1. GLOW Interferometry School, Bielefeld 2011 1.Fourier optics 2.Fourier theorems 3.Visibility function 4.Aperture synthesis 5.Coordinates 6.Correlation interferometer 7.Imaging 8.Cleaning Radio Interferometry Basics U. Klein

2. GLOW Interferometry School, Bielefeld 2011 Fundamental understanding of the functionality of radio interferometry

3. GLOW Interferometry School, Bielefeld 2011 1. Fourier optics Aperture with constant extent perpendicular to screen, with extent a in this y-direction, constant current G y (x) in y-direction. Infinitesimal far-field element dE resulting from an E-field component E(x) along infinitesimal element dx and integrating over y and x leads to This is Huygens’ Principle for the far-field: superposition of field components along the aperture Far-field: R  x  (… ties up to Jackson, Chapt. 14)

4. GLOW Interferometry School, Bielefeld 2011 Normalising by wavelength Lumping constants and R dependence into leads to simple Fourier intergral (x = x /, direction cosine ξ = sin )

5. GLOW Interferometry School, Bielefeld 2011 slit or uniformly illuminated aperture: sinc function circular aperture: Bessel function antenna diagram: P( ,  ) = |E( ,  )| 2

6. GLOW Interferometry School, Bielefeld 2011 2. Fourier theorems Fourier transform F(s) of function f(x) with inverse transform

7. GLOW Interferometry School, Bielefeld 2011 Convolution:

8. GLOW Interferometry School, Bielefeld 2011 Consider the two domains: x s application timet [s]frequency [Hz] acoustics angle [rad]spatial frequ. u [rad -1 ] radio astronomy Oboe: wavesOboe: spectrum LMC: radio brightnessLMC: spatial frequ. spectrum

9. GLOW Interferometry School, Bielefeld 2011 Let F f(x) = F(s), F g(x) = G(s) addition theorem example: an aperture has beam size   /D similarity theorem

10. GLOW Interferometry School, Bielefeld 2011 shift theorem example: electronic steering

11. GLOW Interferometry School, Bielefeld 2011 convolution theorem example: beam smearing convolution with broader beam multiplication with narrower ‘filter‘

12. GLOW Interferometry School, Bielefeld 2011 autocorrelation theorem example: antenna diagramme

13. GLOW Interferometry School, Bielefeld 2011 convolution and auto-/cross-correlation theorem: reception pattern of an interferometer

14. GLOW Interferometry School, Bielefeld 2011 sampling theorem: how to properly convert continuous into discretised functions comb or „Sha“ function: multiplication f(x)  · III (x) converts continuous into discretised function

15. GLOW Interferometry School, Bielefeld 2011 A function whose Fourier transform is zero for |s| > s c is fully specified by values spaced at equal intervals not exceeding ½ s c -1 save for any harmonic term with zeros at the sampling points. ‘aliasing‘!

16. GLOW Interferometry School, Bielefeld 2011

17. 3. Visibility function Correlated power received from source with brightness B where (D = D/ ) D is the baseline. Integration yields

18. GLOW Interferometry School, Bielefeld 2011 With it is convenient to introduce complex visibility such that

19. GLOW Interferometry School, Bielefeld 2011 Since A N  1 (the single-antenna diagramme is very broad), and writing we obtain This is the basic relation for aperture synthesis: Comparing with 1 st version of P we arrive at In order to retrieve the brightness B(ξ,η), we need to measure the visibility V(u,v) for as many baselines D λ = (u,v) as possible and Fourier-transform it.

20. GLOW Interferometry School, Bielefeld 2011 example: double source (schematically): apply convolution theorem source brightness = convolution of  -function with two  -functions visibility = multiplication of cosine with sinc-function Now observe with different base- lines

21. GLOW Interferometry School, Bielefeld 2011 astronomical telescope = spatial filter: single dish: low-pass filter interferometer: high-pass filter

22. GLOW Interferometry School, Bielefeld 2011 4. Aperture synthesis need many D = (u,v) consider 4  4 apertures, producing n (n-1)  2 = 120 baselines, most of which are redundant, while we need only 24 of them to cover the (u,v) plane  aim: preferably little redundancy!

