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Lecture 16 - Interferometry 1 (Aperture arrays (Some applications (radio…
Lecture 16 - Interferometry 1
Interferometers as parts of filled apertures
Conventional telescopes reflect the light of a distant object from a parabolic furface to a focus
the shape of the surface differentially delays the signals arriving across the aperture so as to bring them to the point of focus - and everything work as the speed of light
The reflecting surfaces do not need to be part of the same surface. Suppose we cover up most of the surface of the mirror
We can still combine the radiation from the uncovered section to create an image of the distant object, if we arrange the path lengths to the focus to be the same and then combine the results from many separately uncovered sections. Can thus synthesise an aperture.
Mimicking a filled aperture with a phased array
Split aperture into an array of small collecting elements
signals at each point in the aperture are (electronically or via cables) brought together in phase at the antenna output (the virtual focus) and added together
Like a dish, an array connected like this forms and nstantaneous singe beam pointing upwards
Diagram
This is direct imaging with an interferometer array
Delay Compensation
When the pointing direction is at an angle to the baseline between two antennas the signals must travel the same electrical distance before they are combined. Add the path delay to the signal from A
maths
Aperture arrays
Some applications (radio astronomy, remote sensing systems, radar etc.) require many intantaneous, electronically steerable, beams formed from the addition of the signals from many small collector elements forming the phased array
These are also called "aperture arrays" since they constitute the whole receiving aperture. They do no require mechanical pointing as does the dish
All elements (not just 2 as shown) are co-added with an appropriate phase/delay gradient across the array for the chosen pointing derection.
The sum is the square-law detected
Different pointing directions can be simultaneously acheived electrically with different phase gradients -just needs more electronics
In practice the arrays are 2d
Diagram
Two element adding interferometer
Two antennas of equal dimention d at fixed locations and pointing direction (here vertical) and separated by a baseline vector b
A distant source of quasi-mon0ochromatic (very narrow band) plane wave passes overhead along the direction of the baseline. At any moment s is the unit vector p[ointing at moving source source.
The purpose of the antenna, receiver and associated electronics is to convert the incoming electric field to a voltage, preserving its amplitude and phase (i.e behaving linearly) even though the signal has been through heterodyne stages (not shown)
In the receiver, the outputs from two antennnas are added and the sum is square-law detected
In this basic arrangement the two signal paths are equal, fixed, lengths of cable which restricts us to looking at only a relatively small angle in the sky around the zenith
Maths
The 2-element adding interferometer i a direct analogue of young's slits in optics
Point spread function (reception beam) of a phased array
Diagrams and maths
Convolution theorum and Weiner-Khinchine theorem
Phased Array Pattern
Main lobe: the reception pattern of the whole aarray sets the maximum resolution Angular width
Sidelobes due to finite extent of array i.e. sharp cut-off at bmax
periodic repetitions due to periodic spacing Angular spacing
Direct imaging: an instantaneous beam (the "point pread function") is formed.
If a source is smaller than the main lobe its flux density can be measured
if the source is bigger than the main lobe that its brightness distribution could be built up by scanning the entire beam across the source - in this case electronicalls - but just as for a single dish, but this is not the main use of phased arrays
Phased Array Beam Structure
The pattern corresponds to a "sparse " array which has large spacings between the collecting antennas giving rise to the large periodic sidelobes on either side of the main lobe
In practical arrays the antennas will be closer together (very close in aperture arrays) and there may be very many array beams (now dominated by the main lobes) within the reception pattern of the individual antennas
the low level sidelobes have been suppressed here for clarity
Reception pattern of individual antennas diagram
Independent main lobes from the array antennas combined with different phase gradients in 2.D
Multiplying interferometer
Geometry as before, but now multiply the voltages and integrate
refering back to algebra - on the the VA and VB terms, which both come from the source itself, correlate.
All other terms wash out after integration
Main advantage: total power terms don't correlate so output is insensitive to receiver gain variation and to variations in atmospheric noise
Signal multiplication diagram - in-phase, quadrature phase, anti-phase
Fringe pattern- diagram. Primary element beamwidth, fringe spacing=wavelength on sky, angular frequency of fringes on sky.
Reception pattern diagram
Interferometer responce
Diagram
Responce will be 2 D in reality but in 1d for simplicity maths
The sky brightness distribution is not an even function. If we want to reconstruct it from its Fourier components then we need both the cos and sin terms
Van Cittert-Zernicke THeorum
The 2d lateral cohernce function of the radiation field in space is the fourier transform of the 2d brightness (or intensity) distribution of the soure
The visibility function is therefore another name for the spatial correlation function
Interferometer responce
Can define the complex fringe visibility for a particular baseline
Maths