A regularization strategy for inverse source problems with applications in optics

In this work, we provide a stable algorithm for the inverse source problem where the region corresponds with a circle centered on the origin. The algorithm is obtained using an operational equation, which is ill-posed in the Hadamard sense due to the following points: firstly, many sources produce the same measurement and, secondly, due to the presence of numerical instability. If the operational equation is restricted to the Hilbert space of harmonic functions, the inverse source problem's uniqueness is guaranteed. The algorithm considers two regularization parameters to handle the numerical instability: the Tikhonov regularization parameter and the term N, where the series expansion is truncated. To illustrate the proposed algorithm, we developed one numerical example. Furthermore, we apply the algorithm to solve one inverse optical problem associated with the Intensity Transport Equation when the intensity distribution is considered almost uniform for the case in which the wavefront, which propagates in the direction of the optical axis, is considered within the paraxial approximation. The case where the source is not harmonic has no unique solution without a priori information. This work presents the case where the source belongs to functions that take two values. For this type, it is possible to recover the source completely. We give examples of the same application considering two cases for the source of the right side of the Intensity Transport Equation: when the source is a harmonic function and when it belongs to the above-mentioned type.