23. GLOW Interferometry School, Bielefeld 2011 Earth-rotation synthesis need good sampling of (u,v) plane with many (non- redundant) baselines earth-rotation synthesis helps a lot due to Sir Martin Ryle (Nobel Prize 1974 with A. Hewish)

24. GLOW Interferometry School, Bielefeld 2011 144 m 9  144 m 36 m 300 m180 m 9  144 m 012349ABCD east-west array: e.g. WSRT

25. GLOW Interferometry School, Bielefeld 2011 earth rotation produces a good transfer function: (u,v) tracks e.g. pure east-west interferometer FT  antenna diagramme („dirty beam“)

26. GLOW Interferometry School, Bielefeld 2011 for the kth ring, the grating rings have functional form where sinc 1/2 implies the “half-order derivative“ of the sinc function derívative theorem: and from this: (intrigued?  Bracewell & Thompson, Ap.J. 188, 77, 1973)

27. GLOW Interferometry School, Bielefeld 2011 actually need max. 12 hrs. to obtain full (u,v) coverage VLA: 8 hrs. (Y-shape) V(u,v) is Hermitean, since the brightness is a real quantity. … we get the other half for free!

28. GLOW Interferometry School, Bielefeld 2011 5. Coordinates Complex visibility reads Define coordinate system (u,v,w), which are related to unit vectors and baseline: We may have to deal with non-zero w-component!

29. GLOW Interferometry School, Bielefeld 2011 , ,  are direction cosines and the visibility becomes

30. GLOW Interferometry School, Bielefeld 2011 Coordinate system ( ,  ) corresponds to projection of celestial sphere onto a tangential plane at the origin of the field centre. possible projections: tan: optical astronomy arc: Schmidt telescopes, single dishes sin: aperture synthesis (more: Calabretta & Greisen, A&A 395, 1077, 2002)

31. GLOW Interferometry School, Bielefeld 2011 direction cosines and their relation to other coordinate systems: spherical coordinates ,  equatorial coordinates , 

32. GLOW Interferometry School, Bielefeld 2011 Using polar coordinates , , the solid-angle element d  reads The releation between the infinitesimal elements is given by where |J| is the determinant of the Jacobi matrix which leads to

33. GLOW Interferometry School, Bielefeld 2011 Antenna coordinates and baseline components defines a double ellipse:

34. GLOW Interferometry School, Bielefeld 2011 Coverage of (u,v) plane e.g. VLAe.g. LOFAR a bit weird: component L Z gives rise to non-coplanar baselines  not simple Fourier integral anymore!

35. GLOW Interferometry School, Bielefeld 2011 6. Correlation interferometer has power response with phase lag due to geometric time delay

36. GLOW Interferometry School, Bielefeld 2011 heterodyne principle: down-convert from high (RF or HF) to intermediate frequency (IF) produces frequency spectrum including signal frequ.(USB) image frequ.(LSB)

37. GLOW Interferometry School, Bielefeld 2011 Effect of finite bandwidth (1): Delay compensation  i indispensible in order to avoid ‚fringe washing‘. We measure correlation of two voltages V 1, V 2 : By definition, this is also the Fourier transform of the bandpass |H( )| Calculate power over the band , assuming rectangular bandpass and (roughly) constant power as a function of frequency: T 

38. GLOW Interferometry School, Bielefeld 2011 Hence finally: So the correlated signal is modulated with a sinc function!  = 50 MHz D = 1 km tracking source over 1.6‘ without delay compensation produces ~1% of loss needs  g  1.6 ns

39. GLOW Interferometry School, Bielefeld 2011 the phases from telescopes 1 and 2 are hence the correlated power is where  =  g -  i and  ‘  ‘ refers to USB and LSB. Measure complex visibility with quadrature network:

40. GLOW Interferometry School, Bielefeld 2011 Fringe rotation/stopping: earth‘s rotation modulates correlated power quasi- sinusoidally with natural fringe rate where  e = dh  dt = angular rotation frequency of earth,  = declination. At the VLA, the fringe frequency may exceed 150 Hz. Not interested in fringe rate, but rather in change of  V ; hence compensate fringe rotation by modulating LO phase. In the IF section, we have So control LO phase such that vanishes. This now requires a complex correlator in order to measure the amplitude and phase separately!

