Interferometric ometric imaging imaging

A complex correlator computes the sine

A complex correlator computes the sine and cosine components simultaneously andand cosine components simultaneously and introduces

introduces a a compensating compensating instrumental instrumental delay delay (see (see ﬁgure ﬁgure below) below) in in one one of of the the antennasantennas which

which is is a a good good (but (but imperfect) imperfect) estimate estimate of of the the true true geometrical geometrical delay delay (the (the various various geo- geo-physcia

physcial effects are hard to predict - see earlier l effects are hard to predict - see earlier in this lecture).in this lecture).

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Correlator Correlator

In a modern

In a modern interferointerferometer themeter the

instrumental delay is introduced via a instrumental delay is introduced via a digital phase shift. In the early days of digital phase shift. In the early days of interferometry.

interferometry.

The difference between the

The difference between the geometricalgeometrical delay and the instrumental delay is the delay and the instrumental delay is the delay tracking error,

delay tracking error,

Let us consider a

Let us consider a 2-element interferomete2-element interferometer observing atr observing at frequency,

frequency, vv, , a radio source with a brightna radio source with a brightness distriess distribution,bution, IIv v (s),(s), then the output of a complex correlator is the powerthen the output of a complex correlator is the power received per unit bandwidth, dv, from an element of the received per unit bandwidth, dv, from an element of the radio

radio source,source, dsds is: is:

For parabolic antennas

For parabolic antennas A(s) is usually conside A(s) is usually considered to bered to be zero

zero outside of the fwhm ooutside of the fwhm of the antenna primary beam. f the antenna primary beam. SoSo in practice the integration is restricted to this area.

in practice the integration is restricted to this area.

The total response is

The total response is obtained by integrating overobtained by integrating over the solid angle subtended by

the solid angle subtended by the source:the source:

ds

For

For an an extended extended source source and and noting noting that that that that ssooand and are are essentially essentially perpendicular perpendicular toto one another on the celestial sphere, we can write:

one another on the celestial sphere, we can write:

Substituting this in eqn[9] we can write:

where:

The integral above,

The integral above, V(B)V(B) is the is the visibility functionvisibility function and from its form you can probably see already it is and from its form you can probably see already it is the Fourier

the Fourier TTransform (FT) of thransform (FT) of the source brightness die source brightness distribustribution. tion. In principlIn principle, e, we mawe may thereforey therefore recover the source brightness distribution by performing the inverse FT on the visibility function.

recover the source brightness distribution by performing the inverse FT on the visibility function.

and and is the

is the normalised beam pattern with Anormalised beam pattern with Aoo being the response at the being the response at the beam centre in the direction ofbeam centre in the direction of ssoo

[10]

In order to make use of eqn 10., it is useful to introduce a cartesian coordinate system.

B B

The baseline vector is speciﬁed The baseline vector is speciﬁed by

by (u, v, w)(u, v, w) wherewhere ww is chosen tois chosen to be in the

be in the direction of the sourcedirection of the source direction (known as the phase direction (known as the phase centre), normal to the u,v plane.

centre), normal to the u,v plane.

uu and and v v form a plane with u form a plane with u

orientated towards the east and orientated towards the east and v orientated towards the north.

v orientated towards the north.

This plane, known as the This plane, known as the u-v u-v plane is perpendicular to the plane is perpendicular to the source di

Another look at the

Another look at the sky - uv-plane - sky - uv-plane - telescope baseline geometry :telescope baseline geometry :

u,v,w

u,v,w coordinates specify the baseline vector. coordinates specify the baseline vector.

(in units of wavelength): B=(u,v,w) (in units of wavelength): B=(u,v,w) Positions on the sky, are deﬁned with Positions on the sky, are deﬁned with direction cosines,

direction cosines, x x and and y y , , and meand measuredasured w.r.t. the

w.r.t. the uu and and v v axes (bottom). axes (bottom).

