J. Sánchez Almeida
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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
– No proxi-magnetometry (jargon for magnetic field
measurements which are no based on polarization)
–No extrapolations of photospheric magnetic fields to
the Corona)
– No chromospheric & coronal magnetometry
– No in-situ measurements (solar wind)
–Devoted to the magnetometry of the photosphere.
Summary – Index (1):
Radiative Transfer for Polarized Radiation.
– 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.
Instrumentation:
– Polarimeters, including magnetographs
– Instrumental Polarization
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):
Advanced Solar magnetometry.
– Hanle effect based magnetometry
– Magnetometry based on lines with hyperfine structure
– He 1083nm chromospheric magnetometry
– Polarimeters on board Hinode
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
y
x
r
r r
iw j t
e ( r , t ) = Re ∑ e
E j = Re
j
A x ( t ) cos[ wt
r
iwt
Re e E ( t ) = A y ( t ) cos[ wt
0
{
}
r
iwt
i( w j − w )t
E j =
e ∑ e
j
− φ x ( t )]
− φ y ( t )]
Quasi-monochromatic means that the ellipse change with time
τ
1/τ
ex(t)
time (t)
Frequency (1/t)
w/2π
2π/w = 10-15 s, in the visible (5000 A)
τ : coherency time, for which the ellipse keeps a shape
− τ = 10-8 s, electric dipole transition in the visible
− τ = 5 x 10-10 s, (multimode) He-Ne Laser
− τ = 5 x 10-10 s, high resolution spectra (∆λ/λ=200000)
Integration time of the measuremengts: 1 s (<< τ << 2π/w),
ellipse changes shape some 108 -109 times during the measurement
rr
iwt r
e(r,t) = Re e E(t)
{
r
J
}
r J x Ex (t )
J = =
J
E
(
t
)
y
y
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
r
r
Jin →Jout
Most known optical systems are linear (from a polarizer sheet to a
magnetized atmosphere)
r
r
Jout = m Jin
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 x2 ( t ) + e y2 ( t )
1
f (t ) =
T
(T: integration time of
the measurement)
T
∫
0
f ( t ) dt
r
Jin
r
J out
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
2
I = Jx + Jy
2
Q = Jx − Jy
{
2
2
U = 2 Re J x J *y
{
}
V = −2 Im J x J
M ij
Stokes Parameters, that completely
characterize the properties of the light
from an observational point of view
*
y
}
*
Z
is the complex conjugate of
describes the properties of the optical system
m xx
m =
m yx
m xy
m yy
2
2
2
2
M11 = mxx + myy + mxy + myy / 2
2
2
2
2
M12 = mxx − myy − mxy + myy / 2
M13 = Re mxxm*xy + myxm*yy
M14
{
= Im{m m
xx
*
xy
}
+m m }
yx
*
yy
Z
(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 that 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
Sout = M Sin
M 11
M 21
M =
M 31
M
41
M 12
M 13
M 22
M 32
M 42
M 23
M 33
M 43
M 14
M 24
M 34
M 44
I
Q
S =
U
V
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
1 0 a11 a12
with
+
m =
0 1 a21 a22
1
0
M =
0
0
0 0 0 ε I
1 0 0 εQ
+
0 1 0 εU
0 0 1 εV
εQ
εI
− ϕV
ϕU
a ij ⟨⟨ 1
εU
εV
ϕV − ϕU
εI
ϕQ
− ϕQ ε I
then
ε I = Re {a 11 + a 22 }
ε Q = Re {a 11 − a 22 }
εU
εV
ϕV
ϕU
ϕQ
= Re {a 12 + a 21 }
= − Im {a 12 − a 21 }
= Re {a 12 − a 21 }
= Im {a 12 + a 21 }
= Im {a 11 − a 22 }
- Mueller Matrix for an optical
system producing selective
absorption
εI
εQ
M = 1+
ε
U
ε
V
ε I = −ε Ia
εQ = −ε Qa
εU = −ε Ua
εV = −ε Va
ϕU = 0
ϕQ = 0
ϕV = 0
εQ
εI
− ϕV
ϕU
εU
ϕV
εI
− ϕQ
Ia
Qa
U
a
V
a
ε
εV
− ϕU
ϕQ
εI
Stokes Vector de type of
absorbed light
Change of amplitude produced by the
selective OS
linear polarizer transmitting the vibrations in the x-axis
Ia 1
Qa −1
U = 0
a
V 0
a
0
0
1− ε ε
0
ε 1− ε 0
M =
0
0 1− ε 0
0
0
0 1− ε
Then for unpolarized input light one ends up with
I
Q
U
V
1 − ε
ε
=
0
0
- Mueller Matrix for an optical
system producing selective
retardance
εI
εQ
M = 1+
ε
U
ε
V
εI =0
εQ =0
εU =0
εV =0
ϕQ =ϕ Qa
ϕU =ϕUa
ϕV =ϕVa
εQ
εI
− ϕV
ϕU
εU
ϕV
εI
− ϕQ
εV
− ϕU
ϕQ
εI
Ia
Qa
U
a
V
a
ϕ
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 =∏M
j
j
if the chain is formed by weakly polarizing optical systems,
then the order of the different elements is irrelevant
M = ∏ M i = ∏ (1 + ∆ M i ) ≈ 1 + ∑ ∆ M i
i
i
i
Equation of Radiative Transfer for Polarized Light
S+∆S
observer
∆z
∂M
S + ∆S = 1 + ∆z∑ i
∂z
∂ S em
∆z
∂z
(
∆ z ∂ M i ∂z
S
i
line-of-sight
layer of atmosphere
S + ∂ S em ∆ z
∂z
I
Q
S =
U
V
Emission produced by the layer
)
Mueller matrix of i-th process changing the polarization
∂M i
S + ∆S = 1 + ∆z ∑ i
∂z
∆ S
=
∆ z
∑
i
εI
εQ
∂ M i ∂z ∆z =
ε
U
ε
V
(
)
∂ M
∂z
i
εQ
εI
− ϕV
ϕU
ε i = (∂Ai ∂z ) ∆z
ϕi = (∂Pi ∂z ) ∆z
(Iai
Qai Uai Vai )
t
S + ∂ S em ∆ z
∂z
S + ∂ S em
∂z
εU
ϕV
εI
− ϕQ
ε V ε i I ai
− ϕU
ε i Qai
=−
ϕ Q ε i U ai
ε I ε i Vai
ε i Qai
ε i I ai
ϕ i Vai
− ϕ i U ai
ε i U ai
− ϕ i Vai
ε i I ai
ϕ i Qai
change of amplitude
change of phase
Stokes vector of the selective
absorption + retardance
ε i Vai
ϕ i U ai
− ϕ i Qai
ε i I ai
I
d Q
=
dz U
V
ηI
ηQ
−
ηU
η
V
ηI = ∑(∂Ai ∂z ) I ai
i
.
