Vectorial aerial-image computations of three-dimensional objects based on the extended Nijboer-Zernike theory

Abstract

We present details of a novel imaging algorithm based on the extended Nijboer-Zernike (ENZ) theory of diffraction. We derive integral expressions relating the electric field distribution in the entrance pupil of an optical system to the electric field in its focal region. The evaluation of these integrals is made possible by means of a highly accurate and efficient series expansion similar to those occurring in standard ENZ theory. Based on these results an ENZ imaging scheme is constructed and evaluated in detail with attention to the convergence properties and computational complexity of the new method.

Fresnel transform as a projection onto a Nijboer–Zernike basis set

“…(6). Here we use a Zernike decomposition which deviates from the one presented in [12]. The termūρ; ϕ is written as…”

“…where J m is the Bessel function of the first kind and of order m. By using the basic integral result of the ENZ-theory [12],…”

Image formation in a multilayer using the extended Nijboer-Zernike theory

“…In an earlier publication [20] we have shown that the Extended Nijboer-Zernike (ENZ) theory can be used to compute the through-focus aerial image of an extended object. This work already included the possibility to have a different medium in both object and image space, but did not account for a non-uniform or layered configuration in image space.…”

“…The optical system, represented by its complex transmission function, T I , and magnification, M, transforms the field captured by the entrance pupil into a field distribution in the exit pupil. So far, the approach remains the same as in [20], but now, instead of constructing an image in an uniform image space we are faced with image formation in a focal region that consists of several layers with different refractive index n h . The problem with such a configuration is that, at the interface between two layers having a different refractive index, the light will be partly refracted and reflected, and thus gives rise in every layer to both forward and backward travelling electric field components, t h and r h .…”

“…The computation beforehand and the subsequent storage in look-up tables of basic spread-functions greatly enhances the numerical speed once a real aberrated system needs to be analyzed with respect to its imaging properties in the focal volume. In previous papers, we have shown how the ENZ diffraction theory can also be used to obtain an efficient and accurate imaging algorithm for extended objects with general shape and for arbitrary illumination conditions in the case of a homogeneous medium between the exit pupil and the focal region [19,20]. In the present paper, we extend the ENZ diffraction theory to obtain the three-dimensional field distribution in media that are often encountered in practice, in particular in an image space that is built up of various thin parallel layers (stratified image space).…”