Introduction to Interferometry. Michelson Interferometer. Fourier Transforms. Optics: holes in a mask. Two ways of understanding interferometry

Transcription

1 Introduction to Interferometry P.J.Diamond MERLIN/VLBI National Facility Jodrell Bank Observatory University of Manchester ERIS: 5 Sept 005 Aim to lay the groundwork for following talks Discuss: General concepts: resolution, fringes, delay, terminology A simple interferometer Generation of source brightness distribution Limitations Useful references: Interferometry and Synthesis in Radio Astronomy by Thompson, Moran and Swenson Synthesis Imaging eds Perley, Schwab and Bridle Very Long Baseline Interferometry and the VLBA eds Zensus, Diamond and Napier Very Long Baseline Interferometry eds Felli and Spencer Principles of Optics by Born and Wolf Michelson Interferometer Plane wavefront propogating perpendicular to the line joining the two elements (the baseline) will strike each element simultaneously and constructively interfere Maxima in light intensity occur at angles θ for which the difference in the path lengths is an integral number of wavelengths If angular width of star is << Fringe Visibility Θ spacing in θ the image of star at detector is crossed by series of dark Fig 1.4, T M & S and light V= brightness of maxima brightness of minima bands interference fringes brightness of maxima + brightness of minima If stellar width comparable with spacing in θ then image formed by superposition of images from points across star -> maxima & minima from points do not coincide and fringes merge into a smooth continuum Michelson & Pease demonstrated relationship between fringe visibility and brightness distribution. Fourier Transforms crucial to interferometry Used circular disk model to interpret their observations and determine stellar diameters Their determination of V required great skill since fringes vibrated across image in a random manner as a result of atmospheric fluctuations and mechanical instability Modern connected element interferometry solves this problem through the use of physical phase paths (cables, waveguides, microwave links, fibre optics) + external calibration VLBI isn t connected so has to overcome the problem in a different manner. Fourier Transforms Fig 1.5: TM&S Marcaide et al SN1993 Two ways of understanding interferometry Optics Put a mask over an aperture: still works ~ fine More holes allow more information through Distribution of holes affects quality of image Image quality improves as number of holes in mask increases Physics Radio sources emit random signals: noise but no signal Correlation of voltage far from the source contains information about the source Measure spatial correlation function of voltages at antennas Derive image of sky from sampled correlation function Build a big reflector lens Measure power on the focal plane: get an image of the radio sky 1

2 Resolution ~ wavelength / Diameter Optical telescope has resolution ~ 1 arcsec (1/3600 degrees) At wavelength 0cm, need Diameter ~ 35km! Largest steerable ~ 100m Largest fixed ~ 300m Use smaller antennas to synthesize ~ 35km telescope Can fill an area up to ~ 1km Do not need to fill an area VLA D configuration (1km maximum) Imagine a lens of e.g. a camera Can still get an good image even when a mask is placed in front of the len How does the image quality change with the number of holes? Demonstration Choose a representative image of a source Add holes to a mask Start with two holes and double in every additional frame Basic interferometer geometry Double number of holes from frame to frame:, 4, 8, 16, 3, 64, 18, 56, 51, 104 r r 1 Radio antennas widely separated. There will be a delay, τ, in the receipt of the wavefront at one of the antennas wrt the other. Since the Earth rotates there will also be the time derivative of the delay (delay rate or fringe rate) Basic assumptions: Source is in the far field, i.e. incoming waves are parallel D R s >> λ Source is spatially incoherent, i.e. radiated signals from any two points on the source are uncorrelated

