shows strong variability andcharacteristic statistical properties (e.g., Park & Trippe2012; Kim & Trippe 2013). The radiation from AGNcrudely falls into two physical regimes. At low energiesranging roughly from radio to ultraviolet frequencies,the emission is dominated by synchrotron radiation. Athigher energies, the radiation is probably produced byinverse Compton scattering of low-energy synchrotronphotons.

As AGN are synchrotron sources, their emission islinearly polarized; see also the example provided byFig. 4. Accordingly, AGN polarization has been stud-ied extensively for several decades and has been used toaddress the geometries of magnetic fields and the mat-ter distributions (notably particle densities via Faradayrotation) in and around active nuclei (see, e.g., Saikia& Salter 1988 for an overview). Degrees of linear po-larization are mL ∼< 20%, with typical values around

mL ≈ 5% (e.g., Trippe et al. 2010, 2012a). Circularpolarization has been observed in a handful of sourceson levels mC ∼< 1% (e.g., Agudo et al. 2010).

The outflows of matter from AGN, especially theformation of collimated jets which extend over severalmegaparsecs in extreme cases, are intimately linked tothe immediate (tens of Schwarzschild radii) environ-ment of the central black hole and the geometry of themagnetic fields located there (e.g., Narayan & Quataert2005). AGN jets are – largely – optically thin emittersof synchrotron radiation best observable at radio fre-quencies. Accordingly, linear polarization is used totrace the orientation and strength of magnetic fields

along the jets.‖ Observations at multiple wavelengthspermit the use of Faraday rotation and Faraday de-polarization as a probe of magnetic fields and matterdistributions (e.g., Macquart et al. 2006; Taylor et al.2006; Trippe et al. 2012b). A somewhat unexpectedproperty of AGN jets was the discovery of inverse de-polarization – higher degrees of linear polarization atlonger wavelengths – in some sources which has been in-terpreted as a “conspiracy” of spatial small-scale struc-ture and Faraday rotation (e.g., Homan 2012). Occa-sional observations of circular polarization in AGN jets,most notably in 3C 84 with polarization levels up tomC ≈ 3%, have been attributed to polarization con-version (Homan & Wardle 2004).

Historically, polarimetric observations of the activeSeyfert galaxy NGC 1068 helped to establish the –nowadays standard – viewing angle unification schemeof AGN. Spectropolarimetry at optical wavelengthsshows that the total flux received from the galaxy isactually composed of two components: one – unpolar-ized – from directly observable gas with narrow emis-sion lines, one – linearly polarized – from gas with muchbroader emission lines located within a dust torus andvisible only indirectly via Thomson scattering towardthe observer (cf., e.g., Baek et al. 2007; Lee 2011). Thisobservation eventually removed the distinction betweennarrow and broad emission line galaxies which werefound to be different realizations of AGN (Miller &Antonucci 1983; Miller, Goodrich & Mathews 1991).

5.8 Galactic Magnetic Fields

Disk galaxies and clusters of galaxies are permeatedby large-scale (many kpc) magnetic fields with fieldstrengths B on the order of µGauss. In case of diskgalaxies, these fields are aligned with the galactic planeand follow closely the galactic structure, especially spi-ral arms (see, e.g., Fletcher et al. 2011 for an impres-sive example). The fields are supposed to be generatedvia amplification of primordial cosmic magnetic fields– with B ∼ 10−20G – by “galactic dynamos” drivenby galactic rotation. The most widely applied modelis the α–Ω disk dynamo which comprises as parame-ters (i) the angular speed Ω of galactic rotation and(ii) the quantity α = −τ(v · ∇ × v)/3, with τ be-ing the decorrelation time of plasma turbulences andv being the plasma velocity (the expression in bracketsis also referred to as “kinematic helicity”). In case ofgalaxy clusters, the fields supposedly originate from ex-tended AGN jets (§ 5.7) which carry strong magneticfields into the intragalactic medium and where theseare dissolved over time (Wielebinski & Krause 1993;Kulsrud & Zweibel 2008).

‖At this point it is important to note that the observed polar-ization of extended sources is a function of angular resolution:if several individual emitters of polarized radiation fall withinthe same resolution element (the point spread function or beamof the instrument), the polarization signal can be averaged outpartially – a phenomenon known as beam depolarization.

