# Field of View

VLA capabilities February 2014 - September 2014

# 1. Introduction

At least four different effects will limit the field of view. These are: primary beam; chromatic aberration; time-averaging; and non-coplanar baselines. We discuss each briefly:

# 2. Primary Beam

The ultimate factor limiting the field of view is the diffraction-limited response of the individual antennas. An approximate formula for the full width at half power in arcminutes is: θ_{PB} = 45/ν_{GHz}. More precise measurements of the primary beam shape have been derived and are incorporated in AIPS (task PBCOR) and CASA (clean task and the imaging toolkit) to allow for correction of the primary beam attenuation in wide-field images. Objects larger than approximately half this angle cannot be directly observed by the array. However, a technique known as "mosaicing," in which many different pointings are taken, can be used to construct images of larger fields. Refer to References 1 and 2 in Documentation for details.

Guidelines for mosaicing with the VLA are given in the Guide to Observing with the VLA

# 3. Chromatic Aberration (Bandwidth Smearing)

The principles upon which synthesis imaging are based are strictly valid only for monochromatic radiation. When visibilities from a finite bandwidth are gridded as if monochromatic, aberrations in the image will result. These take the form of radial smearing which worsens with increased distance from the delay-tracking center. The peak response to a point source simultaneously declines in a way that keeps the integrated flux density constant. The net effect is a radial degradation in the resolution and sensitivity of the array.

These effects can be parameterized by the product of the fractional bandwidth (Δν/ν_{0}) with the source offset in synthesized beamwidths (θ_{0}/θ_{HPBW}). Table 8 shows the decrease in peak response and the increase in apparent radial width as a function of this parameter. Table 8 should be used to determine how much spectral averaging can be tolerated when imaging a particular field.

(Δν/ν_{0})*(θ_{0}/θ_{HPBW}) | Peak | Width |
---|---|---|

0.0 | 1.00 | 1.00 |

0.50 | 0.95 | 1.05 |

0.75 | 0.90 | 1.11 |

1.0 | 0.80 | 1.25 |

2.0 | 0.50 | 2.00 |

- Note: The reduction in peak response and increase in width of an object due to bandwidth smearing (chromatic aberration). Δν/ν0 is the fractional bandwidth; θ
_{0}/θ_{HPBW}is the source offset from the phase tracking center in units of the synthesized beam.

# 4. Time-Averaging Loss

The sampled coherence function (visibility) for objects not located at the phase-tracking center is slowly time-variable due to the motion of the source through the interferometer coherence pattern, so that averaging the samples in time will cause a loss of amplitude. Unlike the bandwidth loss effect described above, the losses due to time averaging cannot be simply parametrized, except for observations at δ = 90^{°}. In this case, the effects are identical to the bandwidth effect except they operate in the azimuthal, rather than the radial, direction. The functional dependence is the same as for chromatic aberration with Δν/ν_{0} replaced by ω_{e}Δt_{int}, where ω_{e} is the Earth's angular rotation rate, and Δt_{int} is the averaging interval.

For other declinations, the effects are more complicated and approximate methods of analysis must be employed. Chapter 13 of Reference 1 (in Documentation) considers the average reduction in image amplitude due to finite time averaging. The results are summarized in Table 9, showing the time averaging in seconds which results in 1%, 5% and 10% loss in the amplitude of a point source located at the first null of the primary beam. These results can be extended to objects at other distances from the phase tracking center by noting that the loss in amplitude scales with (θΔt_{int})^{2}, where θ is the distance from the phase center and Δt_{int} is the averaging time. We recommend that observers reduce the effect of time-average smearing by using integration times as short as 1 or 2 seconds (also see the section on Time Resolution and Data Rates) in the **A** and **B** configurations.

Amplitude loss | |||
---|---|---|---|

Configuration | 1.0% | 5.0% | 10.0% |

A |
2.1 | 4.8 | 6.7 |

B |
6.8 | 15.0 | 21.0 |

C |
21.0 | 48.0 | 67.0 |

D |
68.0 | 150.0 | 210.0 |

- Note: The averaging time (in seconds) resulting in the listed amplitude losses for a point source at the antenna first null. Multiply the tabulated averaging times by 2.4 to get the amplitude loss at the half-power point of the primary beam. Divide the tabulated values by 4 if interested in the amplitude loss at the first null for the longest baselines.

# 5. Non-Coplanar Baselines

The procedures by which nearly all images are made in Fourier synthesis imaging are based on the assumption that all the coherence measurements are made in a plane. This is strictly true for E-W interferometers, but is false for the EVLA, with the single exception of snapshots. Analysis of the problem shows that the errors associated with the assumption of a planar array increase quadratically with angle from the phase-tracking center. Serious errors result if the product of the angular offset in radians times the angular offset in synthesized beams exceeds unity: θ > λB/D^{2}, where B is the baseline length, D is the antenna diameter, and λ is the wavelength, all in the same units. This effect is most noticeable at λ90 and λ20 cm in the larger configurations, but will be notable in wide-field, high fidelity imaging for other bands and configurations.

Solutions to the problem of imaging wide-field data taken with non-coplanar arrays are well known, and have been implemented in AIPS (IMAGR) and CASA (clean). Refer to the package help files for these tasks, or consult with Rick Perley, Frazer Owen, or Sanjay Bhatnagar for advice. More computationally efficient imaging with non-coplanar baselines is being investigated, such as the "W-projection" method available in CASA; see EVLA Memo 67 (http://www.aoc.nrao.edu/evla/geninfo/memoseries/evlamemo67.pdf) for more details.