J. Sánchez Almeida

Instituto de Astrofísica de Canarias

Magnetometry: set of techniques and procedures to determine the physical properties of a magnetized plasma (magnetic field and more ...)

Main Constraints:

– No in-situ measurements are possible; inferences have to be based on interpreting properties of the light.

– Interpretation not straightforward. The resolution elements of the observations are far larger than the magnetic structures

(or sub-structure)

Needed Tools:

– Radiative transfer for polarized light

– Instrumentation: telescopes and polarimeters

– Inversion techniques (interpreting the polarization through many

simplifying assumptions)

Purpose: –To give an overview of all ingredients that must be considered, and to

illustrate the techniques with examples taken form recent research.

–It is not a review since part of the techniques used at present are not

covered (not even mentioned). Explicitly

–Devoted to the magnetometry of the photosphere.

– No proxi-magnetometry (jargon for magnetic field

measurements which are no based on polarization)–No extrapolations of photospheric magnetic fields to

the Corona)– No in-situ measurements (solar wind)

Summary – Index (1):

– Stokes parameters, Jones parameters, Mueller matrixes and Jones matrixes– Equation of radiative transfer for polarized light– Zeeman effect– Selected properties of the Stokes profiles, ME solutions, etc.

– Polarimeters, including magnetographs– Instrumental Polarization

Instrumentation:

Radiative Transfer for Polarized Radiation.

Inversion Techniques:– General ingredients– Examples, including the magnetograph equation

Examples of Solar Magnetometry:– Kitt Peak Synoptic maps– Line ratio method– Broad Band Circular Polarization of Sunspots – Quiet Sun Magnetic fields

Summary – Index (2):

– Hanle effect based magnetometry– Magnetometry based on lines with hyperfine structure– He 1083nm chromospheric magnetometry– Polarimeters on board Hinode

Advanced Solar magnetometry.

goto end

Stokes parameters, Jones parameters, Mueller Matrixes and Jones Matrixes

– The light emitted by a point source is a plane wave

– Monochromatic implies that the EM fields describe

elliptical motions in a plane

– The plane is quasi-perpendicular to the direction of

propagation

– Quasi monochromatic implies that the ellipse

changes shape with time

x

y

0

)](cos[)(

)](cos[)(

)(Re

ReRe),( )(

twttA

twttA

tEe

EeeEetre

yy

xxiwt

jj

twwiiwt

jj

tiw jj

Quasi-monochromatic means that the ellipse change with time

time (t)

ex(t)

Frequency (1/t)

w/2

1/

2/w = 10-15 s, in the visible (5000 A)

: coherency time, for which the ellipse keeps a shape

= 10-8 s, electric dipole transition in the visible

= 5 x 10-10 s, (multimode) He-Ne Laser

= 5 x 10-10 s, high resolution spectra (

Integration time of the measuremengts: 1 s (<< << 2/w),

ellipse changes shape some 108 -109 times during the measurement

)(Re),( tEetre iwt

)(

)(

tE

tE

J

JJ

y

x

y

x

J Jones Vector, complex amplitude of the electric field in the

plane perpendicular to the Line-of-Sight (LOS). It completely describes the radiation field, including its polarization.

Consider the effect of an optical system on the light. It just transfoms

outin JJ

Most known optical systems are linear (from a polarizer sheet to a

magnetized atmosphere)

inout JmJ

: m Jones Matrix (Complex 2x2 matrix)

T

yx

dttfT

tf

teteI

0

22

1)()(

)()( (T: integration time of the measurement)

The polarization of the light can be determined using intensity detectors (CCDs, photomultipliers, etc.) plus linear optical systems.

m

outJ

inJ

VMUMQMIMIout 14131211

VMUMQMIMIout 14131211

*

*

Im

Re

yx

yx

yx

yx

JJV

JJU

JJQ

JJI

2

2

22

22

*Z Zis the complex conjugate of

Stokes Parameters, that completely characterize the properties of the light from an observational point of view

ijM describes the properties of the optical system

yyyx

xyxx

mm

mmm

**

**

Im

Re

/

/

yyyxxyxx

yyyxxyxx

yyxyyyxx

yyxyyyxx

mmmmM

mmmmM

mmmmM

mmmmM

14

13

2222

12

2222

11

2

2

(Some) Properties of the Stokes Parameters

– Two beams with the same Stokes parameters cannot be

distinguished

– Which kind of polarization is coded in each Stokes parameter?

– The Stokes parameters of a beam the combines two

independent beams is the sum of the Stokes parameters of the

two beams

– Any polarization can be decomposed as the incoherent

superposition of two fully polarized beams with opposite

polarization states

– A global change of phase of the EM field does not modify the

Stokes parameters

(Some) Properties of the Linear Optical Systems

– Only seven parameters characterize the change of polarization

produced by any optical system. A Jones matrix is characterized by 4

complex numbers (8 parameters) minus an irrelevant global phase. – The modification of the Stokes parameters produced by one of

these systems is linear

inout SMS

V

U

Q

I

S

44434241

34333231

24232221

14131211

MMMM

MMMM

MMMM

MMMM

M

Stokes vector

Mueller Matrix

–The Mueller matrix contain redundant information. It has 16 elements, but only seven of them are independent. The relationships bewteen the elements are not trivial, though.