POLARIZATION AND POLARIMETRY 21

The analysis of large-scale magnetic fields is basedon signatures of their interaction with the interstellaror intergalactic medium, specifically:

(i) Faraday rotation (§ 3.8.3) of radiation from pul-sars or extragalactic background sources (cf. Clarke2004; Kronberg 2004; Kronberg & Newton-McGee2011).

(ii) Weak Zeeman effect line splitting, especially theV (ν) profiles in H i emission and absorption lines(§ 3.5; cf. Heiles & Robishaw 2009).

(iii) Polarized synchrotron radiation (§ 3.4; cf. Heald2009).

(iv) Polarization arising from scattering at magneti-cally aligned dust grains (§ 3.1.2; cf. Pavel 2011).Notably, this method provided detailed insightinto the magnetic field geometry within the centerof the Milky Way (Nishiyama et al. 2010) and inparts of the Large Magellanic Cloud (J. Kim et al.2011).

(v) The Chandrasekhar-Fermi effect (§ 3.8.6; Chan-drasekhar & Fermi 1953)

As should be clear from the discussion provided in §§ 3and 3.8, (i) and (ii) provide information on magneticfield components along the line of sight, whereas (iii),(iv), and (v) provide information on field componentsperpendicular to the line of sight.

5.9 Gamma Ray Bursts

Gamma-ray bursts (GRB; e.g., Piran 2005; Gehrelset al. 2009) are short, intense pulses of soft (hundredsof keV) γ rays of cosmological origin occurring a fewtimes per day. GRBs last from fractions of a secondto hundreds of seconds; with luminosities up to about1046 W they are among the most luminous (transient)sources of radiation in the universe. According to theirduration, GRBs fall into either of two groups:

Long GRBs typically last tens of seconds and are asso-ciated with type Ib/c supernovae. They are assumed tooriginate from collapsars, massive evolved stars (prob-ably Wolf-Rayet stars) whose cores collapse into blackholes.

Short GRBs usually last less than one second. They areassumed to be caused by mergers of compact objects inbinary systems, like two neutron stars or one neutronstar and one stellar black hole.

In either case, the outflowing plasma is collimatedinto relativistic jets with opening angles of a few de-grees; this explains the very large apparent isotropicluminosities of GRBs. The GRB emission results fromsynchrotron and inverse Compton radiation from rela-tivistic electrons. The combination of synchrotron radi-ation (§ 3.4) and non-isotropic geometry should lead tosubstantial linear polarization, and, indeed, degrees ofpolarization up to 30% have been reported (Gotz et al.2013; Mundell et al. 2013). Sufficient measurement ac-curacies provided, the polarization can be used to probe

the plasma-physical conditions and magnetic fields inGRBs similar to the procedures for AGN (§ 5.7).

As noted by, e.g., Toma et al. (2012), polarized high-energy emission from cosmological sources like GRBscan be used to probe a vacuum birefringence arisingfrom a violation of the Lorentz invariance of Einstein’stheory of relativity.

5.10 Cosmic Background Radiation

The cosmic microwave background (CMB) is supposedto originate from the hot plasma filling the universe ap-proximately 400 000 years after the big bang. To firstorder, the CMB corresponds to thermal emission froma black body with a temperature of ≈2.7K. Plasmadensity fluctuations imprint characteristic fluctuationswith amplitudes on scales of µK into the angular distri-bution of the CMB. In addition to fluctuations in thetotal intensity, one may expect localized linear scatter-ing polarization if the radiation propagating throughthe plasma shows quadrupole anisotropies – differencesin intensities at angles of 90 in the sky plane. De-pending on the underlying geometry, two signatures ormodes of polarization have to be distinguished (Zal-darriaga & Seljak 1997; Kamionkowski, Kosowsky &Stebbins 1997).

E mode polarization. A polarization geometrywhere the orientations of polarization are perpendicu-lar to the gradient of a local perturbation of the CMBis referred to as electric-field like (hence E) or gradientmode (G) polarization. By construction, such a polar-ization pattern does not show a handedness. E modepolarization can be attributed to local energy densityfluctuations, also known as scalar perturbations.

