Performance of beam size monitor based on Kirkpatrick–Baez mirror at SSRF

Performance of beam size monitor based on Kirkpatrick–Baez mirror at SSRF 1 2 first-author 1 1 2 1 1 corresp Corresponding author: E-mail address: caojs@ihep.ac.cn Corresponding author: E-mail address: caojs@ihep.ac.cn caojs@ihep.ac.cn 3 3 3 3 Key Laboratory of Particle Acceleration Physics and Technology, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China; Key Laboratory of Particle Acceleration Physics and Technology, Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, China; University of Chinese Academy of Sciences, Beijing 100049, China; University of Chinese Academy of Sciences, Beijing 100049, China; Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China. Shanghai Institute of Applied Physics, Chinese Academy of Sciences, Shanghai 201800, China. High Energy Photon Source (HEPS) is a 6 GeV ultralow-emittance storage ring light source to be built in Beijing, China. Both the horizontal and vertical beam sizes of the HEPS storage ring are below 10 µm. It is a challenge to measure such a small beam size in both directions. To this end, measurement by a Kirkpatrick–Baez (KB) mirror imaging system was evaluated. A test KB system for the Shanghai Synchrotron Radiation Facility storage ring was designed and tested. Two crossed cylindrical mirrors were used to image the dipole source point. Both mirrors can be moved in and out so that the monitor is interchangeable with the original X-ray pinhole monitor. The aberration and point spread function, which would cause image blur, were evaluated. A beam-based calibration scheme was used by varying the beam size with different quadrupole settings and fitting them with the corresponding theoretical values. We updated the original X-ray camera with a new camera having a 5-μm-thick LuAG:Ce scintillator, and the imaging result shows greatly decreased image blur. KB mirror Beam size measurement Beam diagnostic Emittance Synchrotron radiation 1. Introduction 1 The demand for highly brilliant X-ray synchrotron radiation (SR) motivates designers to produce low-emittance electron beams. High Energy Photon Source is a 6 GeV light source with a circumference of approximately 1300 m and is going to be built in Beijing, China [ 1 1 ] . This future light source, with a multiple-bend achromat lattice design for constructing a so-called diffraction-limited storage ring, is expected to achieve an ultralow emittance of 60 pm·rad [ 2 2 ] . As an R & D project for HEPS, the HEPS test facility was established in 2016 to resolve key issues in accelerator physics and technology. Measuring the electron beam emittance is one of the most important issues to resolve for HEPS. Beam size measurement is generally used to obtain the emittance and the coupling. To this end, we proposed a Kirkpatrick–Baez (KB) mirror imaging method to measure the horizontal and vertical beam sizes, which are both approximately 5–10 µm. Here we present the test KB beam size monitor, which was designed and tested specifically for the Shanghai Synchrotron Radiation Facility (SSRF) storage ring to measure a beam size of approximately 78 μm (RMS) (horizontal) and 20 μm (RMS) (vertical). We compared some beam size measurement methods that can be roughly classified as imaging methods and interferometry methods. Imaging methods include visible light imaging [ 3 3 ] , X-ray pinhole imaging [ 4 4 , 5 5 ] , X-ray imaging with a KB mirror [ 6 6 ] , the use of Fresnel zone plates [ 7 7 ] , the use of compound refractive lenses [ 8 8 ] , and also