.
ηV = ∑(∂Ai ∂z ) Vai
i
φQ = ∑(∂Pi ∂z ) Qai
i
.
.
φV = ∑(∂Pi ∂z ) Vai
i
ηQ
ηI
− ρV
ρU
ηU
ρV
ηI
− ρQ
ηV
− ρU
ρQ
ηI
I ?
Q ?
U + ?
V ?
em
Emission term ? Simple assuming emitted radiation field is in
LTE (Local Thermodynamic Equilibrium). In TE
d
dz
=0
and
I B
Q 0
U = 0
V 0
with B the Planck function
then
ηU
ηV B
ηI ηQ
?
ρV − ρU 0
ηQ ηI
?
0
? = η − ρ
η
ρ
V
I
Q
U
?
0
η
ρ
−
ρ
η
U
Q
I
em V
ηV I − B
I ηI ηQ ηU
ρV − ρU Q
d Q ηQ ηI
= −
ηU − ρV ηI
ρQ U
dz U
V η
V
ρ
ρ
η
−
U
Q
I
V
η I = ∑ (∂Ai ∂ z ) I ai
φ Q = ∑ (∂Pi ∂z ) Qai
i
i
.
.
.
.
ηV = ∑ (∂Ai ∂z ) V ai
i
φV = ∑ (∂Pi ∂z ) Vai
i
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
1
I
d Q
−1
= − (∂ A ∂ z )
dz U
0
0
V
Ia 1
Qa −1
U = 0
a
V 0
a
−1
1
0
0
0
0
1
0
and no emission (B=0)
0 I
0 Q
0 U
1 V
I
d Q
dz U
V
1
−1
= − (∂ A ∂ z ) 0
0
d
(I + Q) = 0
dz
⇒
−1
1
0
0
0
0
1
0
0 I
0 Q
0 U
1 V
z ∈ [0 , L ]
( I + Q ) out = ( I + Q ) in
L
− ∫ 2 (∂A ∂z )dz
d
( I − Q) = −2(∂A ∂z )( I − Q) ⇒ ( I − Q)out = ( I − Q)in e 0
≈0
dz
d
U = − 2 (∂ A ∂ z )U
dz
I
1
1 1
Q
U = 2 0
V
0
out
1 0 0 I
1 0 0 Q
0 0 0 U
0 0 0 V in
⇒
U
out
= U
in
L
2 (∂ A
∫
0
e
−
∂ z )dz
≈ 0
Typical Mueller matrix of a linear
polarizer
Zeeman Effect
Purpose: work out the η´s and ρ´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
Spectral line absorption
The wave function characterizing eigenstate of theses Hydrogenlike atoms can be written down as
ψ ( r ,α , β , t ) = ψ
β
0
(r , β ) e
iM α
e
i
E
t
h
r
r
where M is the magnetic quantum number
and E is the energy of the level.
