Laboratorio di Astrofisica (laboratorio radio)

Transcription

1 Daniele Dallacasa Laboratorio di Astrofisica (laboratorio radio) Basic Theory: 1. Fraunhofer diffraction & Fourier Transforms why radio telescopes are diffraction limited. Antenna concepts (as specific to radio astronomy) 3. Radio interferometres (i.e. the quest for angular resolution) 4. Present day and future instruments (Single dish, interferometres,...) 5. How to handle interferometric data (practical stuff) From a set of complex measurements to a sky radio image daniele.dallacasa@unibo.it ricevimento: Martedi' & Giovedi' 15:30 17:00

2 Why bother radio waves LOFAR EVLA ALMA

3 Fraunhofer diffraction screen aperture H R a Q O Pa r

4 Fraunhofer diffraction aperture dr x sin a dx x O

5 Fraunhofer diffraction Point source at infinite distance from aperture parallel wave to the (far away) screen

6 Fraunhofer diffraction The electric field of a randomly located source in a given point of the aperture is g x =F x e i x e i t F(x), (x) and = c/ are amplitude, phase and frequency of e.m. radiation. This is also known as grading. Aim: compute the electric field in any point of the screen at a given time to.

7 Fraunhofer diffraction Each element dx contributes at a given point Pa with: E P a ~ g x e i r / dx = F x e i x e i t e i r / dx In each point of the screen we must: 1. evaluate the contribution of each point which has a distance r from the screen. integrate dx over the whole aperture. 1. r = R + x sin ~ R + x (small angle approximation) therefore E P a ~ g x e i R / e i x / sin dx E P a ~ g x e i R / e i x / dx

8 Fraunhofer diffraction. Let's integrate over dx to describe the whole aperture. It is possible to write the electric field in a given point P at a distance from the axis of the screen. E ~ e i R / E ~ e i R / a/ i x / sin g x e dx a/ i x / sin g x e dx Let's use sin ~ and then u = x/ : E ~ g u e i u du Formalization of Huygens Fresnel principle (each point in the aperture is a source of spherical waves with the same v of the incoming radiation; in the image plane at a distance from the axis each contributes with a vector with amplitude g(u) and phase u sin ~ u )) The electric field in a generic point P in the screen is the FT of the grading It does NOT depend on either x or, but from their ratio it is characteristic of the optical system

9 Fraunhofer diffraction The approximation holds when: 1. Monochromatic (unpolarized) radiation (u does make sense). Far field R a 3. Plane aperture E has various names: Field radiation pattern, Angular spectrum of the radiation field, Far Field. The intensity of the signal on the screen to is the image of the aperture. It varies between positive and negative amplitudes. In order to have a signal that can be accumulated we must consider I = E

10 Fraunhofer diffraction I = E is known as Power Pattern, Beam, PSF. From FT theory I = E = g u g u The Beam represents the FT of the autocorrelation function of the grading. g u g u is known as transfer function of the instrument. and now... FT festival!

11 Fourier Transforms Any periodic function with period = T, can be represented by means of a trigonometric (Fourier) series It can be considered a representation of a periodic function in the frequency domain. 1 F t = T k = e ik t T 0 f x e ik x dx Also non periodic functions can be arranged with FT: Fourier spectrum f(t) 0 over a given interval, =0 elsewhere f(t) non periodic T The Fourier Series can be seen as the limiting case of the periodic function. if k = u F t = e iut dt f x e iux dx

12 Fourier Transforms For a given function f(t) defined over ℝ we have: F u = f t = f t e iut dt iut F u e du F(u) is the FT of f(t) & f(t) is the anti transform of F(u). If t is a time length (space) Radio astronomy: then u is a frequency spatial frequency F(u) is complex function of a real f(t) distribution. Maths: F(u) FT of f(t) exists in case: b a f x dx 1. exists. Discontinuities in f(t) are finite (in amplitude and number) 3. In a given (finite) interval, the number of maxima and minima are finite. In physics these mathematical requirements are much relaxed

