M ain C onstraints:
– 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)
N eeded T ools:
– 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
– 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)
–Devoted to the magnetometry of the photosphere .

Summary – Index (1):
R adiative T ransfer for P olarized R adiation.
– 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.
I nstrumentation:
– Polarimeters , including magnetographs
– Instrumental Polarization
I nversion T echniques:
– General ingredients
– Examples, including the magnetograph equation
E xamples of S olar M agnetometry:
– Kitt Peak Synoptic maps
– Line ratio method
– Broad Band Circular Polarization of Sunspots
– Quiet Sun Magnetic fields

Summary – Index (2):
A dvanced S olar magnetometry.
– Hanle effect based magnetometry
– Magnetometry based on lines with hyperfine structure
– He 1083nm chromospheric magnetometry
– Polarimeters on board Hinode
goto end

S
p
J
p
M
M
J
M
– 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

y x
 e (
 r ,
Re
 t ) e iwt

Re

E ( t )



 j

 
 e iw j t
A x
A y

E
( t )
( t ) j




Re


 e iwt cos[ wt cos[ wt


 x
 y
0
 j
( t )] e i ( w j
( t )] 

 w ) t

E j




Quasi-monochromatic means that the ellipse change with time

t
1/ t e x
(t) time (t) Frequency (1/t) w/2 p
2 p
/w = 10 -15 s , in the visible (5000 A) t
: coherency time , for which the ellipse keeps a shape
 t
= 10 -8 s, electric dipole transition in the visible
 t
= 5 x 10 -10 s , (multimode) He-Ne Laser
 t
= 5 x 10 -10 s , high resolution spectra (
Dl/l200000)
Integration time of the measuremengts: 1 s (<< t
<< 2 p
/w), ellipse changes shape some 10 8 -10 9 times during the measurement


J e

( r

, t )

Re
 e iwt
E

( t )

J





J
J y x







E
E y x
( t
( t )
)



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
J
 in

J
 out
It just t ransfoms
Most known optical systems are linear (from a polarizer sheet to a magnetized atmosphere)
J
 out
 m J
 in m :
Jones Matrix (Complex 2x2 matrix)

The polarization of the light can be determined using intensity detectors
(CCDs, photomultipliers, etc.) plus linear optical systems .
I
 e x
2
( t )
 e y
2
( t ) f ( t )

1
T
T

0 f ( t ) dt
J
 out
( T : integration time of the measurement )
J
 in m
I out

M
11
I

M
12
Q

M
13
U

M
14
V

I out

M
11
I

M
12
Q

M
13
U

M
14
V
I

J x
2 
J y
2
U
V
Q
 2  2
J x
J y


2

2
  x
  x y y
Stokes Parameters , that completely characterize the properties of the light from an observational point of view
Z
* is the complex conjugate of
Z
M describes the properties of the optical system ij m


 m xx m yx m xy m yy


M
11



 m xx
2  m yy
2  m xy
2  m yy
2



/ 2
M
12
M
13
M
14





 m xx
2  m yy
Re
Im

 m m xx xx m m
* xy
* xy
2  m xy

 m yx m
* yy m yx m
* yy


2  m yy
2



/ 2

– 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

– 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
S out

M S in
S




I
Q
U

 Stokes vector
V
M




M
M
21
M
11
31
M
41
M
12
M
22
M
32
M
42
M
13
M
23
M
33
M
43
M
14
M
24
M
34


M
44
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 m




1
0
0
1 





 a
11 a
21 a
12 a
22


 with
M





1
0


0
0
0
1
0
0
0
0
1
0
0
0
0
1 















Q

U

V
I

Q


I

V

U a ij

1 then

U

V


I

Q

V


Q


U
I








I

Q

U

V

V

U

Q







Re
Re

 a
11 a
11
Re
 a
12
Im



 a
12

Re
Im

 a
12 a
12
Im
 a
11 a
22 a
22

 a
21
 a
21




 a
21 a
21 a
22




Mueller Matrix for an optical system producing selective absorption
M

1








Q

U
I

V



Q
I

V

U

U

V

I
 
Q


V

U

Q

I






I

Q


 
 
I a
Q a

U

V

U

Q
  
  

0

0
U a
V a

V

0






Q
U
V
I a a a a







Stokes Vector de type of absorbed light
Change of amplitude produced by the selective OS

linear polarizer transmitting the vibrations in the x-axis






I
Q
U
V a a a a













1

1
0




0 

M







1


0
0
 
1
 
0
0
Then for unpolarized input light one ends up with
0
0
1
 
0
0
0
0
1
 









I
Q
U





1

0



V 0

Mueller Matrix for an optical system producing selective retardance
M

1








Q

U
I

V



Q
I

V

U

U

V

I
 
Q


V

U

Q

I






I

Q

0

0

U

V

U

V

Q

0

0
 
Q a
 
U a
 
V a







Q
U
V
I a a a a






Stokes 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
M
  j
M j if the chain is formed by weakly polarizing optical systems , then the order of the different elements is irrelevant
M
  i
M i
  i
( 1
 D
M i
)

