Origins of decoherence in coherent X-ray diffraction experiments

doi:10.1016/S0030-4018(03)01558-X

Optics Communications

Volume 222, Issues 1–6, 1 July 2003, Pages 29-50

https://doi.org/10.1016/S0030-4018(03)01558-X Get rights and content

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.

Introduction

Current advances in experimental facilities (ESRF, APS, and SPRING-8) provide high-energy, high-brightness hard X-ray beams with relatively high degrees of coherence. The X-ray coherence lengths achievable with these latest synchrotron radiation sources are in the range of a few microns. The unique properties of these modern synchrotron sources have the potential to open new fields in X-ray physics such as fluctuation correlation dynamics [1], [2], [3], [4], [5], [6], phase imaging [7], [8], [9], [10], and coherent X-ray diffraction (CXD) [11], [12], [13], [14]. All these techniques utilize the coherency properties of the synchrotron radiation [15].

As was shown in recent CXD experiments [17], it is possible to image crystals of nanometer size. Illuminated by coherent beam with transverse and longitudinal coherence lengths bigger than the size of the particle they produce a continuous interference diffraction pattern. The diffraction from such nanocrystals is no longer comprised of sharp Bragg peaks and broad diffuse background as in conventional incoherent scattering but have a complicated intensity distribution centered at each reciprocal-lattice point.

It was shown in the same papers the possibility to invert this continuous diffraction pattern into a real space image. This new type of X-ray microscope may have certain advantages compared with commonly used techniques. In principle, it does not need lenses; its resolution depends on the available coherent flux from the synchrotron source (at the moment a resolution better than 0.1 μm can be achieved) and 3D image of the sample can be obtained conveniently from several adjacent scans of reciprocal space, without need of the 180° rotation of the sample as in tomography [16]. What is most important, due to the high penetration of X-rays, this new X-ray microscope can image the inner parts of the crystal (with all possible inhomogeneities and holes) and additionally with the potential of imaging the strain field inside the crystal [18].

In the first reconstructed images we have seen some additional regions of high intensity that cannot be associated with the nanocrystal structure. Computer simulations assuming partial coherence rather than pure coherence of the incoming beam have shown [19] that additional features observed on the images can be attributed to this partial coherence of the incoming beam. In most previous studies it was assumed that the partial coherence of the incoming beam is associated mainly with a finite size of the source. The same approach was used in our model calculations. However, simple estimations show that for the parameters of the hard X-ray beamline where CXD experiments were performed (APS storage ring) the finite size of the source is not the only origin of the coherence loss. Optical elements present in the beamline can contribute significantly to the possible degradation of coherence of the beam. In addition, the transverse coherence of the beam can change non-uniformly and can contain sharp features on the more uniform background. From this preliminary analysis it has become clear that more detailed analysis of the coherency properties of the beam passing different optical elements is necessary.

It was appreciated from the beginning that partial coherence of the incoming beam can change the apparent scattered intensity distribution from a sample. A comprehensive theoretical study of partial coherence effects on the observed intensity distribution in the far-field as well as in the near-field X-ray scattering was made by Sinha et al. [20]. In a recent paper [21] partial coherence effects on the topography measurements were analyzed. There were even proposals [22] to manipulate with coherency properties of the beam in order to obtain a diffraction pattern from just one protein molecule in a crystal. These different applications of X-ray coherent scattering motivate further theoretical and experimental attempts to understand the coherency properties of the beam on third generation storage rings.

During last decade several attempts were made to measure the coherency properties of the X-ray beams. It is not a trivial problem for X-ray wavelengths of the order of angstrom to measure coherence with a two-slit Young type experiment that is routinely used for the visible light [23]. However, in the soft X-ray region, an interferometric Young type measurement of the spatial coherency properties of the beam was performed [24], [25], [26]. To measure spatial coherence in the hard X-ray region different approaches were proposed: to use dynamical diffraction effects to measure the visibility of Pendellösung fringes [27], to measure interference pattern from double reflecting mirrors in a kind of two-slit Young experiment [28], to use gratings as a phase object to measure coherence utilizing Talbot effect [29], to make high precision measurements of intensity distribution on well-defined objects as slits in the far-field [30] and more recently slits and fibers in the near-field [31]. Quite different approach of the characterization of transverse coherence by intensity interferometry technique was proposed in [32]. First two-slit Young experiment to measure spatial coherence of the synchrotron beam in hard X-ray energy range was reported only recently [33] (a specially prepared phase mask was used to measure the coherence at APS storage ring [34]). It is important to note here that coherency properties can change essentially from one beamline to another and even on the same beamline they depend on the optics present in experiment.

