Truncation of the series expressions in the advanced ENZ-theory of diffraction integrals

The point-spread function (PSF) is used in optics for design and assessment of the imaging capabilities of an optical system. It is therefore of vital importance that this PSF can be calculated fast and accurately. In the past 12 years, the Extended Nijboer-Zernike (ENZ) approach has been developed for the purpose of semi-analytic evaluation of the PSF, for circularly symmetric optical systems, in the focal region. In the earliest ENZ-years, the Debye approximation of the diffraction integral, by which the PSF is given, was considered for the very basic situation of a low-NA optical system and relatively small defocus values, so that a scalar treatment was allowed with a focal factor comprising a quadratic function in the exponential. At present, the ENZ-method allows calculation of the PSF in low- and high-NA cases, in scalar form and for vector fields (including polarization), for large wave-front aberrations, including amplitude non-uniformities, using a quasi-spherical phase focal factor in a virtually unlimited focal range around the focal plane, and no limitations in the off-axis direction. Additionally, the application range of the method has been broadened and generalized to the calculation of aerial images of extended objects by including the finite distance of the object to the entrance pupil. Also imaging into a multi-layer is now possible by accounting for both forward and backward propagation in the layers.In the advanced ENZ-approach, the generalized, complex-valued pupil function is developed into a series of Zernike circle polynomials, with exponential azimuthal dependence (having cosine/sine azimuthal dependence as special cases). For each Zernike term, the diffraction integral reduces after azimuthal integration to an integral that can be expressed as an infinite double series involving spherical Bessel functions, accounting for the parameters of the optical system and the defocus value, and Jinc functions comprising the radial off-axis value. The contribution of the present paper is the formulation of truncation rules for these double series expressions, with a general rule valid for all circle polynomials at the same time, and a dedicated rule that takes into account the degree and the azimuthal order of the involved circle polynomials to significantly reduce computational cost in specific cases. The truncation rules are based on effective bounds and asymptotics (of the Debye type) for the mentioned spherical Bessel functions and Jinc functions, and show feasibility of computation of practically all diffraction integrals that one encounters in the ENZ-practice. Thus it can be said that the advanced ENZ-theory is more or less completed from the computational point of view by the achievements of the present paper

Advanced analytic treatment and efficient computation of the diffraction integrals in the extended Nijboer-Zernike theory

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 areexplicit Zernike expansion of the focal factor in the DI,Clebsch-Gordan coefficients for the omnipresent problem of linearizing products ofZernike circle polynomials,recursions for Bessel functions, binomials and for the coefficients of algebraic functionsthat 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-Gordon 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-Gordon coefficients and spherical Besselfunctions, 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

Low-temperature thermochronology of the Indus Basin in central Ladakh, northwest India: implications of Miocene–Pliocene cooling in the India-Asia collision zone

The India‐Asia collision zone in Ladakh, northwest India, records a sequence of tectono‐thermal events in the interior of the Himalayan orogen following the intercontinental collision between India and Asia in early Cenozoic time. We present zircon fission‐track, and zircon and apatite (U‐Th)/He thermochronometric data from the Indus Basin sedimentary rocks that are exposed along the strike of the collision zone in central Ladakh. These data reveal a post‐depositional Miocene–Pliocene (~22–4 Ma) cooling signal along the India‐Asia collision zone in northwest India. Our ZFT cooling ages indicate that maximum basin temperatures exceeded 200 °C but stayed below 280–300 °C in the stratigraphically deeper marine and continental strata. Thermal modeling of zircon and apatite (U‐Th)/He cooling ages suggests post‐depositional basin cooling initiated in Early Miocene time by ~22–20 Ma, occurred throughout the basin across zircon (U‐Th)/He partial retention temperatures from ~20–10 Ma, and continued in the Pliocene time until at least ~4 Ma. We attribute the burial of the Indus Basin to sedimentation and movement along the regional Great Counter thrust. The ensuing Miocene–Pliocene cooling resulted from erosion by the Indus River that transects the basin. An approximately coeval cooling signal is well documented east of the study area, along the collision zone in south Tibet. Our new data provide a regional framework upon which future studies can explore the possible interrelationships between tectonic, geodynamic and geomorphologic factors contributing to Miocene–Pliocene cooling along the India‐Asia collision zone from NW India to south Tibet

High-NA aberration retrieval with the extended Nijboer-Zernike vector diffraction theory - Erratum

In the paper ”High-NA aberration retrieval with the Extended Nijboer-Zernike vector diffraction theory” by S. van Haver, J.J.M. Braat, P. Dirksen and A.J.E.M. Janssen, published in J. Europ. Opt. Soc. Rap. Public. 1, 06004 (2006), some regrettable notation errors are present in Eq.(10), page 06004-3. In this Erratum, the correct expression is given

Energy and momentum flux in a high-numerical-aperture beam using the extended Nijboer-Zernike diffraction formalism

We describe the energy and momentum flux in the case of an aberrated optical imaging system with a high numerical aperture (NA). The approach is based on the extended Nijboer-Zernike diffraction theory, that, in its high-NA version, yields an accurate analytic representation of the electromagnetic field vectors in the focal region of imaging systems that suffer from aberrations and/or transmission defects. In an earlier publication, we have derived the electromagnetic energy density from the field vectors. In this paper, we expand our analysis to the energy flow (Poynting vector) and to the quantities related to the linear and angular momentum of the radiation. Several examples of the energy and momentum flow are presented. In particular, we show how the linear and angular momentum distribution in the focal region depend on the initial polarization state and on the parameters describing the wavefront shape of the converging beam. For the angular momentum flow, we show how the separation between spin and orbital momentum is modified when going from the paraxial case to a high-NA focused beam

Experimental techniques for aberration retrieval with through-focus intensity images

Measurement techniques to determine the aberration of an optical system, by obtaining through-focus intensity images that are produced when the object is a point source at infinity, are shown. The analysis of the aberrations is made using the extended version of the Nijboer-Zernike diffraction theory. This theory provides a semi analytical solution of the Debye diffraction integral and thus a direct relation between the intensity distribution of the field at the focal region and the exit pupil of the optical syste