X-ray coded-aperture imaging [ 9 9 ] . Interferometry methods include visible light double-slit interferometry [ 10 10 , 11 11 ] , the π polarization method [ 12 12 ] , and X-ray Fresnel diffractometry [ 13 13 ] . Each type of monitor has advantages but also limitations. The X-ray pinhole camera is widely used for its simple setup and high practical reliability. The resolution of pinhole optics is a tradeoff between the diffraction limit (hole too small) and geometric blurring (hole too large). Therefore, it is difficult to find much room to improve the resolution. The typical resolution of X-ray pinhole cameras is greater than 10 μm. The KB mirror monitor does not have this limitation, and it has some specific advantages. First, it has no chromatic aberration; therefore, it is not necessary to use a monochromator, and a higher flux is obtained, resulting in a higher signal-to-noise ratio. Second, the KB mirror can operate in the hard X-ray region, so diffraction makes a smaller contribution to the spatial resolution. Further, it can acquire a direct image of an electron beam, which can provide considerable information on the beam. Not only can the beam size in both directions be acquired, but also the operator can easily observe the beam motion, broadening, or tilt status. 2. Beamline Design 1 2.1 SSRF storage ring and its parameters 2 The SSRF is a third-generation synchrotron light source operating at 3.5 GeV and located in Shanghai in China. It has been open to users since 2009 and serves 13 beamlines with nine insertion devices. The basic parameters of the ring are listed in Table 1 Table 1 . The SSRF has two diagnostic beamlines, both of which are extracted from the second dipole in the first cell. One beamline is at the 3° end and is used for visible light diagnostics, whereas the other, at the 0.8° end, is used for X-ray pinhole imaging. The pinhole in the X-ray pinhole camera system, which has a magnification of 1.5, is located in air; therefore, it is possible to install the KB mirror chamber between the pinhole and the X-ray camera. 1 1 Parameters 1 1 Value 1 1 Beam energy (GeV) 1 1 3.5 1 1 Beam current (mA) 1 1 200-300 1 1 Circumference (m) 1 1 432 1 1 RF frequency (MHz) 1 1 499.654 1 1 Natural emittance (nm·rad) 1 1 3.9 1 1 Magnetic field of dipole (T) 1 1 1.2726 1 1 Critical photon energy (keV) 1 1 10.4 1 1 Horizontal RMS beam size (μm) 1 1 78 1 1 Vertical RMS beam size (μm) 1 1 20 2.2 Kirkpatrick–Baez mirror 2 In 1948, Kirkpatrick and Baez [ 14 14 ] first designed a system of two crossed spherical mirrors to eliminate the astigmatism of a single mirror used at glancing incidence. This method essentially eliminates the astigmatism and would eliminate spherical aberration using 1:1 imaging when the object and image both lie on the Rowland circle of the mirror. Hence, the residual aberration is the only factor limiting the resolution. The focusing equation of the two mirrors is 1/ p + 1/ q = 1/ƒ = R sin θ, where p is the object distance, q is the image distance, f is the focal length, R is the radius of curvature, and θ is the grazing incidence angle. We use 1:1 imaging in our KB system with two cylindrical mirrors. The front mirror, which is placed horizontally, is called the vertical focusing mirror (VFM), and the back mirror, which is placed vertically, is called the horizontal focusing mirror (HFM). The parameters are shown in detail in Table 2 Table 2 . Both mirrors are fabricated by Thales SESO and a ZYGO interferometer is used to measure the slope error and radius of curvature. Three lines (-5 