Exact vectorial model for nonparaxial focusing of freeform wavefronts

Abstract

We present a new formalism, based on Richards–Wolf theory, to rigorously model nonparaxial focusing of radially polarized electromagnetic beams with freeform wavefront. The beams can be expressed in terms of Zernike polynomials. Our approach is validated by comparing known results obtained by Richards–Wolf theory. Our integral representation is compliant with diffraction theory, is thoroughly discussed and solved for various freeform wavefront that, so far, have not been treated analytically. The extension of the method to other polarization states is straightforward.

General diffraction integral for converging freeform vectorial fields

“…In the same work, we mentioned that the extension of the method to other polarization states is straightforward which in fact is done with equations (6) and (9). The creativity and abstract reasoning to get equations (6) and (9) were not trivial, but now we can confirm that we have extended the method [38] to other states.…”

“…The same applies to the apodization function. In our last work, we presented an integral that computes the diffraction pattern of the freeform wavefront with rotationally polarized [38]. The mentioned integral can be seen as a special case of equation (6) when θ = 0.…”

Focusing of an aberrated radially polarized beam by microscope objectives

“…Reference 25 considers several approximations that also we do not consider, such as their Eqs. ( 20), ( 21), (22), and (43). Also, it is not clear and the author does not mention in Refs.…”

“…A new diffraction integral has been published in Refs. 22 and 23 and this integral can compute the diffraction pattern of vectorial aberrated wavefronts with radial polarization. Here, the model presented in Ref.…”