Fundamentals ofRadio Interferometry

Adapted from NRAO SynthesisImaging Summer SchoolLecturesBy Rick Perley

Interferometry in Action

Interferometry in Action cont…

Michelson Interferometer

• The movable mirroris used to adjust theextra path-length sothat the split beam iseither in or out ofphase with theoriginal beam

• The fringe pattern isthe resultinginterference patternfor a monochromaticsignal

Beam Pattern Origin

• An antenna’sresponse is aresult of coherentphase summationof the electric fieldat the focus.

• First null willoccur at the anglewhere one extrawavelength of pathis added acrossthe full width of theaperture:

θ ~ λ/D

On-axisincidence

Off-axisincidence

The Standard Parabolic Antenna Response

The power responseof a uniformlyilluminated circularparabolic antenna of25-meter diameter,at a frequency of 1GHz.

Getting Better Resolution

• The 25-meter aperture of a VLA antenna provides insufficientresolution for modern astronomy.– 30 arcminutes at 1.4 GHz, when we want 1 arcsecond or better!

• The trivial solution of building a bigger telescope is not practical.1 arcsecond resolution at λ = 20 cm requires a 40 kilometeraperture.– The world’s largest fully steerable antenna (operated by the NRAO

at Green Bank, WV) has an aperture of only 100 meters ⇒ 4 timesbetter resolution than a VLA antenna.

• As building a 40-kilometer wide antenna is not feasible, we mustconsider a means of synthesizing the equivalent aperture,through combinations of elements.

• This method, termed ‘aperture synthesis’, was developed in the1950s in England and Australia. Martin Ryle (University ofCambridge) earned a Nobel Prize for his contributions.

If the source emission isunchanging, there is noneed to collect all of theincoming rays at one time.

One could imaginesequentially combiningpairs of signals. If we breakthe aperture into N sub-apertures, there will beN(N−1)/2 pairs to combine.

This approach is the basisof aperture synthesis.

Aperture Synthesis – Basic Concept

The Stationary, MonochromaticInterferometer

A small (but finite) frequency width, and no motion.Consider radiation from a small solid angle dΩ, from direction s.

X

s s

An antennab

cg /sb ⋅=τ

)cos(2 tVV ω=])(cos[1 gtVV τω −=

2/])2cos()[cos(21 gg tVV ωτωωτ −+

2/)]/2cos([2/])cos([ 2121 cVVVVR gc sb ⋅== πυωτ

multiply

average

V is the voltageresponse to thesource

τg is the lag

In Phase:τg = nλ/c

Quadrature Phase:τg = (2n+1)λ/4c

Anti-Phase:τg = (2n+1)λ/2c

Examples of the Signal Multiplications

The two input signals are shown in red and blue.The desired coherence is the average of the product (black trace)

Signal Multiplication, cont.

• The averaged signal is independent of the time t, but is dependenton the lag, τg – a function of direction, and hence on the distributionof the brightness.

• V denotes the voltage of the signal which depends upon the sourceintensity by:

so the term V1V2 is proportional to source intensity, Iν.(measured in Watts.m−2.Hz−2.ster−2).

• The product depends on antenna areas and electronic gains, butthese factors can be calibrated for.

• To determine the dependence of the response over an extendedobject, we integrate over solid angle.

• Provided there is no spatial coherence between emission fromdifferent directions, this integration gives a simple result.

IEV ∝∝

The ‘Cosine’ Correlator Response

• The response from an extended source is obtained byintegrating the response over the solid angle of the sky:

for no frequency dependence and no spatial coherence.

Key point: the vector s is a function of direction, so thephase in the cosine is dependent on the angle of arrival.

This expression links what we want – the sourcebrightness on the sky) (Iν(s)) – to something we canmeasure (RC, the interferometer response).