mm, center, +5 mm) are measured in each mirror with a frame. For the VFM, the average RMS slope error is 0.293 μrad, and the radius of curvature is 2.599 km. For the HFM, the average RMS slope error is 0.381 μrad, and the radius of curvature is 2.55 km. 1 1   1 1 VFM 1 1 HFM 1 1 Shape 1 1 cylindrical 1 1 cylindrical 1 1 Radius of curvature (km) 1 1 2.57 1 1 2.57 1 1 Grazing angle (mrad) 1 1 3 1 1 3 1 1 Substrate 1 1 silicon 1 1 silicon 1 1 Coating 1 1 Rh 1 1 Rh 1 1 Acceptance angle (μrad) 1 1 122 1 1 117 1 1 Size (L ×W ×γH) (mm 3 ) 1 1 320 ×40 ×40 1 1 320 ×40 ×40 1 1 Clear aperture (L × W) (mm 2 ) 1 1 300 ×10 1 1 300 ×10 1 1 RMS roughness (nm) 1 1 <0.2 1 1 <0.2 1 1 RMS slope error (μrad) 1 1 <0.3 1 1 <0.3 1 1 Distance to source (m) 1 1 7.36 1 1 7.72 1 1 Distance to image (m) 1 1 8.08 1 1 7.72 1 1 Magnification 1 1 1.1 1 1 1 1 1 Hitting power (W) 1 1 1.083 1 1 0.251 1 1 Absorbed power (W) 1 1 0.832 1 1 0.058 Because of the reflective optics design, there is no chromatic aberration in the KB optics. According to an analysis by Susini and Wulff [ 15 15 ] , the third-order spherical aberration, first-order coma, and third-order coma are the main aberrations contributing to focus broadening. In purely geometrical terms, the spot size can be expressed as follows [ 16 16 ] : <math display="block" id="M1"><mrow><mi>F</mi><mo>=</mo><mfrac><mn>3</mn><mrow><mn>16</mn></mrow></mfrac><msup><mi>L</mi><mn>2</mn></msup><mfrac><mrow><msub><mi>θ</mi><mi>i</mi></msub></mrow><mi>p</mi></mfrac><mfrac><mrow><mn>1</mn><mo>−</mo><msup><mi>M</mi><mn>2</mn></msup></mrow><mi>M</mi></mfrac><mo>+</mo><msub><mi>S</mi><mtext>Z</mtext></msub><mi>M</mi><mo>+</mo><msub><mi>S</mi><mtext>Z</mtext></msub><mo stretchy="false">(</mo><mi>M</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mfrac><mi>L</mi><mrow><mn>4</mn><mi>p</mi></mrow></mfrac><mo>,</mo></mrow></math> where F is the full width at half-maximum (FWHM) focus size, L is the illuminated length on the mirror, M = q / p is the magnification, θ i is the grazing angle, and S z is the vertical source size; p and q are the source–mirror and mirror–focus distances, respectively. The first term corresponds to the third-order spherical aberration, and the two other terms of Eq. (1) are related to the first-order coma and third-order coma. As M ≈ 1 in this case, there is no spherical aberration. Coma becomes the major contribution, which can also be ignored after the calculation because it broadens the FWHM focus size by no more than 2%. Thus, the optical aberrations are not included in the derivation of the point spread function (PSF). 2.3 System layout 2 A schematic of the KB mirror monitor layout is shown in Fig.1 Fig.1 . SR coming from the dipole first crosses a 1-mm-thick and 2-mm-diameter aluminum window at the front end, which defines the X-ray open angle as 0.35 mrad. The Al window acts as a filter and is used to isolate the vacuum from air; the lower photon energy cutoff is 10 keV. The KB mirror pair is located in an independent vacuum chamber to prevent oxidation, with two entrance slits on its upstream end that define the system’s angular acceptance. To isolate the system from any significant low-frequency vibration, the mirrors are fixed on a 518 × 712 × 857 mm 3 granite block by a UHV adjustment mechanism. The HFM is located equidistant between the source and image for one-to-one imaging; further, the magnification of the VFM is 1.1. In front of the entrance slits, a Cu attenuator is used both to attenuate the photon flux and to protect the mirrors from the long-term high heat load