13 Fourier Transforms: properties FT of a FT function: F(u) is FT of f(t). Let's compute the FT of F(u): F u e iut du = F u e iu t du = f t FT and FT 1 are identical except that reverse the function (inversion of t) furthermore... if u=0 F 0 = if x =0 f 0 = f x dx F u du Equivalent width (figure) Le = f t dt f 0 F 0 1 = = F u du L E

14 Fourier Transforms: properties Linearity: a f 1 t b f t a F 1 u b F u [af 1 t bf t ]e ut dt = a f 1 x e ut dt b bf t e i.e. = a F 1 u b F u this is known as superposition theorem. Parity: a function is even when f(t) = f( t); it is odd when f(t) = f ( t) Any function can be uniquely divided into two complex functions: one even and one odd f t f t p t = even f t f t d t = odd then f t = p t d t F u = P u D u ut dt

15 Fourier Transforms: properties.. Simmetry.. f t F u f at e.. 1 u f at F a a Similarity.. f t F u i u t dt = 1 f at e a i u at a d at = 1 u F a a N.B. f and F change in opposite ways. In case one shrinks/broadens, the other broadens/shrinks. Furthermore f (0) is independent from a, therefore the area of the FT remains constant. Shifting theorem: f t s e i u t f t s F u e dt = f t s e i us i u t s e i u s dt = F u e i u s

16 Fourier Transforms: properties Derivation f ' t i u F u [n] n f t i u F u N.B. A derivation in the t domain corresponds to a multiplication in the u domain (& viceversa) Real & Imaginary parts are swapped, 0 is suppressed, high frequencies are enhanced, while low frequencies are damped Parseval Theorem F u F u du = f t dt is known as Spectral Energy of the function f(t).

17 Fourier Transforms: CONVOLUTION Many operations are gathered by this operation (smearing, blurring, scanning smoothing, cross correlation, running mean, etc.) and can be interpreted as the way and instrument measures a physical quantity. definition: h t = f s g s t ds = f t g t f may represent the physical quantity g may represent the observing tool (e.g. telescope) h may represent the observation The function g is inverted and displaced by a given amount s prior to execute the multiplication. Convolution does not change the peak value of f, but may change its distribution

18 Fourier Transforms: CONVOLUTION Properties of h : h t = f t g t F u G u = H u f s g s t e i u s dt ds = f s g s t e i u s t e i u t dt d s t = F u G u = H u The convolution product becomes simple multiplication in the Fourier space. It is also true that: f t g t F u G u Convolution is commutative: associative f t g t = g t f t f t [ g t h t ] = [ f t g t ] h t distributive f t [ g t h t ] = [ f t g t ] f t h t

19 Fourier Transforms: CROSS CORRELATION Similar to convolution but the cross correlating function is not swapped definition: h t = f t g t = f s g.. s t ds In particular, in case the functions are real, then g*(s+t) = g(s+t) and the autocorrelation becomes h t = f t g t = f s g s t ds It is noteworthy to mention the AUTOCORRELATION of a given function f t f t = f s f The autocorrelation peaks at t=0... s t ds f s ds

20 Fourier Transforms: CROSS CORRELATION Properties: distributive wrt addition f t [ g t h t ] = f t g t f t h t [ f t g t ] h t = f t h t f t h t It is NOT commutative h t = f t g t g t f t = h t g t f t = g x t f x dx = g s f s t ds if s t =x g [ x t ]f x dx = g t f t In case of real functions, cross correlation and convolution are coincident.

