Solving the inverse problem of high numerical aperture focusing using vector Slepian harmonics and vector Slepian multipole fields

Jahn, K.; Bokor, Nándor

doi:10.1016/j.optcom.2012.09.051

Cited by 15 publications

(6 citation statements)

References 21 publications

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“…Numerous studies have been conducted to address this issue. For example, a method based on vector Slepian harmonics and vector Slepian multipole fields is presented for a general treatment of the inverse problem of high numerical aperture (NA) focusing to achieve the prescribed electric field distribution in the focal volume [27]. The closest electromagnetic field to a target vectorial function is obtained by using the geometric projection of the target function to the pupil space of a high NA lens [28].…”

Section: Polarization Gratingsmentioning

confidence: 99%

Vectorial optical fields: recent advances and future prospects

2018

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Section: Polarization Gratingsmentioning

confidence: 99%

Vectorial optical fields: recent advances and future prospects

2018

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“…This can be done in manifold ways, where a truncation of the series, which would be the classical Slepian function approach discussed above in (11), is one out of these possibilities. The more general Ansatz corresponds to the scaling function approach described above in (12), where we replace D ↓ by the kernel…”

Section: Scaling Functions Wavelets Reproducing Kernels and Fredholm ...mentioning

confidence: 99%

“…The result is used as the given right-hand side for an inverse problem, where point masses are reconstructed which approximately generate the corre sponding regional gravitational potential. Examples in other application domains are [6,15,26,33].…”

Section: Introductionmentioning

confidence: 99%

A general approach to regularizing inverse problems with regional data using Slepian wavelets

Michel

Simons

2017

Inverse Problems

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Slepian functions are orthogonal function systems that live on subdomains (for example, geographical regions on the Earth's surface, or bandlimited portions of the entire spectrum). They have been firmly established as a useful tool for the synthesis and analysis of localized (concentrated or confined) signals, and for the modeling and inversion of noise-contaminated data that are only regionally available or only of regional interest. In this paper, we consider a general abstract setup for inverse problems represented by a linear and compact operator between Hilbert spaces with a known singular-value decomposition (svd). In practice, such an svd is often only given for the case of a global expansion of the data (e.g. on the whole sphere) but not for regional data distributions. We show that, in either case, Slepian functions (associated to an arbitrarily prescribed region and the given compact operator) can be determined and applied to construct a regularization for the ill-posed regional inverse problem. Moreover, we describe an algorithm for constructing the Slepian basis via an algebraic eigenvalue problem. The obtained Slepian functions can be used to derive an svd for the combination of the regionalizing projection and the compact operator. As a result, standard regularization techniques relying on a known svd become applicable also to those inverse problems where the data are regionally given only. In particular, wavelet-based multiscale techniques can be used. An example for the latter case is elaborated theoretically and tested on two synthetic numerical examples.

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“…This treatment, beyond the scalar approximation, is necessary in several areas of science, including high-resolution microscopy, [6][7][8][9] optical trapping, 10 electron acceleration, 11 optical vortex knots, 12,13 beam shaping, etc. [14][15][16][17][18][19][20][21] Nevertheless, the Richards-Wolf integral only considers perfect spherical wavefronts 4,5 and real optical systems do not generate perfect spherical wavefronts at the exit pupil, but aberrated wavefronts. A new diffraction integral has been published in Refs.…”

Section: Introductionmentioning

confidence: 99%

Focusing of an aberrated radially polarized beam by microscope objectives

González-Acuña,

Thibault

2024

Opt. Eng.

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A rigorous method is presented to compute the vector diffraction pattern of microscope objectives considering on-axis aberrated and radially polarized wavefronts. For that, the diffraction pattern of the microscope objective is computed considering the polarization and the wavefront at the exit pupil of the optical system. These computations do not assume that at the exit pupil of the microscope objective the wavefront is spherical but aberrated. Examples of real microscope objectives' performance are presented.

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Solving the inverse problem of high numerical aperture focusing using vector Slepian harmonics and vector Slepian multipole fields

Cited by 15 publications

References 21 publications

Vectorial optical fields: recent advances and future prospects

Vectorial optical fields: recent advances and future prospects

A general approach to regularizing inverse problems with regional data using Slepian wavelets

Focusing of an aberrated radially polarized beam by microscope objectives

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