– The Mueller matrix becomes very simple if the optical element is weakly polarizing, i.e., if

2221

1211

10

01

aa

aam with 1ija then

IQUV

QIVU

UVIQ

VUQI

M

1000

0100

0010

0001

2211

2112

2112

2112

2112

2211

2211

aa

aa

aa

aa

aa

aa

aa

Q

U

V

V

U

Q

I

Im

Im

Re

Im

Re

Re

Re

- Mueller Matrix for an optical system producing selective absorption

IQUV

QIVU

UVIQ

VUQI

M

1

0

0

0

V

Q

U

aV

aU

aQ

aI

V

U

Q

I

a

a

a

a

V

U

Q

IStokes Vector de type of

absorbed light

Change of amplitude produced by the selective OS

linear polarizer transmitting the vibrations in the x-axis

0

0

1

1

a

a

a

a

V

U

Q

I

1000

0100

001

001

M

0

0

1

V

U

Q

I

Then for unpolarized input light one ends up with

- Mueller Matrix for an optical system producing selective retardance

IQUV

QIVU

UVIQ

VUQI

M

1

aV

aU

aQ

V

U

Q

I

V

U

Q

0

0

0

0

a

a

a

a

V

U

Q

IStokes Vector de type of

polarization that is retarded

Change of phase produced by the selective OS

- The Mueller matrix of a series of optical systems is the product of the individual matrixes. The order does matter

j

jMM

if the chain is formed by weakly polarizing optical systems, then the order of the different elements is irrelevant

i

i

i

i

i

i MMMM 11 )(

Equation of Radiative Transfer for Polarized Light

zobserver

line-of-sight

layer of atmosphere

S+S S

zzS

SzM

zSS emi

i

1

V

U

Q

I

S

zzSem

Emission produced by the layer

zMz i Mueller matrix of i-th process changing the polarization

zzS

SzM

zSS emi

i

1

z

SS

zM

zS em

i

i

aiiaiiaiiaii

aiiaiiaiiaii

aiiaiiaiiaii

aiiaiiaiiaii

IQUV

QIVU

UVIQ

VUQI

i

IQUV

QIVU

UVIQ

VUQI

zzM

zzAii change of amplitude

zzPii change of phase

taiaiaiai VUQI Stokes vector of the selective absorption + retardance

emIQUV

QIVU

UVIQ

VUQI

V

U

Q

I

V

U

Q

I

dzd

?

?

?

?

iaiiV

iaiiQ

iaiiV

iaiiI

VzP

QzP

VzA

IzA

.

.

.

.

0

0

0

B

IQUV

QIVU

UVIQ

VUQI

em

?

?

?

?

Emission term ? Simple assuming emitted radiation field is in LTE (Local Thermodynamic Equilibrium). In TE

0

dzd

0

0

0

B

V

U

Q

I

and with B the Planck function

then

V

U

Q

BI

V

U

Q

I

dzd

IQUV

QIVU

UVIQ

VUQI

iaiiV

iaiiI

VzA

IzA

.

.

iaiiV

iaiiQ

VzP

QzP

.

.

Radiative transfer equation for polarized light in any atmosphere whose emission is produced in LTE

linear polarizer transmitting the vibrations in the x-axis

0

0

1

1

a

a

a

a

V

U

Q

I

There is just one i which absorbs

V

U

Q

I

zA

V

U

Q

I

dzd

1000

0100

0011

0011

and no emission (B=0)

V

U

Q

I

zA

V

U

Q

I

dzd

1000

0100

0011

0011

inout ))0 QIQIQIdzd (()(

0))2 0

2

inout

LdzzA

eQIQIQIzAQIdzd

(()()(

0UU2 0

2

inout

LdzzA

eUzAUdzd

inoutV

U

Q

I

V

U

Q

I

0000

0000

0011

0011

21 Typical Mueller matrix of a linear

polarizer

],[ Lz 0

Zeeman EffectPurpose: work out the ´s and´s in the absorption matrix in the case of a magnetized atmosphere

– Electric dipole transitions– Hydrogen-like atoms– Linear Zeeman effect

Work out contributions to the change of polarization due to:

2) Continuum absorption

1) Spectral line absorption

Assumptions:

The wave function characterizing eigenstate of theses Hydrogen-like atoms can be written down as

tEi

iM eertr ),(),,,( 0

where M is the magnetic quantum number and E is the energy of the level.

r

The electric dipole of the corresponding distribution of charges will be

dvrqdvolume

2

Spectral line absorption

dvrqdvolume

2

When you have a transition between states b (initial) and f (final), the wave function is a linear combination of the two states

ffbb tctc )()(

tcc

tcc

ff

bb

when 1 and 00

when 0 and 10

)(

)(

dvrcqcddvolume

fbfb ** Re 20

dvreecqcdvolume

MMi

fb

tEE

i

fbfb

fb )(** Re 000 2

0d

constant over the period of the wave

dvreecqcddvolume

MMifb

tEE

i

fbfb

fb )(** Re 000 2

0

1

20

1

21

0

0

ier

ier

rrr ii

sinsincos

cos

cossin

cossin

drdeide

drdeide

drdede

tEE

iMMi

tEE

iMMi

tEE

iMMi

fb

fb

fb

fb

fb

fb

0

1

0

1

1

0

0

2

0

1

2

0

1

2

0

)(

)(

)( 002

0

pdeip

Which leads to the selection rules for E-dipole transitions

each one associated with a polarization

1 ,0 M

r

y

z

x

observer

We are interested in the projection in the plane perpendicular to the line of sight (x-y plane)

sin

cos

0

dd y

0

0

1

dd x

a) For M=0

0

0

0

1

1

0

0

iwt

x ed Re

)cos(sin

sin

cosRe wted iwty

1

1

0

0

x

y

)(td

0

0

1

1

2

sin

V

U

Q

I

There are only three types of polarization

b) For M=Mb- Mf=+1

)cos(Re wtied iwtx

0

0

1

0

1

)sin(cos

sin

cosRe wtied iwty

0

0

1

cos

sin

cos

2

0

12

2

V

U

Q

I

b) For M=Mb- Mf=-1

1

x

y

)(td

cos

1

x

y

)(td

cos

cos

sin

cos

2

0

12

2

V

U

Q

I

If the atom is in a magnetized atmosphere, the energy of each Zeeman sublevel is different, which produces a change of resonance frequency of the transitions between sublevels depending on M,

