Detector Sampling of Optical/IR Spectra: How Many Pixels per FWHM?

Most optical and IR spectra are now acquired using detectors with finite-width pixels in a square array. Each pixel records the received intensity integrated over its own area, and pixels are separated by the array pitch. This paper examines the effects of such pixellation, using computed simulations to illustrate the effects which most concern the astronomer end-user. It is shown that coarse sampling increases the random noise errors in wavelength by typically 10-20 % at 2 pixels per Full Width at Half Maximum, but with wide variation depending on the functional form of the instrumental Line Spread Function (i.e. the instrumental response to a monochromatic input) and on the pixel phase. If line widths are determined, they are even more strongly affected at low sampling frequencies. However, the noise in fitted peak amplitudes is minimally affected by pixellation, with increases less than about 5%. Pixellation has a substantial but complex effect on the ability to see a relative minimum between two closely spaced peaks (or relative maximum between two absorption lines). The consistent scale of resolving power presented by Robertson to overcome the inadequacy of the Full Width at Half Maximum as a resolution measure is here extended to cover pixellated spectra. The systematic bias errors in wavelength introduced by pixellation, independent of signal/noise ratio, are examined. While they may be negligible for smooth well-sampled symmetric Line Spread Functions, they are very sensitive to asymmetry and high spatial frequency sub-structure. The Modulation Transfer Function for sampled data is shown to give a useful indication of the extent of improperly sampled signal in an Line Spread Function. The common maxim that 2 pixels per Full Width at Half Maximum is the Nyquist limit is incorrect and most Line Spread Functions will exhibit some aliasing at this sample frequency. While 2 pixels per Full Width at Half Maximum is nevertheless often an acceptable minimum for moderate signal/noise work, it is preferable to carry out simulations for any actual or proposed Line Spread Function to find the effects of various sampling frequencies. Where spectrograph end-users have a choice of sampling frequencies, through on-chip binning and/or spectrograph configurations, it is desirable that the instrument user manual should include an examination of the effects of the various choices.