Correlation telescope (1) I

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The basic unit of an interferometer is a pair of antennas. Their signal are multiplied with each other, that is, they are cross correlated. A correlation telescope measures the product of the signals U1 (t) and U2 (t) (voltages).

I A cyclic phase shift of π caused to signal U2 (t). I Square law detector: alternating (U1 + U2 )2 and (U1 − U2 )2 I By changing the sign of the difference signal (another phase shift)

before integration we obtain the for the time average of the output: < (U1 + U2 )2 − (U1 − U2 )2 >= 4 < U1 U2 >.

Correlation telescope (2)

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Assume that signals U1 (t) and U2 (t) contain besides noise, correlated harmonic signals X (t) and Y (t), so that they differ only by a phase difference caused by the delay τ .

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Time average of the product: the cross correlation of the signals X and Y . We denote this by RXY (τ ): RXY (τ ) =

1 Ux Uy e−i2πντ 2

The complex number presentation facilitates calculations. The measured signal is the real part of R.

Correlation telescope (3)

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Assume that the signals X (t) and Y (t) originate in a point source. The radiation (a plane wave) arrives at the antennas at an angle θi : τ =

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B sin θi c

B·s τg=B·s/c s _ B

T2

T1

Vahvistin

V2

Jännite− kertoja

V1 τi

Integroiva piiri

Visibiliteetti

The output of the correlation telescope, Uout , is proportional to the power received by the antenna: Uout ∝ Ae Fν e−i2πντ = Ae Fν e−i2πνB/c sin θi

Korrelaattori

Extended source

As in the case of a double slit, the response of an extended source is an integral over the brightness distribution: Z Ae Iν (θi ) e−i2πB/λ sin θi dθi . RXY (B/λ) = source

(the 2 dimensional case is presented below)

Fringe stopping (1)

B·s τg=B·s/c s _ B

T2

T1

Vahvistin

V2

Jännite− kertoja

V1 τi

Integroiva piiri

Korrelaattori

Visibiliteetti

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The projection of the baseline changes when the antennas track a source. The correlator response oscillates wildly because B/λ is a large number.

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The oscillation can be stopped by delaying the signal of the first antenna by the time τi which corresponds exactly to the geometric delay τg

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After fringe stopping the brightness distribution of the source dominates the response of the correlator.

Fringe stopping (2) Assume that the antennas point to the direction θ, and that the geometric delay, τg = B sin θ/c, is compensated by an instrumental delay so that τi = τg . I

In case a point source is lying at θ the amplitude of the signal is ∝ Fν and the phase is Φ = 0.

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The phase of a point source in direction θ + ∆α: Φ ≈ 2πB/λ cos θ∆α (when ∆α  1) The response of the correlator: RXY = Ae Fν e−i2πB/λ cos θ∆α

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Note that B cos θ is the projection of the baseline seen from the source.

Complex correlator

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The correlator described above measures just the real part
of the visibility: RXY ,1 = Re |V |e−iΦ = |V | cos Φ

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The imaginary part can be measured by adding another correlator for which the input signal is phase shifted by
π/2: RXY ,2 = Re |V |e−i(Φ−π/2) = |V | sin Φ

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q R12 + R22 |V | = Φ = arctan

R2 R1

Imaging

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Radio interferometer usually consists of several antennas. The number of antenna pairs and baselines: N(N-1)/2

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The projected baselines change when a celestial source is observed

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One way to deal with the measurement is to view the situation from the source and follow the rotation of the interferometer “below”.

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This point of view call forth the concept of the (u, v ) plane and the techniques called aperture synthesis

Coordinate system

Taivaannapa m I(l,m)

u: the projection of B/λ on the “East-West” axis as seen from the source v : the projection of B/λ on the “North-South” axis w: the projection B/λ on the axis pointing towards the source (compensated by the instrumental delay) I

l

dΩ

s0 s

w

v Bλ u

For an East-West oriented interferometer

u

= B/λ cos H

v

= B/λ sin δ sin H ,

where H is the hour angle and δ is the declination of the source.

Visibility and surface brightness distribution (1) I

ˆx + y e ˆy Coordinates in the sky wrt. phase centre: ~σ = x e

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Projection of the baseline as seen from the source: ~ ˆx + v e ˆy B/λ = ue ~ · ~σ /c = λ(ux + vy )/c Delay: τ = B

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Phase: Φ = 2πντ = 2π(ux + vy )

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Phase-corrected response of the correlation telescope can be written as: Z RXY = Ae (x, y )Iν (x, y )e−i2π(ux+vy ) dxdy , kohde

where Ae (x, y ) = Ae Pn (x, y ) (the effective surface area of antenna × beam pattern).

Visibility and surface brightness distribution (2)

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We re-define visibility as follows: V (u, v ) = RXY /Ae (reponse divide by Ae )

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Then Z V (u, v ) =

Pn (x, y )Iν (x, y )e−i2π(ux+vy ) dxdy

kohde I

Visibility defined above is the two-dimensional Fourier transform of the product Pn Iν

Example 3: Point source visibility

A source lies at an right ascension offset of ∆α from the phase centre. The visibility measured with an East-West interferometer is V (u) = Fν e−i2πu∆α = Fν e−i2πB/λ cos H ∆α where H is the hour angle |V | = Fν , Φ = 2πB/λ cos H ∆α

Example 4: “Resolving out” The source is composed of two equally strong point sources with an R.A. separation of ∆α (in radians). We observe it with an East-West interferometer, so that the source on the Western side (right) has been chosen as the phase centre. The measured visibility is then V (u) = Fν (1 + e−i2πu∆α ) , where u = B/λ cos H. If we choose ∆α = λ/(2B cos H), the visibility is V (u) = Fν (1 + eiπ ) = 0 . The visibility of an extended source can be thought to be composed of the sum of the visibilities of the surface elements. When a uniform source with the angular size θS = λ/B is observed, to each point of its surface corresponds another so that the phases of the two visibilities differ by π. Correlating interferometer misses this kind of source.

Antenna locations and the (u,v) coordinates (1) Z

( δ = 90ο )

Antenna locations are often given in the local ground based Cartesian system (X , Y , Z )

X

(H = 0, δ = 0)

(H = −6h, δ = 0) Y

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The X axis points towards the meridian, the Y points East, and the Z axis points towars the North pole.

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The unit is the wavelength λ.

Antenna locations and the (u,v) coordinates (2)

The relationship between (u, v , w) and (X , Y , Z ) (see, e.g., “Johdatus radioastronomiaan”, Ch. 6.5.) 

    u sin H cos H 0 X  v  =  − sin δ cos H sin δ sin H cos δ   Y  w cos δ cos H − cos δ sin H sin δ Z I

For an East-West oriented interferometer X = 0,Y = B/λ, Z =0

The principle of aperture synthesis

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1. Measure the time varying signals U1 (t) ja U2 (t) coming to antennas 1 and 2, respectively

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2. Compute the time average < U1 (t)U2∗ (t) >

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3. Repeat the measurement with a large number of ~ different baselines B/λ = (u, v ), i.e. measure V (u, v )

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4. Compute the surface brightness distribution by Fourier transform: RR Pn (x, y ) I(x, y ) = V (u, v )ei2π(ux+vy ) dudv . (The beam pattern Pn is known, and the equation can be divided by it.)