α
The electric dipole of the corresponding distribution of charges will be
r
d = q
∫ψ
volume
2
r
r dv
r
d = q
∫
ψ
2
r
r dv
volume
When you have a transition between states b (initial) and f
(final), the wave function is a linear combination of the two
states
ψ = cb (t ) ψ b + c f (t ) ψ
cb (0 ) = 1
c f (0 ) = 0
and
and
c b → 0 when
c
f
t → ∞
→ 1 when
r
r
*
d = d 0 + 2 qc b c f Re ∫ ψ bψ
volume
f
t → ∞
*
f
r
r dv =
Eb − E f
i
t
r
i
(
M
−
M
)
α
r
*
*
b
f
ψ 0 bψ 0 f e
r dv
= d 0 + 2 qc b c f Re e h
∫
volume
r
d0
constant over the period of the wave
r
r
β
α
sin β cosα
0
1
1
r sin β iα r sin β −iα
r
r = r sin β cosα = r cos β 0 +
e −i +
e i
2
2
cos β
1
0
0
Eb − E f
i
t
r r
i ( M b − M f )α r
*
*
h
d = d 0 + 2 qcb c f Re e
ψ 0 bψ 0 f e
r dv
∫
volume
0 Eb −E f
i( Mb −M f )α i h t
{L} = ∫ e
dα 0 e
0
1
2π
1 Eb −E f
2π
i( Mb −M f +1)α i h t
dα − i e
∫e
0
0
1 Eb −E f
i( Mb −M f −1)α i h t
dα i e
∫e
0
0
2π
2π
∫∫Ldrdβ +
ipα
e
∫ dα ≠0 ⇔ p = 0
0
∫∫Ldrdβ +
Which leads to the
selection rules for E-dipole
transitions
∆M = 0, ±1
∫∫Ldrdβ
each one associated with
a polarization
observer
x
We are interested in the projection in the
plane perpendicular to the line of sight (xy plane)
z
θ
There are only three types of polarization
θ
y
1
r
dx = d ⋅0
0
0
r
d y = d ⋅ cos θ
− sin θ
y
a) For ∆M=0
d
x
∝ Re e iwt
d y ∝ Re e iwt
0 1
0 ⋅ 0 = 0
1 0
0 1
0 ⋅ cos θ
1 − sin θ
x
r
d⊥(t)
= − sin θ cos( wt )
Iπ
Qπ
U
π
V
π
1
1
−
2
=
sin
θ
0
0
b) For ∆M=Mb- Mf=+1
1 1
d x ∝ Re e iwt − i ⋅ 0 = cos( wt )
0 0
1 0
iwt
d y ∝ Re e − i ⋅ cos θ
0 − sin θ
= cos θ sin( wt )
Iσ +
Qσ +
U
σ+
V
σ+
1 + cos 2 θ
2
sin θ
=
0
− 2 cos θ
y
x
cos θ
1
r
d⊥ (t )
b) For ∆M=Mb- Mf=-1
y
Iσ −
Qσ −
U
σ−
V
σ−
1 + cos 2 θ
2
sin θ
=
0
2 cos θ
x
cos θ
r
d ⊥ (t )
1
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,
B=0
∆w
B=B0
w0
∆ w ∝ B0
w0
w
w
Associated to each transition there is a absorption profile plus a
retardance profile
w0
w0+∆w
w0
w0+∆w
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)
y
∆M=0
cos θ
y
observer
x
x
φ
φ
θ
r
B
∆M=+1
y
x
y
x φ
r
∆w ∝ B
w0
∆M=-1
absorption
w0+∆w
φ
retardance
w0
w0+∆w
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),
Ia
Q a
U
a
V
a
=
− 10
0
0
r
( B / kG )
1
−5
– 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
ηU
ηV I − B
I ηI ηQ
ρV − ρU Q
d Q ηQ ηI
=
−
ρQ U
dz U ηU − ρV ηI
V η
V
ρ
ρ
η
−
U
Q
I
V
1 + cos2 θ
1 + cos2 θ
ηI
1
1
2
2
kl ησ + − sin θ cos 2φ kl ησ − − sin θ cos 2φ
ηQ
0 kl
2 cos 2φ
=
+
+
sin
κ
η
θ
η c 0 2 π
sin 2φ 2 2 − sin2 θ sin 2φ + 2 2 − sin2 θ sin 2φ
U
0
0
η
− 2 cosθ
2 cosθ
V
same for ρ´s with replacing η´s with ρ´s
I
d Q
=
dz U
V
observer
θ
x
φ
r
B
y
ηI
ηQ
−
η
U
η
V
ηQ
ηI
− ρV
ρU
ησ + + ησ −
2
2
sin
+
(
+
cos
)
1
η
θ
θ
π
2
k
η +η
ηQ = l ηπ − σ + σ − sin2 θ cos 2φ
2
2
k
η +η
ηU = l ηπ − σ + σ − sin2 θ sin 2φ
2
2
k η −η
ηV = l σ + σ −
2
2
k
ηI = κ c + l
2
Unno-Rachkovsky Equations
ηU
ρV
ηI
− ρQ
ηV
− ρU
ρQ
ηI
I − B
Q
U
V
k
2
ρσ+ + ρσ−
k
2
ρσ+ + ρσ−
ρQ = l ρπ −
ρU = l ρπ −
2
2
k ρσ+ − ρσ−
2 2
ρV = l
sin θ cos2φ
2
sin θ sin2φ
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
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(λ) = I(- λ)
Q(λ) = Q(- λ)
U(λ) = U(- λ)
V(λ) = -V(- λ)
λ ≡ wavelength - laboratory wavelength of the spectral line corrected by the
macroscopic velocity
No proof given, but it follows from the symmetry properties of the η´s and ρ´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 ∆λΒ ∝ Β is the Zeeman splitting of a Zeeman triplet, and ∆λD is the
width of the line, it can be shown that (e.g., Landi + Landi 1973)
ηI ≈ ηI 0 + ηI 2 (∆λB / ∆λD )2 + K
ρ Q ≈ ρ Q 2 ( ∆ λ B / ∆λ D ) 2 + K
ηQ ≈ ηQ 2 (∆λB / ∆λD )2 + K
ρ U ≈ ρ U 2 ( ∆λ B / ∆λ D ) 2 + K
ρ V ≈ ρ V 2 ( ∆λ B / ∆λ D ) + K
ηU ≈ ηU 2 (∆λB / ∆λD )2 + K
ηV ≈ ηV 2 (∆λB / ∆λD ) + K
then to
first order in (∆λΒ / ∆λD )
I ηI
d Q 0
= −
0
dz U
V η
V
(a)
(b)
0
ηI
− ρV
0
0
ηV I − B
ρV 0 Q
ηI 0 U
− 0 ηI V
d nQ d nU
= n = 0 ⇒ Q = U = 0 Since there is no polarization at the
n
dz
dz
bottom of the atmosphere
d (I ± V )
= (η I ± η V )( I ± V − B )
dz
dη ( λ )
+L ≈
η I + ηV ≈ k c + klη (λ ) − ∆λ B cos θ
dλ
≈ k c + klη ( λ − cos θ ∆λ B )
η
I
− η V ≈ k c + k l η ( λ + cos θ ∆ λ B )
d (I ± V )
= [k c + k l η ( λ m cos θ ∆ λ B ) ]( I ± V − B )
dz
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 θ ∆λB
If the longitudinal component of the magnetic field is constant then cos θ
∆λB is constant and I+V and I-V are identical except for a shift
I-V
I+V
2 cos θ ∆λB
λ
df ( λ )
cos θ ∆ λ B
dλ
df ( λ )
I − V = f ( λ + cos θ ∆ λ B ) ≈ f ( λ ) +
cos θ ∆ λ B
dλ
1
I = [( I + V ) + ( I − V ) ] ≈ f ( λ )
2
1
df ( λ )
dI ( λ )
V = [( I + V ) − ( I − V ) ] ≈ −
cos θ ∆ λ B ≈ −
cos θ ∆ λ B
2
dλ
dλ
I + V = f ( λ − cos θ ∆ λ B ) ≈ f ( λ ) −
dI ( λ )
V (λ ) ≈ −
cos θ ∆ λ B
dλ
Magnetograph equation: the
Stokes V signal is proportional to
the longitudinal component of the
magnetic field
observer
cos θ ∆λB
V
0
r
B cos θ
λ
θ r
B
The previous argumentation is based on the assumption that
the Zeeman pattern is a triplet (one π component, one σ+
component and one σ- 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
∆λB ∝ geff
g eff
r
B
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).