during operation. To achieve better spatial resolution, we set our KB mirror to operate in the hard X-ray region to decrease the diffraction. The targeting source size is 20 μm (vertical) and 78 μm (horizontal). To set the dipole source point as the zero position, the VFM and HFM are located at 7.36 and 7.72 m, respectively, to make an image 15.44 m from the source. The mirror substrate is Si coated with Rh. Both mirrors have a 3 mrad grazing incidence angle, and most of the hard X-rays above 23.5 keV are absorbed by the first mirror. With the V122 μrad × H117 μrad acceptance angle calculated for the lengths of the two mirrors, the heat load at the first mirror is 1.083 W at a 300 mA beam current; 0.832 W is absorbed and would be removed by water cooling, and 0.251 W is reflected to the second mirror. Fig.2 Fig.2 shows the spectrum of the SR after it is filtered by the Al window and limited by the aperture slits; the SR has a spectrum from approximately 12 to above 60 keV and peaks around 24 keV. After the VFM and HFM, the narrowed spectrum is from 12 to 23.5 keV and peaks around 20.5 keV. 2.4 Detector 2 We use two X-ray scintillator cameras to acquire a two-dimensional image of the beam profile; the cameras can be interchanged with each other by using motorized translation stages.  The first X-ray camera (cam1) is the original camera used as the detector in the X-ray pinhole system [ 17 17 ] . It has a 400-μm-thick YAG:Ce scintillator, which converts X-rays into visible light at a peak wavelength of 530 nm; visible light is reflected 90° by a flat mirror. A macro lens (Componon 2.8/50 from Schneider-Kreuznach) is used to image the screen of a compact IEEE 1394 CCD camera (AVT Guppy F-080B, pixel size 4.65 μm) with a magnification of 2. An application based on LabVIEW and a shared memory IOC core technique was developed to control the camera and communicate with the control system through the EPICS CA protocol. The second camera (cam2) is an updated X-ray camera. It has better resolution using a 5-μm-thick LuAG:Ce crystal. The scintillator is a 10 × 0.005 mm 2 LuAG:Ce crystal produced by Crytur and glued onto a quartz substrate. A microscope connected to a CCD camera is used to view the image on the scintillator. To prevent hard X-rays from impinging on the CCD camera, visible light is reflected at 90° by a flat mirror. The CCD camera (Kodak full-frame KAF-8300) has a pixel size and spatial resolution of 5.4 μm, and the total number of pixels is 3358 × 2536. With 20× magnification microscope objectives, the effective pixel size is 0.27 μm. A Jima X-ray test by Crytur proved that the spatial resolution was better than 1.5 μm ( Fig.3 Fig.3 ). A control software package from Crytur is used to acquire images and control the focus. 3. Point Spread Function (PSF) 1 The accuracy of the KB mirror monitor is determined by the RMS PSF. The obtained image on the camera is the convolution of the source profile with the PSF of the entire system, which includes several independent terms: the PSF of the diffraction, the PSF of the X-ray camera, and the image blur caused by the mirror slope error. We calculate the PSF assuming that the source and the PSF are Gaussian. Let us denote the RMS Gaussian size of the image as Σ; then it can be expressed as follows: <math display="block" id="M2"><mtable