On the diffraction limit for lensless imaging

The diffraction limit for lensless imaging, defined as the sharpest possible point image obtainable with a pinhole aperture, is analyzed and compared to the corresponding limit for imaging with lenses by means of theoretical considerations and numerical computations using the Fresnel-Lommel diffraction theory for circular apertures. The numerical result (u = pi) obtained for the best configuration parameter u which defines the optical setup is consistent with the quarter-wave criterion, and is the same as the value reported in a classical paper by Petzval but smaller than the Value (u =1.8 pi) found by Lord Rayleigh. The smallest discernible detail (pixel) in a composite image is defined by an expression found by Rayleigh on applying the half-wave criterion and is shown to be consistent with the Sparrow criterion of resolution. The numerical values of other measures of image size are reported and compared to equivalent parameters of the Fraunhofer-Airy profile that governs imaging with lenses.