Lens to interferometerSuppose the small boxes are very small, then the phase shift Introduced by the lens is constant across the box and the same on both holes, so irrelevant. Thus the cos/sinc pattern we saw in the last slide of lecture 2 is the same if there is no lens, i.e. we just have a standard “Young type” interferometer.

We don’t measure the electric field in the image plane, but its average square, the received power, which in the case of two small holes looks like Sinc^2*cos^2… For more complicated apertures (remember the Besel Functions) we remember from Fourier transformations that the FT of an absolute square is the autocorrelation function of the FT itself. Thus the power psf is the FT of the autocorrelation function of the aperturepattern.

Power versus correlationUp until now we have assumed that our detection equipmentmeasures the total light power received in the image plane.In other words I(p) = <E(p)E*(p)>

)cos()Re( *

EEEEIEEEEI

EEE

2122

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2122

21

21

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Where is the phase angle between E1 and E2. The high resolution part of this signal is the last term which oscillates like cos (kpX/f) while the first two terms represent the total powercoming through the 1st and 2nd slits. These may contain very largeterms due to sky radiation that have nothing to do with the target,so it would be nice to get rid of them. We can do this if we havea Correlator rather than a detector. A correlator measures theaverage produce of two signals:

EEC *21

I’ll describe later how some correlators work.

CorrelationSo now we can abstract our optical system even further , throw awaythe focal system behind the aperture and replace it with a correlator.Then, if I have two small slits looking at a point source, then the correlated flux is:

Where D is the Optical Path Delay(OPD), the difference in path lengthfrom the source to slit 2 versus slit 1.

If the two “slits” are separated by a baseline vector B, andthe source is in the direction n, then:

)/cos( DSC 2

nBD

or: )cos(BD

is the angle between the baseline and the source.Note that C does not depend on the position of thereceivers on the ground but only on their separationvector.

CorrelationNow let’s consider what happens if we have more than one sourceof radiation on the sky. Then antenna 1 receives not only electricfield E1 from one source but also, say, F1 from the other source,with similar E2 and F2 at the other receiver. Then the formulasfor I and C should contain complicated terms like: FE *

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reflecting the difference in position between the two sources.This would be very messy, but fortunately astronomical sourcesare incoherent, that is, the phase difference between two unrelatedsources is never completely constant, but drifts quickly or slowlywith time (to be described later). So a term like

))cos()(cos()cos()cos( FEFE nBnBEFnBFnBEc 22

)cos( EnBEF

actually shows up as )cos( EnBEF where is a random number

and so the cos averages to zero and can be ignored. This would not be true if there were coherent sources (like lasers) distributed overthe sky.

Correlation

So, surprisingly or not, if I have a complicated source on the sky,the response of the interferometer is determined by the sum of theintensities from the individual components rather than the electric fields.Symbolically:

Where is the emitted intensity as a function of the spatial position. This looks sort of like a Fourier Transform but not quite. The term is non-linear in the position n on the sky because of the complicated spherical trigonometry. But in a small piece of sky near some reference point Lo this can be linearized:

ndBnknIC 2)cos()(

)(nI

BnD

MDV

LDULdLUkLIVUC

,;)cos()(),( 2

is the interferometric delay as a function of position in the field, L is the vector position relative to Lo. (U,V) are the “UV”coordinates of the baseline and equal the physical vector baselineprojected onto the sky at Lo. For narrow band (e.g. radio)measurements the wavenumber is included in thedefinition of U.

/2k

LdLUikLIVUC 2)exp()(),(̂

If we are smart enough to design a fully complex correlator,that also measures then we can write moredirectly:

LdLUkLI 2)sin()(

which looks exactly like a fourier transform. The easiest way toget the imaginary part of the correlation is to insert a quarterwave (/4) extra delay in the path, although this can be trickyfor a wide band system.

Ideally this would be all there is to interferometry: if you measurea whole lot of baselines you get an estimate of C over a completepiece of the UV-plane, and by a simple numerical Fourier Transformyou can reconstruct I(L). This is called Aperture Synthesis. Thegoal of measuring many UV points is partly achieved by sittingdown and letting the rotation of the Earth change the projectionof the baseline on the sky, and partly by having many telescopesat different positions to form into pairs, or moving around thetelescopes you have.