research papers
Exploring beam size measurement based on the Talbot effect at BEPCII
^{a}Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, People's Republic of China, and ^{b}University of Chinese Academy of Sciences, Beijing 100049, People's Republic of China
^{*}Correspondence email: zhangwan@ihep.ac.cn, syf@ihep.ac.cn
Vertical beam size measurements were carried out at Beijing Electron–Positron Collider II (BEPCII) using a phase grating and an absorption grating based on the Talbot effect. The transverse coherence of synchrotron radiation is closely related to beam size. Due to the partial coherence of the synchrotron radiation source, the coherence length can be calculated by measuring the visibility decay of interferograms recorded at different distances behind the gratings. A vertical beam size of 68.19 ± 2 µm was obtained based on the relationship between the coherence length and beam size at the 3W1 beamline of BEPCII. A comparison of the vertical emittance derived from the grating Talbot method and the synchrotron radiation visible light interferometer method was presented. The vertical emittances from the two methods are 1.41 nm rad and 1.40 nm rad, respectively. The 0.1% difference indicates that the grating Talbot method for beam size measurement is reliable. This technique has great potential for small beam size measurement of fourthgeneration synchrotron radiation light sources, considering its small diffraction limitation and simple experimental setups.
Keywords: Talbot effect; interferograms visibility; coherence length; beam size measurement; synchrotron radiation light source.
1. Introduction
The requirements of synchrotron radiation facilities for high coherence, high et al., 2009; Yang et al., 2013; Sakdinawat & Attwood, 2010). Beam size measurement plays a critical role in beam diagnostics and beam dynamics studies (Samadi et al., 2020; Koopmans et al., 2019). Accurate measurements of small beam size are becoming increasingly necessary.
and small beam sizes have been proposed in some Xray beamlines for applications such as Xray phase contrast imaging, coherent Xray diffraction imaging and Xray holography (PogoreliyRadiationbased techniques for measuring electron beam sizes are widely used as beam diagnostics at existing synchrotron sources (Garg et al., 2018; Mitsuhashi, 1998). Thanks to their short wavelength, Xrays can greatly improve the imaging resolution to meet small beam size measurements. Xray pinhole imaging, as a common method, has been applied at Diamond Light Source, European Synchrotron Radiation Facility and so on (Thomas et al., 2010; Garg et al., 2014; Leitenberger et al., 2003). It is characterized by realtime and high measurement accuracy. However, it does not work for extremely small beam sizes due to the diffraction limitation. Xray Fresnel zone plate imaging and Kirkpatrick–Baez mirror focusing imaging show advantages in micrometrescale beam size measurement. However, they suffer from special beamline design and high machining accuracy (Suzuki et al., 2001; Alatas et al., 2011). In recent years, some new measurement systems have been applied to beam size measurement. A method for simultaneously measuring the vertical position and vertical divergence angle was proposed by means of Xray diffraction and materials absorption edges at Canadian Light Source (Samadi et al., 2019). However, a smaller beam size is difficult to measure due to diffraction effects. The following year, an Xray tandem doubleslit optical system was proposed to measure beam size and divergence angle at SPring8 (Kagoshima et al., 2020). Although this method can measure beam size in micrometres, the tandem doubleslit needs to be aligned, which is difficult.
For the case of synchrotron radiation sources with high coherence, it is feasible to derive the beam size by spatial coherence. One of the Talbot effect applications is focused on spatial coherence measurements of Xrays in synchrotron radiation sources. The spatial coherence of Xrays emitted from a bending magnet has been measured using the Talbot effect of a π/2 phase checkerboard grating and a π/2 phase circular grating at Advanced Photon Source (Marathe et al., 2014; Shi et al., 2014). In these papers, the Talbot effect is mainly used to study the coherence of Xrays. Given the relationship between spatial coherence and source size, the grating Talbot method can also be used to measure beam size owing to its small diffraction limitation. Most importantly, the experimental setup is simple, needing only gratings, a displacement platform and an Xray camera without special design in the beamline frontend.
At BEPCII, we measured the vertical beam size at 3W1 beamline by employing the grating Talbot effect. Partially coherent quasimonochromatic Xrays have been employed in this experiment. The vertical beam size is calculated successfully from the selfimage interference fringes of a grating interferometer. Then the vertical emittance of the storage ring is calculated by the vertical beam size and β function. As a contrast, the vertical emittance from a bending magnet is derived using the visiblelight interference method. An extremely small difference between the two vertical emittances is presented, which illustrates that the grating Talbot effect method has great potential for measuring beam size.