w

w0

B=0

w

w0

B=B0w

0Bw

Associated to each transition there is a absorption profile plus a retardance profile

w0w0w w0

w0w

In short: for an electric dipole atomic transition, only three kinds of polarizations can be absorbed. They just depend on M (with M the difference of magnetic quantum numbers between the lower and the upper levels)

x

observer

B

y

x

y M=0

x

y

M=-1

Bw

w0w0w

cos

x

y

M=+1

w0w0w

absorption retardance

Continuum Absorption

Although, no details will be given, it is not difficult to show that the continuum absorption has a characteristic polarization for selective absorption of the order of (Kemp 1970),

)/( kGBV

U

Q

I

a

a

a

a

510

0

0

1

– For the solar magnetic fields (1kG magnetic field strengths), the

continuum absorption is unpolarized unless you measure degrees

of polarization of the order of 10-5.– In white dwarfs, B ~ 106 G, leading to large continuum

polarization (~ 1%)

Radiative Transfer Equation in a Magnetized Atmosphere

The equation is generated considering the four types of polarization that are possible

V

U

Q

BI

V

U

Q

I

dzd

IQUV

QIVU

UVIQ

VUQI

cos2

2sinsin

2cossin

cos1

22

cos2

2sinsin

2cossin

cos1

22

0

2sin

2cos

1

sin2

0

0

0

1

2

2

2

2

2

2

2 lllc

V

U

Q

I

kkk

same for ´s with replacing ´s with ´s

x

observer

B

y

V

U

Q

BI

V

U

Q

I

dzd

IQUV

QIVU

UVIQ

VUQI

22

222

222

122

2

2

22

lV

lU

lQ

lcI

k

k

k

k

sinsin

cossin

)cos(sin

22

222

222

2

2

lV

lU

lQ

k

k

k

sinsin

cossin

Unno-Rachkovsky Equations

Zeeman triplet

general Zeeman pattern

effect of a change of macroscopic velocity

effect of a change of magnetic field strength

weak magnetic field strength regime

Selected Properties of the Stokes Profiles

Stokes Profiles representation of the four stokes parameters as a function of wavelength within a spectral line

Stokes Profiles

1.- Symmetry with respect to the central (laboratory) wavelength of the spectral line. If the macroscopic velocity is constant along the atmosphere, then

I() = I(- )Q() = Q(- )U() = U(- )V() = -V(- )

wavelength - laboratory wavelength of the spectral line corrected by the macroscopic velocity

No proof given, but it follows from the symmetry properties of the ´s and ´s of the absorption matrix

these symmetries disappear the velocity varies within the resolution elements (asymmetries of the Stokes profiles)

Symmetries and asymmetries Stokes Profiles

2.- Weak Magnetic Field Approximation,

the width of the absorption and retardance coefficients of the various Zeeman components are much smaller than their Zeeman splittings

if is the Zeeman splitting of a Zeeman triplet, and D is the width of the line, it can be shown that (e.g., Landi + Landi 1973)

)/(

)/(

)/(

)/(

DBVV

DBUU

DBQQ

DBIII

2

22

22

220

)/(

)/(

)/(

DBVV

DBUU

DBQQ

2

22

22

then to first order in ( / D )

V

U

Q

BI

V

U

Q

I

dzd

IV

IV

VI

VI

00

00

00

00

00 UQdz

Ud

dz

Qdn

n

n

n

Since there is no polarization at the bottom of the atmosphere

))(()(

BVIdz

VIdVI

)cos(

)(cos)(

Blc

BlcVI

kkd

dkk

(a)

(b)

)cos( BlcVI kk

I+V and I-V follow to equations that are identical to the equation for unpolarized light except that the absorption is shifted by cos B

)()cos()(

BVIkkdz

VIdBlc

If the longitudinal component of the magnetic field is constant then cos B is constant and I+V and I-V are identical except for a shift

I-VI+V

2 cos B

BB

BB

BB

d

dI

d

dfVIVIV

fVIVII

d

dfffVI

d

dfffVI

cos)(

cos)(

)()(

)()()(

cos)(

)()cos(

cos)(

)()cos(

2

12

1

BddI

V cos

)()(

Magnetograph equation: the Stokes V signal is proportional to the longitudinal component of the magnetic field

V cos B

0

observer

B

cosB

The previous argumentation is based on the assumption that the Zeeman pattern is a triplet (one component, one + component and one - component). If the pattern is more complex but the magnetic field is weak, one can repeat theargumentation to show that everything remains the same except that the full Zeeman pattern has to be replaced by a equivalent Zeeman triplet whose splitting is

BgeffB

effg Is the so-called effective Landé factor, and it equals one for the classical Zeeman effect

4.- Stokes profiles of an spatially unresolved magnetic structure (2-component magnetic atmosphere).

resolution element

non-magnetic magnetic

VVVV

UUUU

QQQQ

III

magnonobs

magnonobs

magnonobs

magnonobs

)1(

)1(

)1(

)1(

filling factor, i.e., fraction of resolution element filled by magnetic fields

:area total

area red

VV

IIIIII

obs

magnonmagnonobs

) (if )1(

Effect on the magnetograph equation cos

)()( B

d

dICV

cos with )(

)( BBBd

dIV effeff

obsobs

Bsd

observer

pixelpixel

eff dssdB /

B

Magnetic flux density

4.- Milne-Eddington solution of the Radiative Transfer Equation for Polarized Light (RTEPL).