21 Fourier Transforms Some examples The pulse: (x xo )

22 Fourier Transforms Some examples... {. The rectangle: (x)= x elsewhere FT sinc u = sin u u

23 Fourier Transforms Some examples... u 3. The Gaussian: f x = A e a x FT 1 a F u = e a

24 Fourier Transforms Some examples The double pulse: g(u) = (x xo ) + (x + xo)

25 Fourier Transforms

26 Elements of a Radio Telescope 1 Detection of low energy radiation: use the wave formalism (not enough energy for photoelectric effect) Hardware: A (main) mirror/collector, a secondary mirror (subreflector) and possibly more, drive the radiation on a given place (focus). A detector must be sensible to an incoming electromagnetic wave. In particular, the easiest component to detect is the electric field of the wave.

27 Elements of a Radio Telescope The mirror(s) must collect most of the radiation (E field). Shape and surface are related to the science driving the construction of the RT Caltech Submillimeter Observatory Westerbork Synthesis Radio Telescope

28 Elements of a Radio Telescope F is small! Most detectors at secondary focus Effelsberg

29 Elements of a Radio Telescope Surface accuracy: solid panels, mesh, wires. Parkes has a parabolic dish antenna, 64 m in diameter with a collecting area of 3,16 m. The dish is made up of aluminium panels supported by a lattice work of supporting struts. To incoming radio waves from space, the dish surface acts in the same manner as a smooth mirror. The waves are reflected and focused into a feed horn in the base of the telescope's focus cabin. The dish has a mass of 300 tonnes and distorts under its own weight as it points to different parts of the sky. Due to clever engineering design, however, this distortion is accounted for so that the radio waves are always reflected to the focus cabin.. The same applies to any radio telescope. The surface type and telescope characteristics are critical to the main scientific drivers for which a given instrument has been built.

30 Antenna basics surface accuracy Deviations from the ideal shape can be measured and corrected either with olographic techniques or with laser ranging, as a function of elevation (and weather conditions). Actuators are supposed to provide close loop real time correction to each mirror panel. Irregularities in the reflecting surface cannot be corrected for and lead to an unrecoverable signal loss the radiation reflected from the hatched area gets into the focus at a later time (with some phase delay, ) leading to a reduction of the coherence.

31 Antenna basics surface accuracy the signal reflected from the bowl acquires a phase = which turns into an additional phase at the focus. The contribution of the hatched portion of the mirror becomes g u = g o u e i u reducing the amplitude of the total signal. If (u) is small, then e i u 1 i /... and now let's consider the reduction of the effective area originated by an irregular reflecting surface Ae A oe g u du = g u du o

32 Antenna basics which becomes surface accuracy Ae A o e g u [1 i /]du = g u du o = o = 1 for a given irregularity, the effect in reducing the effective area depends on : the efficiency of a mirror becomes smaller and smaller with since increases. The other way round: for a given surface, we can define a minimum operating wavelength min Ae min Aoe = = 1 e = 1rad min therefore, conservatively, one can chose min = 0.

33 Antenna basics surface accuracy Northern Cross, Medicina Westerbork Synthesis Radio Telescope Effellsberg Alma

34 Elements of a Radio Telescope Detection of low energy radiation: use the wave formalism (not enough energy for photoelectric effect) Hardware: A (main) mirror/collector, a secondary mirror (subreflector) and possibly more, drive the radiation on a given place (focus). A detector must be sensible to an incoming electromagnetic wave. In particular, the easiest component to detect is the electric field of the wave.

35 Elements of a Radio Telescope A detector must be sensible to an incoming electromagnetic wave. In particular, the easiest component to detect is the electric field of the wave.

36 Elements of a Radio Telescope Mixer; down conversion to IF

37 Elements of a Radio Telescope Each detector can reveal radiation within a given bandwidth wrt the reference frequency. In general, the total available bandwidth is a small fraction of the reference frequency Modern receivers can have larger bandwidths. Future radio telescopes aim at a continuous (but not simultaneous) frequency coverage