1
  i
D
M i

S+
D
S
S line-of-sight observer
D z
S
 D
S

1
 D z
 i

M
 z i
S layer of atmosphere


S em
 z
D z
S
I



Q

U
V


D z

S em
D z


 z
M i
 z
)
Emission produced by the layer
Mueller matrix of i-th process changing the polarization

S
 D
S

1
 D z
 i

M
 z i
S


S
 z em
D z
D
S
D z
  i

M
 z i
S


S em
 z


M i
 z
)
D z










U
V
I
Q


Q
I
 
V

U

I ai
 i
 i




A i
 z
)
D z


P i
 z
)
D z
Q ai
U ai
V ai
) t

U

V



I
Q


V

U

Q

I





 





 i
I ai
 i
Q ai

 i i
U
V ai ai

 i i
Q ai
I ai


 i i
V ai
U ai

 i
 i
U ai
V ai

 i i
I ai
Q ai
 i
V ai


 i i
U ai
Q ai
 i
I ai




 change of amplitude change of phase
Stokes vector of the selective absorption + retardance

I d dz


Q
U


V
 







Q

U

V
I

Q

I
 
V

U

U

V



I
Q


V

U


Q
I






 
I
Q
U
V


?



?
?


?
em
.

I
 i
 

A i
 z
)
I ai
.
.

V

Q

 i
 i



A i
 z
)
V ai


P i
 z
)
Q ai
.

V
 i
 

P i
 z
)
V ai

Emission term ?
Simple assuming emitted radiation field is in
LTE (Local Thermodynamic Equilibrium). In TE d dz













0 and




I
Q


U
V 












B
0
0




0 
 with B the Planck function then






?
?




?
?

 em












I
Q
U
V

Q

I
 
V

U

U

V


I

Q


V

U

Q

I




 







B
0
0




0 


d dz




I


Q
U
V 





 










U

Q
V
I
.

I
  i


A i
 z
)
I ai
.

V
  i


A i
 z
)
V ai

Q

I
 
V

U

U

V


I

Q


V

U

Q

I












I
Q
U
V
B






.

Q
  i


P i
 z
)
Q ai
.

V
  i


P i
 z
)
V ai
Radiative transfer equation for polarized light in any atmosphere whose emission is produced in LTE

linear polarizer transmitting the vibrations in the x-axis
There is just one i which absorbs






I
Q
U
V a a a a













1

1
0




0 
 and no emission ( B =0)
I 1 d dz


Q
U


 


A
 z
) 


1
0
V 0

1
1
0
0
0
0
1
0
0 I
0
0




Q
U


1   V

I d dz


Q
U


1
 


A
 z
) 


1
0
V 0

1
1
0
0
0
0
1
0
0 I
0
0




Q
U


1   V z

[
, L ] d dz
( I

Q )

0

( I

Q ) out

( I

Q ) in d dz
( I

Q )
 
2


A
 z
)
( I

Q )

( I

Q ) out

( I

Q ) in e
 
0
L
2


A
 z
) dz 
0 d dz
U
 
2


A
 z
)
U

U out

U in e
 
0
L
2


A
 z
) dz 
0
I 1


Q

U
V

 out

1
2


1
0
0
1
1
0
0
0
0
0
0
0 I
0
0




Q
U


0   V in
Typical Mueller matrix of a linear polarizer

Z
E
Purpose: work out the
 ´s and  ´s in the absorption matrix in the case of a magnetized atmosphere
Work out contributions to the change of polarization due to:
1) Spectral line absorption
Assumptions:
– Electric dipole transitions
– Hydrogen-like atoms
– Linear Zeeman effect
2) Continuum absorption

The wave function characterizing eigenstate of theses Hydrogenlike atoms can be written down as

( r ,

,

, t )
 
0
( r ,

) e iM
 e i
E t

  r where M is the magnetic quantum number and E is the energy of the level.

The electric dipole of the corresponding distribution of charges will be
 d
 q
  volume
2 dv

 d
 q
  volume
2 dv
When you have a transition between states b (initial) and f
(final), the wave function is a linear combination of the two states
  c b
( t )
 b
 c f
( t )
 f c b c f
( 0 )
( 0 )

1 and c b

0 and c f


0 when
1 when t t




 d d

0


 d
0 d

0


2 qc b c
* f
2 qc b c
* f
Re



  volume

E
Re

 e i
E b
 f t b

* f
  volume r dv



0 b

0
* f
 e i ( M b

M f constant over the period of the wave
)
 r
 dv

  r r

 r


 sin sin

 cos cos
 cos





0 r cos
 

0
1


 r sin

2
1 e i



 i
0


 r sin

2
1 e
 i


 i
0


 d

 d

0

2 qc b c
* f
Re

 e i
E b

E f
 t   volume
0 b

0
* f




2 p
0
 e i ( M b

M f
)
 d




0
0
1









 e i
E b

E f
 t



2 p

0 e i ( M b

M f

1 )
 d










1
0 i 



 e i
E b

E f
 t



2 p

0 e i ( M b

M f

1 )
 d



 i
1
0









 e i
E b

E f
 t

 drd
 


 drd

 drd

 e i ( M b

M f
)
 r
 dv

0
2 p e ip
 d
 
0
 p

0
Which leads to the selection rules for E-dipole transitions
D
M

0 ,

1 each one associated with a polarization

x

 observer z
There are only three types of polarization
We are interested in the projection in the plane perpendicular to the line of sight (xy plane) y d x