I obs = αI + (1 − α ) I non − mag
Qobs = αQ + (1 − α )Qnon − mag = αQ
U obs = αU + (1 − α )U non − mag = αU
V = αV + (1 − α )V
non-magnetic magnetic
non − mag = αV
obs
red area
α=
: filling factor, i.e., fraction of
total area resolution element filled by
magnetic fields
resolution element
dI ( λ )
B cos θ
Effect on the magnetograph equation V ( λ ) ≈ − C
dλ
I obs = αI + (1 − α ) I non − mag ≈ I
(if I ≈ I non − mag )
Vobs = αV
dI obs ( λ )
V obs ( λ ) ≈ −
B eff
dλ
with B eff = α B cos θ
observer
Beff
r
ds
r
B
r r
= ∫∫ B ⋅ ds /
pixel
∫∫ ds
pixel
Magnetic flux density
4.- Milne-Eddington solution of the Radiative Transfer
Equation for Polarized Light (RTEPL).
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
I
d Q
=
dz U
V
ηI
ηQ
−
η
U
η
V
ηQ
ηI
− ρV
ρU
ηU
ρV
ηI
− ρQ
ηV
− ρU
ρQ
ηI
I − B
Q
U
V
RTEPL: first order linear differential equation. Admits an analytic
formal solution of the coefficients are constant (basic maths)
I
d Q
=
dz U
V
I
r
Q
S =
U
V
ηI
η Q
−
η
U
η
V
ηQ
ηI
− ρV
ρU
ηU
ρV
ηI
− ρQ
1
r
0
1 =
0
0
r
r
r
dS
= K (S − B 1 )
dτ
ηV
− ρU
ρQ
ηI
I − B
Q
U
V
dτ = −κ c dz
τ≡
ηI
1 η Q
K =
κ c η U
η
V
ηQ
ηI
− ρV
ρU
continuum optical depth
ηU
ρV
ηI
− ρQ
ηV
− ρU
ρQ
ηI
Compact form of the RTEPL
r
r
r
dS
= K (S − B 1 )
dτ
Assumptions:
the ratio line to continuum absorption coefficient does not depend
on optical depth
κ l / κ c = constant with optical depth
The source function depends linearly on continuum optical depth
B = B0 + τ B1
Broadening of the line constant (both Doppler and damping)
Magnetic field vector constant with depth
… all them together lead to constant absorption matrix
K = constant with optical depth τ
r
r
r
dS
= K (S − B 1 )
dτ
r r r
r
r
try solutions S = S 0 + S1τ , with both S 0 and S1 constant
r
r
r
r
r
S1 = K (S 0 − B0 1) + τ K (S1 − B1 1)
r
r r
r
r
K (S1 − B1 1) = 0 ⇒ S1 = B1 1
r
r
r
r
r
r
−1
S1 = K (S 0 − B0 1) ⇒ S 0 = K S1 + B0 1
r
r
r
r
−1
S(τ = 0) = S 0 = B0 1 + B1 K 1
[
]
I = B0 + B1 η I (η I2 + ρ Q2 + ρU2 + ρV2 ) / ∆
[
U = − B [η η
V = − B [η η
]
+ η ρ ) ]/ ∆
+ η ρ ) ]/ ∆
Q = − B1 η I2η Q + η I (ηV ρU − ηU ρV ) + ρ Q (η Q ρ Q + ηU ρU + ηV ρV ) / ∆
1
2
I U
+ η I (ηQ ρV − ηV ρ Q ) + ρU (ηQ ρ Q + ηU ρU
1
2
I V
+ η I (ηU ρ Q − ηQ ρU ) + ρV (η Q ρ Q + ηU ρU
V
V
V
V
∆ = η I2 (η I2 − ηQ2 − ηU2 − ηV2 + ρ Q2 + ρU2 + ρV2 ) − (η Q ρ Q + ηU ρU + ηV ρV ) 2
Milne-Eddington solutions of the RTEPL (e.g., Landi Degl´Innocenti, 1992)
Free parameters:
1.
