Interferometry

Question 1

Total power telescopes connected with

Answer 1

stability of high-gain electronics have a drawback

solution ] use interferometer

Question 2

Simple 2-element interferometer

Answer 2

We can think in terms of the classic Youngs Slits Experiment:

Take a point source angle ∝ from the normal

Question 3

Diffracted amplitude

Answer 3

𝚿(θ) ∝ 1+exp[ik(θ-α)y]

Question 4

Diffracted Intensity

Answer 4

I(θ) = Io cos^2[k/2(θ-α)y]

Question 5

a point source of flux density S will produce fringes on a distant screen of the form

Answer 5

I(θ) ∝ S cos^2[ky/2(θ-α)]

Question 6

For an extended source - if the source is incoherent

Answer 6

I(θ) ∝ ∫ B(α) cos^2[ky/2(θ-α)] dα

I(θ) ∝ ∫ B(α) dα ∫ B(α) cos ky(θ-α) dα

I(θ) ∝ 1 + Re [ exp[ikyθ] ( ∫ B(α)exp[-ikyα] dα)/( ∫ B(α) dα)]

now simplifying by defining the complex fringe visibility Γ(y)

Γ(y) = ( ∫ B(α)exp[-ikyα] dα)/( ∫ B(α) dα) = |Γ(y)| exp[iΦ(y)]

Now

I(θ) ∝ 1+ |Γ(y)| cos(kyθ + Φ(y))

Question 7

If we define the normalised sky brightness the visibility becomes

Answer 7

Bn(α) = B(α)/ ∫B(α’)dα’

then

Γ(y) = ∫ Bn(α)exp[-ikyα] dα

Question 8

The van Cittert - Zernike Theorem

Answer 8

The complex fringe visibility is the fourier transform of the normalised sky brightness

Question 9

You can recover the sky brightness from measurements of the complex fringe visibility

Answer 9

Bn(α) = ∫ Γ(y) exp[ikyα] dy

Question 10

The previous expression for I(θ) can be expressed as the average product of the two signals

Answer 10

Γ(y) = 2<𝚿1𝚿2> / (<|𝚿1|^2> + <|𝚿2|^2>) ∝ <𝚿1𝚿2>

Question 11

The resolution of the interferometer is

Answer 11

λ/rmax

Question 12

<v1v2*> is a measure of the

Answer 12

spacial coherence of the radiation and is proportional to the correlation coefficient between v1 and v2.

Question 13

The schematic diagram for a correlating interferometer is

Answer 13

see notes

Question 14

For very high resolutions we need

Answer 14

Very Long Baseline Interferometry

VLBI

Question 15

Requirements for VLBI

Answer 15

Source must be very compact otherwise fringe visibility Γ(D) → 0.

No correlated flux means the source has been resolved

timekeeping

LO stability

Question 16

The Van Cittert-Zrtnikr theorem

Answer 16

gives the FT relationship between the complex fringe visibility on a baseline x and the sky brightness distribution. The inverse transform exists, so an image of the sky can be recovered from these observations