38 Elements of a Radio Telescope SEFD = T sys G [Jy ]

39 Plan for the remaining theory... Elements of a Radio Telescope (RT) collecting area & the PSF (resolution) the detector (receiver) how the data look like frequency coverage (&... the proper receiver for each frequency) Why RT have little resolution Where to go then (interferometer) Adding interferometer Simple (cross correlation) interferometer geometric delay how the data look like many element interferometer Earth rotation synthesis and the uv coverage Modern (and future) interferometers (WSRT, VLA, ATCA, GMRT, LOFAR, ALMA, SKA)

40 Plan for the remaining theory Fourier Inversion the effect of the uv coverage & the need of cleaning the clean & restore (various methods) the final image Image analysis

41 Antenna basics (normalized) PSF / beam / power pattern P n, = P, P max where P max =P 0,0 beam solid angle A = 4 P n, d = 0 0 P n, sin d d if P n,, for a dipole isotropic antenna A =4 A dipole = sin sin d d = 3

42 Antenna basics Main beam solid angle M = main lobe P n, d Directivity D = 4 A Effective aperture Ae = A g A Efficiency A = Ae Ag 1 Various features limit the antenna efficiency: surface deformation of the mirror(s) [gravity, wind, temperature, etc.], surface accuracy, blocking, etc.

43 Antenna basics Ae a = S oss = S true A W = m A e B,, P n o, o d = kt A TA = TB TA = T B,, P n o, o d P n o, o d P n o, o d P n o, o d m A e S k = G S G = mae k o Antenna GAIN. K /Jy

44 Antenna basics Antenna gain is both a measure of the collecting area and of the efficiency of the mirror, namely of the effective area. It is function of frequency Gains for a small sample of (old) radio telescopes at = 6 cm Antenna Onsala VLA (1) Medicina OVRO Effelsberg Arecibo Diam (m) Gain ( K/Jy) Often related to SEFD (system equivalent flux density) to take into account also the contribution of electronics to the signal on the detector (see further)

45 Antenna basics What the antenna measures is T SYS = T A T R and the system temperatures is largely dominated by the receiver temperature (tens of K, depending on the observing frequency). This also determines the noise level of an observation. The detector reveals N wave packets in a time t and over a bandwidth and integrated for. N = = t being t= 1 The system temperature represents the average of many measures N, and it holds T SYS T rms =

46 Antenna basics and the signal to noise ratio is TA TA TA S = = N T rms T SYS T A T R weak sources TA TR to increase the SNR, larger TA = GS( ) = S( )mae /k areas are needed, implying larger mirrors. strong sources (high TA ) the SNR does not depend on the size of the mirror.

47 Elements of a Radio Telescope SEFD = T sys G [Jy ]

48 Antenna basics Resolving power Assuming a 1 D telescope of size D, its grading is represented by a rectangle function D/ u D/ g u = u = 1 0 elsewhere { whose FT is the far field E = sinc D = sin D D and the modulus of the far field provides the antenna pattern (beam, PSF) I = sinc D = sin D D whose first nulls occur when D = 1 = = D D rad

49 Antenna basics Resolving power For a D (3 D) telescope, the grading is more complex and its FT is a modified Bessel function, which goes to zero when 1. D = = 1. D D rad and the same happens to the PSF. It is a very bad case, since lambdas are large in the radio domain. Exercise: Which is the PSF size (HPBW, FWFN, etc) of a telescope like Effellsberg (100m) operating at = 0 cm? How it is possible to increase the resolution of a RT?

50 Antenna basics Resolving power For a D (3 D) telescope, the grading is more complex and its FT is a modified Bessel function, which goes to zero when 1. D = = 1. D D rad

51 Confusion noise A contribution to the noise arises from unresolved sources falling within the beam of a RT ( A ) It is related to the distribution P(S) of the sources as a function of the flux density (and of the frequency ) The confusion limit is defined as N min A 1 75 namely, to reveal a given source with a SNR=5, it is necessary that within the A fall less than 75 sources brighter than Slim. It is a function of v, slope in the source counts and A

