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Solar Acoustic Holograms

This study explores the feasibility of applying optical holography to solar acoustic waves and active regions, focusing on the principles and challenges of acoustic holography. Analogies, differences, and model studies are discussed to understand the construction of 3-D wave fields from acoustic holograms. The interplay between coherent time, wavelength, and parameters of the perturbed region is examined, along with the effects of incident wave characteristics on hologram visibility. Key insights into the construction of wave fields from holograms and advantages of digital holograms are highlighted.

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Solar Acoustic Holograms

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  1. Solar Acoustic Holograms Dean-Yi Chou Tsing Hua University, Taiwan January 2008, Tucson

  2. Motivation Is it feasible to apply the principle of optical holography to a system of solar acoustic waves and active regions?

  3. 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.

  4. Hologram (interference pattern) (time average)

  5. diffraction field Construction of Waves (Gabor’s in-line holgram) hologram

  6. Acoustic waves on the Sun

  7. Solar Acoustic Waves + Active Region interference pattern (acoustic power map) solar surface perturbed region

  8. 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)

  9. 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?

  10. 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

  11. 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

  12. 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

  13. 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.

  14. 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.

  15. 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

  16. Solution of Scattered Wave wave equation total solution scattered wave with Green’s function and Born approximation expressed in terms of Fourier components

  17. Hologram Intensity of the hologram is the time average of interference Interference term Need a model for spatial dependence of

  18. 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)

  19. 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

  20. 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

  21. Effects of Coherent Time line width 0.2 mHz (FWHM = 0.47 mHz)

  22. 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

  23. 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.

  24. 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

  25. Construction of Wave Fields from Holograms Illuminate the hologram by a vertically-propagatingmonochromatic wave. hologram on the surface

  26. 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.

  27. Diffraction waves are computed by the Kirchhoff intergral replaced by hologram on the surface

  28. Constructed wave field depth = 30 Mm Incident angle = 0 Mm 30 Mm 205 Mm

  29. Constructed wave field Incident angle = 0 deg. Depth = 30 Mm Incident angle = 0 deg. Depth = 12 Mm

  30. Constructed wave field Incident angle = 0 deg. Depth = 30 Mm Incident angle = 10 deg. Depth = 30 Mm

  31. Effects of Multiple Incident Waves 1. Weaken holograms 2. Distort and weaken constructed wave fields

  32. 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.

  33. magnetic field Power vs. B field 1024 MDI FD images power map before correction power map after correction

  34. 1024 MDI FD images magnetic field power map phase-velocity-filtered power map phase-velocity-filtered power map (3.3mHz/300) (3.3mHz/400)

  35. magnetic field Power vs. B field 512 MDI HR images power map before correction power map after correction

  36. 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?

  37. 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

  38. The End

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