Astronomical Interferometryin a nutshell

1) Why Interferometry?2) Basic Theory3) Interferometers4) The influence of the atmosphere5) Image reconstruction

1. The need for resolution

The need for resolutionSize of BLRin nearby AGN

Schwarzschildradius of BH innearby AGN

Size of NLRin nearby AGN

8kpc ingalaxy at z=1

Galaxy clusterscales

Angularsize ofBetelgeuse

ExoplanetHD209458bstar->planet

The need for resolutionSize of BLRin nearby AGN

Schwarzschildradius of BH innearby AGN

Size of NLRin nearby AGN

8kpc ingalaxy at z=1

Galaxy clusterscales

Angularsize ofBetelgeuse

ExoplanetHD209458bstar->planet

Jodrell Bankat 5GHzGround-based

seeing limitVLT singletelescopediffraction limit

HSTSingleTelescopes visible light

Eye

The need for resolutionSize of BLRin nearby AGN

Schwarzschildradius of BH innearby AGN

Size of NLRin nearby AGN

8kpc ingalaxy at z=1

Galaxy clusterscales

Angularsize ofBetelgeuse

ExoplanetHD209458bstar->planet

Jodrell Bankat 5GHzGround-based

seeing limitVLT singletelescopediffraction limit

HSTSingletelescopes

VLA at5GHz

MERLIN at 5GHz

GlobalVLBI

SpaceVLBI VLTI (IR)

Interferometers

VLTI(optical)

ALMA

2. Basic theory of interferometry

Point source Fringes ofseparationλ/d

d

Young's slits revisited

Larger source

Source subtends anangle 0.4 λ/d

Fringes move by0.4 λ/d. Incoherentsources -> addintensities, fringesstart to add outdestructively

Define contrast |fringe visibility|=(Imax-Imin)/(Imax+Imin)

Still larger source

Source sizegets to λ/d

No fringes remain(cancellation). Littlefringing seen forlarger sources thanλ/d either.

Effect of slit size

Same size source,but smaller slit

Increased fringespacing, so fringesvisible again

Baseline length

Baseline length

Young's slits: summaryVisibility of interference fringes

•Decreases with increasing source size•Goes to zero when source size goes to λ/d•For given source size, increases for decreasing separation•For given source size and separation, increases with λ

It's a Fourier transform!

The fringe visibility of an interferometer gives informationabout the Fourier transform of the sky brightness distribution.

Long baselines record information about the small-scalestructure of the source but are INSENSITIVE to large-scalestructure (fringes wash out)

Short baselines record information about large-scale structureof the source but are INSENSITIVE to small-scale structure(resolution limit)

Van Cittert-Zernicke theorem

OPD – optical phase delay

The u-v plane

If we could measure FV for all u,v, transform ­> image

Location of the 8m Unit Telescopes (UTs) and the 1.8m Auxiliary Telescopes(ATs) of the ESO Very Large Telescope Interferometer (VLTI) 

Earth rotation aperture synthesis (ERAS)

Over a day,can measuremany points inu-v plane witha single baseline

Locus is an ellipse;the longer the baseline,the larger the u-v (higherresolution)

b-vector plotted in brown

Image: A. Gunn/University of Manchester

Exact form of u-v track

D=declination of sourced=declination of point on sky pointed to by baseline

Resolution given by maximumextent of tracks

ERAS imaging of sources atdeclination D=0 is hard!

projected base line

[u,v] coordinates

Actual fringe visibility

Double sourceeach component 1Jy1Jy =10-26 W m-2 Hz-1

separation calculablefrom baseline length

Correlated flux vs. time

Limits on the field of view1. Finite range of wavelengths

Bigger range – smaller field of view (FT again)FOV = (λ/∆λ) x (λ/L) i.e. λ/∆λ resolution elementsCure – observe in multiple channels

Fringe pattern OK at field centre but different colours out of phase at higher relative delay

Limits on the field of view2. Too big integration time per data point3. Non-flat sky over large FOV

Rather technical, and only a problem for wide-field imaging

4. Primary beam

maximum field ofview set by thetelescope aperture

3. InterferometersRadio/Thermal Infrared

Non-photon-limited: electronic, relatively straightforward can “clone” and combine signals (heterodyne detection) “correlation” (multiplication+delay) can even record signals and combine later

Optical/Near-Infrared

Photon-limited case: use classical Michelson/Fizeau arrangements delay lines for manipulation cannot “clone” photons