52 Confusion noise

53 From single dish to interferometry From the Fraunhofer Fourier theory, the simpliest interferometer can be represented by two point like slits in 1 D (e.g. just the detectors!). The grading is a double pulse The far field is a cosine The antenna pattern is a square cosine The resolution is given by the separation between the two. + as the separation increases, the PSF gets narrower and narrower...but it is not possible to know where the radio source is! the collecting area remains small

54 Interferometry

55 Interferometry the geometric delay

56 Interferometry Signal on the elements of an interferometer A = E o sin t B = E o sin[ t g ] where g = D sin D = c c adding interferometer A B = A B AB AB = E o sin t sin[ t g ] = 1 = E o {cos [ t g ] cos g } A B rapidly varying, average is 0 depends on the geometric delay, slowly varying its the only term surviving time averaging, measures the phase difference on the two antennas. D cos g = cos D sin /c = cos sin

57 Interferometry [ P = A B AB = E o 1 cos 1 cos P n = where D sin ] D = sin cercare altra figura (the first) null[s] occur(s) when D cos = 1 namely sin = 1 1 HPBW = D sin B

58 Interferometry However, the continuous term is often disturbed by RFI, while the product is not, and therefore the switching interferometer has been introduced. It can be considered the progenitor of modern correlation interferometers A B A B = = A B AB A B AB = 4AB 1 = 4E o {cos[ t g ] cos g } P n = cos where = D sin the first nulls occur when cos = 0 namely D sin = HPBW 1 1 = D sin B

59 From single dish to interferometry Real interferometers are made of small apertures (compared to the maximum baseline) D a new expression of the grading g u = d [ x D / x D / ] d the far field becomes d D E = sinc cos = sinc D cos a and the beam (antenna pattern) is P = sinc d cos D

60 From single dish to interferometry The cosine term is modulated (and quenched) by the sinc and now maxima are not identical anymore. P = sinc d cos D

61 From infinity... to here! (adapted from Buzz Lightyear, 1995) Each antenna (ith single dish) measures a Voltage, i.e. the E field of the incoming monochromatic radiation is converted into V, to be sampled i V = V o sin t

62 Everybody knows... life is more complicated! hands / linears for polarization (together they fully sample incoming radiation) i V L = V L sin t L i V R = V R sin t R... well there is the noise as well!!!! i V L = V L sin t L n L t i V R = V R sin t R n R t We are not interested in most of the signal collected at each antenna [ n(t) ]

63 radio Interferometry: the signals from two antennas _i and _ j are (cross)correlated i i i V = V sin t n t j j j V = V sin t n t wavefront gets in j with some delay

64 The signal arrives at one of the antennas first and then, after g, gets to the other. In case it gets into i first, then in j the signal is V i V j = V sin t i sin t g j after some (boring) algebra and approximations we obtain: D D V V V cos g =V cos sin =V cos sin c i j vary with earth rotation baseline length ( units!!!)

65 Indeed the CORRELATOR performs a more complicated operation (i.e. the true cross correlation) to deliver VISIBILITIES: ij i j V g = V V = limt T / i j V t V t g dt T / In the ( D) uv plane each visibility samples the FT of the ( D) B( ) Modern correlators: Are special computing devices Handle and deliver a HUGE amount of data Visibilities are: Complex numbers (amplitude & phase), with ancillary information Computed over the integration time = T

66 Correlator throughput X N (N 1)/ baselines X4 (RR LL RL LR if full polarization mode) X sub bands (sensitivity requires large bandwidths, which are arranged into a number of sub bands) X channels (each sub band is divided into a number of spectral channels, i.e. small widows in frequency, over which the data are averaged) X total observing time / integration time Nowadays N is a few tens, the number of sub bands is a few tens, the number of channels can reach a few thousands, and the integration time is of the order of 1 sec. This makes A LOT of measurements!!!