3 Spatial Coherence Function Distant radio source in direction R radiates and produces a time variable electric field E(R,t) Correlation of field at points r 1 and r (i.e. antenna locations) and at frequency ν is: * V ν ( r1, r ) = E( r1 ) E ( r ) From this equation, and utilizing the assumptions that the source is spatially incoherent and lies in the far field, we can derive: V ( r, r ) = ν 1 I ( s) e ν πi νs ( r1 r )/ c dω Where I ν (s) is the observed intensity of the radiation field; s is the unit vector in the direction of R; dω is the solid angle subtended by the radio source. Note that the equation depends only on the separation vector r 1 r not on the absolute locations. Therefore we can learn all about the correlation properties by holding one observation point fixed and moving the other about. V ν is called the spatial coherence function. An interferometer measures the spatial coherence function. This equation also demonstrates the van-cittert Zernicke theorem which states that the spatial coherence function is the Fourier Transform of the source brightness distribution (1) () Some definitions In practical terms we measure the cross-correlation function between two antennas: 1 T * R( τ ) = lim E1( t) E ( t τ ) dt T T + T The visibility function V is defined as the cross-correlation of the signal normalized by the correlation at zero baseline length. i.e. the flux measured on the baseline between two antennas divided by the flux measured by a single dish. V is often expressed as a phasor: iϕ V = Ae Where A is the signal amplitude and φ is the phase In reality we measure a set of fringe visibilities V ij which are corrupted by atmospheric, electronic, pointing effects etc. These all have to be determined and corrected. V ' ij = GiG jvij Also, we gather information from N telescopes, means N(N-1)/ baselines Simple Interferometer Fringe frequency: Definitions πd F = cos sin θ λ Signal reaches RH antenna τ g = (D/c) sin θ before it reaches the LH antenna. τ g is the geometrical delay. Voltages from two antennas are multiplied together and lowpass filtered or timeaveraged (this combination is called a correlator) As earth rotates, the two antennas experience differing radial velocities with respect to source. If source were emitting monochromatic wave the signal would have a different Doppler shift in frequency at the two antennas Output of lowpass filter would be a sine wave whose frequency equalled the difference between the two Doppler-shifted frequencies This fringe frequency varies as the geometry of the interferometer changes. Delay: b ^ ^ τ = bs c Fringe Phase: πb ^ ^ ϕ = bs λ Computer A Practical Interferometer LO phase rotator τ variable delay τ correlator x F Σ Fringe rotator Accumulator mixer Incoming signals are mixed down to baseband by local oscillator Phase rotation needed because delay corrector is not operating at RF frequency The use of a finite bandwidth means that the path lengths on each arm should also be equal and so the geometric delay is calculated and removed. The fringe frequency can be high, 100s of Hz in VLBI, compared with the reciprocal of the averaging time (seconds) so fringes can be smoothed out completely. The fringes are stopped by means of fringe rotator which uses information on the interferometer s geometry to determine the fringe frequency and remove it. The computer knows the source position, the antenna locations, the observing frequency, the LO frequencies, atmospheric information, antenna axis offsets etc. It can then determine the geometry of the interferometer and calculate the geometric delay and the fringe frequency and remove them before correlation and/or accumulation. time Residuals A fringe plot: each sinusoid represents the real part of the correlation function versus time for a particular delay offset. Oscillations result from residual fringe frequency. i.e. the interferometer model is not perfect. 3

4 The u-v plane A u-v plot for a 1 hour MERLIN synthesis of a source at declination 560 Imaging With some manipulation we can rewrite the spatial coherence function (eqn ) in terms of u and v: Vν (u, v) = Iν ( x, y )e πi ( ux +vy ) dxdy A u-v plot for an 8 hour VLBA synthesis of a source at declination 50 Since above is a Fourier transform, it may be formally inverted: Iν ( x, y ) = Vν (u, v)e πi ( ux + vy ) dudv In practice, the spatial coherence function V is not known everywhere, but is sampled at particular places on the u-v plane. The sampling can be described by a sampling function S(u,v), which is zero where no data have been taken. Eqn above must therefore be rewritten as: IνD ( x, y ) = Vν (u, v) S (u, v)e πi ( ux + vy ) dudv This is often referred to as the dirty image UV Sampling Point Spread Function The dirty image is related to the true intensity distribution Iν(x,y) through the Fourier convolution theorem: IνD = Iν B Where the * denotes convolution and: B ( x, y ) = S (u, v)e πi ( ux + vy ) dudv is called the synthesized beam, point spread function or dirty beam. Note that it is the Fourier transform of the sampling function. Point Spread Function Dirty image Deconvolved image 4

5 Real interferometers - I Real Interferometers - II Connected-element, E-W interferometers: ATCA WSRT Real Interferometers - III Variations: the Altacama Large Millimeter Array Observing wavelength short ~ mm Need high, dry site Antenna field of view small Must patch together different pointings Mosaicing Real Interferometers - IV Variations: Optical Interferometry Observe at optical or infra-red Very difficult technically Tolerances tiny Signals very weak Stars twinkle First arrays now coming online 5