columnalign="left"><mtr><mtd><msup><mtext>Σ</mtext><mn>2</mn></msup><mo>=</mo><msup><mrow><mo>(</mo><mrow><mi>σ</mi><mo>×</mo><mi>M</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msubsup><mi>S</mi><mrow><mtext>diff</mtext></mrow><mn>2</mn></msubsup><mo>+</mo><msubsup><mi>S</mi><mrow><mtext>slope</mtext></mrow><mtext>2</mtext></msubsup><mo>+</mo><msubsup><mi>S</mi><mrow><mtext>camera</mtext></mrow><mtext>2</mtext></msubsup></mtd></mtr><mtr><mtd><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mtext> </mtext><mo>=</mo><msup><mrow><mo>(</mo><mrow><mi>σ</mi><mo>×</mo><mi>M</mi></mrow><mo>)</mo></mrow><mn>2</mn></msup><mo>+</mo><msubsup><mi>S</mi><mrow><mtext>sys</mtext></mrow><mtext>2</mtext></msubsup><mo>,</mo></mtd></mtr></mtable></math> where σ is the RMS size of the image of the photon source at the bending magnet, M is the magnification of the KB mirror, S diff is the diffraction introduced by the aperture, S slope is the RMS image blur induced by the mirror slope error, S camera is the RMS spatial resolution of the X-ray camera, and S sys is the effective RMS PSF of the entire system. 3.1 Diffraction limit 2 To calculate the image smear induced by aperture diffraction, a Gaussian is force-fitted to the diffraction pattern of a circular or rectangular aperture, and the width of the Gaussian is treated as a smearing term, S diff , to be taken in quadrature with the beam size [ 18 18 ] . In this case, <math display="block" id="M3"><mrow><msub><mi>S</mi><mrow><mtext>diff</mtext></mrow></msub><mo>≈</mo><mn>0.4</mn><mfrac><mi>λ</mi><mrow><mn>2</mn><mi>N</mi><mi>A</mi></mrow></mfrac><mo>,</mo></mrow></math> where NA is the numerical aperture, which specifies the light-gathering power of the imaging system and is the half open angle of the aperture, and λ is the wavelength of the X-rays. We use a peak wavelength of 0.06 nm (20.5 keV) here for the calculation. There is a small difference in the aperture between the vertical direction and horizontal direction. Table 3 Table 3 lists the crucial apertures that determine the X-ray entrance angle. 1 1   3 1 Vertical 3 1 Horizontal 1 1   1 1 Size 1 1 Distance 1 1 Aperture angle 1 1 Size 1 1 Distance 1 1 Aperture angle 1 1 Al window 1 1 2mm 1 1 5.69m 1 1 351μrad 1 1 2mm 1 1 5.69m 1 1 351μrad 1 1 Dissector slits 1 1 450μm 1 1 7m 1 1 64.3μrad 1 1 450  μm   1 1 7m 1 1 64.3μrad 1 1 X-ray opening angle 1 1 - 1 1 - 1 1 36.1μrad@20.5keV 1 1 - 1 1 - 1 1 - 1 1 Mirror aperture 1 1 300mm 1 1 7.36m 1 1 122μrad 1 1 300mm 1 1 7.72m 1 1 117μrad For the horizontal direction, the entrance angle is determined by the dissector slits, which can be switched between 450 and 900 μm. In our experiment, we always use 450 μm slits, so the entrance angle is 64.3 μrad. According to Eq. (3), the diffraction limit of the HFM is <math id="M4"><mrow><msubsup><mtext>S</mtext><mrow><mtext>diff</mtext></mrow><mrow><mtext>HFM</mtext></mrow></msubsup><mtext> </mtext><mo>≈</mo><mtext> </mtext><mn>0.37</mn><mtext> μm</mtext></mrow></math> , which is proportional to λ . For the vertical direction, the entrance angle is determined by the synchrotron light’s vertical opening angle, σ SR : <math display="block" id="M5"><mrow><msub><mi>σ</mi><mrow><mtext>SR</mtext></mrow></msub><mo>=</mo><mfrac><mrow><msub><mi>E</mi><mn>0</mn></msub></mrow><mi>E</mi></mfrac><msup><mrow><mrow><mo>(</mo><mrow><mfrac><mi>λ</mi><mrow><mn>3</mn><msub><mi>λ</mi><mtext>c</mtext></msub></mrow></mfrac></mrow><mo>)</mo></mrow></mrow><mrow><mfrac bevelled="true"><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>,</mo></mrow></math> where E 0 = 0.51 MeV is the electron rest mass energy; E = 3.5 GeV is the electron energy of the SSRF storage ring; and λ c = 0.326 nm is the critical wavelength of the SSRF dipole. From Eqs. (3) and (4), the VFM diffraction limit can be expressed as <math display="block" id="M6"><mrow><msubsup><mi>S</mi><mrow><msub><mrow/><mrow><mtext>diff</mtext></mrow></msub></mrow><mrow><mtext>VFM</mtext></mrow></msubsup><mo>≈</mo><mn>0.4</mn><mfrac><mi>E</mi><mrow><msub><mi>E</mi><mn>0</mn></msub></mrow></mfrac><msup><mrow><mrow><mo>(</mo><mrow><mn>3</mn><msub><mi>λ</mi><mtext>c</mtext></msub><mo>⋅</mo><mi>λ</mi></mrow><mo>)</mo></mrow></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>∝</mo><msup><mi>λ</mi><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>.