Astronomical Data Analysis, Lecture 9: Speckle Imaging

Lecture 9: Speckle Interferometry

Outline

1 Full-Aperture Interferometry2 Labeyrie Technique3 Knox-Thompson Technique4 Bispectrum Technique5 Differential Speckle Imaging6 Phase-Diverse Speckle Imaging

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

Full-Aperture Interferometry

Diffraction Limittheoretical angular resolution of telescope proportional to λ/Dλ wavelengthD diameter of telescope

Diameter Wavelength Diffraction Limit10 cm 500 nm 1.0′′

100 cm 500 nm 0.′′1100 cm 5000 nm 1.0′′

800 cm 500 nm 0.′′01254000 cm 500 nm 0.′′0025

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Index of Refraction of Airindex at sea level: 1.0003wavelength dependence of index of refraction about 1 · 10−6/λ2

air is largely achromatic1 K temperature difference changes n by 1 · 10−6

temperature of atmosphere is not uniformvariation of 0.01 K along path of 10 km: 104 m∗10−8 = 10−4 m =100 waves at 1µmrefractive index of water vapour is less than that of air⇒ moistair has smaller refractive index

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Seeing

light from astronomical source travels instraight line through spaceNonuniform refractive index fluctuation -masses of warm or cold air refract lightdifferentlyDifferent parts of wavefront interfere witheach other in image plane on the ground, itlooks like the astronomical object is atseveral places in the sky at the same timetemperature fluctuations change about ahundred times per secondblurred image when telescope couldprovide much better angular resolution

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Seeingresolution limited by Earth’satmosphere to ≈ 0.′′5independent of Datmosphere is turbulent mediumwith small-scale temperaturefluctuationsSpeckle-Interferometry: suitableobservational method andpost-facto reconstructionteqchnique to eliminate theangular smearing due to seeing

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Interferencetelescope with focal length fworking at wavelength λcombines light that passedthrough different parts ofatmospheredescribe seeing as manysmall telescopes that aredifferently affected by theatmospherequality of seeing described byFried’s Parameter r0, diameterof telescopes that would bediffraction-limited undercurrent seeing conditionstypical values in the visible r0:5–30 cm

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Specklesimage of point source iscloud of small dots:specklesspeckle diameter λ/Dspeckle cloud diameterλ/r0

speckle life time in visibleabout 10 ms, longer atlarger wavelengths

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Seeing for Single and Binary Stars

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Seeing and Solar Granulation

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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 imageo true object, constant in times point spread function∗ convolution

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Fourier Domainafter Fourier transformation:

I (t) = O · S (t)

I observed imageO true object, constant in timeS optical transfer function describing seeing and instrument

isoplanatic patch: constant PSF over an area of 3–10′′

extended objects as sum of point sources convolved with PSFFourier frequency fobject in Fourier space has amplitude, phase:O(f ) = |O(f )|e−Ψ(f )

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Labeyrie Technique

Basic Ideadeveloped in 1970 by AntoineLabeyrieexposure time shorter than timeconstant of seeing (≤ 20 ms inthe visible)clever average of many imagesthat is free from atmosphericinfluencesassumption: PSF constant duringexposure timeimage of binary star consists oftwo identical, overlapping speckleclounds

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Average AutocorrelationAutocorrelation can detect identical,but shifted speckle cloudsLabeyrie technique: averageautocorrelation containsdiffraction-limited informationautocorrelation in image spacecorresponds to power-spectrum inFourier Space⟨

|I|2⟩

= |O|2⟨|S|2

⟩speckle transfer function

⟨|S|2

⟩from

calibration point sourceLabeyrie technique allowsreconstruction of Fourier amplitudesaverage auto-correlation is not image

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Autocorrelation 6= Image

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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:⟨

I(~f )I∗(~f − ~δf )⟩

= O(~f )O∗(~f − ~δf )⟨

S(~f )S∗(~f − ~δf )⟩

Ψ(

O(~f)

O∗(~f − ~δf

))= −Ψ

(O(~f))

+ Ψ(

O(~f − ~δf

))∼ ∂Ψ

∂~f

phase of⟨

S(~f )S∗(~f − ~δf )⟩

is zero

integrate phase differences in two directions to recover objectphasesiterative approach minimizes sum of squares of phasedifferences

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Average and Best Frame of Image Series

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Average and Knox-Thompson Reconstruction

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Comparison of Techniques

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Bispectrum Technique

Weigelt 1977

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 componentsbispectrum is 4-dimensional⇒ large computational efforttoday most-often used speckle technique

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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 Ideanarrow-band filter images do not have enough signal to do directspeckle imagingmeasure PSF in broad-band channel (b) to deconvolvesimultaneous exposures in narrow-band channel (n)In = OnS, Ib = ObSapproximation of On:

O′n =

⟨InS

⟩=

⟨InIb

⟩Ob.

avoid division by 0 via:

O′n =

⟨(In/Ib)|Ib|2

⟩⟨|Ib|2

⟩ Ob =

⟨InI∗b⟩⟨

|Ib|2⟩Ob

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Application to Solar Zeeman Polarimetry

BUT, in reality In = OnS + Nn, Ib = ObS + Nb

account for random photon noise with appropriate optimum filteraccount for anisoplanatism by overlapping segmentation withsegment sizes on the order of the isoplanatic patchhigh spatial resolution at good spectral resolution can beachieved

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Phase-Diverse Speckle

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Raw DataReconstruction

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Adaptive Optics Real-Time Wavefront Correction

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Evolution of Small-Scale Fields in the Quiet Sun

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