Assumptions: all those needed to get an analytic solution of the of the radiative transfer equations for polarized light

Importance: Used for measuring magnetic field properties

RTEPL: first order linear differential equation. Admits an analytic formal solution of the coefficients are constant (basic maths)

V

U

Q

BI

V

U

Q

I

dzd

IQUV

QIVU

UVIQ

VUQI

V

U

Q

BI

V

U

Q

I

dz

d

IQUV

QIVU

UVIQ

VUQI

V

U

Q

I

S

0

0

0

1

1

IQUV

QIVU

UVIQ

VUQI

c

K

1

dzd c

)( 1SKS

Bd

d

continuum optical depth

Compact form of the RTEPL

)( 1SKS

Bd

d

Assumptions:

the ratio line to continuum absorption coefficient does not depend on optical depth

The source function depends linearly on continuum optical depth

depth opticalith constant w / cl

10 BBB Broadening of the line constant (both Doppler and damping)

Magnetic field vector constant with depth

… all them together lead to constant absorption matrix

depth opticalith constant w K

constant and both with , solutionstry 1 100 SSSSS

)( 1SKS

Bd

d

)()( 11001 1SK1SKS

BB

1S01SK

1111 )( BB

1SKS1SKS 1

010001 )( BB

1K1SS 10

10)0( BB

Milne-Eddington solutions of the RTEPL (e.g., Landi Degl´Innocenti, 1992)

222222222

21

21

21

222210

)()(

/)()(

/)()(

/)()(

/)(

VVUUQQVUQVUQII

VVUUQQVUQQUIVI

VVUUQQUQVVQIUI

VVUUQQQVUUVIQI

VUQII

BV

BU

BQ

BBI

Free parameters: 1. Magnetic field strength2. Magnetic field azimuth3. Magnetic field inclination4. B0

5. B1

6. Macroscopic velocity7. Doppler broadening8. Damping 9. Strength of the spectral line

IDL

5.- 180o azimuth ambiguity (exact)

x

observer

B

y

x

observer

o180

B

y

These two magnetic fields produce the same polarization, therefore, one cannot distinguish them from the polarization that they generate.

IDL

6.- Stokes V reverses sign upon changing the sign of the magnetic field component along the line-of-sight (approximate).

cos)180cos( and cos since o V

x

observer

B

yx

o180

observer

B

y

)()180( o VV IDL

7.- Q=U = 0 for longitudinal magnetic fields. V=0 for transverse magnetic fields. (Approximate.)

x

0

observer

B

yx

o90

observer

B

y

IDLQ=U=0

V=0

Polarimeters

– Modulation package– Intensity detector– Calibration package– Instrumental polarization

Basic elements:

VpMUpMQpMIpMpI jjjjjout~)(~)(~)(~)()( 14131211

optics modulator (pj)

calibration optics

telescope+ optics

optics

V

U

Q

I

V

U

Q

I

~

~

~

~

Intensity detector

out

out

out

out

V

U

Q

I

V

U

Q

I

V

U

Q

I

Telescope

Matrix

Mueller

~

~

~

~

)(

)(

)(

)(

)(

~

~

~

~

4

3

2

1

1

pI

pI

pI

pI

pM

V

U

Q

I

out

out

out

out

kij

Modulation package

Optical system whose Mueller matrix can be (strongly) varied upon changing a set of control parameters.

Usually the last element is an optical element that fixes the polarization state of the exit beam, but this is not always the case.

Example

fixed linear polarizer rotating retarder (/4)

VUQI

V

U

Q

I

out

out

out

out

~)sin(~)cos()sin(~)(cos~ 2222

0

0

1

1

2

Intensity detector for example a CCD

Calibration package

Optical system whose exit polarization is known. It allows to determine the (linear) relationship bewteen the intensities measured by the intensity detector and the input polarization.

2

22

2

12

sin

cossin

cos

out

out

out

out

V

U

Q

I

Examplefixed linear polarizerrotating retarder (/4)

Instrumental Polarization

Ideally, one would like to place calibration optics in front of the optical system used to measure, including the telescope. Unfortunately, this is not possible (there are not high precision polarization optics with the size of a telescope). This causes that the solar polarization is modified (by the telescope etc.) before we can calibrate the system: instrumental polarization.

It is an important effect

(mostly) produced by oblique reflections (e.g. folding mirrors, and windows (stress induced birefringence of the vacuum windows)

GCT Obs. Teide

SPh, 134, 1

Techniques to overcome the instrumental polarization

a) carring out the analysis (the calibration) in the optical axis of the telescope (before the optical system loses axi-symmetry). Specially designed telescopes like THEMIS (Obs. Teide).

b) modeling (and correcting for) the Mueller matrix of the telescope.

The theoretical expression for the Mueller matrixes of all individual optical elements forming the telescope are known (given the geometry the light path, complex refractive indexes of the mirrors, specific retardances of the windows, and the like). It is possible to write down a theoretical Mueller matrix than can be confronted with observations. One can use this Mueller matrix to correct the measurements