1
 d
 

0
0

 d y

0 d

 

 cos
 sin


 a) For
D
M=0 d x

Re


 e iwt
0


0
1


 

1
0
0






0 d


(t ) d y



Re
 e iwt
0


0
1


1
 

 cos
 sin




 



  sin
 cos( wt ) y x
I p


Q p
U p


 sin
2

V p
1



0
1 

0

b) For
D
M=M b
- M f
=+1 d x



Re
 e iwt 

1

0 i 

 

0
0
1





 cos( wt ) d y



Re
 e iwt 

1
 i
0


 


0 cos sin







 cos
 sin( wt ) b) For
D
M=M b
- M f
=-1
I
 


Q
U






V
 





 cos
2 sin
2


2
0 cos





 cos
 cos
I
 


Q
U







V
 




1
 
 sin
2 cos
0
2
 cos
2






 y
 x
1 d


(t ) y
1 d


(t ) x

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
D
M ,
B=0 B=B
0
D w
D w

B
0 w
0 w
0 w w
Associated to each transition there is a absorption profile plus a retardance profile w
0 w
0
D w w
0 w
0
D w

x
In short : for an electric dipole atomic transition , only three kinds of polarizations can be absorbed . They just depend on
D
M (with M the difference of magnetic quantum numbers between the lower and the upper levels) observer cos
 y
 D
M=0 y  x x

B
 D
M=+1 y  x y
 D
M=-1
D w


B absorption retardance w
0 w
0
D w w
0 w
0
D w

C ontinuum A bsorption
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),





I
Q
U a
V a a a












10

5
1
0
(
0

B /



 kG ) 
– 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 ~ 10 6 G, leading to large continuum polarization (~ 1%)

R adiative T ransfer E quation in a M agnetized A tmosphere
The equation is generated considering the four types of polarization that are possible d dz




I


Q
U
V 





 











V
I
Q
U

Q

I
 
V

U

U

V


I

Q


V

U

Q

I












I
Q
U
V
B







I




U
Q

V


1
  c


0
0



0 k l
2
 p
1 sin
2


 cos sin
2

2



 k l
2
0

 
2







1 sin sin

2
2
2 cos
2

 cos

 cos sin
2

2






 k l
2

 
2







1 sin sin


2
2
2 cos
2

 cos

 cos sin
2

2





 same for
 ´s with replacing  ´s with  ´s

x observer
 
B y

I d dz


Q
U


V
 








V
Q

U
I

Q

I
 
V

U

U

V


I

Q


V

U

Q

I







I
Q
U
V
B




Q


I
U
V
 


 k l
2 k l
2 k l
2 c
 k l



 p
2


 p


 sin
2


2






 p


 


 
 

2

 

2




 



 


 sin
2


 sin
2

2



  cos sin
( 1

2

2
 cos
2

)


Unno-Rachkovsky Equations

Q

U

V


 k l
2 k l k l
2
2



 p


 

2

 



 p


 


 
 

2

 

2




 


 sin 2



 sin
2  cos 2
 sin 2


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



1.Symmetry with respect to the central (laboratory) wavelength of the spectral line . If the macroscopic velocity is constant along the atmosphere, then
I( l
) = I(l
)
Q(
U( l
) = Q(l
) l
) = U(l
)
V( l
) = -V(l
) l  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 )

2.-
, the width of the absorption and retardance coefficients of the various Zeeman components are much smaller than their
Zeeman splittings if
Dl

  is the Zeeman splitting of a Zeeman triplet, and
Dl
D is the width of the line, it can be shown that (e.g., Landi + Landi 1973)

I

Q
 
I 0
 
Q 2
 
I
(
D l
B
2
(
D l
B
/
/
D l
D
)
2
D l
D
)
2  
 

U

V
 
U 2
(
D l
B
 
V 2
(
D l
B
/
/
D l
D
D l
D
)
)
2  
 


U

V
Q then
Dl

Dl
D
 
Q 2
(
D l
B
/
D l
D
)
2  



U

V
2
(
D l
B
2
(
D l
B
/
/
D l
D
D l
D
)
)
2  
 

d dz




I
Q


U
V 





 





0
0



V
I
0


I

V
0
0

V

I

0

V
0
0

I







I
Q
U




V
B






(a) d n
Q dz n
 d n
U dz n

0

Q

U

0 Since there is no polarization at the bottom of the atmosphere
(b) d ( I

I


V dz

V


) k c
 k c

(

I
 k l
 k l

 
V
( l
( l
)

)( I
D l
B
 cos


V cos


D l
B
)
B d

)
( l d l
)
  

