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Determination of Crystal Structures

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In the late 1800s, the revelation that light extended beyond visible wavelengths led to the discovery of X-rays by Wilhelm Roentgen. Recognized as high-energy electromagnetic radiation with short wavelengths, X-rays prompted exploration into their interaction with crystals. Max von Laue proposed in 1912 that the periodic arrangement of atoms, ions, or molecules in crystals would cause them to diffract X-rays, a hypothesis confirmed through experiments with copper sulfate and zinc sulfide...

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The size of the unit cell and the arrangement of atoms in a crystal may be determined from measurements of the diffraction of X-rays by the crystal, termed X-ray crystallography.
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Diffraction is the change in the direction of travel experienced by an electromagnetic wave when it encounters a physical barrier whose dimensions are comparable to those of the wavelength of the light. X-rays are electromagnetic radiation with wavelengths about as long as the distance between neighboring...

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Updated: May 13, 2026

Measurement of X-ray Beam Coherence along Multiple Directions Using 2-D Checkerboard Phase Grating

Measurement of X-ray Beam Coherence along Multiple Directions Using 2-D Checkerboard Phase Grating

Published on: October 11, 2016

Numerical solution of an inverse diffraction grating problem from phaseless data.

Gang Bao¹, Peijun Li, Junliang Lv

¹Department of Mathematics, Zhejiang University, Hangzhou, China.

Journal of the Optical Society of America. A, Optics, Image Science, and Vision

|March 5, 2013

Summary

This summary is machine-generated.

This study presents a new continuation method to reconstruct periodic grating profiles from phaseless diffraction data. The method efficiently recovers Fourier coefficients, enabling accurate grating shape determination.

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Area of Science:

Optics and Photonics
Computational Electromagnetics
Applied Mathematics

Background:

Diffraction gratings are crucial optical components.
Reconstructing grating profiles from scattered field data is a challenging inverse problem.
Phaseless measurement limitations complicate traditional inverse scattering methods.

Purpose of the Study:

To develop an efficient numerical method for solving the inverse diffraction grating problem.
To reconstruct a periodic grating profile using phaseless diffracted field measurements.
To recover the Fourier coefficients representing the grating profile.

Main Methods:

An efficient continuation method is developed.
The method proceeds by varying the wavenumber.
Updates are derived using the Landweber iteration at each step.

Main Results:

The proposed continuation method effectively recovers Fourier coefficients of the grating profile.
Numerical simulations demonstrate the method's capability.
Accurate reconstruction of the grating shape is achieved.

Conclusions:

The developed continuation method provides an effective solution for the inverse diffraction grating problem.
The technique is robust for reconstructing periodic grating profiles from phaseless data.
This work advances numerical techniques in computational electromagnetics and optics.