We propose an innovative method to extend the utilization of the phase space downstream of a synchrotron light source for X-ray transmission microscopy. Based on the dynamical theory of X-ray diffraction, asymmetrically cut perfect crystals are applied to reshape the position–angle–wavelength space of the light source, by which the usable phase space of the source can be magnified by over one hundred times, thereby “phase-space-matching” the source with the objective lens of the microscope. The method’s validity is confirmed using SHADOW code simulations, and aberration through an optical lens such as a Fresnel zone plate is examined via matrix optics for nano-resolution X-ray images.
© 2015 Optical Society of America
The performance of synchrotron-radiation-based X-ray microscopes is strongly dependent on the brilliance of the light source. It is well known that the use of increasingly brilliant light sources generally leads to better performance of such microscopes in terms of the resolution and image quality. However, because of the inequality of phase space utilization along the vertical and horizontal directions of the synchrotron light source, this aspect of the light-source brilliance has not been fully exploited in X-ray microscopy, and thus, it requires further analysis, particularly for those microscopes adopting concentric optical devices such as Fresnel zone plates and tapered capillaries.
The brilliance of a given synchrotron light source is by definition the photon flux density in the spatial and temporal phase spaces, and it is defined practically in units of photons/s/mm2/mrad2/0.1%bandwidth. This brilliance can be further understood by the definition of emittance. The emittance of a synchrotron light source is defined as the product of the source size and source divergence, which are conventionally defined along the vertical and horizontal directions, respectively; this emittance thus characterizes the spatial phase space. The coupling between the vertical and horizontal emittance (which is limited by the lattice of magnets) is less than 1% in a modern synchrotron source, thereby resulting in a very asymmetrical light source.
This “inequality” in the spatial phase space does not significantly affect the performance of a scanning-type microscope, since non-concentric optical devices such as Kirkpatrick–Baez mirrors can be utilized instead of concentric ones; the ultimate focal capability is constrained by the diffraction limit within the spot area, which limit is proportional to square of wavelength. The problem of phase-space inequality is more complex for full-field microscopes as a uniform illumination is required for acquiring undistorted images, particularly when acquiring high-quality images using a Fresnel zone plate (FZP) as the magnifying lens [1,2]. The efficiency of a zone plate (ZP)-based X-ray microscope is strongly dependent on the matching of the light-source phase space with the acceptance of the ZP (which is defined as the product of the field of view (FOV) and twice the numerical aperture (NA)). High-performance ZPs used in modern X-ray microscopes (for example, a ZP with FOV of 40 μm and NA of 2.5 mrad corresponds to an acceptance of 200 nanometer-rad along both the vertical and horizontal directions) can suitably match the phase space of the X-ray beam produced by the bending magnet of a third-generation synchrotron radiation light source along the horizontal direction, while this matching is less than 1% along the vertical direction. Consequently, the resulting oval light source leads to either a flux loss or poor uniformity in image resolution.
Figure 1(a) shows the schematic layout of a full-field transmission X-ray microscope. In order to match the vertical phase space with the ZP acceptance, a diverging-optics device is positioned downstream of the light source in order to expand the usable phase space (solid lines) “delivered” further downstream to the condenser and objective lens. In general, diverging optics such as wobbling or spirally moving condensers or Codling slits  are often employed. In this study, we focus on improving the low energy resolution of a high-spatial-resolution X-ray microscope; we propose an alternative approach that utilizes the dispersive characteristics of the dynamical diffraction of X-rays from asymmetrically cut perfect crystals to extend and “reshape” the usable phase space from the light source by effectively combining the spatial and spectral phase spaces with the phase space volume . Figure 1(b) illustrates the dispersive effects of an asymmetrically cut crystal. Perfect crystals are generally used as monochromators in synchrotron X-ray beamlines. The energy “selected” by the crystals can be graphically understood by means of the DuMond diagram . In the case of monochromatically symmetric diffraction, not only the area of the phase space but also the source size and divergence are preserved. In contrast, in the case of monochromatically asymmetric diffraction, only the area of the phase space is preserved. The source size and the source divergence can be reshaped according to Liouville’s theorem, which elucidates the conservation of phase space .