Ω⋅= ∫∫ dcIRC )/2cos()( sbs πνν

A Schematic Illustration

The COS correlator ‘casts’ a sinusoidal fringe pattern of angularscale λ/b (b is antenna separation) radians, onto the sky. Thecorrelator multiplies the source brightness by this wave pattern,and integrates (adds) the result over the sky.

Orientation set by baselinegeometry.Fringe separation set by baselinelength and wavelength.

− + − + − + − Fringe Sign

λ/B rad.

Sourcebrightness

Odd and Even Functions

• But Rc, is insufficient – it is only sensitive to the ‘even’part of the brightness, IE(s).

• Any real function, I(x,y), can be expressed as the sum oftwo real functions which have specific symmetries:

An even part: IE(x,y) = (I(x,y) + I(-x,-y))/2 = IE(-x,-y)

An odd part: IO(x,y) = (I(x,y) – I(-x,-y))/2 = -IO(-x,-y)

= +I IE IO

Recovering the ‘Odd’ Part: The SIN Correlator

The integration of the cosine response, Rc, over the sourcebrightness is sensitive to only the even part of the brightness:

since the integral of an odd function (IO) with an even function(cos x) is zero.

To recover the ‘odd’ part of the intensity, IO, we need an ‘odd’coherence pattern. Let us replace the ‘cos’ with ‘sin’ in theintegral:

since the integral of an even times an odd function is zero. Toobtain this necessary component, we must make a ‘sine’ pattern.

Ω⋅=Ω⋅= ∫∫∫∫ dcIdcIR EC )/2cos()()/2cos()( sbssbs πνπν

Ω⋅=Ω⋅= ∫∫∫∫ dcIdcIR OS )/2(sin)s()/2(sin)( sbsbs πνπν

Making a SIN Correlator

• We generate the ‘sine’ pattern by inserting a 90 degree phaseshift in one of the signal paths.

X

s s

An antennab

cg /sb ⋅=τ

)cos(2 tVV ω=])(cos[1 gtVV τω −=

2/])2sin()[sin(21 gg tVV ωτωωτ −+

2/)]/2sin([2/])sin([ 2121 cVVVVR gs sb ⋅== πυωτ

multiply

average

90o

Define the Complex Visibility

We now DEFINE a complex function, V, to be the complex sum of thetwo independent correlator outputs:

where

This gives us a beautiful and useful relationship between the sourcebrightness, and the response of an interferometer:

Although it may not be obvious (yet), this expression can be inverted torecover I(s) from V(b).

φiSC AeiRRV −=−=

=

+=

−

C

S

SC

R

R

RRA

1

22

tanφ

Ω=−= ⋅−∫∫ dsIiRRV ciSC e /2)()( sbb νπ

ν

Picturing the Visibility

• The intensity, Iν, is in black, the ‘fringes’ in red. The visibility isthe net dark green area.

RC RS

LongBaseline

ShortBaseline

• The Visibility is a function of the source structure andthe interferometer baseline.

• The Visibility is NOT a function of the absoluteposition of the antennas (provided the emission istime-invariant, and is located in the far field).

• The Visibility is Hermitian: V(u,v) = V*(-u,-v). This isa consequence of the intensity being a real quantity.

• There is a unique relation between any sourcebrightness function, and the visibility function.

• Each observation of the source with a given baselinelength provides one measure of the visibility.

• Sufficient knowledge of the visibility function (asderived from an interferometer) will provide us areasonable estimate of the source brightness.

Comments on the Visibility

Examples of Visibility Functions

• Top row: 1-dimensional even brightness distributions.

• Bottom row: The corresponding real, even, visibility functions.

Geometry – the perfect, and not-so-perfect

To give better understanding, we now specify the geometry.

Case A: A 2-dimensional measurement plane.

Imagine the measurements of Vν(b) to be taken entirely on a plane.Then a considerable simplification occurs if we arrange thecoordinate system so one axis is normal to this plane.