B mode polarization. A local curl pattern of po-larization with distinct handedness is referred to asmagnetic-field like (hence B) or curl mode (C) polariza-tion. The amplitudes of those patterns are supposed tobe roughly one order of magnitude weaker than thoseof E mode signatures. By geometry, B mode polariza-tion requires tensor perturbations of the CMB. Thoseperturbations occur due to the propagation of gravi-tational waves through the CMB plasma; accordingly,measurements of B mode polarization are a key exper-iment for probing primordial gravitational waves andcosmic inflation theories.

In the past decade, E mode polarization has beenobserved by a variety of ground based CMB telescopesin the approximate frequency range 30–150GHz (e.g.,Leitch et al. 2002; Kovac et al. 2002; Park & Park 2002;Readhead et al. 2004; Takahashi et al. 2010). A typ-ical CMB telescope is designed as an interferometerwith multiple receivers located on a common carrierplatform spanning a few meters in diameter. By de-sign (using Rayleigh’s criterion for angular resolution)CMB telescopes are sensitive to structures on angu-lar scales of about 1–2, i.e. the characteristic sizescale of E mode polarization patterns. More recently,the Planck satellite has begun a polarization monitor-

22 S. TRIPPE

ing program aimed at both E and B modes (see, e.g.,Lamarre et al. 2003 for technical details), and an obser-vation of B mode polarization by a ground-based CMBtelescope has been reported (Hanson et al. 2013).

6. CONCLUSIONS

Polarization of light and polarimetry play fundamen-tal roles in astrophysics. Polarization is fundamentallylinked to the internal geometry of sources of radiation:the strengths and orientations of magnetic fields, thedistribution and orientation of scattering particles likedust grains, the microscopic structure of reflecting sur-faces, or intrinsic anisotropies of the primordial plasmafilling the early universe. Accordingly, polarimetry hasfound application in a vast variety of astrophysicalfields of study ranging all the way from solar physicsto cosmology, comprising even a “personal touch”: Un-derstanding the interplay between circularly polarizedstarlight and the interstellar medium might help to un-derstand the formation of life on Earth.

Reviewing its applications, it is evident that po-larimetry is a powerful tool for astrophysics; it pro-vides rich information on the physics of targets thatcannot be obtained in any other way. Consequently,a large number of dedicated observational instrumentshas been constructed and progress is fast. One impor-tant current trend is the development of instrumentsdedicated to polarimetry at the high-energy end ofthe electromagnetic spectrum, at X and γ ray wave-lengths (§ 4.7); another one could be investigations ofoptical polarimetric interferometry (Elias 2001). Eachnew technical development eventually opens new win-dows for observational astronomy. Likewise, more tra-ditional polarimetric techniques profit from the generalprogress in instrumentation technologies; a key aspectis the improvement of instrumental sensitivities – whichhave always been harmed by polarimetry measuringrelatively small differential fluxes by definition. Thisbeing said, we may conclude that polarimetry has thepotential for new and exciting astrophysical discoveriesin the future.

ACKNOWLEDGMENTS

I am grateful to Myungshin Im, Masateru Ishiguro,Junghwan Oh, Taeseok Lee, Jong-Ho Park, and Jae-

Young Kim (all at SNU) for valuable discussion. I ac-knowledge financial support from the Korean NationalResearch Foundation (NRF) via Basic Research Grant2012-R1A1A2041387. Last but not least, I am gratefulto an anonymous referee for valuable comments.

REFERENCES

Agudo, I., Thum, C., Wiesemeyer, H. & Krichbaum, T.P.2010, A 3.5mm polarimetric survey of radio-loud activegalactic nuclei, ApJSS, 189, 1

Angel, J.R.P. 1974, Mechanisms that produce linear andcircular polarization, in: Gehrels, T. (ed.), Planets,

stars, and nebulae studied with photopolarimetry, U.Arizona Press, Tucson, 54

Arzoumanian, D., et al. 2011, The contribution of star-spotsto coronal structure, MNRAS, 410, 2472

Bachor, H.-A. & Ralph, T.C. 2004, A guide to experimentsin quantum optics, 2nd edn., Wiley-VCH, Weinheim

Baek, K.M., et al. 2007, Polarization of Thomson scat-tered line radiation from broad absorption line outflowsin quasars, JKAS, 40, 1

Beckmann, V. & Shrader, C. 2012, Active Galactic Nuclei,Wiley VCH, Weinheim

Bellazzini, R. & Spandre, G. 2010, Photoelectric polarime-ters, in: Bellazzini, R., et al. (eds.), X-ray polarimetry:a new window to astrophysics, Cambridge Univ. Press,Cambridge, 19