In the case that polychromatic X-rays are applied, the asymmetrical Bragg diffraction from a crystal further extends the region of divergence due to energy dispersion. Like an optical prism, the asymmetrical Bragg diffraction excites additional momentum transfer at the dispersion surface, and disperses X-rays with different energies . The resultant divergence eventually extends the total phase space area from the polychromatic source, as schematically shown in Fig. 1(c), which will be explained by DuMond diagram in the following section.
In the following section, we briefly describe our proposed method. We focus on the angular dispersion of the diffracted beam from asymmetric cut crystal, which causes the spatial phase space area to “expand” within the permitted spectral bandwidth (). For an FZP, the resolving power, , is dependent on the number of zones, whose typical value is under one thousand. In other words, the spatial phase space reshaping within a bandwidth of under 0.1% will yield no detectable chromatic aberration, and this can be used to obtain the equivalent “expanded” emittance [Fig. 1(c)]. This paper is organized as follows. In section 2, we demonstrate the abovementioned effect by examining the vertical beam properties and describing the theoretical principles of dynamical diffraction with regard to phase space. Section 3 presents the simulation data. Section 4 focuses on the results and discussion, while section 5 concludes the paper.
A synchrotron radiation source in phase space can be described by using the three Cartesian axes, wherein the -axis is normal to the electron orbit plane and the -axis is tangential to the orbit. The angles between the X-rays and the -axis along the horizontal () and vertical () planes are denoted as and , respectively. We employ the space as the vertical phase space to study the effects of various optical elements in the vertical plane. For a Gaussian beam, the phase space is distributed as an ellipse with an equal-intensity X-ray contour. The area of the ellipse is proportional to the product of the beam waist and the beam divergence.
The dynamical effect of crystal diffraction can be explained by examining the dispersion surface in reciprocal space, which is excited due to the difference in the refractive index within and outside the crystal . For the case of symmetric Bragg reflections, the angular acceptanceof the incident beam corresponding to the region of total reflectivity, which is referred to the Darwin width (), equals the angular spread of the exit beam, i.e., . The relationship between the angular acceptance and angular spread is different for the asymmetric reflection case due to the dynamical effect of diffraction. For the case of asymmetric reflection, the asymmetry factor b is expressed using the miscut angle () between the atomic plane and the surface of the crystals and the Bragg angle ()  as below:Eq. (1), we observe that the two asymmetric cases are described according to the sign of the miscut angle: for and for . The angular acceptance and angular spread are linked by the asymmetry factor as
This relationship can be demonstrated by means of the DuMond diagram, which graphically represents Bragg’s law that is expressed as , with the breadth of the curve characterizing the Darwin width deduced by means of the dynamical diffraction theory.
Figure 2 illustrates the principle of phase space volume conservation. For a monochromatic incident beam with waist and divergence , the exit beam divergence is given by the asymmetry factor b as
For a beam with a finite bandwidth, we introduce the angle–wavelength correlation [10–13]. The correlation between the energy bandwidth and the incident angle can be derived by means of Bragg’s law and its derivative, i.e.,. From Figs. 2(b) and 2(e), we directly obtain the following expression:
Using a similar procedure for the monochromatic cases, we can directly obtain the projection area in the () space as in Fig. 2(f), wherein the orange area represents the phase space for the monochromatic case. The additional divergence is given by the difference between the angular acceptance and spread, i.e.,. Using Eq. (2), we have
Depending on the miscut, the above difference can have a negative sign, and the sign indicates the direction of dispersion. While the additional divergence has been added to the resultant exit divergence , is same as in the monochromatic cases. The calculation of is still not the direct additive sum of and . Instead, when the diffraction intensity is considered (as will be clarified later) by ray-tracing simulation, is calculated as the convolution of and . Under the assumption of Gaussian-distributed beam, the resultant divergence of the exit beam can be formulated as below:Figs. 2(d) and 2(f). We define the transfer ratio TR of the incident and exit spatial phase spaces asEq. (8), TR is a quantity no less than unity. We next plot the angular dispersive contour of the TR as a function of the asymmetry factor and the incident beam divergence (Fig. 3). It is to be noted that the dashed line corresponding to the non-dispersive case of equals the identity as expected. The contour disperses away from the identity as TR becomes larger.