j

jMM

Teslecope

matrix

Mueller

V

U

Q

I

V

U

Q

I

Telescope

Matrix

Mueller

~

~

~

~

V

U

Q

I

V

U

Q

I

~

~

~

~1

Telescope

Matrix

Mueller

Instrumental Polarization: removing I V crosstalk

V

U

Q

I

V

U

Q

I

Telescope

Matrix

Mueller

~

~

~

~

VMUMQMIMV 44434241

~

V and ,,UQIsince

VMIMV

IMI

4441

11

~

~

cc

cc

IMV

IMI

41

11

~

~

cc

c

c

cc

I

V

M

MI

I

IVV

IIII

11

44~/~

~~~

/~

/~

at continuum wavelengths V=0

CCD

(longitudinal) Magnetograph

2 states modulator/4-plate + linear

polarizer

V

U

Q

I

Narrow-band color

filter

then2

1

)()(

)()(

VICtI

VICtI

out

out

)()()()(

)()(

)()(

21

21

21

21

IV

and

tItItItI

tItII

tItIV

outout

outout

outout

outout

Magnetogram : just an image of Stokes V in the wing of a spectral line.

Order of magnitude of the degree of polarization to be expected in the various solar magnetic structures (for a typical photospheric line used in magnetic studies):

regionsnetwork -interin 10

regionsnetwork in 1

regions plagein 10

sunspotsin 30

%.

%

%

%

IV

IV

IV

IV

Instrumental Polarization: Seeing Induced Crosstalk

Important bias of any high angular resolution observation, although it is easy to explain in magnetograph observations.

If the two images whose difference should render Stokes V are not taken strictly simultaneously (within a few ms, the time scale that characterizes atmospheric turbulence variations) then Stokes I Stokes V

)()()(

)()()(

222

111

xVxItI

xVxItI

out

out

2with

)

210

00210

212121

/)(

()()(

)()()()()()(

xxx

xVxVttdtxd

xI

xVxVxIxItItI outout

(Lites 1987)

Seeing Induced Crosstalk

How to solve the problem?

1. Using high frequency modulation, so that the atmosphere is frozen during a modulation cycle. (ZIMPOL like.)

2. Using simultaneous spatio-temporal modulation. Preferred technique in ground based observations.

3. Applying image restoration before demodulation. (SST approach.)

4. Going to space (e.g. Hinode), but then you have jitter from the satellite.

Techniques to deduce physical properties of the magnetic atmosphere upon the interpretation of the polarization that it produces.

Ingredients:

model atmosphere (assumptions on the properties of

atmosphere whose magnetic field will be inferred)

polarized spectral synthesis code

fitting technique (e.g., 2 minimization techniques)

All solar magnetic fields measurements (magnetometry) need, and are based on, these ingredients and assumptions. Frequently the assumptions are implicit and people tend to think that they do not exit. The inferred magnetic field depends, sometimes drastically, on the asumptions.

Longitudinal magnetograph

It is just an image showing the degree of circular polarization in the flank of spectral line.

– If the solar atmosphere where the polarization is produced has a discrete number of magnetic component– If the magnetic field of this component does not vary, neither along the line-of-sight nor across the line-of-sight– If the temperature and pressure of the atmosphere does not depend on the magnetic field– If the velocities is constant in the resolution element

Model atmosphere:

Synthesis Code:

– Multi component atmosphere– Weak magnetic field approximation

Fitting technique:

– No sophistication; one observable and one free parameter

resolution

icomponents

iiii

i

icomponents

iiii

resolution

components

iii

sdBd

dICBf

ddI

C

ddI

BfCVfdsxVV

)(cos

)(

cos)(),()(

#

##

resolution

i

resolution

components

iii sdIIfdsxII )()(),()(

#

resolution

resolution

sd

sdB

cIV

)()()(

d

IdCc i )(ln

)( A calibrated magnetograph gives the longitudinal component of the magnetic flux density (mag flux per unit surface)

Milne-Eddington fitting technique

– If the solar atmosphere where the polarization is produced has two components: one magnetic and one non-magnetic–If the magnetic field of this component does not vary, neither along the line-of-sight nor across the line-of-sight– If the line to continuum absorption coefficient ratio does not vary with height in the atmosphere– If the source function varies linearly with continuum optical depth

Model atmosphere:

Synthesis Code:

– Milne Eddington analytic solution of the radiative transfer equations for polarized light

Fitting technique:

– Non-linear least squares minimization

(e.g. Skumanich & Lites 1987)

data

syntheticobserved StokesStokes22

Input model atmosphereB,,, ...

new atmosphere B,,, ...giving a smaller 2

,,,

22

2

B

,,,

V

U

Q

I

V

U

Q

I

B

V

U

Q

I

synthesis

Observed I,Q,U & V

2 small

enough?

2 m

inim

izat

ion

B,,, ...

NONO

YESYESobserved

B,,

Sunspot observation

Skumanich & Lites 1987

MISMA inversion code

– complex, having many different magnetic fields, velocities, temperatures, etc.

Model atmosphere:

Synthesis Code:

– numerical solution of the radiative transfer equations for polarized light

Fitting technique:

– Non-linear least squares minimization

data

syntheticobserved StokesStokes22

Observations

Synthetic

PCA inversions

(PCA: principal component analysis)

Important, since they are extremely fast, and so, they are bound to become popular in the next future.