When the first phase contrast images with coherent X-rays were obtained, it became clear that the quality of the optic elements present in the beamline can be important for this imaging technique. The height profile of the mirror [35] and the inhomogeneities of the Be window [36] illuminated by coherent beam introduce additional distortion of the wavefront and can produce unavoidable artifact images in the form of speckle pattern and as a result can distort the image of the object. What is more important and was not investigated in detail up to now is that all the optics in the beamline can change the coherency properties of the beam. So it will be desirable to have some method to calculate or at least to estimate effects introduced by an optical element on the coherency properties of the X-ray beam. This is the main goal of this paper.

In visible light optics it is well established that coherency properties of the beam can be described by the so-called mutual coherence function (MCF) that measure correlation between two beams separated in space and time [23], [37], [38]. The propagation of this MCF in the free space is governed by two wave equations similar to the wave propagation equation obtained directly from Maxwell equations. This general approach can be in principle applied to propagation of any electromagnetic radiation including hard X-rays. The theoretical description is simplified if “quasi-monochromatic” conditions are fulfilled which means that the path length difference between the two beams is less than the longitudinal coherence length. The statistical properties of the beam in the plane across its propagation direction can then be well described by the mutual intensity function (MIF) that gives correlations of two field amplitudes at different points in this plane and the same time. The general propagation laws of this MIF in free space are obtained directly from the corresponding equations for MCF. These main definitions and equations are briefly summarized in Section 2.

In this paper we will be interested in applying this general theory of propagation of the MIF to the special case of propagation of the hard X-rays starting from a conventional “insertion device” (wiggler or undulator) on a third generation X-ray storage ring. It is well known that such source can be well described as an incoherent source of X-ray radiation because each electron is an independent radiator. Typical beamlines at high energy synchrotron radiation sources are built from components which contain optical elements. These are either intended to adapt the qualities of the beam to the needs of the experiment or can do so inadvertently. Important categories of components would be slits, mirrors, monochromators, lenses, and windows. Each of these elements can change in a different way the coherency properties of the beam. We will consider an idealized beamline configuration with only one optic element on the way from the source to the sample. The radiation incoming on this element can be of any state of coherence. Then by applying the general propagation law for the MIF, the coherency properties of the beam at the sample position will be calculated (Section 3). If the detailed structure (slit size, microstructure of the window or height function of the mirror) of the element is known this general approach gives, in principle, the possibility to calculate the MIF and consequently the coherency properties of the beam at the sample position. However, exact knowledge of the optical elements microstructure is often limited. This makes it difficult to predict the coherency properties of the beam at the sample position and different approaches have to be applied.

The following situation is often realized on the beamline: because of the divergence of the beam, the illuminated area on an optical element is much bigger than the coherence length of the incoming radiation. That is especially true for the last element that is usually an exit Be window. The incoming beam is rescattered and refracted due to all inhomogeneities present in the window that cause an unpredictable change of an optical pathlength of X-rays transmitted through that window. In this situation an optical element can be modeled as a random object and its structure can be characterized by statistical parameters such as roughness of the surface and correlation length of the height–height fluctuations. It will be shown (Section 4) that in this case MIF will split into two parts. The first one will describe the propagation of radiation from the source to the sample without rescattering on the optics and will preserve high coherence. The second one takes into consideration the effects of rescattering on the optics and has reduced coherence lengths. The detailed calculation of this additional contribution is given in Section 5. The short coherence lengths can be explained to a first approximation in the far-field limit by a large effective source size on the window due to divergence of X-ray beam. However, as was discussed in detail in [20], the far-field conditions are easily violated in the case of hard X-rays. It will be shown in the same section, that for the typical parameters of the CXD beamline, the ‘sharp’ contribution to MIF has to be calculated in the near-field rather then far-field limit.

In the last section such a two-component MIF, calculated for elements with various statistical properties, will be applied to calculate the intensity distribution from a small crystalline particle in a CXD experiment. This intensity distribution is similar to one observed in experiment [17]. Then applying reconstruction procedure discussed in [19] this intensity distribution will be used to obtain the real image of the particle. Additional features that can appear in the image of a nanoparticle due to a reduced coherence of the incoming beam will be discussed.