We simulated several cases to examine the feasibility of our concept by adopting asymmetric diffractions of silicon crystals using the SHADOW software package. This package forms an extension code of the popularly used XOP (X-ray Oriented Programs, developed by the European Synchrotron Radiation Facility, ESRF, and the Advanced Photon Source, APS [14,15]). The SHADOW code is often used for performing simulations and calculations for beamline optics design and ray tracing. The change in the transverse phase space by asymmetric Bragg diffraction has also been examined using the SHADOW code . The following section describes our simulation conditions: the X-ray photon source utilizes a bending magnet source with parameters identical to those of the Taiwan photon source (TPS, Table 1). The source size and divergence are 96 μm × 38 μm and 1300 μrad × 200 μrad (H × V, FWHM), respectively. The bandwidth of the source is set to 0.1% around 10 keV, and thus, the beam’s vertical emittance at 10 keV is 7.6 × 10−9 mrad. Ten thousand sigma-polarized rays are generated and distributed in the phase space, as shown in Fig. 4.
Our setup consists of a source, collimating mirror, and Si(111) crystal. The collimating mirror and crystal are positioned 30 m downstream of the source. For a finite source size, an elliptical mirror shape is more suitable than a parabolic one. The divergence of the beam after passing through the collimating mirror is reduced to 1.2 μrad. The phase space at this point is shown in Fig. 4(d). A screen [located at position (c) in Fig. 4(a)] is set between the source and the mirror to monitor the phase space ellipse tilting while the beam propagates. Figure 4(e) shows the phase space of the X-ray beam after diffracting from the asymmetrically cut crystal. For a symmetric crystal, the beam is subject to an optical reflection, and the phase space does not change; only the bandwidth and direction of propagation change. For the asymmetric case, the result of ray tracing can be clearly observed. The beam area is expanded due to dispersion, and both the beam size and divergence increase. It is to be noted that the 0.1% bandwidth () exceeds the Darwin width of the crystal [Eq. (6)], which is typically 0.014% () for Si(111). Consequently, the reflectivity of the diffracting atomic plane as calculated by dynamical theory has to be considered with regards to the X-ray intensity [see Fig. 5(d)]. The angular width of the reflectivity is actually the convolution of the exit beam divergence with the additional divergence, that is, Dt in Eq. (8), as mentioned above. Figures 5(a)-5(c) show the phase space areas “delivered” by asymmetric cut crystals with α = −4.8°, −7.6°, and −9.5°, respectively. With reference to Fig. 3, we observe that the smaller is the b factor, the broader is the bandwidth for a given incident beam divergence. The resultant beam divergence Dt and resolving power are subsequently estimated, and we note that these values agree well with our calculations (Table 2). The observed difference between the calculation and simulation results is due to statistical errors.
DuMond analysis is a simple approach to estimate the width of the Bragg diffraction, which is expected to be broader than the calculated width obtained using dynamical theory by a factor of 1.5 to 2 ; however, the convolution of the exit beam divergence with dispersive divergence diminishes this broadening effect. In principle, the amplification of the emittance can be as large as required for higher asymmetric geometry within the limitation imposed by the ZP in the form of its resolving power. Figure 6 displays the “reshaping” of the phase space volume, which is consistent with our prediction (Fig. 2). The reshaping can be explained as due to the following two-step processes: 1) the “rod-shaped” phase space volume gradually “flattens” as the beam propagates, and 2) the asymmetric Bragg diffraction converts this flattened profile into a sheet that stretches out in the angle domain, and consequently, the beam’s projection onto spatial phase space is expanded.