I
 
V
 k c
 k l

( l  cos
 D l
B
)

d ( I

V )

 k c dz
 k l

( l
 cos
 D l
B
)

( I

V

B )
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
 Dl
B
If the longitudinal component of the magnetic field is constant then cos

Dl
B is constant and I+V and I-V are identical except for a shift
I-V l
I+V
2 cos
 Dl
B

I
I

V

V

 f ( l  cos
 f ( l  cos

D l
B
)

D l
B
)
 f f
( l
)

( l
)
 df ( l df d
( l l d l
)
) cos
 cos

I
V


1
2

( I
1
2

( I

V )

( I

V )

( I

V )



V )

 f ( l
)
 df ( d l l
) cos
 D l
B
D l
B
D l
B
  dI ( d l l
) cos
 D l
B
V ( l
)
  dI ( d l l
) cos
 D l
B
: the
Stokes V signal is proportional to the longitudinal component of the magnetic field observer cos
 Dl
B
V
0

B cos


B
 l

The previous argumentation is based on the assumption that the Zeeman pattern is a triplet (one p component, one

+ component and one

- component). If the pattern is more complex but the magnetic field is weak, one can repeat the argumentation 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
D l
B
 g eff

B g eff
Is the so-called effective Landé factor , and it equals one for the classical Zeeman effect

4.-
non-magnetic magnetic





I
Q obs
U
V obs obs obs





I


Q

V

U


( 1

( 1

( 1
( 1





)


I non
 mag
) Q non
 mag
) U non
 mag
) V non
 mag




Q


V
U
 
filling factor, i.e., fraction of resolution element filled by magnetic fields resolution element

Effect on the magnetograph equation V ( l
)
 
C dI ( d l l
)
I obs
V obs
 
I
 
V

 
I non
 mag

I
I

I non
 mag
B cos

V obs
( l
)
  dI obs d l
( l
)
B eff
with B eff
 
B cos
 d s
 observer

B
B eff
  pixel

B
 d s

/
 pixel ds
Magnetic flux density

4.-
Importance : Used for measuring magnetic field properties
Assumptions : all those needed to get an analytic solution of the of the radiative transfer equations for polarized light d dz
I


Q

U
V


 








V
Q

U
I

Q

I
 
V

U

U

V


I

Q


V

U

Q

I






 
I
Q
U
V
B


RTEPL: first order linear differential equation. Admits an analytic formal solution of the coefficients are constant (basic maths)

I d dz


Q
U



V









U

V
I
Q

Q

I
 
V

U

U

V



I
Q


V

U


Q
I




 


I
Q
U
V
B

 d t    c dz t  continuum optical depth
I

S



Q
U


V
1

1



0
0


0
K

1
 c









V
Q
U
I

Q

I
 
V

U

U

V



I
Q


V

U


Q
I




 d

S d t

K

( S

B

1 ) Compact form of the RTEPL

d

S d t

K

( S

B

1 )
Assumptions :
 the ratio line to continuum absorption coefficient does not depend on optical depth  l
 c

 The source function depends linearly on continuum optical depth
B

B
0
 t
B
1
 Broadening of the line constant (both Doppler and damping)
 Magnetic field vector constant with depth
… all them together lead to constant absorption matrix
K

t

d

S d t try

K

( S
 solutions
B

1 )

S


S
0


S
1 t
, with both

S
0 and

S
1 constant

S
1

K

( S
0

B
0

1 )
 t
K

( S
1

B
1

1 )
K (

S
1

B
1

1 )


0


S
1

B
1

1

S
1

K (

S
0

B
0

1 )


S
0

K

1

S
1

B
0

1

S ( t 
0 )


S
0

B
0

1

B
1
K

1

1

I
Q
U
V




B
0




B
1
B
1


B
1


B
1




I
2

I
2

V
I
Q
I
2

U
(

I
2
 
I
 
I
 
I
 
Q
2
(

V
(

U

U
(

Q

V

Q
 
U
2  
V
2
) /

 
U

V
)

D

Q
(

Q

Q
 
V

Q
)
 
U
(

Q

Q
 
Q

U
)
 
V
(

Q

Q
 
U

U
 
U

U
 
U

U
 
V
 
V
 
V

V

V

V
)
)


/
/
) /

D
D
D
D  
I
2
(

I
2  
Q
2  
U
2  
V
2  
Q
2  
U
2  
V
2
)

(

Q

Q
 
U

U
 
V

V
)
2
Milne-Eddington solutions of the RTEPL (e.g., Landi Degl ´Innocenti, 1992)
Free parameters :
1.
Magnetic field strength
2.
Magnetic field azimuth
3.
Magnetic field inclination
4.
B
0
5.
B
1
6.
Macroscopic velocity
7.
Doppler broadening
8.
Damping
9.
Strength of the spectral line
IDL

5.-
o
 observer

B
 y
B

 observer
 
180 o y
These two magnetic fields produce the same polarization, therefore, one cannot distinguish them from the polarization that they generate.
IDL

6.-
observer observer

B


180 o   y
 y
B

V ( 180 o  
)
 