Berdyugina, S.V. & Fluri, D.M. 2004, Evidence for theHanle effect in molecular lines, A&A, 417, 775

Berdyugina, S.V. 2009, Stellar magnetic fields across theH-R diagram: observational evidence; in: Strassmeier,K.G., et al. (eds.), Cosmic magnetic fields: from planets,to stars and galaxies, Proc. IAU Symp. 259, CambridgeUniv. Press, Cambridge, 323

Beskin, V.S., Gurevich, A.V. & Istomin, Y.N. 1993, Physicsof the pulsar magnetosphere, Cambridge Univ. Press,Campbridge

Bloser, P.F., et al. 2010, The Gamma-RAy Polarimeter Ex-periment (GRAPE) balloon payload, in: Bellazzini, R.,et al. (eds.), X-ray polarimetry: a new window to astro-physics, Cambridge Univ. Press, Cambridge, 314

Born, M. & Wolf, E. 1999, Principles of optics, 7th edn.,Cambridge Univ. Press, Cambridge

Bowell, E. & Zellner, B. 1974, Polarizations of asteroids andsatellites; in: Gehrels, T. (ed.), Planets, stars, and neb-ulae studied with photopolarimetry, U. Arizona Press,Tucson, 381

Bradt, H. 2008, Astrophysics processes, Cambridge Univ.Press, Cambridge

Brosseau, C. 1998, Fundamentals of polarized light: A sta-tistical optics approach, John Wiley & Sons, New York

Buenzli, E. & Schmid, H.M. 2009, A grid of polarizationmodels for Rayleigh scattering planetary atmospheres,A&A, 504, 259

Burn, B.J. 1966, On the depolarization of discrete radiosources by Faraday dispersion, MNRAS, 133, 67

Campbell, B.A. 2002, Radar remote sensing of planetarysurfaces, Cambridge Univ. Press, Cambridge

Cawthorne, T.V. & Wardle, J.F.C. 1988, Kinematic con-straints on models of the BL Lacertae object OJ 287 fromVLBI polarization observations, ApJ, 332, 696

Cellino, A., et al. 2005, A polarimetric study of asteroid25143 Itokawa, Icarus, 179, 297

Cenacchi, E., Kraus, A., Orfei, A. & Mack, K.-H. 2009, FullStokes polarimetric observations wth a single-dish radiotelescope, A&A, 498, 591

Chandrasekhar, S. & Fermi, E. 1953, Magnetic fields inspiral arms, ApJ, 118, 113

Cho, J. & Lazarian, A. 2007, Polarization of FIR emissionfrom T Tauri stars, JKAS, 40, 113

Clarke, T.E. 2004, Faraday rotation observations of mag-netic fields in galaxy clusters, JKAS, 37, 337

Clarke, D. 2010, Stellar polarimetry, Wiley-VCH,Weinheim

POLARIZATION AND POLARIMETRY 23

Das, H.K., Voshchinnikov, N.V. & Il’in, V.B. 2010, Inter-stellar extinction and polarization – a spheroidal dustgrain perspective, MNRAS, 404, 265

De Marcellus, P., et al. 2011, Non-racemic amino acid pro-duction by ultraviolet irradiation of achiral interstellarice analogs with circularly polarized light, ApJL, 727,L27

Dinh-V-Trung 2009, On the theory of astronomical masers– II. Polarization of maser radiation, MNRAS, 399, 1495

Dirac, P.A.M. 1958, The principles of quantum mechanics,4th edn., Oxford Univ. Press, Oxford

Donati, J.-F. & Landstreet, J.D. 2009, Magnetic fields ofnondegenerate stars, ARA&A, 47, 333

Draine, B.T. 2003, Interstellar dust grains, ARA&A, 41,241

Dyson, J.E &Williams, D.A. 1997, The physics of the inter-stellar medium, 2nd edn., Taylor & Francis, New York

Edlund, E. 1860, Uber die Polarisation des Lichtes derCorona bei totalen Sonnenfinsternissen, AN, 52, 305

Einstein, A. 1905, Uber einen die Erzeugung und Verwand-lung des Lichtes betreffenden heuristischen Gesicht-spunkt, Ann. Phys. (Leipzig), 322, 132