For example, they may allow to process, on line, the huge data flux produced by the new synoptic magnetographs (e.g., SOLIS, see http://solis.nso.edu)

It belongs to the class of Prêt-à-porter inversions as opposed to the classical Taylor-made inversions.

Which synthetic profiles are closest to the

observed profiles?

If # i are the closest onesthen

Pre-computed data base

nnnnnnn

iiiiiii

VUQIBn

VUQIBi

VUQIB

,,,),,( # model

,,,),,( # model

,,,),,( #1 model 1111111

Prêt-à-porter inversions

iiiBB ,,,, observed

Obser ved I,Q,U & V

VUQI ,,, Observed

Eigenfaces

Reconstructed faces

Rees et al., 2000

# of eigenfaces used in the reconstruction

i

ii eigenface eeigen valuface

Fitting technique for PCA:

Rees et al. (2000)

eigenvalueth -i :

reigenvectoth -i :

vectorStokes :

i

i

iii

s

e

S

seS

Only a few eigenvalues are needed to characterize the Stokes profiles

Forward modeling (which is an inversion technique!!!)

– Resulting from the solutions of the MHD equations under ´realistic´ solar conditions.

Model atmosphere:

Synthesis Code:

– numerical solution of the radiative transfer equations for polarized light

Fitting technique:

– Not well defined (yet?) The synthetic spectra have to reproduce the observed spectra in some statistical sense.

Tu

rbu

len

t D

ynam

o S

imu

lati

ons

by

Cat

tan

eo &

Em

onet

1´´ seeing

clu

ste

r an

aly

sis

cla

ssifi

catio

n

The case of the large magnetic flux concentrations

Observed

¨A measurement process is regarded as precise if the dispersion of values is regarded as small. A measurement process is regarded as accurate if the values cluster closely about the correct value¨ (definition; e.g., Cameron 1960)

Caveats to keep in mind:

– The simplest the model atmosphere in which the inversion code is based, the higher the precision of the measurement (e.g., no problems of uniqueness in magnetographic observations).

– However precision is not the aim of solar magnetometry; accuracy is more important since it is more difficult to achieve.

– It makes no sense oversimplifying the model atmospheres to end up with magnetic field determinations that are very precise but very inaccurate.

Applications of the tools and techniques developed in the notes to specific problems of solar physics.

Understanding Real Magnetograms, e.g., Kitt Peak Synoptic Maps

README_1

Jones et al., 1992, Solar Phys. 139, 211

README_2

Coelostat Instrumental polarization

Noise 7G

max @flux solar 21

Mx 1024 SuraceSolar G 7 23 .

Line Ratio Method, or the field strength of the network magnetic concentrations

The network magnetic concentrations have very low flux density (say, less than 100 G) but a large magnetic field strength similar to that of sunspots (larger than 1 kG).

network

This fact is known thanks to the so-called line-ratio method (Stenflo 1973)

Pre-line-ratio-method situation (late 60´s and early 70´s): magnetograms of a network region taken using different spectral lines showed inconsistent results.

This is due to the fact that in network regions the magnetograph equation is not valid, implying network magnetic field strength of kG even though the magnetograms show a flux density of a few hundred G.

Stenflo took simultaneous magnetograms in two selected lines,

Fe I 5247 (geff=2.)Fe I 5250 (geff=3.)

These two lines are almost identical if there no magnetic field in the atmoshere (same log(gf) same excitation potential, same element and ionization state), however, they have (very) different magnetic sensitivity.

field magnetic no is thereif )()( 52505247 II

If weak field (sub-kG):

d

dIkBV

d

dIkBV

z

z

)(3/)(

)(2/)(

205250

205247

13/)(

2/)(

5250

5247

V

V

,...)(13/)(

2/)( 2

5250

5247

fBV

Vz

If strong field (sub-kG):

Line ratio obseved in network

kG 1zB0zB

Fe I 5247 Fe I 5250

resolution element

Broad Band Circular Polarization of Sunspots (BBCP)

Clues on the fine-scale structure of the Sunspot´s magnetic fields

Observational facts:

– Sunspots produce (large) Broad-Band circular polarization (

V/I10-3 ,Illing et al. 1974a,b)– It is produced by the individual spectral lines in the band-pass

(i.e., it is not continuum polarization: Makita 1986)– It is maximum produced in to the so-called neutral line, where

the magnetic field is supposed to be perpendicular to the line-of-

sight. (Makita 1986.)– In the neutral line Stokes V is never zero but shows the cross-

over effect

IDL

Broad Band Imaging - Polarimetry

d )Filter( )Signal(Signal

neutral line

Sun

sunspot

neutral line

solar limb

solar center

we

typical resolution element

a) The BBCP is produced by gradients along the line-of-sight, i.e., the magnetic field, velocity etc. change in the sunspot over scales of less than 150 km, i.e., much smaller than the resolution element of typical observations (1” or 1000 km). Why?

0gradient LOS nofor since

LOS thealong gradients are thereunless 0

BBCP

widthband

i

resolution widthband

ii

widthband resolutionii

widthband

dV

dVf

dVfdV

)(

)(

)()(

b) it is produced by gradients of inclination along the LOS. They are present since Stokes V is never zero in the neutral line (i.e., there is no point where the magnetic field is perpendicular to the line-of-sight).

SA & Lites, 1992, ApJ, 398, 359

Cross-over effect, Grigorjev and Kart, 1972, SPh, 22, 119

Sto

kes

V

