Vector Slepian basis functions with optimal energy concentration in high numerical aperture focusing

Jahn, K.; Bokor, Nándor

doi:10.1016/j.optcom.2011.11.107

Cited by 36 publications

(26 citation statements)

References 23 publications

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“…These functions can be regarded as the plane-wave amplitude spectra of new multipole-like fields, the vector Slepian multipole fields (VSLMFs) that not only exhibit the desired directionality but also enable us to reduce the dimension of the parameter space in optimization problems, compared to an expansion in terms of the original multipole fields. 8 In this paper we demonstrate the applicability of the novel vector Slepian bases to the inverse problems in high NA focusing when only the three-dimensional intensity distribution in prescribed in some reasonably chosen volume around the focus. Three examples from contemporary literature are considered: the optical needle, 9 tube 10 and bubble 11 focal spots.…”

Section: Introductionmentioning

confidence: 95%

“…Following an alternative formulation 6, 12 of the multipole theory of focusing, 5 the focused field can be expanded in terms of divergence-free solutions of Eq. (1), the vector multipole fields (VMF) M lm (r) and N lm (r), 8 as…”

Section: The Multipole Expansion Of Focused Fields and Their Plane-wamentioning

confidence: 99%

“…8 As already indicated, by solving the concentration problem, we also obtain a whole set of orthonormal functions, which we call the vector Slepian harmonics U mj (θ, φ). Here j indexes VSLHs of the same order m so that 1 > η m,1 ≥ η m,2 ≥ · · · ≥ η mNm > 0, where N m = L − max(1, |m|) + 1 denotes the maximal number of VSLHs corresponding to a given order m. As an illustration, the concentration ratios η mj are depicted in Fig.…”

Section: The Concentration Problem Of Tangential Vector Fields Withinmentioning

confidence: 99%

“…Therefore, such an approximation using the VSLHs usually involves less expansion coefficients than an approximation using the VSHs, 8 which is advantageous for the inverse problem, as a reduction in the dimension of the parameter space can be expected.…”

Section: The Concentration Problem Of Tangential Vector Fields Withinmentioning

confidence: 99%

“…This suggests that it is possible to construct vectorial basis functions more suitable than the multipole expansion, 5 which automatically exhibit directionality, as also demonstrated by Moore and Alonso. 6 We took an alternative approach and adapted the concentration problem of band-limited functions within a spherical cap 7 to the case of vector fields, 8 resulting in novel basis functions, the vector Slepian harmonics (VSLHs). These functions can be regarded as the plane-wave amplitude spectra of new multipole-like fields, the vector Slepian multipole fields (VSLMFs) that not only exhibit the desired directionality but also enable us to reduce the dimension of the parameter space in optimization problems, compared to an expansion in terms of the original multipole fields.…”

Section: Introductionmentioning

confidence: 99%

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Vector Slepian functions and the inverse problem of high numerical aperture focusing

Jahn

Bokor

2012

SPIE Proceedings

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The vector Slepian multipole fields are recently published basis functions that are naturally suitable for approximating the focal electric field of a high numerical aperture lens, because their plane-wave amplitude spectrum shows an angular energy distribution that is highly confined to the spherical cap corresponding to the numerical aperture of the lens. Thus when solving inverse problems, no additional constraints are needed to ensure the directionality of the focused field, and the number of degrees of freedom can be reduced, as opposed to a representation using conventional multipole fields.In this paper we demonstrate the applicability of this new basis to an inverse problem of focusing where the three-dimensional intensity distribution is prescribed in some reasonably chosen volume around the focus. Using numerical optimization, an approximation of the focused electric field in terms of the vector Slepian multipole fields is obtained and the illuminating field is calculated in a straightforward way. Three examples from recent literature have been chosen to illustrate the method.

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Section: Introductionmentioning

confidence: 95%

Section: The Multipole Expansion Of Focused Fields and Their Plane-wamentioning

confidence: 99%

Section: The Concentration Problem Of Tangential Vector Fields Withinmentioning

confidence: 99%

Section: The Concentration Problem Of Tangential Vector Fields Withinmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Vector Slepian functions and the inverse problem of high numerical aperture focusing

Jahn

Bokor

2012

SPIE Proceedings

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High‐resolution local magnetic field models for the Martian South Pole from Mars Global Surveyor data

Plattner

Simons

2015

JGR Planets

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We present two high-resolution local models for the crustal magnetic field of the Martian south polar region. Models SP130 and SP130M were derived from three-component measurements made by Mars Global Surveyor at nighttime and at low altitude (<200 km). The availability area for these data covers the annulus between latitudes −76 ∘ and −87 ∘ and contains a strongly magnetized region (southern parts of Terra Sirenum) adjacent to weakly magnetized terrains (such as Prometheus Planum). Our localized field inversions take into account the region of data availability, a finite spectral bandlimit (spherical harmonic degree L = 130), and the varying satellite altitude at each observation point. We downward continue the local field solutions to a sphere of Martian polar radius 3376 km. While weakly magnetized areas in model SP130 contain inversion artifacts caused by strongly magnetized crust nearby, these artifacts are largely avoided in model SP130M, a mosaic of inversion results obtained by independently solving for the fields over individual subregions. Robust features of both models are magnetic stripes of alternating polarity in southern Terra Sirenum that end abruptly at the rim of Prometheus Planum, an impact crater with a weak or undetectable magnetic field. From a prominent and isolated dipole-like magnetic feature close to Australe Montes, we estimate a paleopole with a best fit location at longitude 207 ∘ and latitude 48 ∘ . From the abruptly ending magnetic field stripes, we estimate average magnetization values of up to 15 A/m.

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Scalar and Vector Slepian Functions, Spherical Signal Estimation and Spectral Analysis

Simons

Plattner²

2013

Handbook of Geomathematics

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Vector Slepian basis functions with optimal energy concentration in high numerical aperture focusing

Cited by 36 publications

References 23 publications

Vector Slepian functions and the inverse problem of high numerical aperture focusing

Vector Slepian functions and the inverse problem of high numerical aperture focusing

High‐resolution local magnetic field models for the Martian South Pole from Mars Global Surveyor data

Scalar and Vector Slepian Functions, Spherical Signal Estimation and Spectral Analysis

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