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Abstract
Fourth-generation synchrotron radiation delivers x-ray sources with unprecedented coherence and brilliance, which enables the development of many advanced coherent techniques taking advantage of the inherent high coherence of the x-ray beams. Simple and accurate measurement of two-dimensional (2D) coherence is of utmost importance for the applications of these coherent experimental techniques. Here, we propose a novel approach based on diffraction of aperture array mask (AAM) to obtain accurate 2D spatial coherence with a single-shot measurement. We utilize a coherent mode decomposition algorithm to simulate the diffraction of AAM illuminated by Gaussian-Schell model beam and demonstrate that spatial coherence function of the incident light beam can be accurately and robustly retrieved. We expect that this new approach will be applied into transverse coherence measurements for the new-generation synchrotron radiation source and relevant coherent experimental techniques.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
New-generation synchrotron radiation sources, fueled by advancements in accelerator technology, are capable of delivering x-ray beams with unprecedented brilliance and coherence [1,2]. The inherent high coherence of the x-ray beams from new generation synchrotron radiation sources significantly enhances the performance of many experimental techniques, such as x-ray interference lithography [3], x-ray coherent diffraction imaging [4,5,6] and x-ray photon correlation spectroscopy [7,8], among others. A comprehensive knowledge of the transverse coherence of x-rays is essential for understanding the source properties, and also a prerequisite for carrying out these advanced experimental techniques. Moreover, the coherence of x-ray is impacted by vibration and surface error [9,10] of beamline optics. Therefore, simple and accurate measurement of the complex coherence factor (CCF) is greatly demanded in not only the pursuit of diffraction-limited and coherence-preserving beams, but also development of coherence-based x-ray optics techniques.
Various approaches have been proposed to measure coherence properties of x-ray beams. One such method involves mapping the transverse coherence of the x-ray beam with a grating by measuring visibility of the interferogram at multiple distances behind the grating [11,12]. However, this interferogram-based technique is time-consuming and impractical because of the multiple measurement and long travel distance. More generalized approaches were proposed for two-dimensional measurement by exploiting the heterodyne near-field speckle, which is arisen from the interference between weak scattering field by colloidal suspension and the transmitted field of the illuminating partially coherent beam [13]. Two-dimensional CCF can be obtained from the 2D power spectrum of the heterodyne near-field speckle. The colloidal suspension consisted of silica spheres of diameter 450 nm suspended in water and these spheres must have a narrow size distribution (<5%). Otherwise, secondary scattering of spheres could increase errors. This heterodyne based method can be easier to implement using a cellulose acetate membrane filter [14] which has pores with average size of 0.8 µm. The heterodyne signal is significantly weaker than the background signal, resulting in a poor signal-to-noise ratio (SNR). Although only one signal is needed in principle, repeated measurements are necessary to improve SNR. Furthermore, the talbot oscillation of power spectrum results in zero-zone, causing the loss of information. Another limit is that near-field zone is too short to satisfy a soft x-ray source with short coherence length. For example, for photon energy of $E = 800\textrm{ }eV$ and coherence length of ${8 \mu m}$, the near-field condition is ${0}{.26 m}$.
Full 2D CCF can be also measured by analyzing the Fourier spectrum of magnetic speckle pattern [15]. This speckle pattern is obtained from ferromagnetic $C{o_{35}}P{d_{65}}$ alloy film using X-ray resonant magnetic scattering (XRMS). The magnetic domain pattern has a quasi-flat autocorrelation function. So, this autocorrelation in Fourier spectrum can be neglected and CCF can be obtained from Fourier spectrum. However, XRMS works at a specific wavelength, which limits its generality. To overcome this limitation, The XRMS is replace by small-angle X-ray scattering from a disordered array of nanodots which have diameter of 20 nm [16,17]. The nanodots array was fabricated out of a homogeneous metallic multilayer ${({C{o_{1.64\textrm{ }nm}}/P{t_{2\textrm{ }nm}}} )_2}$ deposited on a 200 nm thick $\textrm{S}{\textrm{i}_\textrm{3}}{\textrm{N}_\textrm{4}}$ membrane. Nevertheless, there is high approximation error for quasi-flat autocorrelation that leads to low accuracy, and nanodot array must satisfy a specific statistical distribution to separate the scattering signal from the direct beam, which is not completely controllable in fabrication.