2.
3.
4.
5.
6.
7.
8.
9.
Magnetic field strength
Magnetic field azimuth
Magnetic field inclination
B0
B1
Macroscopic velocity
Doppler broadening
Damping
Strength of the spectral line
IDL
5.- 180o azimuth ambiguity (exact)
observer
θ
x
φ
observer
r
B
r
B
θ
φ + 180 o
y
y
x
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).
observer
observer
θ
x
φ
r
B
φ
y
180
o
−θ
y
x
r
B
V (180 − θ ) ≈ −V (θ )
o
IDL
since V ∝ cos θ and cos(180o − θ ) = − cos θ
7.- Q=U = 0 for longitudinal magnetic fields.
V=0 for transverse magnetic fields. (Approximate.)
observer
observer
r
B
x
φ
θ =0
φ
Q=U=0
y
θ = 90
o
y
x
V=0
r
B
IDL
Polarimeters
Basic elements:
– Modulation package
– Intensity detector
– Calibration package
– Instrumental polarization
optics
modulator (pj)
~
I
~
Q
U~
~
V
Iout
Qout
U
out
V
out
Intensity
detector
telescope
+ optics
optics
I
Q
U
V
calibration
optics
~ + M ( p ) U~ + M ( p ) V~
I out ( p j ) = M 11 ( p j ) I~ + M 12 ( p j ) Q
13
j
14
j
~
I
~
Q
U~
~
V
= M
[
ij
−1
( pk )
]
I out
I out
I out
I out
(
(
(
(
p1 )
p2 )
p3 )
p 4 )
~
I
~
Q
U~
~
V
Mueller
= Matrix
Telescope
I
Q
U
V
Modulation package
Optical system whose Mueller matrix can be (strongly) varied
upon changing a set of control parameters.
Example
fixed linear polarizer
rotating retarder (λ/4)
α
I out
Q out
U
out
V
out
1
− 1 ~
~ + sin( 2α ) cos( 2α )U~ + sin( 2α ) V~
2
I
+
cos
(
2
α
)
Q
=
0
0
[
Usually the last element is an optical element that fixes the polarization
state of the exit beam, but this is not always the case.
]
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.
rotating retarder (λ/4)
Example
fixed linear polarizer
α
I out
Q out
U
out
V
out
1
2
− cos 2α
= − sin 2α cos 2α
sin 2α
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 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
= M =∏M
j
Teslecope
~
I
~
Mueller
Q
=
Matrix
U~
Telescope
~
V
I
Q
U
V
j
⇒
I
Q
U
V
Mueller
= Matrix
Telescope
−1
~
I
~
Q
U~
~
V
Instrumental Polarization: removing I V crosstalk
~
I
~
Q
~
U~
V
Mueller
=
Matrix
Telescope
I
Q
U
V
~
V = M 41 I + M 42Q + M 43U + M 44V
~
I ≈ M 11 I
since I >> Q , U , and V ⇒
~
V ≈ M 41 I + M 44V
at continuum wavelengths V=0
~
I c ≈ M 11 I c
~
Vc ≈ M 41 I c
~ ~
I / Ic ≈ I / Ic
~
~ ~ I ~ M 44 V
V − Vc ~ / I c ≈
M 11 I c
I
c
(longitudinal) Magnetograph
CCD
2 states modulator
λ/4-plate + linear
polarizer
I out ( t1 ) = C ( I + V )
I out ( t 2 ) = C ( I − V )
then
Narrowband color
filter
I
Q
U
V
V ∝ I out ( t1 ) − I out ( t 2 )
I ∝ I out ( t1 ) + I out ( t 2 )
and
V I out ( t1 ) − I out ( t 2 )
=
I
I out ( t1 ) + I out ( t 2 )
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
≈ 30%
I
in sunspots
V
≈ 10%
I
in plage regions
V
≈ 1%
I
in network regions
V
≤ 0.1%
I
in inter - network regions
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
r
r
Iout (t1 ) = I ( x1 ) +V ( x1 )
r
r
Iout (t2 ) = I ( x2 ) −V ( x2 )
r
r
r
r
I out ( t1 ) − I out ( t 2 ) = I ( x1 ) − I ( x 2 ) + V ( x1 ) + V ( x 2 )
r
d
x
r
[t1 − t 2 ] + V ( xr0 ) ≠ V ( xr0 )
≈ ∇ I ( x0 ) ⋅
dt
r
r
r
with x 0 = ( x1 + x 2 ) / 2
(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., χ2 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.
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
r
V ( λ , x )ds ≈
∫
V (λ ) =
# components
∑ f V ( λ ) ≈ −C ∑
i
i
i
resolution
r
dI
f i Bi cos θ i i
dλ
# components
i
r r
dI i ( λ ) # components r
dI i ( λ )
θ
= −C
≈
−
f
B
cos
C
B
ds ≈
∑
i
i
i
∫
dλ
dλ resolution
i
I (λ) =
∫
r
I (λ, x)ds ≈
# components
∑ fi Ii (λ) ≈ Ii (λ)
i
resolution
∫
r
ds
resolution
r r
∫ Bds
V (λ )
= c ( λ ) resolution r
I (λ )
∫ ds
resolution
d ln I i ( λ )
c ( λ ) = −C
dλ
A calibrated magnetograph gives the
longitudinal component of the magnetic
flux density (mag flux per unit surface)
Milne-Eddington fitting technique
(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
χ =
2
∑ Stokes
∀ data
observed
− Stokes
2
synthetic
Observed I,Q,U & V
χ2 minimization
Input model atmosphere
B,θ,ϕ, ...