V (

)
IDL
V


o  
 


7.-
observer observer
B
  
0

 
90 o y
 y
B

IDL

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

Intensity detector optics modulator ( p j
) optics




I
Q out out


U
V out out








I
~

 calibration optics telescope
+ optics
I


Q
U


V
I out
( p j
)

M
11
( p j
) I
~ 
M
12
( p j
)

M
13
( p j
)

M
14
( p j
)






M ij
( p k
I out
( p
1
)
)


1


I out
(
I out
( p
2 p
3
)
)


I out
( p
4
)




Mueller
 

Matrix
Telescope





I
Q
 
U
V



Optical system whose Mueller matrix can be (strongly) varied upon changing a set of control parameters.
Example fixed linear polarizer rotating retarder ( l
/4)

I out


Q
U out out





1

0
1


 
V out
0
 cos 2 ( 2

)
 sin( 2

) cos( 2

) U
 sin( 2

)

Usually the last element is an optical element that fixes the polarization state of the exit beam , but this is not always the case.

for example a CCD
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 .
rotating retarder ( l
/4) fixed linear polarizer
Example
I out


Q
U out out


V out
1




 sin cos
2
2

2
 cos 2
 sin 2





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 axisymmetry ). 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
Mueller

 matrix
Teslecope



M
  j
M j



~


Mueller
 

Matrix
Telescope





I
Q
U
 
V



I
Mueller


Q
U


V
 

Matrix
Telescope



1


U









U
~
V
~
I
~
Q
~





Mueller
 

Matrix
Telescope





I
Q
U
 
V


~
V

M
41
I

M
42
Q

M
43
U since I

Q , U , and V

I
~

~
V


M
11
I
M
41
I

M
44
V

M
44
V at continuum wavelengths V=0

V
I
~
~ c c

M
11
I c

M
41
I c


~
V

~
V c
I
~
I
I
~ c
/
~
I
~ c

 /

I
~ c
I / I c

M
44
M
11
I
V c

CCD
2 states modulator l
/4-plate + linear polarizer
I out
t
1

C
I
I out
t
2

C
I

V

V
then
Narrowband color filter
V
I


I
I out
( t
1
)

 out
( t
1
)
I out
( t
2
I out
( t
2
)
) and
V

I I
I out
( t
1
)

 out
( t
1
)
I out
( t
2
)
I out
( t
2
)




I


Q
U
V 






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):
V
I

30 % in sunspots
V
I

10 % in plage regions
V
I

1 % in network regions
V
I

0 .
1 % in inter network regions

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
I
I out out
(
( t t
1
2
)
)


I
I
(
( x

1
 x
2
)
)


V
V
(
( x
1
 x
2
)
)
I
 out

( t
1
I with
(
)

0

0

)
I

 out
 d x
( t
2
( dt x
1


) t
1



2
)
I t
/
2
(

 x
1
2

)
V

(
I (

0
 x
2
)

)

V
V
(
(

0
 x
1
)
)

V (
 x
2
)
(Lites 1987)

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.

It is just an image showing the degree of circular polarization in the flank of spectral line.
Model atmosphere :
–
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
Synthesis Code :
– Multi component atmosphere
– Weak magnetic field approximation
Fitting technique :
– No sophistication; one observable and one free parameter

V ( l
)
  resolution
V ( l
,
 
C
) ds
 dI ( i d l l
)
# components  i f i

B i
# components  i f i
V i
( l
)
 
C
# components  i f i

B i cos
 i
 
C dI ( i d l l
) 

B resolution d
 cos
 i

I ( l
)
  resolution
I ( l
, x

) ds

# components i
 f i
I i
( l
)

I i
( l
)
 d s
 resolution dI d l i
V ( l
I ( l
)
)
 c ( l
)
 resolution


B d d s resolution c ( l
)
 
C d ln I d l i
( l
) A calibrated magnetograph gives the longitudinal component of the magnetic flux density (mag flux per unit surface)

(e.g. Skumanich & Lites 1987)
Model atmosphere :
–
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
Synthesis Code :
–
Milne Eddington analytic solution of the radiative transfer equations for polarized light
Fitting technique :
– Non-linear least squares minimization

2 

 data
observed

synthetic
2

Input model atmosphere
B,

,

, ...
Observed I,Q,U & V

2
,
 

B
2
,
 
 
2
, 
B,

,

, ...
NO new atmosphere B,

,

, ...
giving a smaller

2
D
2 small enough?
YES synthesis
I


Q
U


 ,


B
I


Q
U


 ,

 
I


Q
U


 , 
V V V observed
B,

,


Sunspot observation
Skumanich & Lites 1987

Model atmosphere :
– complex , having many different magnetic fields, velocities, temperatures, etc.
Synthesis Code :
– numerical solution of the radiative transfer equations for polarized light
Fitting technique :
– Non-linear least squares minimization

2 

 data
observed

synthetic
2

Synthetic
Observations

(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)

Observed ,
I,Q,U & V
, , V
Pre-computed data base model #1 ( B
1
,

1
,

1
 )

I
1
, Q
1
, U
1
, V
1
 model # i ( B i
,
 i
,
 i
 )