</mo></mrow></math> The calculated value is <math id="M7"><mrow><msubsup><mtext>S</mtext><mrow><mtext>diff</mtext></mrow><mrow><mtext>VFM</mtext></mrow></msubsup></mrow></math> ≈ 0.66 μm, which is proportional to λ 1/2 . 3.2 Slope error 2 Short-length-scale variations from an ideal mirror surface are referred to as surface roughness. The surface roughness of the mirror may affect its reflectivity. Long-length-scale deviations from an ideal mirror surface are referred to as slope errors. The slope error can be amplified by the mirror-to-image distance q when X-rays are reflected away from the surface. The image blur due to the slope error depends on the focal length of the mirror and the RMS slope error: S slope = 2 × σ slope × q , where σ slope is the RMS slope error. Fig.4 Fig.4 shows the RMS slope error of the mirror center line measured by the ZYGO interferometer. Table 4 Table 4 shows the average RMS slope error of the center line, +5 mm line, and -5 mm line, and the calculated image blur of the VFM and HFM. 1 1   1 1 Slope error ( σ slope ) 1 1 Mirror-to-image distance ( q ) 1 1 Blur function ( S slope ) 1 1 VFM 1 1 0.29μrad 1 1 8.08m 1 1 4.69  μm   1 1 HFM 1 1 0.38μrad 1 1 7.72m 1 1 5.87μm 3.3 Camera resolution 2 To measure the PSF of the X-ray camera composed of a scintillator screen, macro lens, and camera, we used an opaque mask to cover the left part of the screen in front of the X-ray camera. The opaque mask, which has a sharp edge, is made of a tungsten bar. Fig.5 Fig.5 shows the measurement results of cam1 and cam2 in our system. Cam2 has a greatly improved resolution of 1.53 μm. Table 5 Table 5 summarizes the results. 1 1 Camera 1 1 Scintillator 1 1 Thickness 1 1 Magnification 1 1 Pixel size 1 1 Pixel binning 1 1 Resolution 1 1 Cam1 1 1 YAG:Ce 1 1 400 μm 1 1 2 1 1 4.65 μm 1 1 1×1 1 1 19.73 ± 0.43μm 1 1 Cam2 1 1 LuAG:Ce 1 1 5 μm 1 1 20 1 1 5.4 μm 1 1 4×4 1 1 1.53   ± 0.04μm 3.4 System PSF evaluation 2 In this section, we briefly summarize the total width of the PSF S sys of the KB system. It is calculated by using the quadratic sum of all the terms. The diffraction limit item is in the submicron range, and the slope error term is 4.69 μm for the VFM and 5.87 μm for the HFM. Table 6 Table 6 summarizes the PSF in the vertical and horizontal directions for different scintillator cameras. It shows that when cam1 is used, the camera resolution is the dominant term, contributing the largest component to the image extension. After installation of the updated camera, cam2, with 1.5 μm resolution (which can be interchanged with cam1), the largest contribution becomes the slope error term. It is obvious that a smaller PSF width is better. Nevertheless, deconvolution can easily be done by quadratic subtraction, as given by Eq. (2) when the PSF is below the beam size to be measured. To obtain the image size with an error of less than 10%, the distortion S sys should be less than half the beam size; hence, for our system, a beam size of ~10 μm can be measured directly using cam2. 