χ 2,
∂χ
∂χ
,
,L
∂B
∂θ
2
B,θ,ϕ, ...
2
synthesis
I
I
Q
∂ Q ∂
U , ∂B U , ∂ θ
V
V
I
Q
U , L
V
new atmosphere B,θ,ϕ, ...
giving a smaller χ2
NO
NO
∆χ2
small enough?
YES
YES
observed
B,θ,ϕ
Sunspot observation
Skumanich & Lites 1987
MISMA inversion code
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
χ =
2
∑
∀ data
Stokes
observed
− Stokes
2
synthetic
Synthetic
Observations
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.
Prêt-à-porter inversions
Obser
Observed
I , Q ,ved
U ,V
I,Q,U & V
Pre-computed data base
model#1(B1,θ1,φ1K) →I1,Q1,U1,V1
M
model#i (Bi ,θi ,φi K) →Ii ,Qi ,Ui ,Vi
M
Which synthetic profiles
are closest to the
observed profiles?
If # i are the closest ones
then
model#n(Bn,θn,φn K) →In,Qn,Un,Vn
observed B, θ , φ = Bi , θ i , φi
Fitting technique for PCA:
Eigenfaces
# of eigenfaces used in the reconstruction
Reconstructed faces
face = ∑ eigen value i × eigenfacei
i
Rees et al., 2000
S ≈ ∑ ei si
i
S : Stokes vector
ei : i - th eigenvecto r
si : i - th eigenvalue
Only a few eigenvalues are needed to
characterize the Stokes profiles
Rees et al. (2000)
Forward modeling (which is
an inversion technique!!!)
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.
Turbulent Dynamo Simulations by Cattaneo & Emonet
cluster analysis classification
1´´ seeing
The case of the large magnetic
flux concentrations
Observed
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.
¨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.
Understanding Real Magnetograms, e.g., Kitt
Peak Synoptic Maps
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
solar flux @ max
2
Line Ratio Method, 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 (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.
I (λ ) 5247 = I (λ ) 5250 if there is no magnetic field
If weak field (sub-kG):
dI (λ )
V5247 (λ ) / 2 ≈ − Bzαkλ
dλ ⇒ V5247 (λ ) / 2 ≈ 1
(
)
dI
λ
V5250 (λ ) / 3
V5250 (λ ) / 3 ≈ − Bzαkλ20
dλ
2
0
If strong field (sub-kG):
V5247 (λ ) / 2
≈ 1 + Bz2 f (λ ,...)
V5250 (λ ) / 3
Line ratio obseved in network
Fe I 5247
Bz ≈ 0
Bz ≈ 1 kG
resolution element
Fe I 5250
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/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-ofIDL
sight. (Makita 1986.)
– In the neutral line Stokes V is never zero but shows the crossover effect
+∞
Signal ∝
∫ Signal(
λ ) Filter( λ ) d λ
−∞
Broad Band Imaging - Polarimetry
neutral line
we
sunspot
Sun
solar limb ↑
neutral line
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 ∝
∫ V ( λ )dλ =
band − width
∑
resolution
λ
f
V
(
)
dλ =
∑ i i
∫
band − width resolution
f i ∫ Vi ( λ ) d λ = 0 unless there are gradients along the LOS
band − width
since for no LOS gradient
∫ V ( λ )dλ = 0
i
band − width
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).
and θ = 90
o
then
dV
= 0 and V = 0
dz
Stokes V
r
if B is constant
SA & Lites, 1992, ApJ, 398, 359
Cross-over effect, Grigorjev and Kart, 1972, SPh, 22, 119
c) The BBCP cannot be due to smooth well-organized vertical
variations of magnetic fields inclination.
150 km
750 km
Resolution element
Sanchez Almeida (2005)
∇B = 0 = ρ
−1
∂
ρ
∂ρ
Bρ
∂
+
∂z
B
z
The BBCP has to be due to very intermitent variations of magnetic
field inclinations.
150 km
Resolution element
750 km
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
r
B1
r
B2
V2 = -V1 ⇒ V1+V2 = Vobs = 0
r
B1
r
B2
Q2 = -Q1 ⇒ Q1+Q2 = Qobs = 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.
Inter-Network Quiet Sun
Domínguez Cerdeña et al. (03)
1”x1”
aangular resolution mag. ≅ 0.5”
asensitivity ≅ 20 G
aVTT (obs. Teide), speckle reconstructed
aUnsigned flux density ≈ 20 G
12 G x SolarSurface = 7x1023 Mx = solarflux@max
12 G
1.6 G
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
r
r
β
α
sin β cosα
0
1
1
r sin β iα r sin β −iα
r
r = r sin β cosα = r cos β 0 +
e −i +
e i
2
2
cos β
1
0
0
Eb − E f
i
t
r r
i ( M b − M f )α r
*
*
h
d = d 0 + 2 qcb c f Re e
ψ 0 bψ 0 f e
r dv
∫
volume
0 Eb −E f
i( Mb −M f )α i h t
{L} = ∫ e
dα 0 e
0
1
2π
1 Eb −E f
2π
i( Mb −M f +1)α i h t
dα − i e
∫e
0
0
1 Eb −E f
i( Mb −M f −1)α i h t
dα i e
∫e
0
0
2π
2π
∫∫Ldrdβ +
ipα
e
∫ dα ≠0 ⇔ p = 0
0
∫∫Ldrdβ +
Which leads to the
selection rules for E-dipole
transitions
∆M = 0, ±1
∫∫Ldrdβ
each one associated with
a polarization
Hanle Effect Based Magnetometry
A weak magnetic field splits the Zeeman sublevels but … it
is weaker than the natural width of the lines.
