Transcript Diapositiva 1 - Polarisation

The Physics of Polarization
Egidio Landi Degl’Innocenti
Dipartimento di Fisica e Astronomia
Università di Firenze, Italia
COST Working Group Meeting
Warsaw, May 8-11, 2010
Generalities
Polarization = phenomenon connected with the transversality character of
electromagnetic waves.
In principle, polarization can be defined for any kind of transverse waves
(elastic waves in a solid, seismic waves, waves in a guitar string, etc.
Longitudinal waves “have no polarization”.
For studying polarization, more than for any other discipline of physics, the
famous words of Galileo still sound extremely approriate:
“The Universe is written in mathematical language, and its characters
are triangles, circles, and other geometrical figures, without which it is
humanly impossible to understand a single word.”
General challenge of astronomical polarimetry: to lower the sensitivity limits
of the available instruments to smaller and smaller values (10-2, 10-3, 10-4, 105…..). It has to be remarked that this is “a never ending story”…..
Historical Introduction I
Herasmus Bartholinus (~1670) in “Experiments on double-refracting
Icelandic crystals, showing amazing and unusual refraction” gives an
early account of doble refraction, a phenomenon connected with polarization:
it was later recognized that the two rays refracted inside an Iceland spar, a
transparent variety of calcite, have different polarization properties.
Christian Huyghens (~1690) in “Treatise on light” and Isaac Newton
(~1730) in “Optiks” reflecting on Bartholinus’ experiment put forward, though
in a qualitative way, the idea of the transversal character of light (in Newton’s
words “light has sides”).
Louis Malus (~1810) in “Sur une proprieté de la lumière réflechie”
introduces in the lexicon of physics the word “polarization” as an intrinsic
property of light. He observes that reflection on a surface changes the
properties of light beams and starts producing “artificially polarized” light by
means of reflections and refractions on material surfaces. He also proves the
first physical law on polarization, the Malus law, or the cos2θ law.
Historical Introduction II
Augustin Fresnel (~1830) in “Mèmoires sur la réflexion de la lumière
polarisée” gives the definite proof of the transversality of the “luminous
oscillations”, despite the current belief that “ether is a fluid”. He also
publishes the so-called Fresnel equations, still in use today, and constructs
the first retarder (Fresnel prism) to produce circularly polarized light.
Further important contributions are brought about by François Arago (optical
activity in crystals), Jean-Baptiste Biot (optical activity in solutions, David
Brewster (the Brewster angle), William Nicol (building the first polarizer),
Michael Faraday (discovers the Faraday effect).
George Stokes (~1852) in “On the composition and resolution of streams
of polarized light from different sources” gives for the first time a fully
consistent description of polarized radiation by introducing statistical
arguments. He gives the operational definition of 4 real quantities (originally
denoted as (I, M, C, S), today (I, Q, U, V), that are now called the Stokes
parameters.
The Fresnel Equations I
Fresnel equations are at the basis of a large number of physical phenomena
concerning the reflection, refraction, and transmission of light at the
separation surface between two media of different refractive index. In
modern terms, the equations can be deduced as a boundary layer problem
of Maxwell equations. They still provide the necessary tool for the
construction of all the optical devices that are currently used in polarimetry.
The Fresnel Equations II
The Fresnel equations can be written in several different ways. In the
formalism of Muller matrices they acquire a rather compact form that
imply the introduction of four independent, complex quantities, two for
reflection and two for transmission. By writing I’ = M I, where I is the
Stokes vector of the incident beam and I’ is the Stokes vector of the
reflected (or transmitted) beam the Muller matrix M is given by
Scattering polarization
One of the most important phenomena in the physics of polarization
concerns the polarization properties of scattered radiation. Scattering can
take place either on free electrons (Thomson scattering or Compton
scattering, according to the energy of the incident photon, as compared to
the rest mass of the electron), on bound electrons (Rayleigh sctattering on
atoms or molecules).
The physical laws controlling such properties have been initially derived
from classical physics and they have been later generalized to handle
relativistic and quantum effects.
In classical terms, the laws of scattering polarization can be simply derived
by first considering the acceleration of the electron, resulting from the
electric field of the incident, polarized electromagnetic radiation beam, and
by then evaluating the polarization of the radiation emitted by the
accelerated electron according to the standard theory based on the Liénard
& Wiechart potentials. By a similar procedure it is possible to derive the
theoretical polarization of bremsstrahlung, cyclotron and synchrotron
radiation. In these cases the acceleration of the electron is just produced
either by the collision or by the Lorentz force due to the magnetic field.
