interferometry and coherence

Question 1

drawbacks to total power telescopes

Answer 1

the stability of high-gain electronics:

even if Tsys»Tsource, it’s easy to pick out Tsource if the system is stable

but the gain can fluctuate ((eg due to temp variations)

Question 2

2 solutions to overcome the drawbacks of total power telescope

Answer 2

1. beam chopping
2. use an interferometer

Question 3

beam chopping

Answer 3

move the antenna rapidly on and off the source, faster than Tsys is changing and measure the difference

Tsource=Ton-Toff

Question 4

simple 2-element interferometer starting point

Answer 4

1D young slits
take a point source at infinity, illuminating the slits at an anlge a to the normal

Question 5

a point source of flux density s will produce

Answer 5

the familiar cos^2 fringe pattern on a distance screen

Question 6

if an extended source is incoherent (different parts of source radiate independently and can’t interfere) we can just

Answer 6

add the fringe intensities from each part of the source

Question 7

van cittert-zernike theorem

Answer 7

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

Question 8

from the van cittert-zernike theorem, you can recover the sky brightness from

Answer 8

measurements of the complex fringe visibility

Question 9

do we need to actually make fringes to measure the complex fringe pattern

Answer 9

no-
think of two emerging signals
to compute the fringe pattern, introduce a phase delay corresponding to an angle and add up the waves

Question 10

the complex fringe visibility can be computed directly from

Answer 10

the average (conjugate) product of the signals received by the two antennas

Question 11

interferometry - what to do practically (in 1 dimension) steps

Answer 11

1. get two antennas a distance y apart
2. measure two noise voltages (these are measures of the electric fields)
3. determine the mean product
4. repeat for different values of y
5. compute sky brightness using fourier transform

Question 12

the fourier transform relationship can be extended

Answer 12

to two dimensions

Question 13

truncating the complex fringe visibility measurements at some maximum baseline level is the equivalent of

Answer 13

smoothing (convolving) B with a function of width

Question 14

the angular resolution of the interferometer is

Answer 14

lambda/rmax

like a dish of width rmax

Question 15

why are interferometers relatively insensitive to changes in gain and Tsys

Answer 15

because <V1>=<V2>=0 so no large offsets in the system and measurements of <V1V2*> are not affected by small gain variations</V2></V1>

Question 16

what is <v1v2*> a measure of

Answer 16

spacial coherence of the radiation

(the similarity between the field at two spatially separated points)

Question 17

what is spatial coherence proportional to

Answer 17

correlation coefficient between v1 and v2

as a result, this act of multiplicaiton (<v1v2*>) is called correlation

Question 18

how do you perform the correlation

Answer 18

there are several ways, both analogue and digital but the digital correlator is the most common and the most flexible

Question 19

digital correlator - the signal form the antennas are noise-like so

Answer 19

even a simple 1-bit digitisation of v1(t) and v2(t) can be sufficient

Question 20

what does a simple 1-bit digitiation give

Answer 20

tow bitstreams of 1s and 0s

Question 21

EXNOR gate

Answer 21

0 1 =0
1 0 =0
0 0 =1
1 1 =1
(same =1, different=0)

Question 22

1-bit digitisation is efficient in terms of

Answer 22

storage

Question 23

1-bit digitisation con

Answer 23

loses some sensitivity and higher bit levels (eg 8-bit) are preferred when possible

Question 24

correlating interferometer: the phase difference between the two signals is

Answer 24

the phase of the complex fringe visibility

Question 25

correlating interferometer: if we can measure phase to pi/4, we can measure alpha to

Answer 25

1/8 lambda/D

Question 26

correlating interferometer: angular resolution is

Answer 26

lambda/D or slightly better

Question 27

for very high resolutions (D>100km) we need

Answer 27

Very Long Baseline Interferometry

Question 28

what is VLBI

Answer 28

same as collerating interferometer but: two antennas totally isolated from each other separate local oscillators. timing done with clocks signals are recorded onto local disk packs recordings are replayed later into the correlator

Question 29

requirements for VLBI

Answer 29

suitable only for very compact sources, otherwise fringe visibility will tend to zero

Question 30

if the source has been resolved on a baseline, there will be

Answer 30

no correlated flux to measure

Question 31

for VLBI, what sources do we require

Answer 31

generally: compact, high surface brightness sources such as quasars, radiogalaxy cores and pulsars

Question 32

timekeeping requirement for VLBI

Answer 32

if the signal bandwidth is delta v, then its coherence time is 1/delta v recordings must be time-synchronised

Question 33

coherence time

Answer 33

roughly the time over which it has a well-defined pahse

Question 34

the global navigation satellite systems gives the time requirement to be

Answer 34

better than 1 micro second

Question 35

we need to integrate for a time T before fringes are strong enough to see so

Answer 35

the correlated phase must not wander on timescales

Question 36

fractional stability of the oscillatir

Answer 36

delta t /T

Question 37

to enshre delta phi <<2pi, we can only integrate for

Answer 37

T<< 1/v 1/fractional stability