2. Measurement system
2.1. Theory background
In 1836, Talbot found that a monochromatic parallel beam transmitting through a grating vertically will generate a series of grating images at certain distances behind the grating, which is called the Talbot effect (Talbot, 1836; Born & Wolf, 1999). There is a specific relationship between the visibility of the Talbot image and the complex coherence function of the light source (Cloetens et al., 1997; Guigay et al., 2004).
The interference intensity of two beams from an extended source at any point can be written as
where E(p_{1}) and E(p_{2}) are electric fields formed by two points on an extended source, and J_{12} is the mutual intensity function of the two light beams from the two points (Pfeiffer et al., 2005).
The normalized mutual intensity function j_{12}, also called the complex coherence function, is expressed as
Combining equations (1) and (2),
where = Arg(j_{12}). The third term of equation (3) expresses the interference effect. j_{12} takes the value 1 corresponding to the complete coherence of the two beams and takes the value 0 corresponding to the complete incoherence. In the case of partial coherence, it takes a value between 0 and 1.
The complex coherence function can be calculated from the visibility measurement of interference fringes formed by two light beams. The visibility can be expressed as
where I_{max} and I_{min} are the maxima and minima intensity of interference fringes, respectively. They can be found from equation (3),
Inserting equation (5) into equation (4),
For a phase grating, I_{1} = I_{2}, and equation (6) can be rewritten as
That is to say, the visibility of fringes is equal to the coherence degree of the beam.
The intensity of synchrotron radiation shows a good approximation to a Gaussian distribution, and can be written as
where s_{x}, s_{y} are the coordinates in the source plane, and σ_{x} and σ_{y} are the source size along the horizontal and vertical directions, respectively.
The complex coherence function is also a Gaussian distribution according to the propagation theory of the mutual intensity function, which is expressed as
where x and y are the horizontal and vertical axes, respectively, which are perpendicular to the direction of beam propagation, ξ_{x} and ξ_{y} are the coherence length in the x and y directions, respectively, D is the beam propagation distance from the source, and λ is the transmission wavelength.
In this experiment, the measured visibilities are from a series of grating selfimaging interference fringes at different d, the distance between grating and camera. For a π/2 phaseshift grating, the interference fringes are formed by the neighboring diffraction orders, thus the coordinates in equation (9) can be rewritten in terms of d (Zanette et al., 2010),
where p_{i} is the period of the interference fringes along the i direction. As a consequence, equation (9) can be expressed as a function of d,
where δ_{i} is the width of the Gaussian envelop function along the i direction. Combining equations (9) and (11), the coherence length can be expressed as
By inserting the measured coherence length ξ_{exp,i} into equation (10), the source size can be calculated.
2.2. Energy bandwidth of Xrays
The experiment was performed at the 3W1 beamline of BEPCII. Xrays of 15 keV were obtained after passing through a Si(311) doublecrystal monochromator (DCM). The Xray energy bandwidth has an effect on the beam size measurement which has to be considered. It can be measured by scanning the second crystal angle of the DCM (Liermann et al., 2015). An ion chamber as a detector provides a current which is proportional to the absolute incident The incident is changed with the relative angle between the two crystals. Then the current can be converted to a voltage which is a function of the relative angle between the two crystals. The corresponding voltage curve is illustrated in Fig. 1. The horizontal coordinate is the relative angle between the two crystals, and the vertical coordinate is the generated by the Xrays. The full width at halfmaximum (FWHM) of the Gaussian distribution is Δθ_{B} = (2.56 ± 0.014) × 10^{−4} degrees. According to the Bragg diffraction formula 2dsinθ_{B} = mλ [where m is an integer, d = 1/(h^{2} + k^{2} + l^{2})^{1/2}, for Si(311), h = 3, k = 1, l = 1] and the photon energy formula E = hc/λ, an energy bandwidth ΔE/E of (1.7 ± 0.001) × 10^{−5} is calculated according to the equation ΔE/E = Δθ_{B}/tanθ_{B} (Yang et al., 2020).
2.3. Experimental setups
Figure 2 shows a schematic of the experimental setup. A 9 mm × 1.3 mm slit is placed 15.54 m from the source. The DCM is located downstream and 2.25 m from the slit. A beryllium window is 9.48 m away from the DCM. The 1D π/2 phase grating is placed 28.91 m from the source. The distance between the 1D absorption grating and camera is about 0.01 m. The Xray camera consists of a CCD, a lens and a LYSO (lutetiumyttrium oxyorthosilicate) scintillator which is used to convert Xrays into visible light. The CCD pixel size is 6.5 µm × 6.5 µm and the lens magnification is 10. The resolution of the Xray camera is 0.65 µm.