The x-y plane is the projection of the The x-y plane is the projection of the celestial sphere onto a plane (right) with celestial sphere onto a plane (right) with aa tangent point deﬁned by

tangent point deﬁned by ssoo

In this coordinate system,

In this coordinate system, = (x,y,z)= (x,y,z)

x

= dx dy/cos(theta) dx dy/cos(theta)

= dx dy /z = dxdy/(1-x

= dx dy /z = dxdy/(1-x ²²-y -y ²² ) )^1/2^1/2 On a unit sphere the

On a unit sphere the source solid angle issource solid angle is dxdy, but because dx and dy are cartesian dxdy, but because dx and dy are cartesian coordinates (and the solid

coordinates (and the solid angle measured areaangle measured area must be projected on a sphere:

must be projected on a sphere:

In this coordinate system we can therefore re-write eqn[10] as:

- noting

- noting again that Aagain that A^’’(x,y)=0 outside of the primary beam of the (x,y)=0 outside of the primary beam of the antennas.antennas.

Also note, that for an array of telescopes orientated exactly east-west, the baseline vector always Also note, that for an array of telescopes orientated exactly east-west, the baseline vector always rotates within the uv-plane such that

rotates within the uv-plane such that w=0. w=0. Eqn[11Eqn[11] ] then reduces then reduces to:to:

[11]

i.e. the measured visibility function,

i.e. the measured visibility function, V(u,v),V(u,v), is related to the projected brightness on the skyis related to the projected brightness on the sky I(x,y)I(x,y) viavia an exact 2-D Fourier Transform. Eqn[12] is called the van Cittert-Zernike relation after the person an exact 2-D Fourier Transform. Eqn[12] is called the van Cittert-Zernike relation after the person that ﬁrst derived it

that ﬁrst derived it in the in the context of physical optics.context of physical optics.

[12]

More explicitly, we can formulate the inverse FT of [12]:

[13]

where I

where I^//(x,y) is the brightness distribution on the sky modiﬁed by the beam shape (primary(x,y) is the brightness distribution on the sky modiﬁed by the beam shape (primary beam) of the individual antennas. S

beam) of the individual antennas. Since the beam shape is ince the beam shape is well known (and also close to unity inwell known (and also close to unity in the centre of

the centre of the ﬁeld) the ﬁeld) its effect can easily be its effect can easily be removremoved by applying what is ed by applying what is known as a known as a “primary“primary beam co

beam correction”.rrection”.

In any case, the point is that we can make images of the sky by measuring the amplitude and In any case, the point is that we can make images of the sky by measuring the amplitude and phase of the

phase of the visibility function at visibility function at different points in the different points in the (u,v) plane. (u,v) plane. These points in These points in the uv-planethe uv-plane correspond to different projections of the baseline vector, B.

correspond to different projections of the baseline vector, B.

N.B.

N.B. what we have derived so far, is the special case for arrays that oriented East-West. what we have derived so far, is the special case for arrays that oriented East-West.

Westerbork is a good examples of an E-W array:

(u,v) w=0

As a

As a telescope collects telescope collects measuremmeasurements of ents of V(u,v) weV(u,v) we talk of “ﬁlling in” the uv-plane. In this pursuit,

talk of “ﬁlling in” the uv-plane. In this pursuit, astronomers are greatly aided by the rotation of astronomers are greatly aided by the rotation of the Earth!

the Earth!

As the Earth rotates, the projected baseline vector, As the Earth rotates, the projected baseline vector, B changes as seen by the source. In this way we B changes as seen by the source. In this way we collect many measurements of

collect many measurements of V(u,v) - V(u,v) - thesethese measurements are called “visibilities”.

measurements are called “visibilities”.

The visibilities are values for the amplitude and The visibilities are values for the amplitude and phase of the response of the correlator for each phase of the response of the correlator for each interferom

interferometer (baseline) at eter (baseline) at a given time (equivalenta given time (equivalent to a measurement in the uv-plane).

to a measurement in the uv-plane).

In document Basics of Radio Interferometry (Page 35-44)