^{9}and radio interferometry developed after the Second World War. In 1946 Ryle and Vonberg

^{10}constructed a radio analogue of the Michelson interferometer and soon located a number of new cosmic radio sources. The signals from two radio antennas were added electronically to produce interference. Ryle and Vonberg's telescope used the rotation of the Earth to scan the sky in one dimension. Fringe visibilities could be calculated from the variation of intensity with time. Later interferometers included a variable delay between one of the antennas and the detector as shown in Figure 5.

Figure 5 - Radio interferometer |

*l*further in order to reach the left-hand antenna. These signals are thus delayed relative to the signals received at the right hand antenna by a time

*c l*=

*ca*sin[ ] where

*c*is the speed of the radio waves. The signal from the right hand antenna must be delayed artificially by the same length of time for constructive interference to occur. Interference fringes will be produced by sources with angles in a small range either side of determined by the coherence time of the radio source. Altering the delay time

*t*varies the angle at which a source will produce interference fringes. It should be noted that the effective baseline of this interferometer will be given by the projection of the telescope positions onto a plane perpendicular to the source direction. The length of the effective baseline, shown at the bottom of Figure 5, will be

*x*=

*a*cos where

*a*is the actual telescope separation.

An interferometer constructed from two antennas with separation variable in one direction can only provide information about the sky brightness distribution in one dimension. However, a two dimensional map of the sky can be produced if the separation vector is varied in two dimensions. In Figure 6 the separation between two radio antennas is described by the vector (*a*,*b*) constructed from two cartesian co-ordinates. The position of the source in the sky is described using the angles in the plane of the *a* axis and in the plane of the *b* axis. As in Figure 5, the effective baseline (*x*,*y*) will be the projection of the separation vector onto a plane perpendicular to the source direction: (*x*,*y*)=(*a*cos[ ],*b*cos[ ]). Measurements of complex visibility are usually plotted in the Fourier transform plane of the sky brightness distribution using the dimensionless variables *u* conjugate to angle and *v* conjugate to angle . These can be calculated as *u*=*kx* and *v*=*ky*, where *k* is the wavenumber of the radio source defined as . Either the phase of signals from the left-hand antenna can be measured relative to the those from the right hand antenna, or the phase of the signals from the right-hand antenna can be measured relative to those from the left. A measurement of complex visibility for an antenna separation (*a*,*b*) can thus provide values of the complex visibility function at two points in the *u*-*v* plane:

(*u*,*v*)=(*kx*,*ky*)=(*ak*cos[ ],*bk*cos[ ]) and (*u*,*v*)=(-*kx*,-*ky*)=(-*ak*cos[ ],-*bk*cos[ ])

Figure 6 - The telescope separation vector (a,b) |

In order to produce a perfect map of the sky brightness distribution the complex visibility would have to be known for all points in the *u*-*v* plane (Fourier transform plane). The complex visibility must be known at all points in a *n*×*m* rectangular array in the *u*-*v* plane for a portion of the sky to be mapped with resolution equivalent to *n*×*m* pixels. The radio source brightness distribution *B*( , ) is reconstructed by Fourier transforming the array of complex visibility measurements. Figure 7 shows a typical cosmic radio source with brightness distribution *B*( , ). Fourier transforming a 40×40 array of complex visibility measurements in the *u*-*v* plane gives a relatively accurate model of the source brightness distribution, as shown in Figure 8. Figure 9 shows the cruder model formed from a 9×9 array of complex visibility measurements.

Figure 7 - Source brightness distribution | Figure 8 - brightness distribution with 40x40 Fourier components | Figure 9 - brightness distribution with 9x9 components |

**Axes and brightness key**

For direct measurement of the complex visibility at a rectangular array of points in the *u*-*v* plane a large number of different baselines is required. The cost of radio antennas soon led astronomers to try and find methods for calculating the complex visibility throughout the *u*-*v* plane using measurements from only a small number of antennas. The most important of these is the Earth rotation aperture synthesis technique.

If an interferometer is constructed from two antennas with a separation which is not parallel to the Earth's axis of rotation, the effective baseline of the interferometer will rotate. Figure 10 shows an interferometer in the northern hemisphere with antennas located at *A* and *B*. During the day antenna *A* will move to *A'* and then *A''* whilst *B* moves to *B'* and *B''*. Only the relative positions of the two antennas are relevant when constructing a map of complex visibility in the *u*-*v* plane. To an irrotational observer standing beside antenna *A*, antenna *B* would appear to rotate in a circle, and vice-versa. In a twelve hour period the complex visibility can be measured at all points on an ellipse in the *u*-*v* plane. If one of the antennas is mobile, the antenna separation can be altered every day so as to measure complex visibilities in a different part of the *u*-*v* plane. A mathematical function which approximates the complex visibility is created by interpolation from the measurements made. This can then be Fourier transformed to give an approximation to the source brightness distribution.