0 and 0 then 90 andconstant is if o VdzdV

B

150 km

750 km Resolution element

c) The BBCP cannot be due to smooth well-organized vertical variations of magnetic fields inclination.

BB zzB

1

0Sanchez Almeida (2005)

150 km

750 km

Resolution element

The BBCP has to be due to very intermitent variations of magnetic field inclinations.

This is a general feature of the magnetic fields in the penumbrae of sunspots that is inferred from the (careful) interpretation of the circular polarization that it produces, despite the fact that we do not resolve the fine-scale structuring of the magnetic field

Quiet Sun Magnetic Fields

Cancellation of polarization signals in complex (tangled) magnetic fields

1B

2B

Q2 = -Q1 Q1+Q2 = Qobs = 02B

1B

V2 = -V1 V1+V2 = Vobs = 0

This kind of cancellation seems to take place in the quiet Sun

Size of a Network cell (25000 km)

Turbulent Dynamo Simulations by Cattaneo & Emonet

Effect of insufficient angular resolution

1” seeingoriginal

Variation of the Flux Density in the simulations with the angular

resolution and the sensitivity of the synthetic magnetograms.

1”x1”

Do

mín

gue

z C

erd

eña

et a

l. (0

3)Inter-Network Quiet Sun

angular resolution mag. 0.5”sensitivity 20 GVTT (obs. Teide), speckle reconstructedUnsigned flux density 20 G

1.6 G

12 G

12 G x SolarSurface = 7x1023 Mx = solarflux@max

Rabin et. al. 2001

How can we measure the properties of the quiet Sun magnetic fields?

Need to use inversion techniques whose model atmospheres allow for the complications that the quiet Sun field has:

Different polarities in the resolution element (different magnetic field inclinations in the resolution element) Different magnetic field strength in the resolution element …

Quite Sun fields: matter of active research

Techniques and methods employed in the recent literature on solar magnetometry. Used by specialist groups.

Model dependent but with substantial potential.

No realistic inversion techniques exist so far.

– Hanle effect based magnetometry– Magnetometry based on lines with hyperfine structure– He 1083nm chromospheric magnetometry– Polarimeters on board Hinode

dvreecqcddvolume

MMifb

tEE

i

fbfb

fb )(** Re 000 2

0

1

20

1

21

0

0

ier

ier

rrr ii

sinsincos

cos

cossin

cossin

drdeide

drdeide

drdede

tEE

iMMi

tEE

iMMi

tEE

iMMi

fb

fb

fb

fb

fb

fb

0

1

0

1

1

0

0

2

0

1

2

0

1

2

0

)(

)(

)( 002

0

pdeip

Which leads to the selection rules for E-dipole transitions

each one associated with a polarization

1 ,0 M

r

A weak magnetic field splits the Zeeman sublevels but … it is weaker than the natural width of the lines.

The eigenstates involved in the transition are not pure states but combinations of them …Various frequencies are excited at the same time, and they add coherently.

w0-w w0w

Hanle Effect Based Magnetometry

wtiwtiiwt

y

xeee

d

d

21Re UU

In the case that two eigenstates contribute to the dipolar emergent radiation, the resulting electric dipole is .

1. Since non-monochromatic, the radiation is always partly polarized (Hanle effect is said to depolarize)

2. Modifies the state of polarization with respect to the case Δw=0 (Hanle effect rotates the plane of polarization.)

3. Purely non-LTE effect, since the integration of many atoms emitting at random times lead to the incoherent superposition of the two polarization states U1 and U2, and have no effect. In the coherency matrix representation,

wtiyx

wtiyxyxyxyx

wtiyyyyy

wtixxxxx

eUUeUUUUUUJJ

eUUUUJ

eUUUUJ

2*12

2*21

*22

*11

*

2*21

2

2

2

1

2

2*21

2

2

2

1

2

Re2

Re2

)sin(

)cos()cos(0 wt

wtwtU

d

d

y

x

Textbook case: describes linearly polarized in the x axis at t=0.

0/

)2sin(/

)2cos(/

2

0

IV

wtIU

wtIQ

UI

w

w2

unpolarized

w

: coherency time

We

atom

Sun

0B

Sun

0B

Sun

non-magnetic scattering Hanle effect

2w

Bw

For Hanle effect to depend on the field strength (and so to be a useful tool),

)s10/( g

nm) /500( G 70 ||

8eff

2

B

Hanle signals even if tangled fields

Faurobert et al. (2001)

observed

modelled) (known 0 B

Sr I 4607Å Hanle depolarization

Hanle saturation at some 50 G

depolarizing collisions are critical for a proper modeling

general Zeeman pattern

Magnetometry Based on Lines With Hyperfine

Magnetometry Based on Lines With Hyperfine Structure

Hyperfine Structure: due to the interaction between the electron angular momentum and the nuclear angular momentum.

Old theory by Landi Degl’Innocenti (1975), but recently recovered and used for actual observations by López Ariste et al. (2002, ApJ, 580, 519).

What would be a single line becomes a blend of lines. They now undergo regular Zeeman effect, with their π and σ± components. Hundreds of components show up.

When the HFS splitting and the Zeeman splitting become comparable, Zeeman pattern depends on the magnetic field strength (it is not the independent superposition of the Zeeman patterns of the independent components).

σ

Landi Degl’Innocenti (1975)

π

López Ariste et al. (2002)

Stokes V changes shape when the field is several hundred G … good diagnostic tool for hG field strengths.

Despite the apparent complexity, the HFS patterns present several regularities (Landi Deg’Innocenti 1975)