67 Deep radio images implies sensitive continuum observations with: Large bandwidths, often organized in sub bands, in turn sliced into channels Necessary to prevent (minimize) bandwidth smearing [radial] Allow an accurate RFI removal Long on source times [(repeated) full tracks of the target(s)] Short integration times prevent (minimize) time smearing [tangential] Effective to remove time variable (intermittent) RFI All this also allow to image wide fields [surveys!] (large primary beams at low frequencies) (a huge amount of significant pixels on the sky!)

68 external a single antenna

69 Warnings: The Visibility Function is not sampled in a uv point, but it is rather averaged over an area (depending on integration time and bandwidth/frequency) this leads to image distortions known as smearing Old type continuum datasets are nowadays out of date. Nonetheless still useful to understand how CALIBRATION works in practice Modern datasets perform a sort of frequency synthesis: the field of view may appear rather different at the edges of the observing bandwidths Many effects are (strongly) frequency dependent e.g. the FoV is small / large at high / low frequencies and the data handling must take this into account More appropriate presentations will be given by other lecturers!

70 Real interferometry: An interferometer samples the Visibility Function as transmitted by the atmosphere and the instrumentation (antenna, receiver, electronics, cables, correlator, etc.) V ij obs = V ij true i GG j With a number of fair assumptions, CALIBRATION is the process to determine Gi aiming at transforming the observed quantities to the proper scale. All the quantities are COMPLEX, and therefore we need to find two values, AMPLITUDE and PHASE, for each antenna, polarization, sub band, channel,... possibly as a function of time

71 Real interferometry: ij ij i V obs = V true G G j We need to know the true visibility in order to determine the complex gains Simplest (ideal) case: Point like source of known flux density S, observed at the centre of the field of view. ALL amplitudes are identical to S ALL phases are 0 (zero) warnings: the number of such ideal sources is ridiculously small (0) sources (and FoV) are different as a function of frequency [high (10s GHz)/ low (a few 100s MHz)] interferometer (baseline length, also depends on frequency) sources are often variable in both flux density and polarization Let's try the simplest approach

72 Real interferometry: ij ij i V obs = V true G G j The complex gain G can be generally split into two terms: Amplitude Phase a and the new relationship can be written as: A ij obs e i ijobs = A ij true Calibration means to find appropriate i j aa e i ijtrue i j a and for the raw data.

73 We can term the modification of the true signal into the observed measure as a corruption of the information. Basic assumption (1st order) all the signal corruption can be determined and corrected solving an element/antenna based system Each interferometric element will have a correction for AMPLITUDE (t) PHASE (t) to be applied (in combination) to ALL the measurements delivered by the correlator. This operation MUST be done prior of Fourier inversion.

74 The complex gain Gi contains many components (along the signal path): F = ionospheric Faraday rotation T = tropospheric effects P = parallactic angle (altaz mounts) E = antenna voltage pattern D = polarisation leakage J = electronic gain B = bandpass response K = geometric compensation Gi = K i Bi J i Di E i Pi T i F i They are either additive (phases) or multiplicative (amplitudes). In most cases, when performing calibration we can forget the origin of the contribution to be removed. Some of them are specific to each type of observation (VLBI, Spectral line, wide field) and of the observing frequency.

75 The complex gain Gi contains many components (along the signal path): F = ionospheric Faraday rotation T = tropospheric effects P = parallactic angle (altaz mounts) E = antenna voltage pattern D = polarisation leakage J = electronic gain B = bandpass response K = geometric compensation Gi = K i Bi J i Di E i Pi T i F i Each term on the right has matrix form. The full matrix equation Gi is very complex, but usually only need to consider the terms individually or in pairs, and rarely in open form Existing software does the job (... more or less) but it is software...!

