Affordable real-time diffraction compensation in the spatial domain

Diffraction is a source of distortion in optical imaging, and can be nearly eliminated by linear compensating components such as lenses. However, in many applications such components are impractical, and thus the ongoing interest in simple low-cost solutions. Since standard detection is based on power measurements (rather than coherent field measurements) a simple linear filter cannot compensate for the distortions. However, if the variations in the image are relatively small the process is approximately linear and can be compensated by applying a linear filter. This approach has been developed, however, the analysis was always done in the spectral domain, in which case electrical spectral filters cannot be easily implemented on a 2D matrix without digital processing. We present a spatial filter for diffraction compensation in the spatial domain, based on a filter which was originally developed by us for dispersion mitigation. To the best of our knowledge, this is the first time that an analytical expression is derived for a diffraction compensating filter in the spatial domain. Moreover, in previous works a small imaginary parameter was introduced to eliminate divergence. In the present paper a simple analytical evaluation of this parameter is given, which eliminates the need for iterative computations.