π and σ components are normalized to one (there is no net circular polarization).

When the magnetic field is weak enough, the Stokes V signal follow the weak magnetic field approximation.

BddI

V cos

)()(

The centers of gravity of the π and σ components is independent of the HFS.

He I 1083nm Chromospheric Magnetometry

The need for a simple but quantitative diagnostic of upper chromospheric magnetic fields is keenly felt (Rüedi et al. 1995, 293, 252).

Popular in chromospheric magnetometry.

It is a bend of 3 He I lines sharing the same lower level (19.79 ev).

Optically thin. Bend modeled using ME profiles given line strengths and Zeeman splittings. Need incomplete Pashen-Back effect to carry out the calculations.

Entirely formed in the chromosphere in standard 1D model atmospheres (Fontenla et al. 1993). Formed by recombination.

Rüedi et al. (1995)

Incomplete Pashen-Back effect required for a proper analysis (Socas-Navarro et al. 2004)

blend of 3 linesME fitME fit

Creates NCP by saturation

Hinode, satellite ideal for polarimetry. 50 cm diffraction limited optical telescope (λ/D~0.26’’ @ 6302 Å)

Polarimeters on board Hinode

Open data policy! Every one is welcome to use them

Launched, end of 2006

Japanese (ISAS), in cooperation with US (NASA) and Europe (PPARC, ESA).

Hinode European Data Center here in Oslo.

SOT-SP

SOT-FG

SOT: Solar Optical Telescope

GOTO Summary -- Index:

Selected referencesref_magnetometry.pdf

Exercises on solar magnetometry

Sutterlin et al, 1999, DOT, G-band, speckle reconstructed

Volume averaged in one pixel of a typical photospheric observation

The cartoon shows the right scale for the horizontal and vertical smearing

SST, Scharmer et al. 20020.12 arcsec, spatial resolution

1´´ x 1´´

Point Source

kk

/

A

B

Observers A and B receive exactly the same signal, which is constant in the plane perpendicular to k

constantrk

sc

zz

yy

xx

zz

yy

xx

zz

yy

xxrkwti

AwtAwt

A

A

A

wt

A

A

A

wt

wtA

wtA

wtA

rkEetre

sincos

sin

sin

sin

sin

cos

cos

cos

cos

)cos(

)cos(

)cos(

)(Re),( )(

Monochromatic means plane Elliptical Motion

z

y

x

iz

i

y

ix

eA

eA

eA

E

z

y

x

z

y

x

rk

rk

rk

)( 1te

)( 2te

)( 3te 321 ttt

)(Re),( )( rkEetre rkwti

Inserting monochromatic solutions of the kind

into the wave equation derived from the Maxwell equations ,one finds

1 LEE //||

:

:

:

:||

L

E

E

Component in the direction of

Transverse component

Characteristic scale for the variation of

Wavelength

k

E

wc /2

0

0

0

1

V

U

Q

Ix

y

)(te

If Jx(t) and Jy(t) vary at random, then the light Unpolarized Light

2222 IVUQ

p is the degree of polarization> p=0 represents unpolarized light> p=1 corresponds to fully polarized light

x

y

)(te

Monochromatic wave

1222

pI

VUQIn general

x

y

)(te

0

0

1

1

V

U

Q

I

x

y

)(te

0

0

1

1

x

y

)(te

0

1

0

1

x

y

)(te

0

1

0

1

1

0

0

1x

y

)(te

1

0

0

1x

y

)(te

21

*22

*11

*22

*11

*

21*

21

22

21

22

Im2Im2Im2Im2

Re2

VVJJJJJJJJJJV

UUJJU

QQJJQ

IIJJI

yxyxyxyxyx

yx

yx

yx

Re

ReReRe),(),(),(

21

21

2

2

1

1

2

2

1

121

yy

xxiwt

y

x

y

xiwt

y

xiwt

y

xiwt

JJ

JJe

J

J

J

Je

J

Je

J

Jetretretre

21

21

yy

xx

y

x

JJ

JJ

J

J

*22

*12

*21

*11

*2

*121

*yxyxyxyxyyxxyx JJJJJJJJJJJJJJ

0*12

*21 yxyx JJJJ

*22

*11

*yxyxyx JJJJJJ

(because the two beams are incoherent)

V

U

Q

pI

pp

V

U

Q

pI

pp

V

U

Q

I

21

21

IVUQp /222

x

y

Decomposition of any polarization in two fully polarized beams

x

y

y

x

J

JJ1

*

*

2

x

y

J

JJ

*

*

22

22

Im2

Re2

yx

yx

yx

yx

JJV

JJU

JJQ

JJI

VJJV

UJJU

QJJJJ

IJJ

xy

xy

yxxy

xy

*

*

2222

22

Im2

Re2

The Jones vectors of these two beams are orthogonal

1J

2J

0*21 JJ

yyyx

xyxx

mm

mmm

**

**

**

**

**

**

**

**

**

**

**

**

Re

Im

Im

Im

Im

Re

Re

Re

Im

Re

/

/

Im

Re

/

/

yyxxyxxy

yyxxyxxy

yyxyyxxx

yyxyyxxx

yyxxyxxy

yyxxyxxy

yyxyyxxx

yyxyyxxx

yyyxxyxx

yyyxxyxx

yyxyyyxx

yyxyyyxx

yyyxxyxx

yyyxxyxx

yyxyyyxx

yyxyyyxx

mmmmM

mmmmM

mmmmM

mmmmM

mmmmM

mmmmM

mmmmM

mmmmM

mmmmM

mmmmM

mmmmM

mmmmM

mmmmM

mmmmM

mmmmM

mmmmM

44

43

42

41

34

33

32

31

24

23

2222

22

2222

21

14

13

2222

12

2222

11

2

2

2

2

ijm

From Jones matrix

to Mueller matrix

ijM

1U

2U

1 ; 0 21*21 UUUU

: , 21 UU

For any selective absorption, this set is a base of complex 2D vectors (e.g., the Jones vector)

For any polarization with Jones vector J

2*21

*1 )( ))(1( UUJUUJJmJout

2*21

*1 )( )( UUJUUJJ

The OS just changes the Jones vector as

y

x

yyx

yxx

yyyxx

yxyxx

y

xyyxx

J

J

UUU

UUU

UJUUJ

UUJUJ

U

UUJUJUUJJm

2

11*1

*11

2

12

11*1

*11

2

1

1

1*1

*11

*1

)(

)( )(10

01

2

11*1

*11

2

1

2221

1211

10

01

yyx

yxx

UUU

UUU

aa

aam

0Im

0Im

0Re

Im2Im/

Re2Re/

Re/

)1( Re/

2211

2112

2112

1*112112

1*112112

1

2

1

2

12211

1

2

1

2

12211

aa

aa

aa

VUUaa

UUUaa

QUUaa

IUUaa

Q

U

V

yxV

yxU

yxQ

yxI

weak magnetic field approximation

)( B )( B

)()()( 2 BB

)()( BB

dd

B

)(2 B Zeeman shift

2

22

2

dd

dd B

BB

)()()()(

Band-pass of typical magnetogram observations

continuum

References

•Kemp 1970, ApJ, 162, 169, in connection with the continuumpolarization in a magnetic field•Sanchez Almeida