The grating interferometer, consisting of a phase grating and an absorption grating, is the key part of the experiment. Figure 3 shows photographs and scanning electron microscope images of two gratings fabricated by Microworks, Germany. The black lines in the photographs indicate the direction of the grating lines. In the photographs, there are some horizontal structures. These horizontal structures are the nodes generated during the gratingmaking process. It can be seen in the zoomed image (Fig. 3, top right) that the grating lines are still vertical.
Some fabricating parameters of the gratings are shown in Table 1. The gratings' period is 2.4 µm and the duty cycle is 0.5 for uniform selfimaging fringes. The grating lines of the phase grating and absorption grating are made of polymer and gold, respectively. The substrates for both gratings are made of polyimide. The height of the polymer of the phase grating is 18.6 µm and the height of the gold of the absorption grating is 14 µm, which is determined by the phase shift and Xray energy of 15 keV.

The grating interferometer device is shown in Fig. 4. The phase grating is fixed on an optical platform. The absorption grating and Xray camera move together along the direction of light propagation. The phase grating produces selfimages at fractional Talbot distances (d_{n}) following the equation d_{n} = n(p^{2}/2λ), where n = 0.5, 1.5, 2.5… (Pfeiffer et al., 2005). Each selfimage is superimposed with the absorption grating forming Moire fringes which are detected by an Xray camera. The grating lines of the phase grating are in the horizontal direction, and there is a relatively small angle between the grating lines of the two gratings.
The interference fringes represent the coherence of the source in the vertical direction, and the vertical beam size can be calculated from the coherence. Measurements are performed by acquiring interferograms at multiple detector positions from close to the grating to the maximum distance.
3. Results and analysis
3.1. Grating selfimaging fringes
A selfimage of the phase grating is shown in Fig. 5(a). It is similar to the photograph of the scanning electron microscope of the phase grating. However, the contrast between light and dark stripes is very small. In order to achieve obvious interference fringes, an absorption grating is introduced. Figure 5(b) shows the Moire fringes formed by the phase grating and the absorbing grating. The period can be calculated by the Moire fringe period formula, B = p/sinθ, where p is the grating period and θ is the relative angle of the grating lines of the two gratings. Notably, the interference image has a more significant period benefitting from extraction of the visibility.
A Moire fringe interference image for different angles θ is shown in Fig. 6. A larger angle θ corresponds to a smaller fringe period. According to interference theory, the visibility of the selfimaging fringe generated by a single phase grating is the same as the visibility of the Moire fringes. Thus, in the following experiments, the twograting structure with θ = 2° is selected due to its larger fringe period. In the topleft image of Fig. 6, the fringe highlighted by the red solid lines is the Moire fringe, which has a large period; the fringe highlighted by the blue dotted lines is the selfimage of the grating, which has a small period.
3.2. Visibility and vertical beam size
The visibility as a function of detection distance d is shown in Fig. 7. The sinusoidal oscillation of the visibility is due to the fractional Talbot effect imparted by the phase grating, and it decreases gradually because of the partial coherence of the source. Some images of the interferograms at different distances d are displayed. The images have higher visibilities at d = 40 mm, 96 mm and 164 mm near the fractional Talbot distances. On the contrary, the images at d = 68 mm, 140 mm and 200 mm have lower visibilities due to the weak selfimages effect. The two kinds of images appear alternately with increasing d, which is consistent with theoretical expectations.
The transverse coherence of the Xray wavefront is related to the width of the envelope function, which can be modeled as Gaussian according to equation (10). Figure 8 shows the fitted Gaussian envelope function drawn through the maximum visibility points. The first fractional Talbot distance about 34 mm is eliminated because it is smaller than the minimum distance between two gratings. The R^{2} value of the fitted Gaussian function is greater than 0.99, presenting a high fitting accuracy. The transverse coherence of the Xrays can be calculated by utilizing the width of the envelope function according to the theory in Section 2. The width of the envelope function σ is 162.03 ± 4.6 mm, and a vertical coherence length of 5.58 ± 0.16 µm on the phase grating plane is achieved. The vertical beam size is σ_{y} = 68.19 ± 2 µm by inserting the vertical coherence length into equation (10). In the experiment, the visibility is averaged over multiple measurements, and error bars are introduced. The vertical beam size error is calculated considering the visibility error and the error caused by the energy bandwidth. An error of ±2 µm indicates that this method has a small measurement error. The accuracy of this method is evaluated by comparing the vertical emittances measured by the two methods discussed in the following section.
