Tsing Hua University, Taiwan Solar Acoustic Holograms January 2008, Tucson Dean-Yi Chou
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Tsing Hua University, Taiwan
Solar Acoustic Holograms
January 2008, Tucson
Dean-Yi Chou
Motivation
Is it feasible to apply the principle of optical holography to a system of solar acoustic waves and active regions?
Contents
Principle of optical holography.
Concept of acoustic holography of active regions.
Construct 3-D wave fields of the magnetic region from the acoustic hologram.
Set up a simplified model to compute acoustic holograms of magnetic regions.
1. analogies and differences between two
2. difficulties
Challenges and prospects.
Hologram (interference pattern)
(time average)
Construction of Waves
hologram
diffraction field
(Gabor’s in-line holgram)
Acoustic waves on the Sun
solar surface
interference pattern
Solar Acoustic Waves + Active Region
(acoustic power map)
perturbed region
2
0 sI
Optical Holography Solar Acoustic Holography
reference wave
object
hologram
p-mode wave
magnetic region
acoutsic power map
Analogies
(coming from below)
(near the surface)
(on the surface)
Questions:
1. Can we detect the inference pattern (hologram) due to a magnetic region on the surface?
2. Can we use the observed hologram to construct the 3-D image of the magnetic region?
Optical Holography Solar Acoustic Holography
1. monochromatic
5. far field approximation
4. single reference wave
finite band width
wavelength
~ dimension of object
~ distance to hologram
* multiple incident waves
Differences
2. no boundary trapped in cavities
3. straight ray path curved ray path
If the width of power spectrum of a wave field is , the cohernt time of waves is
coherent time of waves
0
2
1
2
1
0
: central frequency
: period of central frequency
example
0 3.3 mHz
0.2 mHz (FWHM = 0.47 mHz)
2.6
solar surface
trapped in cavitiescurved ray pathmultiple incident waves
2. Waves are approximately vertical near the surface
1. Refracted waves from the lower turning point are ignored.
s
a
λ ～ a ～ s
Multiple Incident Waves
i
i0If incident waves are , total waves are )( 0 sii
i
Intensity of hologram
jijsisjisjsiji
i i iisisii
sjj
jsii
i
*0
*0
**00
*0
22
0
**00
Re2
)()(
cross terms
If different waves are uncorrelated, the contribution from cross terms is small.
Total interference is the sum of interference of individual wave.
interference term
Summation of interferences of different waves reduces the visibility of fringes.
1. Set up a simplified model for scattering of acoustic waves by a magnetic region.
2. Solve for the scattered waves.
3. Compute the interference pattern (hologram) between incident wave and scattered wave.
4. Study the influence of various parameters on the hologram.
5. Compute the constructed wave field by illuminating the hologram with a reference wave.
Model Study
Assume unperturbed medium is uniform, and the wave equation is
Assume the interaction between waves and magnetic regions is described by sound-speed perturbations:
0),(1
),(2
2
22
t
tx
ctx
)()( 10 xccxc
),(
),(12
),(1),(
2
2
20
12
2
22
txS
t
tx
cc
c
t
tx
ctx
time independent
Wave equation becomes
Source of scattering
Wave Equation
Solution of Scattered Wave
'|'
)','(1
'
/)'(
2
1
'')','()',';,(),(
3/''2
02
20
01
3
xdt
tx
cxx
cxc
dtxdtxStxtxGtx
cxxtt
s
),(),(),( 0 txtxtx s
scattered wave with Green’s function and Born approximation
wave equation
total solution
t
tx
cc
ctxS
t
tx
ctx
2
2
20
12
2
22 ),(1
2),(),(1
),(
expressed in terms of Fourier components
dextx ti ),(2
1),( 00
dxdex
cxx
cxctx cxxti
s '),'('
/)'(
2
1),( 3)/'(
020
201
2/3
Hologram
Intensity of the hologram is the time average of 2
*0
22
0**
00
2Re2 ssss
interference
0/'2/
2/
*002
0
2013
*0
2/
2/
*0
),(),'('
/)'('
2
1
),(),(1
cxxiT
T
T
T
ss
exxc
dxx
cxcxd
T
dttxtxT
Interference term
Need a model for spatial dependence of ),(0 x
A Simplified Model for ),(0 x
assumptions:
1. Consider only one upward wave mode and its reflected wave at the surface.
2. Assume the free-end boundary at the surface.
)(0
)(00 )()(),( zkxkizkxki zz eRex
1R
xkiz ezkx
)cos()(2),( 00
interference term
]/')'([2
020
2013*
00)'cos()cos()(
'
/)'('
2),'(),( cxxxxki
zzs ezkzkc
dxx
cxcxd
Ttxtx
normalized interference term (related to fringe visibility)
2
0
]/')'([2
020
2013
2
0
*0
)(
)'cos()cos()('
/)'('
2Re
)(
),(),(Re2
0
d
ezkzkc
dxxcxc
xdT
t
txtx
cxxxxkizz
s
3. Simple dispersion relation:
)( 22
20
2zkkc
Normalized Interference Term (fringe visibility)
2
0
]/')'([2
020
2013
2
0
*0
)(
)'cos()cos()('
/)'('
2Re
)(
),(),(Re2
0
d
ezkzkc
dxxcxc
xdT
t
txtx
cxxxxkizz
s
Effects of parameters on holograms
1. coherent time of incident waves
3. size of the perturbed region
4. depth of the perturbed region
2. wavelength
5. angle of incidence
0
2
1
2
1
0
Effects of Coherent Time of Incident Waves
Setup of incident wave
3. Modes with a Gaussian power spectrum centered at 3.3 mHz, with different widths.
1. Waves propagate vertically: 0k
2. Dispersion relation: 2
22 kc
4. coherent time
Perturbed region
1. Uniform cylinder with 03.0/ 01 cc
2. diameter = 9.6 Mm, vertical extent = 4.8 Mm, depth = 12 Mm
3.3 mHz, 14.7 Mm (l=300), 0c 48.5 km/s
Effects of Coherent Time
0.2 mHz (FWHM = 0.47 mHz)
line width
Effects of Wavelength
0 3.3 mHz,
0.2 mHz
uniform cylinder with 03.0/ 01 cc
diameter = 9.6 Mm, vertical extent = 4.8 Mm, depth = 12 Mm
wavelength
Effects of Angle of Incidence
At 5Mm depth, the angle of incidence is about for at 3.3 mHz.010 100l020 for at 3.3 mHz.200l
Waves with different phase velocities have different angles of incience.
For example:
Effects of Angle of Incidence (cont.)
0 3.3 mHz, 0.2 mHz,
uniform cylinder with 03.0/ 01 cc
diamter = 9.6 Mm, vertical extent = 4.8 Mm, depth = 12 Mm
14.7 Mm (l=300)
incident angle
Construction of Wave Fields from Holograms
Illuminate the hologram by a vertically-propagating monochromatic wave.
hologram on the surface
Advantages of digital holograms
DC signal
2. Disentangling wave fields of virtual and real images.
1. DC signals are removed to enhance the interference pattern.
hologram on the surface
')'('
)'('
'1
'2)(
'
daxxx
xxn
xxk
i
xx
e
i
kx
xxik
Diffraction waves are computed by the Kirchhoff intergral
replaced by
Constructed wave field
205 Mm
30 Mm
7.14
Incident angle = 0
Mm
depth = 30 Mm
Constructed wave field
Incident angle = 0 deg.
Depth = 30 Mm
Incident angle = 0 deg.
Depth = 12 Mm
Constructed wave field
Incident angle = 0 deg.
Depth = 30 Mm
Incident angle = 10 deg.
Depth = 30 Mm
Effects of Multiple Incident Waves
1. Weaken holograms
2. Distort and weaken constructed wave fields
The maximum occurs at . 2/ zyx
dc
c 0
1max
1. Signals of holograms are weak.
Challenges in detecting interference fringes
2. Interference fringes are contaminated by suppression of acoustic power in magnetic region.
Fluctuation of 1000 MDI Dopplergrams is about 10%.
1% for the 2nd and 3rd fringes if
Remove suppression by an empirical relation of power vs. field strength.
Search for interference fringes outside magnetic regions.
3. Find an optimal filter to detect interference fringes.
03.0/ 01 cc
power map before correction power map after correction
magnetic field Power vs. B field
1024 MDI FD images
phase-velocity-filtered power map
magnetic field power map
1024 MDI FD images
phase-velocity-filtered power map
(3.3mHz/300) (3.3mHz/400)
power map before correction power map after correction
magnetic field Power vs. B field
512 MDI HR images
Challenges in Constructed 3D Wave Fields
2. Is there a better way to construct 3D wave fields?
1. How to disentangle wave fields of virtual and real images and obtain the 3D structure of the magnetic region?
Improvement in computing interference fringes
1. A better model to compute scattered waves.
2. Study of simulation data
interaction between waves and B fields
more realistic dispersion relation
Prospects
Better Data
Hinode & HMI
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