In this study, we present a novel approach to measure two-dimensional degree of coherence of x-ray from a single diffraction pattern of a simple aperture array mask (AAM). Unlike existing techniques, the AAM structure is controllable in fabrication which enable CCF retrieval without any approximation. Furthermore, the CCF is completely coded by absence of zero-zone in autocorrelation of AAM, and the diffraction intensity does not contain background by using binary-amplitude mask, which has better SNR. Therefore, the real one-shot measurements can be performed. To validate the effectiveness of our proposed method, we conducted simulations by the coherent mode decomposition algorithm to retrieve CCF of Gaussian Schell Mode (GSM) [18] sources. In addition, the effects of diffraction distance and mask structure are analyzed in detail, which demonstrates the precision and robustness of our proposed method.
2. Theory
The partially coherent X-ray source is described by a GSM field [18], for which mutual intensity can be written as [19,20]
Equation (3) reveals that complete CCF is encoded into the power spectrum and can be retrieved by performing simple division operation if zero-value of $K(\Delta \vec{r})$ is excluded. Therefore, delicate design of screen function $T(\Delta \vec{r})$ to ensure a well-defined and nonzero autocorrelation function $K(\Delta \vec{r})$ is the prerequisite to obtain spatial coherence by single-shot measurement.
To fulfill the specific requirement discussed above, we proposed an AAM, which is consisted of identical hexagonal apertures arranged in a graphene lattice pattern [see Fig. 1(b)]. The marker d and D represent aperture side length and adjacent aperture’s distance. Similar to uniformly redundant array proposed in [20,21], the AAM cover all the spatial frequencies by containing various aperture separations. Because the aperture has a bigger size and hexagonal shape, the single frequency component is also a hexagonal spot, rather than a small dot in grating case [12]. Therefore, once the center distance of spot of neighboring frequency components is less than the spot size of single frequency components, the condition of no zero-zone can be satisfied and entire CCF can be retrieved in a single measurement. The physical condition of no zero-zone can be converted to geometrical condition:
The implication is that the zero-zone would emerge in autocorrelation if $D > {D_{crit}}$. Figure 2 shows how D impacts autocorrelation of the AAM with ${2\ \mathrm{\mu} \mathrm{m}}$ aperture side. The obvious zero-zone presents when $D > {D_{crit}}$ but disappear once $D < {D_{crit}}$. And as D decreases, the lower limit becomes greater, which can be seen from Fig. 2(c). Consequently, the CCF could be completely coded by using such an AAM.
Fig. 2. (a) AAM with different aperture distance (indicated in the top line); (b) autocorrelation of (a), using logarithm colormap; (c) profile of red line in (b), using linear scale.
According to equation (3), the retrieval of CCF can be written as:
Therefore, it is necessary to calculate $K(\Delta \vec{r})$, which requires the complex amplitude $E(\vec{r})$ of the incident light beam. The access to complex amplitude is usually a complicated task but can be easily obtained at the waist of the light beam, in which the complex amplitude can be expressed as $E(\vec{r}) = \sqrt {I(\vec{r})} $,where $I(\vec{r})$ is the illumination intensity. In this case $K(\Delta \vec{r})$ can be obtained with information of illumination intensity $I(\vec{r})$, thus the 2D CCF is able to be obtained by measuring the diffracted intensity and the illumination intensity.
3. Simulations
As a validation of AAM-based spatial coherence measurement, we simulate the diffraction of AAM illuminated by a partially coherent GSM source. The mutual intensity associated with GSM is given as
The photon energy used in the simulation is 1000 eV and intensity RMS is ${\sigma _x} = 1{2 \mu m}$ and ${\sigma _y} = 1{2 \mu m}$. The CCF RMS is ${\xi _x} = 5.{0 \mu m}$ and ${\xi _y} = 8.{0 \mu m}$. The parameters of AAM are chosen as $d = 2.{0 \mu m}$ and $D = 0.9\textrm{ }{D_{crit}}$. Subsequently, the diffracted light intensity is collected at distance of two meters away from the AAM screen. Figure 3(a) presents the optical intensity distribution of the illumination source and Fig. 3(b) gives the pattern of AAM used in the simulation. The pattern of diffracted light intensity in Fig. 3(c) shows a hexagonal diffraction stripe, which is determined by the AAM, and the stripe is more blurred in the horizontal direction due to the horizontal coherence lower than the vertical coherence. Figure 3(d) is the $K(\Delta \vec{r})$ and Fig. 3(e) is Fourier spectrum. According to equation (5), the CCF [Fig. 3(f)] can be retrieved using Fig. 3(d) and Fig. 3(e). Thanks to complete separation of $K(\Delta \vec{r})$ from Fourier spectrum, the retrieval CCF is in excellent agree with theoretical expectations [Fig. 3(g) and (h)]. In Fourier-analysis method [15,16], autocorrelation of sample is approximated to quasi-flat, so $K(\Delta \vec{r})$ is only autocorrelation of illumination intensity. But this approximation has big errors which lead to low accuracy.