Thomson and Rayleigh Scattering I
The result of classical theory for Thomson (and Rayleigh) scattering is
condensed in the equations below. Note that the matrix R (defined by
I = k R I’) is just a “geometrical matrix”. It simply contains the scalar
products between the polarization unit vectors of the scattered and of the
incident beam.
Thomson and Rayleigh Scattering II
The Rayleigh scattering matrix looks rather complicated. However, it can be
brought to a much simpler form by suitably selecting the geometry, like in
this example, where “everything” is referred to the scattering plane.
Rayleigh Scattering III
Unfortunately, when modelling astronomical objects, we are often
(not to say always) confronted with very complicated geometrical
scenarios. Obviously, we do not have the freedom, as usual in
laboratory experiments, to set things in such a way to get what is
generally called the “good geometry”. It is then very useful to
introduce particular quantities that, from one side, can handle in a
compact way even the most complicated geometrical situations,
and, on the other hand, can keep track in the mathematical
description of the relevant symmetries of the problem.
In the physics (better to say in the geometry) of scattering
polarization this is obtained by introducing an irreducible spherical
tensor, called TKQ (Landi Degl’Innocenti: 1984, Solar Physics 91, 1) ,
that has revealed to be particularly suitable both in theory and
applications.
Rayleigh Scattering IV
The tensor is defined, in terms of rotation matrices, by the expression
[R is the rotation that carries the system (e1,e2,Ω) in (x,y,z)].
In thse expressions the index i refers to the i-th Stokes parameter
Rayleigh Scattering V
By means of this tensor the Rayleigh scattering matrix can be
expressed in the form
In as far as scattering in a single spectral line is concerned, and
frequency redistribution problems are neglected, this expression
can be easily generalized to quantum mechanics by introducing
suitable “polarizabilty factors” which depend on the angular
momentum quantum numbers of the particular line considered.

where
Rayleigh Scattering
+ HANLE Effect I
In the presence of a magnetic field, the Rayleigh scattering
matrix is further modified by the introduction of a supplementary
term

where H is the magnetic field expressed in terms of the “critical
field for the Hanle effect”
Finally, if depolarizing collisions are also present, another factor
comes into play

Rayleigh Scattering
+ HANLE Effect II
Just for comparison: here is the matrix expressed via the tensor T KQ
and here is one of the 16 matrix elements expressed through more
conventional methods
Rayleigh Scattering
+ HANLE Effect III
Rayleigh Scattering
+ HANLE Effect IV
The parameter γ appearing in these expressions is H of the former slide
Rayleigh Scattering in the solar corona
Khan, A., Belluzzi, L., Landi Degl’Innocenti, E., Fineschi, S., Romoli, M.:
2011, Astron. & Astrophys. 529, A12
Rayleigh Scattering in the solar corona
+ HANLE Effect
Khan, A., Landi Degl’Innocenti, E.: 2011, Astron. & Astrophys. 532, A70
Rayleigh Scattering
in multiplets I
A further quantum-mechanical generalization concerns the laws
of scattering in multiplets. The structure of the matrix R remains
the same, but the “polarizability factor” WK gets transformed into
a frequency dependent function
The expression of the function is rather complicated. The following
slides give some representative examples. It has to be remarked that
the frequency dependence of the polarizability factor is brought about
by the so called “quantum interference”. When a photon is scattered
within the multiplet we do not know “through which upper level it is
passed”. The situation is similar to the one appearing in the Young
interference experiment: we do not know through which hole the
photon is passed. Remarkably, these interference phenomena in
multiplets have been clearly demonstrated by solar observations
(Stenflo, 1980)
Rayleigh Scattering
in multiplets II
Multiplet n. 1 of Ca II
K line
H line
Rayleigh Scattering
in multiplets III
Multiplet UV n. 1 of Fe II
More general theoretical schemes
The scattering laws previously illustrated are results that can be obtained as
limiting case of a general theoretical framework that has been developed mostly for
the interpretation of solar observations and for the diagnostics of solar magnetic
fields in sunspots, active regions, prominences, etc.
