Fundamental interdependence between resolution and spatial bandwidth of analog optical spatial differentiators

Recently, in [Opt. Express28, 898 (2020)10.1364/OE.379492], two relations have been obtained between the gain and resolution of two ideal cases of spatial differentiators. The resolution was computed using the Rayleigh criterion. The first case is an ideal differentiator in that the magnitude of its transfer function is limited to unity, and the second case is the ideal form of a typical differentiator. The relation corresponding to case II has been used as a figure of merit (FOM) for comparison purposes between different differentiator performances. In this paper, we show that the Rayleigh criterion cannot properly compute the resolution of these ideal differentiators, especially for case II. The correct resolution is much smaller than that computed by the Rayleigh criterion in [Opt. Express28, 898 (2020)10.1364/OE.379492]. Hence, the mentioned relations between gain and resolution, and accordingly, the FOM in [Opt. Express28, 898 (2020)10.1364/OE.379492] are not correct. Herein, we propose three conditions (two obligatory and one optional) to determine the resolution of an edge detector. We mathematically prove that two of the three criteria (one optional and the other obligatory) are always met by both the ideal differentiators. We then demonstrate that the correct value of the resolution is approximately independent of the gain in case II but dependent on the spatial bandwidth of the ideal differentiator. We also show that similar resolution results are obtained when using a Gaussian light beam. Hence, we introduce a new FOM, which is a trade-off between the correct resolution and spatial bandwidth of the ideal differentiator in case II. We then use this new FOM to compare the performances of some recently proposed differentiators.