3.3. Comparing methods – doubleslit interference
The experimental results are assessed by comparing two vertical emittances from two methods – the grating selfimaging method (at beamline 3W1 with wiggler source) and the synchrotron radiation (SR) interferometer method (at a visiblelight diagnostics beamline with bending magnet source). The vertical emittance ɛ_{y} is a constant for an accelerator. It can be calculated by vertical beam size and Lattice parameter β_{y} at the source point according to the formula = ɛ_{y}β_{y}.
A doubleslit interferometer is applied to measure the beam size at the visiblelight diagnostics beamline. The principle of doubleslit interference is based on Van Citterut–Zernike theory. The spatial coherence of a finitesize light source is the Fourier transform of the light source size, expressed as (Wang et al., 2013)
where γ(ν) is the spatial coherence, f(x) is the normalized source distribution and ν is the spatial frequency. A diagram of doubleslit interference is shown in Fig. 9. The intensity of the interference pattern measured in the detector plane is
where a is half of the single slit height, b is the distance between the centers of two slits, R is the distance from the doubleslit to the detector, λ_{0} is the wavelength of the observation, ϕ denotes the fringe phase and I_{0} is the sum of incoherent intensities from the two slits. The spatial frequency ν is expressed as ν = b/λ_{0}R_{0}, where R_{0} is the distance from the source to the double slit.
The spatial coherence γ can be calculated by fitting the intensity distribution of the interference fringes according to equation (15). The crosssectional distribution of the original source is obtained from the inverse Fourier transform of the different spatial coherences measured by the corresponding doubleslit spacing b (meaning different spatial frequency). The crosssectional distribution of the beam in the storage ring approximates a Gaussian distribution. To simplify the complex calculation process, a Gaussian function is substituted into the coherence calculation equation (14), and the relationship between beam size σ and spatial coherence γ can be expressed as
In general, the beam size is calculated with the spatial coherence measured by a fixed slit spacing according to equation (16).
At the visiblelight diagnostics beamline, we measured the vertical beam size with a doubleslit interferometer with a wavelength of 550 nm. Taking into account the relationship between light intensity, singleslit width and doubleslit distance, a slit width of 0.8 mm and a doubleslit distance of 4 mm are selected. The distance from the source to the doubleslit, R_{0}, is 5.85 m.
Figure 10 shows the interference fringes collected in the vertical beam size measurement experiment. The raw data are a 2D matrix with 250 rows and 520 columns. With integration along the ydirection, the intensity distribution curve along the xdirection is shown in Fig. 10. By fitting the curve using equation (15) (where the fitting line is expressed in Fig. 10), γ is found to be 0.41. A vertical beam size of 172 µm is derived by substituting γ into equation (16). The vertical beam size as a function of γ for a period of time is show in Fig. 11.
Finally, a comparison of two vertical emittances from the two methods is presented in Table 2. The measured vertical emittance is 1.41 nm rad at beamline 3W1 and 1.40 nm rad at the visiblelight diagnostics beamline. The extremely small difference, about 0.1%, indicates that the grating selfimaging method can be used to measure beam size at a synchrotron radiation source. Possible future measurements for smaller beam sizes of fourthgeneration synchrotron light sources with higher light source coherence are being considered due to the small diffraction limitation and simple experimental setups of the proposed method.

4. Conclusion
A method based on the grating selfimaging effect for accurate beam size measurement has been implemented at BEPCII. Due to the partial coherence of Xrays, the transverse coherence of the wavefront has a relationship with the visibility of selfimaging interferograms. The beam size can be derived from the visibility.
In this paper, the vertical beam size from a wiggler source is measured by extracting visibilities of interferograms formed by a phase grating and absorption grating. The interferograms, also called Moiré fringes, have a larger fringe period which is conducive to visibility extraction. The transverse coherence of 5.58 ± 0.16 µm in the vertical direction in the plane of the phase grating is obtained from the width of the fitted Gaussian envelope function. Then the vertical beam size is calculated to be 68.19 ± 2 µm. Finally, to evaluate the accuracy of the experimental results, the vertical emittances derived from the grating selfimaging method and SR interferometer method are compared, which are 1.41 nm rad and 1.40 nm rad, respectively. Thanks to the extremely small difference, about 0.1%, the grating selfimaging method for measuring beam size is considered to be reliable and accurate. This technique has great prospects for application in small beam size measurement of fourthgeneration synchrotron light sources, benefiting from the small influence of diffraction limitation.
Acknowledgements
The authors would like to thank and acknowledge Yiming Yang, Lirong Zhen, Gang Li and Jie Zhang from the beamline at BEPCII for their assistance during the experiment preparation.
Funding information
The following funding is acknowledged: The Youth Innovation Promotion Association CAS (award No. Y202005).
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