1 1   1 1 S diff 1 1 S slope 1 1 S camera 1 1   1 1 S sys 1 1 Vertical 1 1 0.66μm 1 1 4.69μm 1 1 Cam1 1 1 19.73μm 1 1 20.29μm 1 1   1 1   1 1   1 1 Cam2 1 1 1.53μm 1 1 4.97μm 1 1 Horizontal 1 1 0.37μm 1 1 5.87μm 1 1 Cam1 1 1 19.73μm 1 1 20.59μm 1 1   1 1   1 1   1 1 Cam2 1 1 1.53μm 1 1 6.08μm 4. Experimental Results 1 4.1 Beam size measurement 2 Here we present measurements of the electron beam size by cam1 in top-up mode at a 240 mA beam current ( Fig.6 Fig.6 , data from July 20, 2017). The transverse beam size was Σ x = 77.5 μm (horizontal) by Σ y = 34.3 μm (vertical), as obtained from the CCD camera output before PSF calibration. After PSF calibration using the data in Table 6 Table 6 , the beam size is found to be σ x = 74.7 μm and σ y = 25.1 μm. 4.2 Beam-based calibration 2 To evaluate the online PSF for cam1, a beam-based calibration method [ 19 19 ] was used. We gradually changed the horizontal beam size σ x at the source point by modifying the power supply current I Q5 of the fifth set of quadrupoles and measured the image size Σ x at each I Q5 setting. It is obvious that σ x is what we want to acquire from Σ x by quadratic subtraction, as given by Eq. (2), with a correct PSF width S sys . However, we can use a model horizontal beam size <math id="M8"><mrow><msubsup><mtext>σ</mtext><mtext>x</mtext><mrow><mtext>model</mtext></mrow></msubsup></mrow></math> , which can be calculated using linear optics from closed orbits (LOCO), to evaluate the PSF. After the linear optics measurements and the optimization procedure using LOCO, the maximum beta function beating of the SSRF storage ring was minimized to less than 1%. In this case, the difference in the beam parameters between the model and the practical machine is also smaller than 1%. The model horizontal beam size σ model x is calculated by <math display="block" id="M9"><mrow><msubsup><mi>σ</mi><mtext>x</mtext><mrow><mtext>model</mtext></mrow></msubsup><mo>=</mo><msqrt><mrow><msub><mi>β</mi><mtext>x</mtext></msub><msub><mi>ε</mi><mtext>x</mtext></msub><mo>+</mo><msup><mrow><mo stretchy="false">(</mo><msub><mi>η</mi><mtext>x</mtext></msub><msub><mi>σ</mi><mtext>e</mtext></msub><mo stretchy="false">)</mo></mrow><mn>2</mn></msup></mrow></msqrt><mo>,</mo></mrow></math> where β x and η x are the betatron and dispersion functions at the source point and in the horizontal plane, respectively, and ε x and σ e are the horizontal emittance and relative energy spread of the electron beam, respectively. The machine parameters and measurement values are shown in detail in Table 7 Table 7 . 1 1 I Q5 (Ratio) 1 1 ε x (nm·rad) 1 1 σ e 1 1 β x (m) 1 1 η x (m) 1 1 <math id="M10"><mrow><msubsup><mi>σ</mi><mi>x</mi><mrow><mi>mod</mi><mtext>el</mtext></mrow></msubsup></mrow></math> (μm) 1 1 Σ x (μm) 1 1   92% 1 1 8.538 1 1 0.9827×10 -3 1 1 1.0998 1 1 0.0998 1 1 137.9 1 1 134.5 1 1 93% 1 1 7.094 1 1 0.9828×10 -3 1 1 1.0426 1 1 0.0901 1 1 123.4 1 1 122.8 1 1 94% 1 1 6.027 1 1 0.9828×10 -3 1 1 0.9907 1 1 0.0818 1 1 111.5 1 1 112.6 1 1 95% 1 1 5.250 1 1 0.9828×10 -3 1 1 0.9432 1 1 0.0746 1 1 101.6 1 1 103.9 1 1 96% 1 1 4.699 1 1 0.9828×10 -3 1 1 0.8995 1 1 0.0682 1 1 93.4 1 1 97.7 1 1 97% 1 1 4.327 1 1 0.9828×10 -3 1 1 0.8589 1 1 0.0626 1 1 86.6 1 1 90.6 1 1 98% 1 1 4.100 1 1 0.9828×10 -3 1 1 0.8210 1 1 0.0576 1 1 81.0 1 1 85.2 1 1 99% 1 1 3.992 1 1 0.9828×10 -3 1 1 0.7855 1 1 0.0531 1 1 76.5 1 1 81 1 1 100% 1 1 3.983 1 1 0.9828×10 -3 1 1 0.7521 1 1 0.0490 1 1 72.9 1 1 77.7 1 1 101% 1 1 4.059 1 1 0.9828×10 -3 1 1 0.7204 1 1 0.0453 1 1 70.1 1 1 74.9 1 1 102% 1 1 4.209 1 1 0.9828×10 -3 1 1 