Differentiable optimization of the Debye-Wolf integral for light shaping and adaptive optics in two-photon microscopy

Control of light through high numerical aperture (N.A.) objectives is a common requirement in microscopy, for example for engineering specific point spread functions for super-resolution imaging [1, 2], for generating target light distributions for optical stimulation [3, 4, 5], for optical tweezers [6, 7], or for aberration corrections in adaptive optics [8, 9, 10]. For controlling light in all these situations, computational modeling is the most versatile approach for finding a phase pattern that, when displayed on a spatial light modulator (SLM), results in the desired target light distribution in the focal volume. Light propagation through a microscope objective with high N.A. can accurately be described with the vectorial Debye-Wolf diffraction integral [11]. The Debye-Wolf integral takes into account the orientation of the electromagnectic field vector (polarization) which contributes to the shape of the focus in high N.A. objectives. Such effects can be exploited for high resolution imaging, for example with diffraction limited objects, such as single molecules or nanostructures [1, 12, 2]. However, inversion of the Debye-Wolf integral does not have a general closed-form solution [13], and one therefore typically resorts to numerical approaches for applications that aim to generate an intended target light distribution. A fast method for calculating the Debye-Wolf integral would therefore be useful across a range of applications, for example vectorial imaging [14], vectorial beam shaping for tight focusing [2] or superresolution computational imaging [15], as well as any light shaping or imaging applications also at lower resolution, that is where polarization effects have less of an impact.