Fig. 3. (a) Illumination light intensity. (b) Aperture array mask. (c) Diffracted light intensity. (d) Full coherent Fourier spectrum. (e) Partially coherent Fourier spectrum (FT of intensity). (f) The recovered CCF. Comparison of recovered and theoretical CCF profiles across (g) in horizontal and (h) in vertical directions.
To further verify the validity of this method, different sources were applied for simulation. Figure 4 shows simulation results of different CCF. It is observed that the intensity pattern is elongated in horizontal direction of Fig. 4(a) because of lower coherence. Correspondingly, more high frequency components in the Fourier spectrum are observed along high-coherence direct (Fig. 4(b) and 4(f)). The simulations exhibit a good agreement with the theoretical predictions, regardless of whether the coherence is high or low. Not like the near-field heterodyne method [14,13], propagation distance must be shorter than 18.6 mm at low coherence to meet near-field condition, our approach can be performed at the same propagation distance for sources with different CCF.
Fig. 4. The simulation with different illumination sources. The upper row is simulated results for source with ${\xi _x} = 1.2\mathrm{\ \mu m}$(10% of ${\sigma _x}$) and ${\xi _y} = 8.4\mathrm{\ \mu m}$(70% of ${\sigma _y}$). The lower row is for source with ${\xi _x} = 3.6{ \mu m}$(30% of ${\sigma _x}$) and ${\xi _y} = 6.0{ \mu m}$(50% of ${\sigma _y}$). (a) (e)The diffraction intensity; (b) (f) Fourier spectrum; (c) (g) Recovered CCF; (d) (h) Comparisons of theoretical and recovered CCF.
To validate above theory analysis about adjacent aperture’s distance D. The simulation with different D is performed and corresponding diffracted light intensity and retrieval CCF is shown in Fig. 5. The CCF exhibits an absent zone when $D > {D_{crit}}$ due to presence of the zero-zone in Fourier spectrum. This phenomenon even persists for $D = {D_{crit}}$ but disappears when $D < {D_{crit}}$ which is consistent with above analysis. But the side lobes of intensity also become darker as D decreases, necessitating the selection of D to match detector dynamical range of detector.
Fig. 5. (a) Diffracted beam intensities. (b) The recovered CCFs from Corresponding column in (a). The aperture spacing is marked in top left of each panel.
In actual experiment, the diffraction intensities suffer from noise and limit dynamic range of detector. The Gaussian noise is added to diffraction intensity, and the SNR is set as 8 dB. The simulated CCF is distorted. Fig. 6(a) and (b) show the noisy intensities by AAM with different adjacent aperture separation ($0.9{D_{crit}}$ and $0.6{D_{crit}}$), and Fig. 6(c) and 6(d) are corresponding CCF. As can be seen from Fig. 6(h), the AAM with aperture separation of $0.6{D_{crit}}$ is more robust to noise. As shown in Fig. 6(e) and (f), the CCF is obviously distorted when dynamic range of detector is set as 8 bits. And the AAM with $0.6{D_{crit}}$ aperture separation [Fig. 6(e)] is less influenced by the detector dynamic range than $0.9{D_{crit}}$ AAM, which can be clearly observed in vertical profile [Fig. 6(g)].
Fig. 6. Noisy diffraction intensity by AAM with adjacent aperture separation of (a) $0.9{D_{crit}}$ or (b)$0.6{D_{crit}}$, and the SNR of both is 8 dB. (c) Retrieval CCF from (a). (d) Retrieval CCF from (b). (e) Retrieval CCF by AAM with $0.6{D_{crit}}$ from intensity with 8-bit dynamic range. (f) Retrieval CCF by AAM with $0.9{D_{crit}}$ from intensity with 8-bit dynamic range. (g) Vertical profile of (e) and (f). (h) Vertical profile of (c) and (d).
