Sensitivity - (Sub)Millimetre Summer School灵敏度（子）毫米暑期学校.pptVIP

Ninth Synthesis Imaging Summer School, Socorro, June 15-22, 2004 Basics of Interferometry and Interferometric Calibration Claire Chandler NRAO/Socorro (with thanks to Rick Perley and George Moellenbrock) Overview Aperture synthesis A simple 2-element interferometer The visibility The interferometer in practice Calibration of interferometric data Baseline-based and telescope-based calibration Calibration in practice VLA example Non-closing errors Tropospheric phase fluctuations Response of a parabolic antenna Origin of the beam pattern The beam pattern (cont.) A voltage V(q) is produced at the focus as a result of the electric field The voltage response pattern is the Fourier Transform of the aperture illumination; for a uniform circle this is J1(x)/x The power response, P(q) μ V2(q) P(q) is the Fourier Transform of the autocorrelation function of the aperture, and for a uniformly-illuminated circle is the familiar Airy pattern, (J1(x)/x)2 FWHP = 1.02 l/D First null at 1.22 l/D Aside: Fourier Transforms A function or distribution may be described as the infinite sum of sines and cosines with different frequencies (Fourier components), and these two descriptions of the function are equivalent and related by Where x and s are conjugate variables, e.g.: Time and frequency, t and 1/t = n Distance and spatial frequency, l and 1/l = u Some useful FT theorems Addition Shift Convolution Scaling Output for a filled aperture Signals at each point in the aperture are brought together in phase at the antenna output (the focus) Imagine the aperture to be subdivided into N smaller elementary areas; the voltage, V(t), at the output is the sum of the contributions DVi(t) from the N individual aperture elements: Aperture synthesis: basic concept The radio power measured by a receiver attached to the telescope is proportional to a running time average of the square of the output voltage: Any measurement with the large filled-aperture telescope can be written as a sum,