Let (u,v,w) be the coordinate axes, with w normal to the plane. Alldistances are measured in wavelengths. Then, the components ofthe unit direction vector, s, are:

and for the solid angle

€

s = l,m,n( ) = l,m, 1− l2 −m2( )

221 mldldmd −−=Ω

Direction Cosines

The unit direction vector sis defined by its projectionson the (u,v,w) axes. Thesecomponents are called theDirection Cosines.

221)cos(

)cos(

)cos(

mln

m

l

−−==

=

=

γ

β

α

The baseline vector b is specified by its coordinates (u,v,w)(measured in wavelengths).

)0,,(),,( vuwvu λλλλλ ==b

u

v

w

s

αβ

γ

l m

b

n

The 2-d Fourier Transform Relation

Then, νb.s/c = ul + vm + wn = ul + vm, from which we find,

which is a 2-dimensional Fourier transform between the projectedbrightness:

and the spatial coherence function (visibility): Vν(u,v).

And we can now rely on a century of effort by mathematicians on howto invert this equation, and how much information we need toobtain an image of sufficient quality. Formally,

With enough measures of V, we can derive I.

dldmeml

mlIvuV vmuli∫∫ +−

−−= )(2

221

),(),( πυ

ν

)cos(/ γνI

dvduevuVmlI vmuli∫∫ += )(2),()cos(),( πνν γ

• Which interferometers can use this special geometry?a) Those whose baselines, over time, lie on a plane (any plane).

All E-W interferometers are in this group. For these, the w-coordinatepoints to the NCP.

– WSRT (Westerbork Synthesis Radio Telescope)

– ATCA (Australia Telescope Compact Array)

– Cambridge 5km telescope (almost).

b) Any coplanar array, at a single instance of time.– VLA or GMRT in snapshot (single short observation) mode.

• What's the ‘downside’ of this geometry?– Full resolution is obtained only for observations that are in the w-

direction. Observations at other directions lose resolution.• E-W interferometers have no N-S resolution for observations at the

celestial equator!!!

• A VLA snapshot of a source will have no ‘vertical’ resolution for objectson the horizon.

Interferometers with 2-d Geometry

VLA Coordinate System

w points to the source, u towards the east, and v towards the NCP.The direction cosines l and m then increase to the east andnorth, respectively.

b s0s0

22 vu +w

v

dldmeml

mlIvuV vmuli∫∫ +−

−−= )(2

221

),(),( πυ

ν

We have shown that under certain (and attainable)assumptions about electronic linearity and narrowbandwidth, a complex interferometer measures the visibility,or complex coherence:

(u,v) are the projected baseline coordinates,measured in wavelengths, on a plane oriented facing thephase center, and (l,m) are the sines of the angles between thephase center and the emission, in the EW and NSdirections, respectively.

Making Images

vu)vu,()cos() )vu(2 ddeV(l,mI mli∫∫ ++= πν γ

This is a Fourier transform relation, and it can be ingeneral be solved, to give:

This relationship presumes knowledge of V(u,v) for allvalues of u and v. In fact, we have a finite number, N,measures of the visibility, so to obtain an image, theintegrals are replaced with a sum:

vu)]vu(2exp[)v,u(V1

)1

n ΔΔ+= ∑=

mliN

(l,mI nn

N

nnn πν

If we have Nv visibilities, and Nm cells in the image, we have~NvNm calculations to perform – a number that can exceed 1012!

Making Images

• The sum on the last page is in general complex, whilethe sky brightness is real. What’s wrong?

• In fact, each measured visibility represents twovisibilities, since V(-u,-v) = V*(u,v).

• This is because interchanging two antennas leaves Rc

unchanged, but changes the sign of Rs.

• Mathematically, as the sky is real, the visibility mustbe Hermitian.

• So we can modify the sum to read:

vu])vu(2cos[A1

)1

ΔΔ++= ∑=

nnn

N

nn ml

N(l,mI φπν

But Images are Real

• Each cosine represents a two-dimensional sinusoidalfunction in the image, with unit amplitude, andorientation given by: α = tan-1(u/v).

