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

Optical HolographySolar 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 HolographySolar 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 : central frequency : period of central frequency example 3.3 mHz 0.2 mHz (FWHM = 0.47 mHz) 2.6

solar surface trapped in cavities curved ray path multiple 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 If incident waves are, total waves are Intensity of hologram 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: time independent Wave equation becomes Source of scattering Wave Equation

Solution of Scattered Wave scattered wave with Green’s function and Born approximation wave equation total solution 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. interference term normalized interference term (related to fringe visibility) 3. Simple dispersion relation:

Normalized Interference Term (fringe visibility) 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

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: 2. Dispersion relation: 4. coherent time Perturbed region 1. Uniform cylinder with 2. diameter = 9.6 Mm, vertical extent = 4.8 Mm, depth = 12 Mm 3.3 mHz,14.7 Mm (l=300),48.5 km/s

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

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 At 5Mm depth, the angle of incidence is about for at 3.3 mHz. for at 3.3 mHz. Waves with different phase velocities have different angles of incience. For example:

Effects of Angle of Incidence (cont.) 3.3 mHz,0.2 mHz, uniform cylinder with 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 Diffraction waves are computed by the Kirchhoff intergral replaced by

Constructed wave field 205 Mm 30 Mm 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. 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.

power map before correctionpower map after correction magnetic fieldPower vs. B field 1024 MDI FD images

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

power map before correctionpower map after correction magnetic fieldPower 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