Section snippets

Laws of propagation of the mutual coherence function. Basic equations

The central concept in the theory of partial coherency is the so-called mutual coherence function (MCF) Γ(P₁,P₂;τ) that defines the correlations between two complex scalar² values of the electric field at different points P₁ and P₂ and different times. It is defined as $Γ(P_{1},P_{2};τ)=〈E(P_{1},t+τ)E^{∗} (P_{2},t)〉_{T},$ where E(P₁,t+τ) and E(P₂,t) are the field values at the points P₁ and P₂ and brackets 〈⋯〉_T mean an

Propagation of the mutual intensity function through the optical element

We will consider now an idealized situation of the beamline when on the way from the synchrotron source to the sample the X-ray beam passes through one optical element (Fig. 2). It can be pair of slits, lens, Be window or a mirror. In the most general way these elements can be characterized by their complex valued amplitude transmittance function $T(u)$ . The actual form of the transmittance function may differ from one element to another and will be defined more explicitly below.

We will also

Propagation of the mutual intensity function through a random optical element

For the perfect Be window the partial coherence of the beam incoming on the sample can be attributed entirely to the final size of the incoherent source. The wavefront of the beam passing such ideal optical element will not be affected and will propagate further to the sample position. However, such ideal conditions can be rarely met in practice. It is well known from the previous experiments [36] that Be windows due to technological problems are highly nonuniform in its internal structure and

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

In this section we will discuss in detail the second term $J_{W} (r_{1}, r_{2})$ (33) in expression for the total MIF $J (r_{1}, r_{2})$ (31). According to definition of the CCF (6) the MIF in the incoming beam is $J_{in} (u_{1}, u_{2})= I_{in} (u_{1}) I_{in} (u_{2}) μ_{in} (u_{1}, u_{2})$ . In the often-used quasi-homogeneous approximation, the CCF of the incoming beam $μ_{in} (u_{1}, u_{2})$ depends only on the difference of the coordinates $Δ u = u_{2} − u_{1}$ and is characterized by its coherence length ξ_in. Further it is assumed that the width of the intensity distribution $I_{in} (u$

Effects of the optical element on the imaging of small crystals

In this part we will apply results obtained in the previous sections to a specific type of CXD experiment – coherent X-ray scattering from small crystalline particles [17]. Applying special iteration techniques originally proposed by Gerchberg and Saxton [47] and then further developed by Fienup [48] and Millane and Stroud [49], the intensity distribution obtained in such experiment can be inverted to give a real space image of the particle itself. It was demonstrated in our previous work [19]

Conclusions

In conclusion we have seen how coherency properties of the X-ray beam can change while propagation through optics present in a typical hard X-ray beamline. It was shown that it is convenient to describe coherency properties of the beam by the MIF which can be obtained through Eq. (11). Detailed knowledge of the structure of an optical element, given by its transmittance function $T(u)$ , is necessary to calculate the coherency properties of the beam. Limits of coherent and incoherent illumination

Acknowledgments

We acknowledge useful and stimulating discussions with K. Nugent, S. Sinha and M. Sutton. This work was supported by NSF DMR98-76610 and US Department of Energy under DEFG02-91ER45439.

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

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Origins of decoherence in coherent X-ray diffraction experiments

Abstract

Introduction

Section snippets

Laws of propagation of the mutual coherence function. Basic equations

Propagation of the mutual intensity function through the optical element

Propagation of the mutual intensity function through a random optical element

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

Effects of the optical element on the imaging of small crystals

Conclusions

Acknowledgments

J. Phys.

Opt. Lett.

J. Synchrotron. Rad.

Phys. Rev. E

Phys. Rev. Lett.

Phys. Rev. E

Phys. Rev. E

Phys. Rev. B

Phys. Rev. Lett.

Rev. Sci. Instrum.

Nature

J. Phys. D

Appl. Phys. Lett.

Nature (London)

Phys. Rev. B

Phys. Rev. B

Europhys. Lett.

Naturwissenschaften

Phys. Rev. Lett.

Phys. Rev. Lett.

Appl. Surf. Sci.

Phys. Rev. B

J. Phys. D

Acta Cryst. A