Tsing Hua University, Taiwan

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Solar Acoustic Holograms
Dean-Yi Chou

• Tsing Hua University, Taiwan

January 2008, Tucson
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Motivation
Is it feasible to apply the principle of optical
holography to a system of solar acoustic waves
and active regions?
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Contents
Principle of optical holography.
Concept of acoustic holography of active regions.
1. analogies and differences between two
2. difficulties
Set up a simplified model to compute acoustic
holograms of magnetic regions.
Construct 3-D wave fields of the magnetic region
from the acoustic hologram.
Challenges and prospects.
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Hologram
(interference pattern)
(time average)
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diffraction field
Construction of Waves
(Gabors in-line holgram)
hologram
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Acoustic waves on the Sun
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Solar Acoustic Waves Active Region
interference pattern
(acoustic power map)
solar surface
perturbed region
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Analogies
Optical Holography
Solar Acoustic Holography
reference wave
p-mode wave
(coming from below)
object
magnetic region
(near the surface)
hologram
acoutsic power map
(on the surface)
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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?
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Differences
Optical Holography
Solar Acoustic Holography
1. monochromatic
finite band width
2. no boundary
trapped in cavities
3. straight ray path
curved ray path
multiple incident waves
4. single reference wave
5. far field approximation
wavelength dimension of object distance
to hologram
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coherent time of waves
If the width of power spectrum of a wave field is
, the cohernt time of waves is
central frequency
period of central frequency
example
3.3 mHz
0.2 mHz (FWHM 0.47 mHz)
2.6
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trapped in cavities
curved ray path
multiple incident waves
? a s
solar surface
s
1. Refracted waves from the lower turning point
are ignored.
a
2. Waves are approximately vertical near the
surface
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Multiple Incident Waves
If incident waves are
, total waves are
Intensity of hologram
cross terms
interference term
If different waves are uncorrelated, the
contribution from cross terms is small.
Total interference is the sum of interference of
individual wave.
Summation of interferences of different waves
reduces the visibility of fringes.
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Model Study
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.
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Wave Equation
Assume unperturbed medium is uniform, and the
wave equation is
Assume the interaction between waves and magnetic
regions is described by sound-speed perturbations
time independent
Wave equation becomes
Source of scattering
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Solution of Scattered Wave
wave equation
total solution
scattered wave with Greens function and Born
approximation
expressed in terms of Fourier components
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Hologram
Intensity of the hologram is the time average of
interference
Interference term
Need a model for spatial dependence of
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A Simplified Model for
assumptions
1. Consider only one upward wave mode and its
reflected wave at the surface.
2. Assume the free-end boundary at the surface.

3. Simple dispersion relation
interference term
normalized interference term (related to fringe
visibility)
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Normalized Interference Term (fringe visibility)
Effects of parameters on holograms
1. coherent time of incident waves
2. wavelength
3. size of the perturbed region
4. depth of the perturbed region
5. angle of incidence
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Effects of Coherent Time of Incident Waves
Setup of incident wave
1. Waves propagate vertically
2. Dispersion relation
3. Modes with a Gaussian power spectrum centered
at 3.3 mHz, with different widths.
3.3 mHz,
14.7 Mm (l300),
48.5 km/s
4. coherent time
Perturbed region
1. Uniform cylinder with
2. diameter 9.6 Mm, vertical extent 4.8 Mm,
depth 12 Mm
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Effects of Coherent Time
line width
0.2 mHz (FWHM 0.47 mHz)
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Effects of Wavelength
3.3 mHz,
0.2 mHz
uniform cylinder with
diameter 9.6 Mm, vertical extent 4.8 Mm,
depth 12 Mm
wavelength
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Effects of Angle of Incidence
Waves with different phase velocities have
different angles of incience.
For example
At 5Mm depth, the angle of incidence is about
for at 3.3 mHz.
for at 3.3 mHz.
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Effects of Angle of Incidence (cont.)
3.3 mHz,
0.2 mHz,
14.7 Mm (l300)
uniform cylinder with
diamter 9.6 Mm, vertical extent 4.8 Mm,
depth 12 Mm
incident angle
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Construction of Wave Fields from Holograms
Illuminate the hologram by a vertically-propagatin
g monochromatic wave.
hologram on the surface
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Advantages of digital holograms
1. DC signals are removed to enhance the
interference pattern.
DC signal
2. Disentangling wave fields of virtual and real
images.
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Diffraction waves are computed by the Kirchhoff
intergral
replaced by
hologram on the surface
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Constructed wave field
depth 30 Mm
Incident angle 0
Mm
30 Mm
205 Mm
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Constructed wave field
Incident angle 0 deg. Depth 30 Mm
Incident angle 0 deg. Depth 12 Mm
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Constructed wave field
Incident angle 0 deg. Depth 30 Mm
Incident angle 10 deg. Depth 30 Mm
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Effects of Multiple Incident Waves
1. Weaken holograms
2. Distort and weaken constructed wave fields
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Challenges in detecting interference fringes
1. Signals of holograms are weak.
The maximum occurs at
.
1 for the 2nd and 3rd fringes if
Fluctuation of 1000 MDI Dopplergrams is about 10.
2. Interference fringes are contaminated by
suppression of acoustic power in magnetic
region.
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.
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magnetic field
Power vs. B field
1024 MDI FD images
power map before correction
power map after correction
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1024 MDI FD images
magnetic field
power map
phase-velocity-filtered power map
phase-velocity-filtered power map
(3.3mHz/300)
(3.3mHz/400)
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magnetic field
Power vs. B field
512 MDI HR images
power map before correction
power map after correction
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Challenges in Constructed 3D Wave Fields

1. How to disentangle wave fields of virtual and
   real images and obtain the 3D structure of the
   magnetic region?

2. Is there a better way to construct 3D wave
fields?
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Prospects
Improvement in computing interference fringes
1. A better model to compute scattered waves.
interaction between waves and B fields
more realistic dispersion relation
2. Study of simulation data
Better Data
Hinode HMI
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The End