0.6902 1 1 0.0420 1 1 67.9 1 1 72.6 1 1 103% 1 1 4.423 1 1 0.9828×10 -3 1 1 0.6615 1 1 0.0389 1 1 66.3 1 1 71 1 1 104% 1 1 4.696 1 1 0.9828×10 -3 1 1 0.6339 1 1 0.0361 1 1 65.1 1 1 69.8 1 1 105% 1 1 5.024 1 1 0.9828×10 -3 1 1 0.6075 1 1 0.0335 1 1 64.3 1 1 68.8 1 1 106% 1 1 5.403 1 1 0.9828×10 -3 1 1 0.5819 1 1 0.0311 1 1 63.9 1 1 68.3 1 1 107% 1 1 5.833 1 1 0.9828×10 -3 1 1 0.5572 1 1 0.0289 1 1 63.7 1 1 67.9 To determine the PSF, a least-square linear fitting method is used to fit the squares of Σx and <math id="M11"><mrow><msubsup><mtext>σ</mtext><mtext>x</mtext><mrow><mtext>model</mtext></mrow></msubsup></mrow></math> with Eq. (2) at each I Q5 setting, where the fitting slope is set to 1 ( Fig.7 Fig.7 ). The fitting value is S sys = 21.9 ± 2.5 μm. It is in good agreement with the off-line calculated value of 20.59 μm in Table 6 Table 6 . Fig.8 Fig.8 compares the model theoretical value and measurement value of the beam size with on-line PSF calibration. The measurement value is in good agreement with the theoretical value. 4.3 High-resolution camera imaging results 2 In January 18, 2018, the new LuAG:Ce camera (cam2) was installed next to the original YAG:Ce camera (cam1) so that both cameras can be used by changing their position using motorized translation stages. Both the Jima X-ray test by Crytur and the slanted edge test at the diagnostic beam line (see 3.3 for details) demonstrated very good performance, indicating that the spatial resolution was approximately 1.5 μm. Fig.9 Fig.9 shows the imaging results of the two cameras. It is obvious that cam2 shows the focal image better. After a Gaussian fit of the raw data, the transverse beam sizes from cam1 are <math id="M12"><mrow><msubsup><mtext>Σ</mtext><mtext>x</mtext><mrow><mtext>cam1</mtext></mrow></msubsup></mrow></math> = 81.1 μm, <math id="M13"><mrow><msubsup><mtext>Σ</mtext><mtext>y</mtext><mrow><mtext>cam1</mtext></mrow></msubsup></mrow></math> = 35.0 μm, whereas the data from cam2 are <math id="M14"><mrow><msubsup><mtext>Σ</mtext><mtext>x</mtext><mrow><mtext>cam2</mtext></mrow></msubsup></mrow></math> = 71.5 μm, <math id="M15"><mrow><msubsup><mtext>Σ</mtext><mtext>y</mtext><mrow><mtext>cam2</mtext></mrow></msubsup></mrow></math> = 28.6 μm. Cam2 shows greatly decreased image blur, as we expected. 5. Conclusion 1 In this paper, we presented the KB mirror beam size monitor with a bending magnet source at the SSRF. It can be interchanged with the original X-ray pinhole monitor. We evaluated the PSF of the system by calculating each term that could cause image blur. 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Sawhney , S. Scott et al ., The Diamond-NOM: A non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability . Nucl. Instrum. Methods Phys. Res. A   616 , 224 - 228 ( 2010 ). doi: 10.1016/j.nima.2009.10.137 http://doi.org/10.1016/j.nima.2009.10.137 The Diamond-NOM: A non-contact profiler capable of characterizing optical figure error with sub-nanometre repeatability 10.1007/s41365-018-0477-y 2018-03-02 2018-04-17 2018-04-26 2018-09-03 2018-10-01 2021-04-11 © China Science Publishing & Media Ltd. (Science Press), Shanghai Institute of Applied Physics, the Chinese Academy of Sciences, ChineseNuclear Society and Springer Nature Singapore Pte Ltd. 2018 This is a post-peer-review, pre-copyedit version of an article published in Nuclear Science and Techniques. The final authenticated version is available online at: http://dx.doi.org/10.1007/s41365-018-0477-y http://dx.doi.org/10.1007/s41365-018-0477-y . 2018 Nuclear Science and Techniques 10 29