This general framework is referred to as the “Theory of the generation and
transfer of polarized radiation”. It aims at giving a self-consistent description of
the polarized radiation field propagating through an astrophysical plasma and of
the statistical properties, in terms of populations and “coherences”, of the atoms
(or molecules) composing the plasma itself.
The “atomic system” is described in terms of its density matrix whereas the
radiation field is described through the Stokes parameters. By standard
procedures of quantum electrodynamics two sets of equations are obtained: the
radiative transfer equations for polarized radition and the statistical equilibrium
equations for the density matrix of the atomic system.
The Second Solar Spectrum: main test-bench for theory
Second Solar Spectrum (Ivanov, 1991): linearly polarized spectrum of the
solar radiation coming from quiet regions close to the limb.
Reference direction
A. Gandorfer, “The Second
Solar Spectrum”, Vol II,
2002.
Observation close to the
south solar pole, 5 arcsec
inside the limb.
Observational Improvements
Wiehr (1975)
Stenflo, Baur, & Elmore (1980)
Stenflo, & Keller (1997)
Description of atomic polarization:
the density matrix formalism
The density operator
A physical system which is in a statistical
mixture of states can be convenientely
described through the density operator
Matrix elements of the density operator in the standard representation
Diagonal elements: populations of the various magnetic sublevels
Non-diagonal elements: quantum interferences (or coherences)
between pairs of magnetic sublevels.
The Non-LTE problem of the 2° kind
Radiative
Transfer
Equations
Collisions
I, Q, U, V
Magnetic
field
Statistical
Equilibrium
Equations
Comparison
with
observations
Strenghts
•Equations and expressions of rates and of radiative transfer coefficients
derived in a self-consistent way from the principles of quantum electrodyn.
•Possibility to take into account lower level polarization
•Possibility to treat multi-level atoms
•Possibility to include HFS
•Possibility to take into account the interaction with a magnetic field
in every regime (from Zeeman to complete Paschen-Back effect regime).
•Possibility to take into account both Zeeman and Hanle effects.
•Possibility to describe within a self-consistent scheme “unfamiliar”
quantum effects like level-crossing and anti-level-crossing.
•Prediction of new polarization mechanisms like the “alignment-toorientation conversion mechanism”
Successes
Modeling of Ca II IR triplet
(Manso Sainz and Trujillo Bueno 2003)
Key-ingredient: lower level polarization
8542 Ǻ
3/2 - 5/2
8662 Ǻ
1/2 - 3/2
Stenflo et al. (2000)
8498 Ǻ
3/2 - 3/2
Upper level cannot carry atomic
alignment (J=1/2): observed signal
can be interpreted only if lower level
polarization is taken into account
Manso Sainz and Trujillo Bueno (2003)
Modeling of Mg I b-lines (Trujillo Bueno (1999, 2005))
Key-ingredient: lower level polarization
5167 Ǻ
1–0
(h = 830 km)
b4
5173 Ǻ
1–1
(h = 950 km)
b2
Stenflo, Keller, and Gandorfer (2000)
5183 Ǻ
1–2
(h = 1010 km)
b1
If lower level polariz. is neglected,
a very different amount of upper
level atomic polarization is required
to reproduce the profiles observed
in the 3 lines (which share the same
upper level, and form at similar
heights!)
If lower level polarization is taken into
account, the observations can be
reproduced by assuming the same
amount of upper level polarization
in the 3 lines, and a given amount of
atomic polarization in the lower levels
(see also Trujillo Bueno 2001).
Interpretation of spectropolarimetric observations
in prominences and filaments
(e.g. Trujillo Bueno et al. 2002)
Key ingredients: Hanle effect, lower level polarization
He I 10830 Å
10829.09 Å: 0 – 1
10830.25 Å: 1 – 1
10834.40 Å: 2 – 1
Filament case (forward scattering)
Observation:
Positive polarization signal observed
in the red components (1 – 1 and 2 – 1)
Negative polarization signal observed in
the blue component (0 – 1).
Theoretical interpretation:
Hanle effect in the presence of an
inclined magnetic field.
Presence of lower level polarization
(anisotropic incident radiation differently
absorbed by lower magnetic sublevels).
Interpretation of the second solar spectrum of Ce II
(Manso Sainz et al. 2006)
key ingredient: multi-level atoms
Line formation model: plane parallel
slab of Ce II ions illuminated from
below by the continuum photspheric
radiation field.
