MTF and integration time versus fill factor for sparse-aperture imaging systems

Telescopes and imaging interferometers with sparsely filled apertures can be lighter weight and less expensive than conventional filled-aperture telescopes. Sparse-aperture systems can be characterized by their fill factor, which is the ratio of the area of a given aperture to the area of a filled aperture having comparable theoretical resolution. We show that the modulation transfer function (MTF) at the midrange spatial frequencies tends to be proportional to the fill factor. Using signal-to-noise ratio (SNR) expressions for various sources of noise, we derive the relationship between the integration time, needed to achieve a given SNR, and the fill factor. For example, for a fixed, sparse array, the integration time is proportional to the inverse cube of the fill factor for photon noise, to the inverse square of the fill factor for readout noise, and to the inverse fourth-power of the fill factor for dark current.