Elias, N.M., II 2001, Optical interferometric polarimetry. I.Foundation, ApJ, 549, 647

Elitzur, M., 1982, Physical characteristics of astronomicalmasers, Rev. Mod. Phys., 54, 1225

Elitzur, M. 1991, Polarization of astronomical maser radia-tion, ApJ, 370, 407

Elitzur, M. 2000, Astronomical masers and their polariza-tion; in: Trujillo-Bueno, J., et al. (eds.), Astrophysicalspectropolarimetry, Cambridge Univ. Press, Cambridge,225

Fletcher, A., et al. 2011, Magnetic fields and spiral arms inthe galaxy M51, MNRAS, 412, 2396

Fowles, G.R. 1975, Introduction to modern optics, 2nd edn.,Dover Publ., New York

Ginzburg, V.L. & Syrovatskii, S.I. 1965, Cosmic magne-tobremsstrahlung (synchrotron radiation), ARA&A, 3,297

Gehrels, N., Ramirez-Ruiz, E. & Fox, D.B. 2009, Gamma-ray bursts in the Swift era, ARA&A, 47, 567

Gotz, D., et al. 2013, The polarized gamma-ray burst GRB061122, MNRAS, 431, 3550

Goldreich, P. & Kylafis, N.D. 1981, On mapping the mag-netic field direction in molecular clouds by polarizationmeasurements, ApJ, 243, L75

Goldreich, P. & Kylafis, N.D. 1982, Linear polarization ofradio frequency lines in molecular clouds and circumstel-lar envelopes, ApJ, 253, 606

Goldstein, D. 2003, Polarized light, 2nd edn., MarcelDekker Inc., New York

Goodrich, R.W., Cohen, M.H. & Putney, A. 1995, Spec-tropolarimetry. II. Circular polarization optics and tech-niques, PASP, 107, 179

Haken, H. & Wolf, H.C. 1990, Atom- und Quantenphysik,4th edn., Springer, Berlin

Hamaker, J.P., Bregman, J.D. & Sault, R.J. 1996, Un-derstanding radio polarimetry. I. Mathematical founda-tions, A&ASS, 117, 137

Hamaker, J.P. & Bregman, J.D. 1996, Understanding radiopolarimetry. III. Interpreting the IAU/IEEE definitionof the Stokes parameters, A&ASS, 117, 161

Hamaker, J.P. 2000, Understanding radio polarimetry. IV.The full-coherency analogue of scalar self-calibration:Self-alignment, dynamic range and polarimetric fidelity,A&ASS, 143, 515

Hamaker, J.P. 2006, Understanding radio polarimetry. V.Making matrix self-calibration work: processing of a sim-ulated observation, A&A, 456, 395

Hanle, W. 1924, Uber magnetische Beinflussung der Polar-isation der Resonanzfluoreszenz, Z. Phys., 30, 93

Hanson, D., et al. 2013, Detection of B-mode polarizationin the cosmic microwave background with data from theSouth Pole Telescope, Phys. Rev. Lett., 111, 141301

Heald, G. 2009, The Faraday rotation measure synthesistechnique; in: Cosmic magnetic fields: from planets, tostars and galaxies, Proc. IAU Symp. 259, CambridgeUniv. Press, Cambridge, 591

Heiles, C., Goodman, A.A., McKee, C.K. & Zweibel, E.1993, Magnetic fields in star forming regions: Observa-tions, in: Levy, E.H. & Lunine, J.I. (eds.), Protostarsand planets III, Univ. Arizona Press, Tucson, 279

Heiles, C. 2002, A heuristic introduction to radio astronom-ical polarization, in: Stanimirovic, S. et al. (eds.), Single-dish radio astronomy: techniques and applications, ASPConf. Series, 278, 131

Heiles, C. & Robishaw, T. 2009, Zeeman splitting in the dif-fuse interstellar medium – The Milky Way and beyond;in: Cosmic magnetic fields: from planets, to stars andgalaxies, Proc. IAU Symp. 259, Cambridge Univ. Press,Cambridge, 579

Homan, D.C. & Wardle, J.F.C. 2004, High levels of circu-larly polarized emission from the radio jet in NGC1275(3C 84), ApJ, 602, L13

