Astronomical Data Analysis, Lecture 9: Speckle Imaging 1 Full-Aperture Interferometry

Outline

1 Full-Aperture Interferometry 2 Labeyrie Technique 3 Knox-Thompson Technique 4 Bispectrum Technique 5 Differential Speckle Imaging 6 Phase-Diverse Speckle Imaging

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 1 Full-Aperture Interferometry

Diffraction Limit theoretical angular resolution of telescope proportional to λ/D λ wavelength D diameter of telescope

Diameter Wavelength Diffraction Limit 10 cm 500 nm 1.000 100 cm 500 nm 000. 1 100 cm 5000 nm 1.000 800 cm 500 nm 000. 0125 4000 cm 500 nm 000. 0025

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 2 Index of Refraction of Air index at sea level: 1.0003 wavelength dependence of index of refraction about 1 · 10−6/λ2 air is largely achromatic 1 K temperature difference changes n by 1 · 10−6 temperature of atmosphere is not uniform variation of 0.01 K along path of 10 km: 104 m∗10−8 = 10−4 m = 100 waves at 1µm refractive index of water vapour is less than that of air ⇒ moist air has smaller refractive index

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 3 Seeing

light from astronomical source travels in straight line through space Nonuniform refractive index ﬂuctuation - masses of warm or cold air refract light differently Different parts of wavefront interfere with each other in image plane on the ground, it looks like the astronomical object is at several places in the sky at the same time temperature ﬂuctuations change about a hundred times per second blurred image when telescope could provide much better angular resolution

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 4 Seeing resolution limited by Earth’s atmosphere to ≈ 000. 5 independent of D atmosphere is turbulent medium with small-scale temperature ﬂuctuations Speckle-Interferometry: suitable observational method and post-facto reconstruction teqchnique to eliminate the angular smearing due to seeing

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 5 Interference telescope with focal length f working at wavelength λ combines light that passed through different parts of atmosphere describe seeing as many small telescopes that are differently affected by the atmosphere quality of seeing described by Fried’s Parameter r0, diameter of telescopes that would be diffraction-limited under current seeing conditions

typical values in the visible r0: 5–30 cm Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 6 Speckles image of point source is cloud of small dots: speckles speckle diameter λ/D speckle cloud diameter λ/r0 speckle life time in visible about 10 ms, longer at larger wavelengths

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 7 Seeing for Single and Binary Stars

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 8 Seeing and Solar Granulation

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 9 Mathematical Description of Seeing image of a point source: Point Spread Function (PSF) image formation due to time-varying PSF

i (t) = o ∗ s (t)

i observed image o true object, constant in time s point spread function ∗ convolution

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 10 Fourier Domain after Fourier transformation:

I (t) = O · S (t)

I observed image O true object, constant in time S optical transfer function describing seeing and instrument isoplanatic patch: constant PSF over an area of 3–1000 extended objects as sum of point sources convolved with PSF Fourier frequency f object in Fourier space has amplitude, phase: O(f ) = |O(f )|e−Ψ(f )

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 11 Labeyrie Technique

Basic Idea developed in 1970 by Antoine Labeyrie exposure time shorter than time constant of seeing (≤ 20 ms in the visible) clever average of many images that is free from atmospheric inﬂuences assumption: PSF constant during exposure time image of binary star consists of two identical, overlapping speckle clounds

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 12 Average Autocorrelation Autocorrelation can detect identical, but shifted speckle clouds Labeyrie technique: average autocorrelation contains diffraction-limited information autocorrelation in image space corresponds to power-spectrum in Fourier Space D E D E |I|2 = |O|2 |S|2

speckle transfer function |S|2 from calibration point source Labeyrie technique allows reconstruction of Fourier amplitudes average auto-correlation is not image Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 13 Autocorrelation 6= Image

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 14 Knox-Thompson Technique

Reconstructing the Fourier Phases Knox and Thompson (1973) phases crucial for extended objects (star clusters, galaxies, Sun) phases Ψ(O) with Knox and Thompson approach: D E D E I(~f )I∗(~f − δ~f ) = O(~f )O∗(~f − δ~f ) S(~f )S∗(~f − δ~f )

∂Ψ Ψ O ~f O∗ ~f − δ~f = −Ψ O ~f + Ψ O ~f − δ~f ∼ ∂~f D E phase of S(~f )S∗(~f − δ~f ) is zero integrate phase differences in two directions to recover object phases iterative approach minimizes sum of squares of phase differences

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 15 Average and Best Frame of Image Series

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 16 Average and Knox-Thompson Reconstruction

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 17 Comparison of Techniques

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 18 Bispectrum Technique

Weigelt 1977 D ~ ~ ∗ ~ ~ E product of three Fourier components I(f1)I(f2)I (f1 + f2) phase of average atmospheric factor is again zero ~ ~ ~ ~ phase of bispectrum: Ψ f1 + Ψ f2 − Ψ f1 + f2 many more correlations between Fourier components bispectrum is 4-dimensional ⇒ large computational effort today most-often used speckle technique Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 19 Dutch Open Telescope Speckle Movie

DOT Blue Continuum Movie of AR10425

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 20 Differential Speckle Imaging

The Idea narrow-band ﬁlter images do not have enough signal to do direct speckle imaging measure PSF in broad-band channel (b) to deconvolve simultaneous exposures in narrow-band channel (n)

In = OnS, Ib = ObS

approximation of On: 0 In In On = = Ob. S Ib avoid division by 0 via:

2 ∗ (In/I )|I | InI O0 = b b O = b O n 2 b 2 b |Ib| |Ib|

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 21 Application to Solar Zeeman Polarimetry

BUT, in reality In = OnS + Nn, Ib = ObS + Nb account for random photon noise with appropriate optimum ﬁlter account for anisoplanatism by overlapping segmentation with segment sizes on the order of the isoplanatic patch high spatial resolution at good spectral resolution can be achieved Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 22 Phase-Diverse Speckle

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 23 Raw Data Reconstruction

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 24 Adaptive Optics Real-Time Wavefront Correction

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 25 Evolution of Small-Scale Fields in the Quiet Sun

Christoph U. Keller, Utrecht University, [email protected] Astronomical Data Analysis, Lecture 9: Speckle Imaging 26