• The cosinusoidal sea on the image plane is multipliedby the visibility amplitude An, and a shifted by thevisibility phase φn.

• Each individual measurement adds a (shifted andamplified) cosinusoid to the image.

• The basic (raw, or dirty) map is the result of thissummation process.

• The actual imaging is performed using FFTs

Interpretation

+

−

+−+

l

m

The rectangle below represents a piece of sky. The solidred lines are the maxima of the sinusoids, the dashed linestheir minima. Two visibilities are shown, each with phasezero.

α

A simple example

For a 1 Jy point source, all visibility amplitudes are 1 Jy, and all phases are zero. The lower panel shows the responsewhen visibilities from 21 equally-spaced baselines are added.

The individualvisibilities are shown in the toppanel. Their (incremental) sumsare shown in thelower panel.

1-d Example: Point-Source

• For a centeredsquare object,the visibilityamplitudesdecline withincreasingbaseline, and thephases are allzero or 180.

• Again, 21baselines areincluded.

Example 2: Square Source

Real interferometers must accept a range of frequencies (amongstother things, there is no power in an infinitesimal bandwidth)!So we now consider the response of our interferometer overfrequency.

To do this, we first define the frequency response functions, G(ν),as the amplitude and phase variation of the signals paths overfrequency.

Then integrate:

The Effect of Bandwidth.

νν

ντπνν

ννν devGvGIV gi2*

21

2

2

)()()(1

∫Δ+

Δ−Δ= s

G

ν ν0

Δν

The Effect of Bandwidth

If the source intensity does not vary over frequency width,we get

where the G(ν) are assumed to be square, real, and ofwidth Δν. The frequency ν0 is the mean frequency withinthe bandwidth.

The fringe attenuation function, sinc(x), is defined as:

ΩΔ= −

∫∫ dIV gig e τνπ

ν ντ 02)(csin)(s

6)(

1

)sin()(csin

2xxx

x

πππ

−≈

=

for x << 1

The Bandwidth/FOV limit

The source emission is attenuated by the function sinc(x), known asthe ‘fringe-washing’ function. Noting that τg ~ (b/c) sin(θ) ~ bθ/λν ~(θ/θres)/ν, we see that the attenuation is small when

The ratio Δν/ν is the fractional bandwidth. The ratio θ/θres is the sourceoffset in units of the fringe separation, λ/b.

In words: the attenuation is small if the fractional bandwidth times theangular offset in resolution units is less than unity. Significantattenuation of the measured visibility is to be expected if the sourceoffset is comparable to the interferometer resolution divided by thefractional bandwidth.

1<<Δ

resθθ

νν

Bandwidth Effect Example

Finite Bandwidth causes loss of coherence at large angles, becausethe amplitude of the interferometer fringes are reduced withincreasing angle from the delay center.

1/

=Δ

Bλθ

νν

• The trivial solution is to avoid observing largeobjects! (Not helpful).

• Although there are computational methods whichallow recovery of the lost amplitude, the loss inSNR is unavoidable.

• The simple solution is to observe with a smallbandwidth. But this causes loss of sensitivity.

• So, the best (but not cheapest!) solution is toobserve with LOTS of narrow channels.

• Modern correlators will provide tens to hundreds ofthousands of channels of appropriate width.

Avoiding Bandwidth Losses

There are still more details…

• Another important consequence of observing with a finitebandwidth is that the sensitivity of the interferometer is notuniform over the sky.

• Because the internal electronics of an interferometer workat a different frequency than the observed frequency, thereare other complications that must be worked out.

• We have assumed everywhere that the values of thevisibility are obtained ‘instantaneously’. This is of coursenot reasonable, for we must average over a finite timeinterval.

• The time averaging, if continued too long, will cause a lossof measured coherence which is quite analogous tobandwidth smearing.