I i
, Q i
, U i
, V i
 model # n ( B n
,
 n
,
 n
 )

I n
, Q n
, U n
, V n
Which synthetic profiles are closest to the observed profiles?
If # i are the closest ones then observed B ,

,
 
B i
,
 i
,
 i

Fitting technique for PCA :
# of eigenfaces used in the reconstruction
Eigenfaces
Reconstructed faces
  i
i

i
Rees et al., 2000

S
  i e i s i
S : Stokes vector e i s i
: i th eigenvecto r
: i th eigenvalue
Only a few eigenvalues are needed to characterize the Stokes profiles
Rees et al. (2000)

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

1´´ seeing

Observed

– 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.
¨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)

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

U nderstanding R eal M agnetograms, e.g., K itt
P eak S ynoptic M aps
README_1

README_2
Jones et al., 1992, Solar Phys. 139, 211
Coelostat

Instrumental polarization
Noise

7G
7 G

Solar Surace

4 .
2

10
23
Mx

1
2 solar flux @ max

L ine R atio M ethod, or the field strength of the network magnetic concentrations network
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 ).
This fact is known thanks to the so-called lineratio 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 (g eff
=2.)
Fe I 5250 (g eff
=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.
I
l
5247

I
l
5250

If weak field (sub-kG):
V
5247
( l
) / 2
 
B z
 k l
2
0
V
5250
( l
) / 3
 
B z
 k l
2
0 dI dI
( l
) d
( l l d l
)






V
5247
( l
V
5250
( l
)
)
/
/
2
3

1
If strong field (sub-kG):
V
5247
( l
V
5250
( l
)
) /
/
2
3

1

B z
2 f ( l
,...)

Line ratio obseved in network
Fe I 5247
B z

0 B z

1 kG resolution element
Fe I 5250

B road B and C ircular P olarization of
S unspots (BBCP)
Clues on the fine-scale structure of the Sunspot ´s magnetic fields
Observational facts :
IDL
– Sunspots produce (large) Broad-Band circular polarization (
V/I

10 -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-ofsight. (Makita 1986.)
– In the neutral line Stokes V is never zero but shows the crossover effect






l
l
l
Broad Band Imaging - Polarimetry

Sun neutral line neutral line sunspot solar limb
 we solar center


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?
BBCP
 band


V width
( l
) d l  band

 width


 f resolution i
V i
( l
)

 d l 
 resolution f i


 band


V i width
( l
) d l




0 unless there are gradients along the LOS since for no LOS gradient band


V i
( width l
) d l 
0

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).
if

B is constant and
 
90 o
then dV dz

0 and V

0
SA & Lites, 1992, ApJ, 398, 359
, Grigorjev and Kart, 1972, SPh, 22, 119

c) The BBCP cannot be due to smooth well-organized vertical variations of magnetic fields inclination .
750 km
150 km
Resolution element

Sanchez Almeida (2005)

B

0

 
1



 

B





 z
B z

The BBCP has to be due to very intermitent variations of magnetic field inclinations.
750 km
150 km
Resolution element
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

Q uiet S un M agnetic
F ields
Cancellation of polarization signals in complex (tangled) magnetic fields

B
1

B
2
V
2
= V
1

V
1
+ V
2
= V obs
= 0
B


B
2
1

Q
2
= Q
1

Q
1
+ Q
2
= Q obs
= 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

original
1” seeing
Effect of insufficient angular resolution

Variation of the Flux Density in the simulations with the angular resolution and the sensitivity of the synthetic magnetograms

 angular resolution mag.
@ 0.5”
 sensitivity
@
20 G
 VTT (obs. Teide), speckle reconstructed
 Unsigned flux density

20 G
1”x1”

12 G x SolarSurface = 7x10 23 Mx = solarflux@max
Rabin et. al. 2001
12 G
1.6 G

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

  r r

 r


 sin sin

 cos cos
 cos





0 r cos
 

0
1


 r sin

2
1 e i



 i
0


 r sin

2
1 e
 i


 i
0


 d

 d

0

2 qc b c
* f
Re

 e i
E b

E f
 t   volume
0 b

0
* f




2 p
0
 e i ( M b

M f
)
 d




0
0
1









 e i
E b

E f
 t



2 p

0 e i ( M b

M f

1 )
 d










1
0 i 



 e i
E b

E f
 t



2 p

0 e i ( M b

M f

1 )
 d



 i
1
0









 e i
E b

E f
 t

 drd
 


 drd

 drd

 e i ( M b

M f
)
 r
 dv

0
2 p e ip
 d
 
0
 p

0
Which leads to the selection rules for E-dipole transitions
D
M

0 ,

1 each one associated with a polarization

H
E
B
M
A weak magnetic field splits the Zeeman sublevels but … it is weaker than the natural width of the lines . w
0
-
D w w
0
D w
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 .