Radio

VLA 30m-36kmMERLIN 6km-250kmEVN 250km-2300kmVLBA 250km-9000kmGlobal VLBI-12000km

Space VLBI-32000km

Very Large Array, NM, USA

VLBI (Very Long Baseline Interferometry)

Limited only by Earth size12000km baselines -> mas resolution

Atacama Large Millimetre Array - ALMA

- 30-950 GHz- max. baseline ~20km- molecules in galaxies at cosmological z- gas in Galactic star- forming regions

Chajnantor, Chile

Optical/IR interferometers

from Monnier 2003

Michelson's interferometer at the Mt. Wilson observatory. The angular diameters of 7 stars could be measured (Betelgeuse, Arcturus, Antares, Aldebaran, Ras Algethi (Alpha Herculis), Scheat (Beta Pegasi), and Mira).

Correcting the OPD difference – Delay Lines

fringe scanning: measurement of thefringe pattern by periodic modulationof the OPD

fringe

VINCI – VLTI Commissioning Instrument

What the atmosphere does

# Corrupts phase and amplitude of incoming signals

# Corruption is different for different telescopes/apertures

# Corruption changes with time (scales ms to mins)

# Corruption varies with position (<size of tel. for optical)

# Sources: water vapour..., ionosphere (low-frequency radio)

4. The real world - Dealing with the atmosphere

How bad is the problem? Phase fluctuations

Waveband Problem Phase variation timescale

Radio <300MHz ionosphere seconds-minutes few GHz water &c minutes >20GHz water sec (site dependent)mm water highly site dependentnear-IR atm cells ~100 millisecondsoptical 1-10 milliseconds

The shorter the wavelength, the more rapid the phasefluctuation and the harder the problem becomes.

Optical: Fried parameter ro – length scale of refractive index fluctuations Timescale = ro/wind velocity

VLA, 8.4GHz, phase fluctuations [degrees] over time

One approach: closure phase

Closure-phase mapping: COAST

Betelgeuse (Young et al. 2004,Proc. Nat. Astr. Meeting)

Capella (www.mrao.cam.ac.uk/telescopes/coast)

Self-calibration in action

Dirty map CLEANed map CLEANed map withphase selfcalibration

Phase calibration

Requirements

a<size of isoplanactic patcht <coherence time of atmosphere at this wavelength

Can nod back and forth, or have target and calibrator in samefield of view (FOV)

Phase calibration

Phase calibrator must be * bright (S:N in reasonable time/atmospheric

coherence time) * close (same wavefront distortion)

(cf. adaptive optics on single telescopes)

If isoplanatic patch is small * calibrator may not exist

Signal-to-noise (wave regime)

Radio interferometernoise level =

Tsys = system temperature, nb = number of baselines, T=integration time, ∆ν=bandwidth in Hz, A=area ofapertures, η=aperture efficiency

linearly as 1/A not (1/A)

In practice you rarely get to this!

1/2

Signal-to-noise (photons)

Fundamental limit set by n photon statistics

system limit is given by number ofphotons in coherence volume (multiplied by loss factors)

Mandatory adaptive optics on individual telescopes (need natural/artificial guide star for wavefront sensing)

Phase referencing for ultimate sensitivity (need bright star in isoplanatic patch)

1/2

Deconvolution

We want the image as a function of (x,y):

But instead we have the “dirty image”

where the sampling function S is 1 in the parts of the uvplane we've sampled and 0 where we haven't.

5. Making the image

Deconvolution (ctd)

We can use the convolution theorem to write

Where B is known as the dirty beam

and is the FT of the sampling function.

Problem is then one of deconvolution.

Hogbom CLEAN deconvolution

Brute-force iterative deconvolutionusing the dirty beam

Effectively reconstructs informationin unsampled parts of the u-vplane by assuming sky is sum ofpoint sources

Hogbom CLEAN in action

Dirty map Dirty beam

Hogbom CLEAN in action

Residual after 1 CLEAN (gain 0.5) CLEAN map (residual+CCs)after 100 CLEANs (gain 0.1)

Summary

Further readingPrinciples of Long Baseline Interferometry Mozurkevich, Michelson Summer School 2000

Synthesis imaging in radio astronomy ASP, Proc NRAO summer school

Optical interferometry in astronomy Monnier, Rep. Prog. Phys, 66, 789, 2003

An Introduction to Optical Stellar Interferometry Labeyrie et al., ISBN 0521828724 Optical Long Baseline Interferometry (OLBIN) website

N. Jackson, ELBA lecture; A. Glindemann, ESO-VLTI

try yourself the Virtual Radio Interferometer (VRI)!