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The computational methods for the diffraction integrals that occur in the Extended Nijboer-Zernike (ENZ-) approach to circular, aberrated, defocused optical systems are reviewed and updated. In the ENZ-approach, the Debye approximation of Rayleigh's integral for the through-focus, complex, point-spread function is evaluated in semi-analytic form. To this end, the generalized pupil function, comprising phase aberrations as well as amplitude non-uniformities, is assumed to be expanded into a series of Zernike circle polynomials, and the contribution of each of these Zernike terms to the diffraction integral is expressed in the form of a rapidly converging series (containing power functions and/or Bessel functions of various kinds). The procedure of expressing the through-focus point-spread function in terms of Zernike expansion coefficients of the pupil function can be reversed and has led to the ENZ-method of retrieval of pupil functions from measured through-focus (intensity) point-spread functions. The review and update concern the computation for systems ranging from as basic as having low NA and small defocus parameter to high-NA systems, with vector fields and polarization, meant for imaging of extended objects into a multi-layered focal region. In the period 2002-2010, the evolution of the form of the diffraction integral (DI) was dictated by the agenda of the ENZ-team in which a next instance of the DI was handled by amending the computation scheme of the previous one. This has resulted into a variety of ad hoc measures, lack of transparency of the schemes, and sometimes prohibitively slow computer codes. It is the aim of the present paper to reconstruct the whole building of computation methods, using consistently more advanced mathematical tools. These tools are-explicit Zernike expansion of the focal factor in the DI,-Clebsch-Gordan coefficients for the omnipresent problem of linearizing products of Zernike circle polynomials,-recursions for Bessel functions, binomials and for the coefficients of algebraic functions that occur as pre-factors of the focal factor in the DI. This results in a series representation of the DI involving (spherical) Bessel functions and Clebsch-Gordan coefficients, in which the dependence of the DI on parameters of the optical configuration, on focal values, on spatial variables in the image planes, and on degree and azimuthal order of the circle polynomials are separated. This separation of dependencies, together with bounds on Clebsch-Gordan coefficients and spherical Bessel functions, facilitate the error analysis for the truncation of series, showing that in the new scheme the DI can be computed virtually without loss-of-digits. Furthermore, this separation allows for a modular implementation of the computation scheme that offers speed and flexibility when varying the various parameters and variables. The resulting scheme is pre-eminently appropriate for use in advanced optical simulations, where large defocus values, high NA and Zernike terms of high order and degree occur.

http://www.jeos.org/index.php/jeos_rp/article/view/07022We present a comparison among several fully-vectorial methods applied to a basic scattering problem governed by the physics of the electromagnetic interaction between subwavelength apertures in a metal film. The modelled structure represents a slit-groove scattering problem in a silver film deposited on a glass substrate. The benchmarked methods, all of which use in-house developed software, include a broad range of fully-vectorial approaches from finite-element methods, volume-integral methods, and finite-difference time domain methods, to various types of modal methods based on different expansion techniques

The judgment of the imaging quality of an optical system can be carried out by examining its through-focus intensity distribution. It has been shown in a previous paper that a scalar-wave analysis of the imaging process according to the extended Nijboer-Zernike theory allows the retrieval of the complex pupil function of the imaging system, including aberrations as well as transmission variations. However, the applicability of the scalar analysis is limited to systems with a numerical aperture (NA) value of the order of 0.60 or less; beyond these values polarization effects become significant. In this scalar retrieval method, the complex pupil function is represented by means of the coefficients of its expansion in a series involving the Zernike polynomials. This representation is highly efficient, in terms of number and magnitude of the required coefficients, and lends itself quite well to matching procedures in the focal region. This distinguishes the method from the retrieval schemes in the literature, which are normally not based on Zernike-type expansions, and rather rely on pointby-point matching procedures. In a previous paper [J. Opt. Soc. Am. A 20, 2281 (2003)] we have incorporated the extended Nijboer-Zernike approach into the Ignatowsky-Richards/Wolf formalism for the vectorial treatment of optical systems with high NA. In the present paper we further develop this approach by defining an appropriate set of functions that describe the energy density distribution in the focal region. Using this more refined analysis, we establish the set of equations that allow the retrieval of aberrations and birefringence from the intensity point-spread function in the focal volume for high-NA systems. It is shown that one needs four analyses of the intensity distribution in the image volume with different states of polarization in the entrance pupil. Only in this way will it be possible to retrieve the "vectorial" pupil function that includes the effects of birefringence induced by the imaging system. A first numerical test example is presented that illustrates the importance of using the vectorial approach and the correct NA value in the aberration retrieval scheme.

We study the image formation by a high-numerical-aperture optical imaging system in the presence of a multilayer structure in the region around the image plane. Earlier references to this subject in the literature use numerical solutions of the diffraction integrals. In this paper, we use a numerical approach based on the semi-analytic Extended Nijboer-Zernike (ENZ) theory to solve the diffraction integrals in the presence of a multilayer structure. The specific ENZ calculation scheme uses the complex Zernike expansion of the complex amplitudes of forward and backward propagating plane wave components in a certain layer of the multilayer stack. By its nature, the ENZ approach enables an accurate and fast calculation of the vector field in the stratified image region. Examples of multilayer imaging that are encountered in high-numerical-aperture optical systems and in optical lithography for semiconductor manufacturing are presented and the accuracy of the ENZ approach is examined.