Homan, D.C. 2012, Inverse depolarization: A potentialprobe of internal Faraday rotation and helical magneticfields in extragalactic radio jets, ApJL, 747, L24

Huard, S. 1997, Polarization of light, Masson, Paris

Hughes, P.A., Aller, H.D. & Aller, M.F. 1985, Polarizedradio outbursts in BL Lacertae. II. The flux and polar-ization of a piston-driven shock, ApJ, 298, 301

Ishiguro, M., et al. 1997, Maximum visible polarization of4179 Toutatis in the apparition of 1996, PASJ, 49, L31

Jackson, J.D. 1999, Classical electrodynamics, 3rd edn.,Wiley & Sons, New York

Jones, R.C. 1941, A new calculus for the treatment of opti-cal systems. I. Description and discussion of the calculus,J. Opt. Soc. Am., 31, 488

Jordan, S. 2009, Magnetic fields in white dwarfs and theirdirect progenitors; in: Strassmeier, K.G., et al. (eds.),Cosmic magnetic fields: from planets, to stars and galax-ies, Proc. IAU Symp. 259, Cambridge Univ. Press, Cam-bridge, 369

Kamionkowski, M., Kosowsky, A. & Stebbins, A. 1997,Statistics of cosmic microwave background polarization,Phys. Rev. D, 55, 7368

Kim, J., et al. 2011, Near-IR polarimetry around 30 Do-radus: I. Separation of the galactic sources, JKAS, 44,135

Kim, K.-T., et al. 2011, 100-GHz band test observations ofthe KVN 21-m radio telescopes, JKAS, 44, 81

Kim, J.-Y. & Trippe, S. 2013, How to monitor AGN intra-day variability at 230GHz, JKAS, 46, 65

24 S. TRIPPE

Kitchin, C.R. 2009, Astrophysical techniques, 5th edn.,CRC Press, Boca Raton

Kovac, J.M., et al. 2002, Detection of polarization in thecosmic microwave background using DASI, Nature, 420,772

Kronberg, P.P. 2004, New probes of intergalactic magneticfields by radiometry and Faraday rotation, JKAS, 37,343

Kronberg, P.P. & Newton-McGee, K.J. 2011, Remark-able symmetries in the Milky Way disc’s magnetic field,PASP, 28, 171

Kulsrud, R.M. & Zweibel, E.G. 2008, On the origin of cos-mic magnetic fields, Rep. Prog. Phys., 71, 046901

Kwok, S. 2007, Physics and chemistry of the interstellarmedium, University Science Books, Sausalito

Kwon, Y.-J & Suh, K.-W. 2012, Properties of OH, SiO, andH2O masers in O-rich AGN stars, JKAS, 45, 139

Kwon, J., et al. 2013, Near-infrared circular polarizationimages of NGC 6334-V, ApJL, 765, L6

Lamarre, J.M., et al. 2003, The Planck High FrequencyInstrument, a third generation CMB experiment, anda full sky submillimeter survey, New Astron. Rev., 47,1017

Landau, L.D. & Lifschitz, E.M. 1997, Klassische Feldtheo-rie, 12th edn., Harri Deutsch, Thun

Lang, K.R. 2006, Astrophysical formulae. Vol. I. Radiation,gas processes, and high energy astrophysics, 3rd edn.,Springer, Berlin

Lee, H.-W. 2011, Linear polarization of a double peakedbroad emission line in active galactic nuclei, JKAS, 44,59

Lee, S.-S., et al. 2011, Single-dish performance of KVN 21-m radio telescopes: simultaneous observations at 22 and43 GHz, PASP, 123, 1398

Leitch, E.M., et al. 2002, Measurement of polarization withthe Degree Angular Scale Interferometer, Nature, 420,763

Lyne, A. & Graham-Smith, F. 2012, Pulsar astronomy, 4thedn., Cambridge Univ. Press, Cambridge

Macquart, J.-P., et al. 2006, The rotation measure and 3.5-millimeter polarization of Sagittarius A*, ApJ, 646, L111

Mandel, L. & Wolf, E. 1995, Optical coherence and quan-tum optics, Cambridge Univ. Press, Cambridge

Matsumura, M., et al. 2011, Correlation between interstel-lar polarization and dust temperature: Is the alignmentof grains by radiative torques ubiquitous?, PASJ, 63, L43