w0-∆w w0+∆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 .
dx
= Re{eiwt U1ei∆wt + U2e −i∆wt }
dy
[
]
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 ∆w=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,
Jx
Jy
2
= U1 x
2
+ U2x
2
= U1 y
2
+ U2 y
2
2
{
+ 2 Re{ U
+ 2 Re U1 xU 2*x ei 2 ∆wt
* i 2 ∆wt
U
1y 2 ye
}
}
J x J *y = U1 xU1*y + U 2 xU 2* y + U1 xU 2* y ei 2 ∆wt + U 2 xU1*y e −i 2 ∆wt
Textbook case: describes linearly polarized in the x axis at t=0.
∆w
2π
w
dx
cos(∆wt )
= U 0 cos(wt )
d
sin(∆wt )
y
I = U 02
Q / I = cos(2 ∆wt )
U / I = sin( 2∆wt )
V /I =0
τ∆ w
τ : coherency time
unpolarized
r
∆w ∝ B
atom
We
For Hanle effect to depend on the field
strength (and so to be a useful tool),
τ ∆ w ≤ 2π
r
(λ/500 nm)2
| B | ≤ 70 G
g eff (τ / 10−8 s)
Sun
Hanle signals even if tangled fields
non-magnetic scattering
Hanle effect
r
B≠0
r
B=0
Sun
Sun
Sr I 4607Å Hanle depolarization
depolarizing collisions are critical
for a proper modeling
r
B = 0 (known = modelled)
observed
Hanle saturation at some 50 G
Faurobert et al. (2001)
Magnetometry Based on Lines With Hyperfine
general Zeeman pattern
Magnetometry Based on Lines With
Hyperfine Structure
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.
dI ( λ )
cos θ ∆ λ B
V (λ ) ≈ −
dλ
The centers of gravity of the π and σ components is
independent of the HFS.
He I 1083nm Chromospheric Magnetometry
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.
ME
MEfitfit
blend of 3 lines
Rüedi et al. (1995)
Incomplete Pashen-Back effect
required for a proper analysis
(Socas-Navarro et al. 2004)
Creates NCP by saturation
Polarimeters on board Hinode
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: Solar Optical Telescope
SOT
-FG
GOTO Summary -- Index:
Selected references
ref_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. 2002
0.12 arcsec, spatial resolution
1´´ x 1´´
A
r r
k /k
B
r
k
Observers A and B receive exactly the same signal, which is
constant in the plane perpendicular to
r r
k ⋅ r = constant
Point Source
A x cos( wt − φ x )
r
r
r
r r
e ( r , t ) = Re e
E ( k ⋅ r ) = A y cos( wt − φ y ) =
A cos( wt − φ )
z
z
A x cos φ x
A x sin φ x
cos wt A y cos φ y + sin wt A y sin φ y =
A cos φ
A sin φ
z
z
z
z
r
r
iα x
A
e
= cos wt A c + sin wt A s
x iα
r
plane motion
E = Ay e y
Choosing t0 properly,
A e iα z
{
rr
i ( wt − k r )
}
r r
2 As ⋅ Ac
tan( 2 wt0 ) = 2
As − Ac2
r
r
r
e( r , t ) = cos( wt − wt0 )a + sin( wt − wt0 )b
r r
where a ⋅ b = 0
φx
φ y
φ
z
z
rr
k
r −α
rr
= kr − α
kr rr − α
x
y
z
r r
e ( r , t ) = Re
{e
rr
i ( wt − k r )
r r r
E (k ⋅ r )
}
r
r
r
e( r , t ) = cos( wt − wt0 )a + sin( wt − wt0 )b
r r
where a ⋅ b = 0
r
e ( t1 )
r
e (t2 )
r
e (t3 )
t1 ⟨ t 2 ⟨ t 3
Monochromatic means plane Elliptical Motion
Inserting monochromatic solutions of the kind
{
rr
r r
r r
i ( wt − k r ) r
e ( r , t ) = Re e
E (k ⋅ r )
}
into the wave equation derived from the Maxwell equations ,
one finds
E || / E ⊥ ≈ λ / L << 1
E|| :
E⊥ :
Component in the direction of
λ = 2πc / w
r
k
Transverse component
λ:
Wavelength
L:
Characteristic scale for the variation of
r
E
y
Monochromatic wave
Q2 + U 2 + V 2 = I 2
x
r
e (t )
Q2 + U 2 + V 2
= p ≠1
I
In general
> p=0 represents unpolarized light
p is the degree of polarization
> p=1 corresponds to fully polarized light
If Jx(t) and Jy(t) vary at random, then
