[Paper Review] Understanding transverse coherence properties of X-ray beams in third generation Synchrotron Radiation sources

Gianluca Geloni, Evgeni Saldin+2 more|arXiv (Cornell University)|Jun 30, 2005

Advanced X-ray Imaging Techniques6 references3 citations

TL;DR

This paper develops a rigorous theoretical framework for calculating the transverse coherence properties of X-ray beams from third-generation synchrotron radiation sources, moving beyond the flawed quasi-homogeneous approximation of the van Cittert-Zernike theorem. It derives an analytical expression for the cross-spectral density valid up to the undulator exit and provides practical formulas for transverse coherence length applicable in beamline design.

ABSTRACT

This paper describes a theory of transverse coherence properties of Undulator Radiation. Our study is of very practical relevance, because it yields specific predictions of Undulator Radiation cross-spectral density in various parts of the beamline. On the contrary, usual estimations of coherence properties assume that the undulator source is quasi-homogeneous, like thermal sources, and rely on the application of van Cittert-Zernike (VCZ) theorem, in its original or generalized form, for calculating transverse coherence length in the far-field approximation. The VCZ theorem is derived in the frame of Statistical Optics using a number of restrictive assumptions: in particular, the quasi-homogeneous assumption is demonstrated to be inaccurate in many practical situations regarding undulator sources. We propose a technique to calculate the cross-spectral density from undulator sources in the most general case. Also, we find the region of applicability of the quasi-homogeneous model and we present an analytical expression for the cross-spectral density which is valid up to the exit of the undulator. For the case of more general undulator sources, simple formulas for the transverse coherence length, interpolated from numerical calculations and suitable for beamline design applications are found. Finally, using a simple vertical slit, we show how transverse coherence properties of an X-ray beam can be manipulated to obtain a larger coherent spot-size on a sample. This invention was devised almost entirely on the basis of theoretical ideas developed throughout this paper.

Motivation & Objective

• To address the inaccuracy of the quasi-homogeneous assumption used in standard coherence estimation for undulator radiation.
• To develop a general method for calculating the cross-spectral density of undulator radiation without restrictive approximations.
• To identify the domain of validity of the quasi-homogeneous model in practical undulator configurations.
• To derive analytical expressions for transverse coherence length that are suitable for beamline design and optimization.
• To demonstrate a method for manipulating transverse coherence using a simple vertical slit to increase coherent spot size on a sample.

Proposed method

• Formulates the cross-spectral density of undulator radiation in the most general case using statistical optics and Fourier analysis.
• Derives an analytical expression for the cross-spectral density valid up to the undulator exit, avoiding far-field and quasi-homogeneous approximations.
• Applies the van Cittert-Zernike theorem in its generalized form but identifies its limitations in undulator sources.
• Uses Fourier-Bessel transforms and the autocorrelation theorem to express the coherence function in terms of the sine integral function.
• Performs numerical interpolation to derive simplified, practical formulas for transverse coherence length across various undulator parameters.
• Proposes a method to engineer coherence by placing a vertical slit in the beamline, enabling control over the coherent spot size.

Experimental results

Research questions

• RQ1 How does the transverse coherence of X-ray beams from third-generation undulators evolve along the beamline beyond the far-field approximation?
• RQ2 In what conditions does the quasi-homogeneous assumption of the van Cittert-Zernike theorem break down for undulator radiation?
• RQ3 What is the exact analytical form of the cross-spectral density for undulator radiation up to the undulator exit?
• RQ4 Can simple, usable formulas for transverse coherence length be derived from numerical calculations for beamline design?
• RQ5 Is it possible to manipulate transverse coherence properties using a simple optical element like a vertical slit?

Key findings

• The cross-spectral density of undulator radiation is independent of the propagation distance in the far field when expressed in normalized angular variables, validating a key analytical result.
• The Fourier transform of the field correlation function is independent of the propagation distance $ ilde{z}_o$, indicating that the coherence function remains invariant under certain transformations.
• An analytical expression for the coherence function $eta( ilde{ heta})$ is derived using the Fourier-Bessel transform, involving the sine integral function: $eta( ilde{ heta}) = rac{1}{2 au} igint_{0}^{ au} au J_0( au ilde{ heta}) ar{eta}( au) d au$, with $ar{eta}( au) = 2[ au - 2 ext{Si}( au^2)]^2$.
• The paper identifies the region of applicability of the quasi-homogeneous model and shows it fails in many practical undulator scenarios.
• Simple, interpolated formulas for transverse coherence length are derived from numerical results and are suitable for beamline design applications.
• A vertical slit can be used to tailor the transverse coherence, enabling the creation of a larger coherent spot size on a sample, as demonstrated by theoretical modeling.