In actual experiment, the available diffraction distance may be limited, necessitating an analysis of its impact. The simulation results, depicted in Fig. 7, illustrates how the propagation distance affects the Fourier spectrums and intensities. Unfortunately, the sidelobes of Fourier spectrum become lower at short distance [Fig. 7(a) and Fig. 7(c)], which is unfavorable for CCF measurement. As the distance increases, this issue gradually diminishes, however, the intensities also become broader. But there is no tradeoff between distance and field of view. As shown in Fig. 8, the diffraction intensity size can be modulated by aperture size, meanwhile the CCF retrieval is still well. So, once diffraction distance is determined, the aperture size can be selected to make diffraction intensity size match field of view.
Fig. 7. Comparison of different diffraction distances. (a) Horizonal profiles of Fourier spectrum with different distances. (b) Diffraction beam intensities. (c) Full coherence Fourier spectrums. (d) Recovered CCFs. Diffraction distance marked in bottom line.
Fig. 8. Comparison of different aperture size of AAM. Each column represents a specific size: ${0}{.5 \mu m}$, ${1}{.0 \mu m}$, ${2}{.0 \mu m}$, ${3}{.5 \mu m}$, from left to right. (a) Diffraction light intensity. (b) Full coherence Fourier spectrum. (c) Recovered CCF.
To investigate the physical mechanism underlying the modulation of Fourier spectrum by propagation distance, the expression for full coherent Fourier spectrum is written as
Since only the magnitude is of concern here, so the only term involving z is $\exp ({ - ik\Delta \vec{r}\cdot \vec{r}/z} )$, which appends one oscillation to integrand. As the value of z decreases, the oscillation frequency increases, leading to a reduction in integral value. Furthermore, this phenomenon emerges from Fresnel propagation phase, indicating that other methods based on Fourier spectrum would also encounter similar issues.
For hard X-rays, the bigger wave number leads to higher frequency Fresnel propagation phase. As an example, wave number of 15 keV hard X-ray is 15 times than that of 1 keV soft X-ray, and the shortest distance for 15 keV is estimated to be 7.5 m, which is 15 times longer as that of 1 keV (0.5 m). Furthermore, it is challenging to fabricate binary mask for hard X-rays, but a phase AAM can also perform well. Fig. 9 shows the retrieval of CCF for 15 keV X-ray using phase AAM at 7.5 m, which proves that the above analysis is effective.
Fig. 9. CCF retrieval for 15 keV using phase AAM. (a) Diffraction light intensity after AAM 7.5 m. (b) Phase distribution of phase AAM, phase shift in the apertures is zero while in the other zone, it is a negative $\pi $. (c) Retrieved CCF.
In real synchrotron beamline, there will be apertures and reflective elements with figure error, and these elements would distort shift-invariance of CCF [23,24]. Fortunately, the distortion can typically be regarded as a perturbation in most experimental situations, and the retrieval CCF by AAM represents the average of various zone CCF. Consequently, in this scenario, the retrieval CCF can be seen as a measure of the overall coherence level of sources. Strict derivation from propagation equation of mutual intensity is needed to develop the CCF measurement when the distortion arisen from the beamline optics cannot be regarded as a perturbation.
4. Conclusions
In conclusion, we proposed a novel method for accurate measurement of two-dimensional CCF using the aperture array mask. With well-designed AAM, the CCF of the incident light beam can be accurately retrieved from the power Fourier spectrum of the diffraction pattern of the AAM. The constraint condition about the geometric parameters of the AAM are derived and demonstrated by numerical simulations. Furthermore, we observed that the size of the Fourier spectrum is modulated by the Fresnel propagation phase, necessitating the selection of an appropriate diffraction distance to match the desired measurement range. By incorporating the measured coherence information, it becomes feasible to comprehensively investigate the characteristics of fourth-generation synchrotron sources and beamlines, thereby enhancing related applications.
Funding
National Key Research and Development Program of China (2020YFB2007501); National Natural Science Foundation of China (U20A20216); joint funding from National Synchrotron Radiation Laboratory.
Acknowledgments
We also thank the supporting of joint funding from National Synchrotron Radiation Laboratory.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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