In the case that two eigenstates contribute to the dipolar emergent radiation, the resulting electric dipole is .

 d d x y



Re
 e iwt

U
1 e i
D wt
 U
2 e
 i
D wt
 
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 U
1 matrix representation, and U
2
, and have no effect. In the coherency
J x
2
J y
2


U
1 x
2
U
1 y
2


U
2 x
2
U
2 y
2


2 Re

U
1 x
U
*
2 x e i 2
D wt
2 Re

U
1 y
U *
2 y e i 2
D wt


J x
J
* y

U
1 x
U
1
* y

U
2 x
U
2
* y

U
1 x
U
2
* y e i 2
D wt 
U
2 x
U
1
* y e
 i 2
D wt

2 p
D w w
Textbook case: describes linearly polarized in the x axis at t=0 .

 d d y x



U
0 cos( wt )

 cos( sin(
D wt )
D wt )


I
Q

/ I
U
0
2
 cos( 2
D wt )
U / I
 sin( 2
D wt )
V / I

0 t D w t
: coherency time unpolarized

atom
Sun non-magnetic scattering

B

We
D w


B
For Hanle effect to depend on the field strength (and so to be a useful tool), t D w

2 p
|

B |

( l
/500 nm)
70 G g eff
( t
/ 10

8
2 s )
Hanle signals even if tangled fields
Hanle effect

B

Sun
Sun

Sr I 4607 Å Hanle depolarization depolarizing collisions are critical for a proper modeling

B


observed
Hanle saturation at some 50 G
Faurobert et al. (2001)

M agnetometry B ased on L
W
H
general Zeeman pattern

M agnetometry B ased on L
W
H
Hyperfine Structure: due to the interaction between the electron angular momentum and the nuclear angular momentum .
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).
Old theory by Landi Degl’Innocenti (1975), but recently recovered and used for actual observations by López Ariste et al. (2002,
ApJ, 580, 519).

Landi Degl’Innocenti (1975)
σ
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 .
V ( l
)
  dI ( d l l
) cos
 D l
B
 The centers of gravity of the π and σ components is independent of the HFS.

H e I 1083nm C hromospheric M agnetometry
Popular in chromospheric magnetometry.
The need for a simple but quantitative diagnostic of upper chromospheric magnetic fields is keenly felt (Rüedi et al. 1995,
293, 252).
It is a bend of 3 He I lines sharing the same lower level (19.79 ev).
Entirely formed in the chromosphere in standard 1D model atmospheres (Fontenla et al. 1993). Formed by recombination .
Optically thin . Bend modeled using ME profiles given line strengths and Zeeman splittings. Need incomplete Pashen-Back effect to carry out the calculations.

Incomplete Pashen-Back effect required for a proper analysis
(Socas-Navarro et al. 2004)
Creates NCP by saturation blend of 3 lines
Rüedi et al. (1995)

P olarimeters on board H inode
Hinode, satellite ideal for polarimetry . 50 cm diffraction limited optical telescope ( λ/D~0.26’’ @ 6302 Å)
Launched , end of 2006
Japanese (ISAS), in cooperation with US (NASA) and
Europe (PPARC, ESA).
Hinode European Data Center here in Oslo .
Open data policy ! Every one is welcome to use them

SOT-
SP
SOT-
FG

Selected references ref_magnetometry.pdf
Exercises on solar magnetometry

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

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

SST, Scharmer et al. 2002
0.12 arcsec , spatial resolution
1 ´´ x 1´´

B
A k

/ k

Point Source
Observers A and B receive exactly the same signal, which is constant in the plane perpendicular to k

 r

 constant k


( , t )

Re
 e i ( wt

 k

)

E (
 k
 cos
 wt cos

 wt
A x cos
 x
A y cos

A z

A c cos

 z sin y


 sin wt

A s wt

t
1


A x cos( wt
)


A x
A y
A z


A y
A z cos( wt cos( wt sin
 x sin
 sin
 z y






 x

 z y
)
)
)




E


 e i

A x
A y e i

A z e i
 x z y






 z y x


 

 k
 k
 k r



 
 
  z y x


( t
3
)
( t
2
) t
1
 t
2
 t
3
Monochromatic means plane Elliptical Motion

Inserting monochromatic solutions of the kind
( , t )

Re
 e i ( wt

 k r )

E (
 k

)
 into the wave equation derived from the Maxwell equations , one finds
E
| |
/ E

 l
/ L

1 l 
2 p c / w
E
||
: l
E

:
:
L :
Component in the direction of k

Transverse component
Wavelength
Characteristic scale for the variation of
E


y
Monochromatic wave
Q
2 
U
2 
V
2 
I
2 x
(t )
Q
2 
U
2 
V
2
 p

1
I
> p=0 represents unpolarized light
> p =1 corresponds to fully polarized light
In general p is the degree of polarization
(t ) y x
If J x
(t) and J y
(t) vary at random, then the light Unpolarized Light



I
Q
U


1



0
0


V 0

y
 e (t ) x
I 1


Q

U
V





1
0


0 y e (t ) y x
1


0
1


0
(t ) x
1


0
0


1 y
(t ) e (t ) y x
1



0
1


0 y x
1



0
1 

0
(t ) x
1


0
0

1



e

( r

, t )

 e
1
(
 r , t )

 e
2
( r

, t )

