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Solar Acoustic Holograms. Dean-Yi Chou. Tsing Hua University, Taiwan. January 2008, Tucson. 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.
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Solar Acoustic Holograms Dean-Yi Chou Tsing Hua University, Taiwan January 2008, Tucson
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. 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.
Hologram (interference pattern) (time average)
diffraction field Construction of Waves (Gabor’s in-line holgram) hologram
Solar Acoustic Waves + Active Region interference pattern (acoustic power map) solar surface perturbed region
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)
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?
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
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
trapped in cavities curved ray path multiple incident waves λ ~a~ s solar surface s 1. Refracted waves from the lower turning point are ignored. a 2. Waves are approximately vertical near the surface
Multiple Incident Waves If incident waves are , total waves are Intensity of hologram interference term cross terms 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.
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.
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
Solution of Scattered Wave wave equation total solution scattered wave with Green’s function and Born approximation expressed in terms of Fourier components
Hologram Intensity of the hologram is the time average of interference Interference term Need a model for spatial dependence of
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)
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
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 (l=300), 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
Effects of Coherent Time line width 0.2 mHz (FWHM = 0.47 mHz)
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
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.
Effects of Angle of Incidence (cont.) 3.3 mHz, 0.2 mHz, 14.7 Mm (l=300) uniform cylinder with diamter = 9.6 Mm, vertical extent = 4.8 Mm, depth = 12 Mm incident angle
Construction of Wave Fields from Holograms Illuminate the hologram by a vertically-propagatingmonochromatic wave. hologram on the surface
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.
Diffraction waves are computed by the Kirchhoff intergral replaced by hologram on the surface
Constructed wave field depth = 30 Mm Incident angle = 0 Mm 30 Mm 205 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
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.
magnetic field Power vs. B field 1024 MDI FD images power map before correction power map after correction
1024 MDI FD images magnetic field power map phase-velocity-filtered power map phase-velocity-filtered power map (3.3mHz/300) (3.3mHz/400)
magnetic field Power vs. B field 512 MDI HR images power map before correction power map after correction
Challenges in Constructed 3D Wave Fields • 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?
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