Herein, we discuss the salient aspects of the results presented in the previous section. Firstly, we note that the case with leads to reduction in the beam waist [Eq. (5)]; consequently, the transfer effect from Dt is negligible since TR ~1. In addition, a narrower bandwidth is unsuitable for the requirement of high flux. Next, we remark that the ZP commonly used for the hard X-ray region has 425 zones, which is considerably smaller than that considered in our study; thus, the output beam will not suffer chromatic aberration as it passes through the ZP. Next, we consider the blurred spot that is formed due to wavelength-dependent divergence for a small defocus. As per the matrix method of geometric optics, imaging by a single lens can be described as follows:Eq. (12) can be rewritten asEq. (12) after substitution of Eqs. (13) and (14) is expressed asEq. (15)]; for a non-wavelength-selective condenser such as a capillary condenser, the illumination is unchanged.
While the large brilliance and coherence of an undulator source can be utilized to obtain full illumination [17,18], our method that reduces the emittance difference by two orders while providing a “balance” of the vertical and horizontal phase spaces may be exploited to more effectively design microscopes that use undulator sources.
The double crystal monochromator (DCM) is commonly employed in synchrotron X-ray beamlines to select the photon energy and fix the beam path. For a two-crystal system, the phase space transfer effect is recalculated by using DuMond's analysis and summarized as below. Here, we assume that the diffractions of both crystals are identical and that the acceptance of the second crystal is larger than the angular spread of the first crystal, i.e., in order to avoid photon loss. Consequently, we haveEqs. (4), (5), (9), and (10). From Eq. (13), we note the small improvement in output beam divergence D due to magnification by the second crystal; however, this improvement will be saturated to the limit of if an n-crystal arrangement is used since as per Bragg’s law or Eq. (6), the output will not exceed the angular width corresponding to the input bandwidth.
The Codling slit design [2,19] has also been proposed to solve this mismatching of the vertical phase space. The solution involves repeated scanning of the beam within the acceptance angle to maximize the illumination on the ZP. The scan frequency of the slit has to be large to average the illumination. In comparison with the slit design approach, our approach using asymmetric crystal diffraction with a single crystal can be flexibly adopted for various experimental setups. Further, our approach is suitable for pulsed sources, which implies that it can be exploited for time-resolved experiments [20,21].
Asymmetrically cut crystals have been proposed as optical elements for coherent X-ray beams . Similar conclusion has been pointed out in the paper that the exit beam divergence (Dt) greatly degrades the transverse coherence of the beam, implying that the asymmetrically cut crystal is of limited efficiency as an optical device for coherent x-ray scattering. A highly coherent illumination produces undesired speckles image, subjective to be eliminated for a full-filed microscope . In our study, the coherence is not considered due to the phase space accepted by a full-field transmission X-ray microscope is several orders of magnitude larger than that of the transverse coherence of the source.
We proposed a new approach to match the vertical and horizontal emittances in a transmission X-ray microscope. The expansion of the vertical emittance under a specific condition, i.e., low energy resolution, of the system is essential; the matching issue forms our point of departure, and the mechanics of phase space reshaping by asymmetric Bragg diffraction is interpreted in terms of the position–angle–wavelength space using the dynamical theory of X-ray diffraction. Our simulation using the SHADOW code matches the results obtained with ray tracing and reflective intensity, thereby indicating the validity of our approach. Our method can be applied in synchrotron facilities, and we believe that it is also applicable to imaging systems operating in other wavelength regions when the core optics of such systems have a relatively low resolving power.
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