the light Unpolarized Light
y
x
r
e (t )
I
Q
U
V
1
0
= 0
0
y
y
x
r
e (t )
I 1
Q 1
U = 0
V 0
x
r
e (t )
y
y
x
r
e (t )
1
0
1
0
x
r
e (t )
y
1
0
− 1
0
y
x
r
e (t )
1
− 1
0
0
1
0
0
1
x
r
e (t )
1
0
0
−1
iwt Jx1 Jx2
iwt Jx1
iwt Jx2
rr
r r
r r
e(r,t) = e1(r,t) +e2 (r,t) = Re e + Re e = Re e + =
Jy1
Jy2
Jy1 Jy2
iwt Jx1 + Jx2
⇒
Re e
Jy1 + Jy2
Jx Jx1 + Jx2
=
J J + J
y y1 y2
(
)
J x J *y = ( J x1 + J x 2 ) J *y1 + J *y 2 = J x1 J *y1 + J x1 J *y 2 + J x 2 J *y1 + J x 2 J *y 2
J x1 J *y 2 = J x 2 J *y1 = 0
(because the two beams are incoherent)
J x J *y = J x1 J *y1 + J x 2 J *y 2
2
I = Jx + J y
2
Q = Jx − J y
2
2
= I1 + I 2
= Q1 + Q2
{ }
V = −2 Im{ J J }= −2Im{ J
U = 2 Re J x J y* = U1 + U 2
x
*
y
}
{
}
{
}
*
*
*
*
J
+
J
J
=
−
2
Im
J
J
−
2
Im
J
J
= V1 + V2
x1 y1
x2 y2
x1 y1
x2 y2
I
Q
U
V
1+ p
= 2p
pI
Q 1− p
+
2p
U
V
pI
−Q
−U
−V
p = Q2 + 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
r
x
r Jx
J1 = ⇒
Jy
r r*
J1 ⋅ J 2 = 0
J1
r
J2
I = J 2 + J 2
x
y
2
2
Q = J x − J y
U = 2 Re J J *
x y
V = −2 Im J J *
x y
{ }
{ }
r − J *y
J 2 = * ⇒
Jx
J 2+ J 2 =I
x
y
2
2
2
2
J y − J x = − J x − J y = −Q
U = 2 Re − J * J = −U
y x
V = 2 Im J * J = −V
y x
{ }
{ }
mxx mxy
m =
m
m
yx yy
M
M
12
M
13
M
14
M
From Jones matrix
[m ]
ij
to Mueller matrix
[M ]
ij
11
21
M
22
M
23
M
24
M
31
M
32
M
33
M
34
M
41
M
42
M
43
M
44
+ m xy
2
+ m yy
2
/ 2
2
2
= m xx − m yy − m xy
= Re m xx m *xy + m yx m *yy
2
+ m yy
2
/ 2
= m xx
2
{
= Im {m
+ m yy
2
m *xy + m yx m *yy
}
}
2
+ m xy
2
− m yy
2
/ 2
2
2
= m xx + m yy − m xy
= Re m xx m *xy − m yx m *yy
2
− m yy
2
/ 2
= m xx
xx
2
− m yy
{
}
= Im {m m − m m }
= Re {m m + m m }
= Re {m m − m m }
= Re {m m − m m }
= Im {− m m + m m }
= − Im {m m + m m }
= − Im {m m − m m }
= − Im {m m + m m }
= Re {− m m + m m }
xx
*
xy
yx
*
yy
xx
*
yx
xy
*
yy
xx
*
yx
xy
*
yy
xy
*
yx
xx
*
yy
xy
*
yx
xx
*
yy
xx
*
yx
xy
*
yy
xx
*
yx
xy
*
yy
xy
*
yx
xx
*
yy
xy
*
yx
xx
*
yy
r
U2
r
U1
r r
U1 ,U 2 :
(
)
r r*
r
r
U1 ⋅ U 2 = 0 ; U1 = U 2 = 1
For any selective absorption, this set is a
base of complex 2D vectors (e.g., the
Jones vector)
For any polarization with Jones vector
r
J
r
r r* r
r r* r
J = ( J ⋅U1 ) U1 + ( J ⋅ U 2 ) U 2
The OS just changes the Jones vector as
r
r
r r* r
r r* r
J out = m J = (1 − ε )( J ⋅U1 ) U1 + ( J ⋅U 2 ) U 2
r r* r
U
1 0 r
*
* 1x
=
J = −ε ( J ⋅U1 ) U1 = −ε ( J xU1x + J yU1 y )
m −
0 1
U1 y
J x U1x 2 + J y U1xU1*y
U1x 2 U1xU1*y J x
= −ε
−ε
2
2
J (U * ⋅U ) + J U
U * ⋅ U
J y
U
1
1
1
1
1
1
x
x
y
y
y
x
y
y
(
1 0 a11
=
m −
0 1 a21
)
2
*
U
U
U
a12
1x
1x 1 y
= −ε *
2
U ⋅ U
a22
U
x
y
y
1
1
1
− ε I / ε = Re {a 11 + a 22 } = U 1 x
2
+ U 1y
2
= I1
− ε Q / ε = Re {a 11 − a 22 } = U 1 x
2
− U 1y
2
= Q1
{
}
( = 1)
− ε U / ε = Re {a 12 + a 21 } = 2 Re U 1 x U 1*y = U 1
{
}
− ε V / ε = − Im {a 12 − a 21 } = − 2 Im U 1 x U 1*y = V 1
ϕ V = Re {a 12 − a 21 } = 0
ϕ U = Im {a 12 + a 21 } = 0
ϕ Q = Im {a 11 − a 22 } = 0
weak magnetic field
approximation
dη (λ ) ∆λ2B d 2η (λ )
+
+L
η (λ + ∆λB ) ≈ η (λ ) + ∆λB
dλ
2
dλ2
η ( λ + ∆λ B )
η ( λ − ∆λ B )
η ( λ + ∆λ B ) + η (λ − ∆λ B ) ≈ 2η ( λ )
λ→
∆ λ
B
≡ Zeeman shift
η(λ + ∆λB ) −η(λ − ∆λB ) ≈ 2 ∆ λ
B
dη (λ )
dλ
Band-pass of typical magnetogram observations
continuum
References
•Kemp 1970, ApJ, 162, 169, in connection with the
continuumpolarization in a magnetic field
•Sanchez Almeida