41. GLOW Interferometry School, Bielefeld 2011 Effect of finite bandwidth (2): Fourier integral is precise only for monochromatic radiation! At the centre frequency 0 we have the precise relation Away from the centre frequency we have Generalised similarity theorem in n dimensions:

42. GLOW Interferometry School, Bielefeld 2011 application to 2-D Fourier relation between visibility and brightness: so that (“~“ indicating the influence of the bandpass) Delay tracking is exact for  0 =  0 at  = 0. For signals at frequeny arriving from direction ( ,  ) there will be a lag error so that the phase is imprecise by an amount

43. GLOW Interferometry School, Bielefeld 2011 Consider point source at location  0, assume rectangular bandpass. Then where we have assumed (  0 ) 2  1. Thence, with du = (  0 ) d u 0  d u 0, we obtain the convolution The Fourier transform of the sinc function is a box function where This effect called bandwidth smearing (chromatic aberration) is irreversible! V(u)Fourier kernel

44. GLOW Interferometry School, Bielefeld 2011 Short version of this: radiation within range of frequencies produces u-v tracks of different size  produces images of different scale if referred to same frequency when doing the Fourier transform. 13‘ to field centre radial smearing u v

45. GLOW Interferometry School, Bielefeld 2011 7. Imaging In practice: convert Fourier integral into a sum and apply an FFT. This requires a retabulation of the measured visibilities onto a 2 N  2 M grid. In what follows, we use the integral notation for the sake of better readability of the equations. The true visibilities are strongly modified owing to: 1.sampling by transfer function: S(u,v) 2.regridding onto regular grid for FFT with grid function: G(u,v) 3.weighting to shape the synthesized beam and control sensitivity: R(u,v) 4.FFT

46. GLOW Interferometry School, Bielefeld 2011 transfer function: S(u,v) gives rise to ‘dirty beam‘ weighting with W(u,v) W = R  T  D R: A eff, T sys, ,  T: taper (shapes the beam) D: density of visibilities brightness (‘dirty image‘) now is Plateau de Bure Interferometer

47. GLOW Interferometry School, Bielefeld 2011 now prepare for FFT by retabulation S(u,v); needs two steps: (i)convolve with appropriate function C(u,v) (ii)multiply with grid function G(u,v) (fakir‘s bed of nails) this is a significant modi- fication of the data  use convolution theorem to see modification in the image domain:

48. GLOW Interferometry School, Bielefeld 2011 can ‘undo‘ gridding by dividing V D by the inverse transform of G:, yielding the ‘grid-corrected‘ dirty image: This is the true brightness distribution, convolved with the dirty beam

49. GLOW Interferometry School, Bielefeld 2011 8. Cleaning Goal: deconvolution in order to get rid of sidelobes

50. GLOW Interferometry School, Bielefeld 2011 basic clean algorithm (Högbom 1974): (i)First compute the dirty map B D ( ,  ) via FT of V D (u,v) and the dirty beam F D ( ,  ) via the FT of S(u,v) · W(u,v). (ii)Find position of maximum brightness and subtract  ·F D ( ,  ) · B max ; “loop gain”  controls speed of the cleaning process ( 0 <  < 1 ); store B max at the corresponding position in a work array (creates “fakir’s bed of nails”). (iii) Go to (ii) until a user-defined level is reached (e.g. number of iterations or rms noise). (iv)Convolve the so obtained “point source model” (“fakir’s bed of nails”) with the “clean beam”, or synthesized beam, e.g. a Gaussian of width ~ / D max. (v)Add this convolved model to the residual map to yield the clean image B C ( ,  ). There are variants with major (check in the u-v domain) and minor cycles (Clark 1980, Cotton-Schwab 1984)

51. GLOW Interferometry School, Bielefeld 2011

52. Zero-spacing problem Missing short spacings: interferometer measures zero total flux! hence since which cannot be measured  “negative bowl”

53. GLOW Interferometry School, Bielefeld 2011 need to add in single-dish measurements (in the u,v domain!)

54. GLOW Interferometry School, Bielefeld 2011 Don‘t forget that the brightness distribution is multiplied by the enveloping ‘primary beam‘, i.e. the beam of the individual antennae. Need to correct for this degradation by dividing the clean image by the primary beam: Naturally gives rise to increasing noise at radial distances larger than the primary HPBW.

55. GLOW Interferometry School, Bielefeld 2011 3C 120

56. GLOW Interferometry School, Bielefeld 2011 T h a n k s f o r y o u r a t t e n t i o n !