Abstract

The propagation of the mutual intensity function from an incoherent synchrotron source to the sample is discussed. It is shown how coherency properties of the beam are changed by propagation through random optical elements, such as Be windows and mirrors present in the beamline. The mutual intensity function in this case cannot be described by one coherence length but will rather have several components with different coherence lengths. With computer simulations it is shown how such multicomponent mutual intensity function can affect the reconstruction of nanoparticles in coherent X-ray diffraction experiments.

Author Keywords: Propagation of coherence; Coherent X-ray diffraction; Phase retrieval

PACS classification codes: 61.10.Dp; 41.50.+h; 42.30.Rx; 81.07.Bc

Article Outline

1. Introduction

2. Laws of propagation of the mutual coherence function. Basic equations

3. Propagation of the mutual intensity function through the optical element

4. Propagation of the mutual intensity function through a random optical element

5. Contribution to the coherency properties of the beam from rescattering from random optical element

6. Effects of the optical element on the imaging of small crystals

7. Conclusions

Acknowledgements

Appendix A. Calculation of undistorted part of MIF J_S(r₁,r₂)

References

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Fig. 1. Propagation of mutual coherence function from surface ₁ to surface ₂. Vectors n₁ and n₂ are the normal vectors to the surface ₁ at points P₁ and P₂.

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Fig. 2. Beamline with one optical element on the way of propagation of X-rays from the synchrotron source to the sample. The source, element, and the sample are described in their "local" 2D coordinate frames perpendicular to the direction of the beam propagation by their coordinates s, u, and r, respectively.

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Fig. 3. Schematic view of propagation of the MIF from the incoherent synchrotron source through a random optical element (here Be window). Upon passing the Be window, the MIF has two components: one broad component with high coherency properties propagating directly from the source and a second one originating from the Be window that gives reduced coherence lengths at the sample position.

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Fig. 4. Horizontal and vertical components of the complex coherence factor (Δx,Δy) calculated for the X-ray radiation with the wavelength Å propagating through a random phase optical element at a distance L₂=6 m in front of the sample. Parameters of the source and statistical properties of an element are given in the text and are summarized in Table 1. Each curve is shifted by one unit for clarity.

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Fig. 6. The MIF (a) used for calculation of diffraction intensity patterns (b) from the crystal shape shown in Fig. 5. Reconstructed real-space images for two different sets of starting random phases are shown in (c). Parameters used for calculations of the A–D MIF are listed in Table 2.

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Fig. 5. Initial crystal shape (left image) used for calculations, its diffraction pattern (central image) in a perfect coherent beam and reconstructed shape of the particle (right image). Another non-linear color gradient is used for reconstructed image to enhance the background contribution. Calculations used an array of 700 × 400 pixels. In the figure, real space images have been cut to a size of 160 × 160 pixels and reciprocal space image to 300 × 160 pixels. The support region in the calculations was a rectangular box with a lateral size of 150 × 150 pixels.

Table 1. Parameters used for the calculation of the CCF (Δx,Δy) at the sample position
Notations A–D are maintained throughout the paper, notably in Fig. 4 and Fig. 6. Here, _x,y are the horizontal and vertical source sizes, _(eff)x,y are an effective source size of the beam on the Be window, ² and are variance and longitudinal correlation length of this window.


Table 2. Transverse coherence lengths of the "broad" _S and "sharp" _S components of the MIF J_in(Δr) on the sample position obtained as a result of fitting with expression (63)
Letters (L) and (G) mean Lorentzian or Gaussian form of CCF _W(Δr). Parameter _Qx,y gives the size of distribution of incoherent intensity in the reciprocal space according to Eq. (73).

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Corresponding author. Tel.: +1-217-333-682; fax: +1-217-244-2278

¹ On leave from: Institute of Crystallography RAS, Leninsky Pr. 59, 117333 Moscow, Russia.