Good qualitative agreement
between calculated and observed
polarization patterns.
The lines showing the largest
theoretical profiles are those
which show the largest observed
signals.
Sign of the Q/I signal reproduced
for most of the observed lines.
Interpretation of the second solar spectrum of Ti I
(Manso Sainz & Landi Degl’Innocenti 2002)
key ingredient: multi-level atoms
Good qualitative
agreement between
calculated and observed
polarization patterns.
The lines showing the
largest theoretical
profiles are those which
show the largest
observed signals.
Sign of the Q/I signal
reproduced for most of
the observed lines.
Differential magnetic sensitivity of Ba II 4554Å line
(Belluzzi et al. 2007)
Key ingredient: HFS + Hanle and Zeeman effects in the
incomplete Paschen-Back effect regime
Observed profile
Theoretical profile
Wavelength (Å)
Stenflo & Keller (1997)
Observation close to the north solar pole,
5 arcsec inside the limb.
Belluzzi et al. (2007)
(see also Stenflo (1997))
Weak fields
Weak fields
Strong fields
Strong fields
Vertical field
Horizontal field
Fit of the observations
Fit of spectropolarimetric
observations in the Ba II D2
line (4554Å) performed on
an active region, 9 arcsec
from the limb.
Date: 13/03/2007
Polarimeter: ZIMPOL
(IRSOL, Locarno)
(Ramelli et al. 2009)
Limitations and open problems
Unexpected failure in the interpretation of the
“MS” Q/I profile of Sc II 4247 Å
(Belluzzi, 2009)
2P o
3/2
1D o
2
4554.0 Å
2S
1/2
7 isotopes
2 is. (18%) I=3/2
5 is. (82%) I=0
4246.8 Å
1D
2
1 isotope
I=7/2
Is the three peak structure of
Ba II 4554 really due to HFS?
Impossibility to take into account PRD effects
Intensity spectrum:
PRD effects result to be important only for a limited number of very strong lines
(e.g. Ca II, Mg II resonance lines, Ly-alpha), mainly in the wings.
Second solar spectrum:
several works, starting with the pioneering one by Dumont, Omont and Pecker
(1973), pointed out that PRD effects might be the key ingredient to model the
typical triplet-peak structure observed in the Q/I profile of the “champion” Ca I
line at 4227 Å (the first one to be observed).
We now believe that without including such effects it would not be possible to
intrepret those linear polarization profiles that have been classified as “M” by
Belluzzi & Landi Degl’Innocenti (2009) (e.g. Na I D2, Ca I 4227Å,…).
Indeed, without invoking PRD effects it seems impossible to explain the
physical origin of such multi-peak structures.
Theoretical approaches with PRD effects
Redistribution matrix for resonance scattering of polarized radiation (two-level
atom with unpolarized lower level)
Omont, Smith & Cooper (1972, 1973)
Domke & Hubeny (1988)
Heuristic approach: metalevel theory
Landi Degl’Innocenti, Landi Degl’Innocenti, & Landolfi (1997)
Landi Degl’Innocenti (1998)
Generalization of the previous theoretical approach including the higher order
terms in the perturbative expansion of the atom-radiation interaction, under the
hypothesis of a two-level atom with unpolarized lower term.
Bommier (1997a, 1997b)
Full generalization of the previous theoretical approach considering the next
terms in the perturbative expansion of the atom-radiation interaction (i.e.
inclusion of second and higher order processes).
Not yet developed
Conclusions
The theoretical approach based on the density matrix formalism, and self-consistently
derived from the principles of quantum electrodynamics is presently one of the most
solid theoretical frameworks in the field of spectropolarimetry.
This theoretical approach has been highly successfull, mainly for the interpretation of
signals which can be described in the limit of CRD (“S” signals). However, it suffers
from the severe limitation that PRD effects cannot be accounted for.
The redistribution matrix derived within the standard scattering theory of spectral
line polarization is the most suitable theoretical tool for taking into account PRD
effects. However, it suffers from the severe limitation of being capable of describing
only two-level atoms with unpolarized lower level.
The interpretation of multi-peak linear polarization profiles is today the most exciting
and complex challenge for the theory of polarimetry.
The crucial step remains the one of succeeding to obtain a good theoretical fit of
the linear polarization profile observed in the Ca I line at 4227 Å, for which the
two-level atom with unpolarized lower level represents an excellent approximation.