McConnell, M.L. 2010, Scattering polarimetry in high-energy astronomy, in: Bellazzini, R., et al. (eds.), X-raypolarimetry: a new window to astrophysics, CambridgeUniv. Press, Cambridge, 11

Michel, F.C. 1991, Theory of neutron star magnetospheres,U Chicago Press, Chicago

Milic, I. & Faurobert, M. 2012, Hanle diagnostics of weaksolar magnetic fields, A&A, 547, A38

Miller, J.S. & Antonucci, R.R.J. 1983, Evidence for a highlypolarized continuum in the nucleus of NGC 1068, ApJ,271, L7

Miller, J.S., Goodrich, R.W. & Mathews, W.G. 1991, Mul-tidirectional views of the active nucleus of NGC 1068,ApJ, 378, 47

Milli, J., et al. 2013, Prospects of detecting the polarimetricsignature of the Earth-mass planet α Centauri B b withSPHERE/ZIMPOL, A&A, 556, A64

Muller, H. 1948, The foundations of optics (abstract), J.Opt. Soc. Am., 38, 661

Mundell, C.G., et al. 2013, Highly polarized light from sta-ble ordered magnetic fields in GRB120308A, Nature,504, 119

Narayan, R. & Quataert, E. 2005, Black hole accretion,Science, 307, 77

Nishiyama, S., et al. 2010, Magnetic field configurationat the Galactic center investigated by wide-field near-infrared polarimetry: Transition from a toroidal to apoloidal magnetic field, ApJL, 722, L23

Oort, J.H. & Walraven, T. 1956, Polarization and compo-sition of the Crab nebula, Bull. Astron. Inst. Neth., 12,285

Ostro, S.J. 1993, Planetary radar astronomy, Rev. Mod.Phys., 65, 1235

Ott, T., Eckart, A. & Genzel, R. 1999, Variable and em-bedded stars in the Galactic center, ApJ, 523, 248

Pacholczyk, A.G. 1970, Radio Astrophysics, W.H. Freeman& Co.

Pacholczyk, A.G. & Swihart, T.L. 1970, Polarization ofradio sources. II. Faraday effect in the case of quasi-transverse propagation, ApJ, 161, 415

Pacholczyk, A.G. 1973, Circular repolarization in compactradio sources, MNRAS, 163, 29P

Park, C.-G. & Park, C. 2002, Simulation of cosmic mi-crowave background polarization fields for AMiBA ex-periment, JKAS, 35, 67

Park, J.-H. & Trippe, S. 2012, Multiple emission states inactive galactic nuclei, JKAS, 45, 147

Pavel, M.D. 2011, Constraining Galactic magnetic fieldmodels with starlight polarimetry, ApJ, 740, 21

Piran, T. 2005, The physics of gamma-ray bursts, Rev.Mod. Phys., 76, 1143

Poincare, H. 1892, Theorie mathematique de la lumiere,vol. II, Georges Carre, Paris

Rankin, J.M. 1983, Toward an empirical theory of pulsaremission. I. Morphological taxonomy, ApJ, 274, 333

Readhead, A.C.S., et al. 2004, Polarization observationswith the Cosmic Background Imager, Science, 306, 836

Reid, M.J. & Moran, J.M. 1981, Masers, ARA&A, 19, 231

Rybicki, G.B. & Lightman, A.P. 1979, Radiative processesin astrophysics, Wiley & Sons, New York

Saikia, D.J. & Salter, C.J. 1988, Polarization properties ofextragalactic radio sources, ARA&A, 26, 93

Saito, H., et al. 2009, Near-infrared imaging polarimetry ofS106 cluster-forming region with SIRPOL, AJ, 137, 3149

Sampoorna, M., Nagendra, K.N. & Stenflo, J.O. 2013, Line-interlocking effects on polarization in spectral lines byRayleigh and Raman scattering, ApJ, 770, 92

Sault, R.J., Hamaker, J.P. & Bregman, J.D. 1996, Under-standing radio polarimetry. II. Instrumental calibrationof an interferometer array, A&ASS, 117, 159

Schrijver, C.J. & Zwaan, C. 2000, Solar and stellar magneticactivity, Cambridge Univ. Press, Cambridge

Secchi, P.A. 1860, On the polarisation of light reflected bythe moon, MNRAS, 20, 70