Re e iwt



J
J x 1 y 1




Re
Re e iwt



J x 1
J y 1


J
J x 2 y 2



J x
J
* y





J
J y x







J x 1
J y 1

J x 2

J y 2




J x 1

J x 2
 )
J
* y 1

J
* y 2
)

J x 1
J
* y 1 e iwt



J x 2
J y 2





Re
J x 1
J
* y 2
 e iwt




 

J
J x 1 y 1







J
J x 2 y 2




 

J x 2
J
* y 1

J x 2
J
* y 2

J x 1
J
* y 2
J x
J
* y


J
J x 2
J
* y 1 x 1
J
* y 1


0
J x 2
J
*
(because the two beams are incoherent) y 2
I

J x
2 
J y
2 
I
1

I
2
U
V
Q



J x
2 
J y
2 
Q
1

Q
2
2

2
Re
Im

J

J x x
J
J
* y
* y




U
1


U
2 Im
2

J x 1
J
* y 1

J x 2
J
* y 2

 
2 Im

J x 1
J
* y 1
 
J x 2
J
* y 2


V
1

V
2




I
Q
U


V

1

2 p p

 pI
Q
U
V pI



1

2 p p



Q

U



V p

Q
2 
U
2 
V
2
/ I y x
Decomposition of any polarization in two fully polarized beams

The Jones vectors of these two beams are orthogonal y x
J

1
J

1

J

*
2

0
J

2
J

1



J
J y x







U
 V

 I


Q




J x
J x
2
2


J y
J y
2
2
2

2
Re J J x
J J y
Im
  x y
J

2



J
J
* x
* y











U
 V
J
J

 y y
2
2
2
2


J
J x x
2
2


I

Re


J
*
J

Im
* y
  x y x

J x


V
2

U

J y
2  
Q

m



 m xx m yx m xy m yy



From Jones matrix
  ij to Mueller matrix
  ij
M
11
 m xx
2  m yy
2  m xy
2  m yy
2
/ 2
M
12
M
13
M
14


 m xx
2  m yy
Re
Im

 m xx m xx m
* xy m
* xy


2  m xy m yx m yx m
* yy m
* yy


2  m yy
2
M
21
 m xx
2  m yy
2  m xy
2  m yy
2
/ 2
/ 2
M
22
M
23
M
24
M
31
M
32
M
33
M
34
M
41
M
42
M
43
M
44










 m xx
2  m yy
Re
Im

 m xx m xx m
* xy m
* xy
Re
Re

 m m xx
Re
Im

 m
 xx xy m m
* yx
* yx m
* yx







Im
Im

 m m m xy xx xx

Im
Re


 m m xy xy m m m
* yx m
* yx
* yx
* yx m
* yx
2 2  m xy
 m m m m m yx yx xy xy xx m
* yy m m
* m
* yy m
* yy
* yy yy









 m xx m
* yy m xy m
* yy m xy m
* yy m xx m
* yy m xx m
* yy




 m yy
2
/ 2


U
2
U

1

U
1


U
2
* 
0 ;

U
1


U
2

1
 
U
1
,

U
2
)
: 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

J


( J



U
1
*
) U

1

( J



U
*
2
) U

2
The OS just changes the Jones vector as
J
 out
 m J


( 1
 
)( J



U
1
*
)

U
1

( J


U

2
*
) U

2



 m



1
0
 


J x
U
1 x
J x
( U
1
* x
0
1





J

2 
J
 y
 
( J


U

1
*
) U

1

U
1 y
)


U
1 x
U
1
* y
)
J y
U
1 y
2



  
( J
* x
U
1 x
 


 U
1
U
* x

1 x
U
2
1 y

J
* y
U
1 y
)


U
U
1
1 x y



U
1 x
U
1
* y
U
1 y
2






J
J y x

 m



1
0
0
1




 a
11 a
21 a
12 a
22


  


U
U
1
* x

1 x
U
2
1 y
U
1 x
U
1
* y
U
1 y
2


 
I
/
 
Re
 a
11
 a
22


U
1 x
2 
U
1 y
2 
I
1
(

1 )
 
Q
/
 
Re
 a
11
 a
22


 
U
/
 
Re
 a
12



V

U

Q

V



/
  
Re
Im

 a
12 a
12
Im
 a
11
Im



 a
12 a
21 a
21 a
22






 a
21

 a
21

0
0
0

U
2
1 x
Re

2
2


U
Im
1

U
1 y x
U
1
* y
U
1
2 x
U

*
1

 y

Q
1
U

1
V
1

weak magnetic field approximation

( l  D l
B
)
 
( l
)
 D l
B d

( l d l
)

D l
2
B
2 d
2

( l d l
2
)
 

( l  D l
B
)

( l  D l
B
)

( l  D l
B
)
 
( l  D l
B
)

2

( l
)
D l
B

Zeeman shift l 

( l  D l
B
)
 
( l  D l
B
)

2
D l
B d
 d
( l l
)

Band-pass of typical magnetogram observations

continuum

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