Figure 10 - Rotation of the Earth |

Information about the fine structural detail of a radio source is found at large values of *u* and *v* due to the reciprocal nature of the Fourier transform plane. In order to produce a radio map of high angular resolution it is therefore necessary to measure fringe visibilities over very long baselines. The radio signal received at an antenna cannot be sent further than a few tens of kilometers by electrical cable due to the signal loss incurred. Electronic amplification en route introduces delays and distortion to the signal. The most effective method for measuring the complex visibility for very long baseline interferometry (VLBI) is to first record the signals received by each antenna along with timing signals from a local atomic clock. The recorded signals from each antenna can then be sent to a laboratory where they are replayed to produce interference. Figure 11 shows the received signals from three antennas being recorded onto magnetic tapes along with timing signals from local atomic clocks. From these tapes the complex visibility can be calculated at six points in the *u*-*v* plane corresponding to the antenna separations *a*_{1}, -*a*_{1}, *a*_{2}, - *a*_{2}, *a*_{3} and -*a*_{3} in Figure 11.

Figure 11 - Recording radio signals for very long baseline interferometry |

Each antenna will be a different distance from the radio source, and as with the short baseline radio interferometer (Figure 5) the delays incurred by the extra distance to one antenna must be added artificially to the signals received at each of the other antennas. The approximate delay required can be calculated from the geometry of the problem. The tapes are played back in synchronous using the recorded signals from the atomic clocks as time references, as shown in Figure 12. If the position of the antennas is not known to sufficient accuracy or atmospheric effects are significant, fine adjustments to the delays must be made until interference fringes are detected. If the signal from antenna *A* is taken as the reference, inaccuracies in the delay will lead to errors e * _{B}* and e

*in the phases of the signals from tapes*

_{C}*B*and

*C*respectively. As a result of these errors the phase of the complex visibility cannot be measured with a very long baseline interferometer.

Figure 12 - Visibility measurements in very long baseline interferometry |

The phase of the complex visibility depends on the symmetry of the source brightness distribution. Any brightness distribution B( , ) can be written as the sum of a symmetric component and an anti-symmetric component . The symmetric component B_{S} of the brightness distribution only contributes to the real part of the complex visibility, while B_{A} only contributes to the imaginary part. To demonstrate the dependence of the phase of the complex visibility on the symmetry of the source I separated the 9×9 array of complex visibility used to produce Figure 9 into real and imaginary parts. Figure 13 was produced using only the real component of the visibility, with the imaginary component set to zero. As the phase of the complex visibility is zero throughout the u-v plane the image is symmetric about its centre. In Figure 14 the real component was removed instead, giving an anti-symmetric image. As the phase of each complex visibility measurement cannot be determined with a very long baseline the symmetry of the corresponding contribution to the source brightness distributions is not known.

Figures 13 - Symmetric components | Figure 14 - Anti-symmetric components |

R. C. Jennison developed a novel technique for obtaining information about visibility phases when delay errors are present, using an observable called the closure phase. Although his initial laboratory measurements of closure phase had been done at optical wavelengths, he foresaw greater potential for his technique in radio interferometry. In 1958^{11} he demonstrated its effectiveness with a radio interferometer, but it only became widely used for long baseline radio interferometry in 1974^{12}. A minimum of three antennas are required. I will initially look at the simplest case, with three antennas in a line separated by the distances *a*_{1} and *a*_{2} shown in Figure 11. The radio signals received are recorded onto magnetic tapes and sent to a laboratory as described above. The effective baselines for a source at an angle will be *x*_{1}=*a*_{1}cos[ ], *x*_{2}=*a*_{2}cos[ ] and *x*_{3}=(*a*_{1}+*a*_{2})cos[ ]. The phases of the complex visibility of the radio source corresponding to baselines *x*_{1}, *x*_{2} and *x*_{3} I will call _{1}, _{2} and _{3} respectively. The phase of interference fringes on each baseline will contain errors resulting from e * _{B}* and e

*in the signal phases. The measured phases for baselines*

_{C}*x*

_{1},

*x*

_{2}and

*x*

_{3}, denoted

_{1},

_{2}, and

_{3}, will be:

* *_{1}= _{1}+e * _{B}*-e

_{C} _{2}= _{2}-e _{B }

_{3}= _{3}-e _{C}

Jennison defined the quantity _{C} for the three antennas as:

_{C}= _{1}+ _{2}- _{3} = _{1}+ _{2}- _{3}

_{C} is often called the *closure phase*^{12}.

The contributions to * _{C}* from errors e

*and e*

_{B}*in the signal phases cancel out allowing accurate measurement. Using measurements of*

_{C}*,*

_{C}_{3}can be written in terms of

_{1}and

_{2}, the unknown phases. If many closure phase measurements are made the complex visibility can be written as a function of several unknown phases. In order to produce an image of the sky the unknown phases must be estimated so that the complex visibility function can be calculated. This is usually done using iterative algorythms

^{13,14,15}which attempt to minimise unphysical properties of the image, such as areas of negative brightness (black areas above and below the source in figures 8 and 9) and large fluctuations in the background radio noise well away from the known location of the source. In radio astronomy visibilities are typically measured on more than three baselines simultaneously, providing more information about the source than Jennison's closure phase technique. The mapping algorithms are designed to retreive the maximum amount of information from the measurements performed without adding artificial detail. Images have been produced with baselines of many thousands of kilometers and resolution higher than one milliarcsecond.

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