POLARIZATION AND POLARIMETRY 25

Serkowski, K., Mathewson, D.S. & Ford, V.L. 1975, Wave-length dependence of interstellar polarization and ratioof total to selective extinction, ApJ, 196, 261

Silver, E. & Schnopper, H. 2010, Bragg crystal polarime-ters, in: Bellazzini, R., et al. (eds.), X-ray polarimetry:a new window to astrophysics, Cambridge Univ. Press,Cambridge, 34

Sparks, W., et al. 2012, Remote sensing of chiral signatureson Mars, Planet. Space Sci., 72, 111

Stenflo, J.O. 1982, The Hanle effect and the diagnostics ofturbulent magnetic fields in the solar atmosphere, SolarPhys., 80, 209

Stenflo, J.O. 2013, Horizontal or vertical magnetic fields onthe quiet sun, A&A, 555, A132

Stokes, G.G. 1852, Trans. Cambridge Phil. Soc. 9, part III,399

Sugitani, K., et al. 2010, Near-infrared imaging polarime-try of the Serpens cloud core: Magnetic field structure,outflows, and inflows in a cluster forming group, ApJ,716, 299

Takahashi, Y.D., et al. 2010, Characterization of the BI-CEP telescope for high-precision cosmic microwave back-ground polarimetry, ApJ, 711, 1141

Taylor, G.B., et al. 2006, Magnetic fields in the centre ofthe Perseus cluster, MNRAS, 368, 1500

Thompson, A.R., Moran, J.M. & Swenson, G.W. 2004, In-terferometry and synthesis in radio astronomy, 2nd edn.,Wiley-VCH, Weinheim

Thum, C., Wiesemeyer, H., Paubert, G., et al. 2008, XPOL– the Correlation Polarimeter for the IRAM 30-m Tele-scope, PASP, 120, 777

Tinbergen, J. 1996, Astronomical polarimetry, CambridgeUniv. Press, Cambridge

Toma, K., et al. 2012, Strict limit on CPT violationfrom polarization of γ-ray bursts, Phys. Rev. Lett, 109,241104

Townes, C.H. 1997, Astronomical masers and lasers, Quan-tum Electron., 27, 1031

Tribble, P.C. 1991, Depolarization of extended radio sourcesby a foreground Faraday screen, MNRAS, 250, 726

Trippe, S., Paumard, T., Ott, T., et al. 2007, A polarizedinfrared flare from Sagittarius A* and the signatures oforbiting plasma hotspots, MNRAS, 375, 764

Trippe, S. Neri, R., Krips, M., et al. 2010, The firstIRAM/PdBI polarimetric millimeter survey of activegalactic nuclei. I. Global properties of the sample, A&A,515, A40

Trippe, S. Neri, R., Krips, M., et al. 2012, The firstIRAM/PdBI polarimetric millimeter survey of activegalactic nuclei. II. Activity and properties of individualsources, A&A, 540, A74

Trippe, S., Bremer, M., Krichbaum, T.P., et al. 2012, Asearch for linear polarization in the active galactic nu-cleus 3C 84 at 239 and 348GHz, MNRAS, 425, 1192

Weisskopf, M.C., et al. 1978, A precision measurement ofthe X-ray polarization of the Crab nebula without pulsarcontamination, ApJ, 220, L117

Wielebinski, R. & Krause, F. 1993, Magnetic fields in galax-ies, A&AR, 4, 449

Wilson, T.L., Rohlfs, K. & Huttemeister, S. 2010, Tools ofradio astronomy, 5th edn., Springer, Berlin

Witzel, G., Eckart, A., Buchholz, R.M., et al. 2011, Theinstrumental polarization of the Nasmyth focus polari-metric differential imager NAOS/CONICA (NACO) atthe VLT, A&A, 525, A130

Wolstencroft, R.D. 1974, The circular polarization oflight reflected from certain optically active surfaces; in:Gehrels, T. (ed.), Planets, stars, and nebulae studiedwith photopolarimetry, U. Arizona Press, Tucson, 54

Zaldarriaga, M. & Seljak, U. 1997, All-sky analysis of po-larization in the microwave background, Phys. Rev. D,55, 1830

Zeeman, P. 1897, On the influence of magnetism on thenature of the light emitted by a substance, ApJ, 5, 332