76 Ionospheric Faraday Rotation F i ionosphere is inhomogeneous (t, ):

77 Ionospheric Faraday Rotation F i ionosphere is inhomogeneous: various directions have different refraction indices: It is birefringent:one hand of circular polarisation is delayed w.r.t. the other, introducing a phase shift: F RL =e i e 0 i 0 sin XY i cos ; F =e i sin cos e progressively relevant at long wavelengths ( ) 0cm could be tens of deg.) + at solar maximum and at sunrise/sunset (high and variable TEC) Distant antennas have very different signal paths across the ionosphere Direction dependent within field of view 81 l B n dl e rad ; =n e (coherence length)

78 The Tropospheric contribution Ti the troposphere causes polarization independent amplitude and phase variations due to emission/opacity and refraction T RL = t relevant above ~15 GHz where water vapour (& oxygen,...) absorbs/emits elevation dependent (path length across the troposphere) (gain curve!) Distant antennas are have very different signal paths across the troposphere May be critical on bad weather (Tsys and noise contribution, very short coherence time)

79 The polarisation leakage Di Polarisers are not ideal: orthogonal polarisations are not perfectly isolated and mix. i V R = V R sin t R D R V L sin t L i V L = V L sin t L D L V R sin t R A geometric property of the feed design & frequency dependent Vital for linear polarization imaging (RL & LR). Plays a role in very high dynamic range Stokes' I Good receivers may have D terms of a few percent or less

80 The Parallactic angle χ = Pi Alt az antennas rotate during tracking (equatorial have a fixed orientation) Imply a rotation of the FoV and of the polarization response (intrinsic + leakage terms) Plays a role in very high dynamic range Stokes' I Good receivers may have D terms of a few percent or less P RL = e i t e 0 t = atan 0 i t ; P XY = cos t sin t sin t cos t cos lat sin[ha t ] sin lat cos Dec cos lat sin Dec cos[ha t ]

81 The antenna voltage pattern Ei Individual antennas have direction dependent gains (non uniform illumination) Relevant when the full FoV (Primary Beam) is imaged (Fields are wider and much more populated at low frequencies) Rotates with azimuth Related to elevation (gain curve) N.B. Changes with AzEl

82 The electronic antenna GAIN Ji It accounts for most of amplitude and phase effects introduced by antenna electronics (amplifiers, mixers, digitizers, samplers,...) and characteristics (collecting area, efficiency,...). It is the dominant term In practice in this term, many other effects (mentioned earlier) can be included No frequency dependence is deliberately considered (B i) Can be considered the tribute to engineers: implies the need to convert to physical units. J RL = jr 0 0 j L ; J XY = jx 0 0 j Y

83 The bandpass response Bi It represents the frequency dependence of the performance of the whole system (mainly electronics) It is dominated by the filter design and performance Disturbances introduced by spurious electronic behaviour (e.g. with temperature) B RL = b R 0 0 L b ; B XY = b X 0 0 b Y A strong (a good SNR per channel is necessary) calibrator, possibly point like, observed by all the antennas

84 The geometric compensation Ki The geometric model (antenna i, antenna j, source position) must be (ideally) perfect so that Synthesis Fourier Transform relation can work in real time; strong dependence on baseline length arises from uncertainties in antenna and source position (geodesy/astrometry) independent clocks and LO are a problem! specific of VLBI, gets worse with frequency K RL = kr 0 0 k L ; K XY = kx 0 0 k Y Specific correlation and data handling techniques are necessary (Fringe Fitting) to recover residual errors. In general it is not relevant for conventional interferometers (EVLA, WSRT, GMRT, ATCA, MERLIN,...)

85 T?

86 Summary of arguments for the Lab test Things to remember: 1. FT of particular functions and its relation to Fraunhofer diffraction. Interferometers:.1 simple interferometer ( elements). multi element interferometer 3. The UV plane 4. Visibility function 4.1 Visibility for a point like source 4. Visibility for an extended source 5. Fourier Inversion 6. Clean & Restore 7. Modern interferometers: WSRT, (E)VLA, ATCA...LOFAR, ALMA 8. The VLBI 9. The VLA and the lab test

87 Interferometry Uv plane 11C Source visibility (point, extended) Primary beam 11B 87