Derivation of closed-form ellipsoidal X-ray mirror shapes from Fermat's principle

Goldberg, Kenneth A.

doi:10.1107/s1600577522005793

Cited by 4 publications

(3 citation statements)

References 14 publications

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“…This mirror-centered coordinate derivation is complementary to previously published works on ellipsoidal (Goldberg, 2022a) and paraboloidal (Goldberg, 2022b) mirrors.…”

Section: Introductionsupporting

confidence: 73%

Off-axis representation of hyperbolic mirror shapes for X-ray beamlines

Goldberg¹,

Río²

2023

J Synchrotron Rad

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Mirror-centered, closed-form expressions for hyperbolic surfaces used in X-ray beamlines have been derived. Hyperbolic mirrors create a virtual focus or source point and can be used to lengthen or shorten the effective focal distance of a compound optical system. The derivations here express off-axis segments of a hyperbolic surface in terms of the real and virtual focal distances and the incident glancing angle at the center of the mirror. Conventional mathematical expressions of hyperbolic shapes describe the surfaces in Cartesian or polar coordinates centered on an axis of symmetry, necessitating cumbersome rotation and translation to mirror-centered coordinates. The representation presented here, with zero slope and the origin at the central point, is most convenient for modeling, metrology, aberration correction, and general surface analysis of off-axis configurations. The direct derivation avoids the need for nested coordinate transforms. A series expansion provides a helpful approximation; the coefficients of the implicit equation are also provided.

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“…This mirror-centered coordinate derivation is complementary to previously published works on ellipsoidal (Goldberg, 2022a) and paraboloidal (Goldberg, 2022b) mirrors.…”

Section: Introductionsupporting

confidence: 73%

Off-axis representation of hyperbolic mirror shapes for X-ray beamlines

Goldberg¹,

Río²

2023

J Synchrotron Rad

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“…Our approach employs rotationally symmetric, hollow ellipsoidal substrates coated with multilayer graphite crystals on their inner concave surface. This geometry enables the capture of a larger X-ray beam cross-section (thereby, a larger solid angle) compared to, e.g., Kirkpatrick-Baez (KB) mirrors or side-by-side Montel mirrors [18,19,31]. Some of the well-known mosaic graphite optics (GrO) today include highly oriented pyrolytic graphite (HOPG) and highly annealed pyrolytic graphite (HAPG), which exhibit total integrated reflectivity an order of magnitude higher than perfect crystals due to their mosaic structure [32,33].…”

Section: Introductionmentioning

confidence: 99%

High-Spatial-Resolution Benchtop X-ray Fluorescence Imaging through Bragg-Diffraction-Based Focusing with Bent Mosaic Graphite Crystals: A Simulation Study

Kumar,

Fachet,

Hoeschen

2024

IJMS

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X-ray fluorescence imaging (XFI) can localize diagnostic or theranostic entities utilizing nanoparticle (NP)-based probes at high resolution in vivo, in vitro, and ex vivo. However, small-animal benchtop XFI systems demonstrating high spatial resolution (variable from sub-millimeter to millimeter range) in vivo are still limited to lighter elements (i.e., atomic number Z≤45). This study investigates the feasibility of focusing hard X-rays from solid-target tubes using ellipsoidal lens systems composed of mosaic graphite crystals with the aim of enabling high-resolution in vivo XFI applications with mid-Z (42≤Z≤64) elements. Monte Carlo simulations are performed to characterize the proposed focusing-optics concept and provide quantitative predictions of the XFI sensitivity, in silico tumor-bearing mice models loaded with palladium (Pd) and barium (Ba) NPs. Based on simulation results, the minimum detectable total mass of PdNPs per scan position is expected to be on the order of a few hundred nanograms under in vivo conform conditions. PdNP masses as low as 150 ng to 50 ng could be detectable with a resolution of 600 μm when imaging abdominal tumor lesions across a range of low-dose (0.8 μGy) to high-dose (8 μGy) exposure scenarios. The proposed focusing-optics concept presents a potential step toward realizing XFI with conventional X-ray tubes for high-resolution applications involving interesting NP formulations.

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“…(see, for example, Refs. [1][2][3][4][5] and references therein). Availability of such mirrors on the market will directly advance the fundamental and applied research performed at x-ray light facilities [6][7][8][9][10].…”

Section: Introductionmentioning

confidence: 99%

Binary pseudo-random array standards for calibration of 3D optical surface profilers used for metrology with aspheric x-ray optics

Munechika¹,

Chao

Dhuey³

et al. 2022

Interferometry XXI

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High-accuracy surface metrology is vitally important in manufacturing ultra-high-quality free-form mirrors designed to manipulate x-ray light with nanometer-scale wavelengths. The current and potential capabilities of x˗ray mirror manufacturing are limited by inherent imperfections of the integrated metrology tools. Metrology tools are currently calibrated with super-polished flat test-standard/reference mirrors. This is acceptable for fabrication of slightly curved xray optics. However, for even moderately curved aspherical x-ray mirrors the flat-reference calibration is not sufficiently accurate. For micro-stitching interferometry developed for surface measurements with curved x-ray mirrors, the tool aberration errors are known to be transferred into the optical surface topography of x-ray mirrors. Our approach to improving metrology is to thoroughly calibrate the measuring tool and apply the results of the calibration to deconvolution of the measured data. Here we explore the application of a recently developed technique for calibrating the instrument transfer function (ITF) of 3D optical surface profilers to metrology with significantly curved x-ray optics. The technique, based on test standards patterned with two-dimensional (2D) binary pseudo-random arrays (BPRAs), employs the unique properties of the BPRA patterns in the spatial frequency domain. The inherent 2D power spectral density of the pattern has a deterministic white-noise-like character that allows direct determination of the ITF with uniform sensitivity over the entire spatial frequency range and field of view of an instrument. The high efficacy of the technique has been previously demonstrated in application to metrology with flat and slightly curved optics. Here, we concentrate on development of an efficient fabrication process for production of highly randomized (HR) BPRA test standards on flat and 500-mm spherical optical substrates. We also compare and discuss the results of the ITF calibration of an interferometric microscope when using the HR BPRA standards on flat and curved substrates.

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Derivation of closed-form ellipsoidal X-ray mirror shapes from Fermat's principle

Cited by 4 publications

References 14 publications

Off-axis representation of hyperbolic mirror shapes for X-ray beamlines

Off-axis representation of hyperbolic mirror shapes for X-ray beamlines

High-Spatial-Resolution Benchtop X-ray Fluorescence Imaging through Bragg-Diffraction-Based Focusing with Bent Mosaic Graphite Crystals: A Simulation Study

Binary pseudo-random array standards for calibration of 3D optical surface profilers used for metrology with aspheric x-ray optics

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