Irradiance transfer function using Hermite-Gaussian beams, with an application to challenging lensless imaging in the Fresnel regime

Modeling the diffraction of a scalar field from a plane of interest to a quadratic detector is a generic topic, and inverting this information flux is the core of digital holography. The purpose of this work is to identify a basis of modes and their associated scalar transfer function, accounting for propagation, finite boundaries, and quadratic detection. This model is relevant to set the detection parameters, evaluate performance, or define an inverse filter. Using Hermite-Gaussian functions (HGFs) to decompose the complex field amplitude in both planes, we numerically compute the resulting irradiance in the detector plane and derive, using a singular value decomposition (SVD), the transfer function and the associated modes in each plane. Closed-form formulas, confirmed by numerical simulations, are derived to compute the effect of acquisition parameters such as collector diameter, spectral width, and wavefront distortion. The SVD leads to a classical triangle-like transfer function, while new irradiance modes (similar to HGFs) have some nice mathematical properties. This model is applied to the design and simulation of a hectometric (∼100 m) holographic system for GEO satellite imaging.