2.1 Principles of X-ray quasi-monochromators

X-ray monochromators are analogous to grating monochromators and spectrometers in the visible spectrum [10]. If the lattice spacing for a crystal is accurately known, the observed angles of diffraction can be used to measure and identify unknown X-ray wavelengths. Because of the sensitive wavelength dependence of Bragg reflection exhibited by materials like silicon, a small portion of a continuous spectrum of radiation can be isolated [10]. Bent single crystals used in X-ray spectroscopy are analogous to the curved line gratings used in optical spectroscopy. After the beam has passed through a high-resolution monochromator, the spectral bandwidth can be as narrow as 10^-4 [10]. A crystal monochromator operates through the diffraction process according to Bragg’s law. Photon energy (wavelength) can be selected by crystal, net planes, and Bragg angle.

The Bragg’s law is presented in Eq. (1):

where d is the lattice spacing of the crystal, θ is the glancing angle, and n is the diffraction order. The lattice spacing is 0.314 nm in the case of an Si(111) crystal. The energy of the diffracted X-rays is determined by the glancing angle θ, which is shown in Fig. 2. For Bragg angles ranging from 3 to 70, the detected energy range is between 2 keV and 40 keV. Energy resolution is a very important parameter for a monochromator system. An Si(111) crystal has an energy resolution ranging from 10^-5 to 10^-3.

Figure 2:Full-screen View

(Color online) Energy and wavelength as a function of glancing angle θ.

2.2 Setup of the X-ray quasi-monochromator

According to the X-ray monochromator principles presented in the previous section, it can be seen that a typical X-ray monochromator generally consists of a silicon crystal and an accurate rotating rail stack.

The selected crystal should be able to cover the main spectral range for synchronous light. Spectra of the extracted X-rays are shown in Fig. 8. The properties and performance of Si(111) crystals are described in the previous section, and Si(111) crystal is able to meet the requirements of this work. In order to realize the selection of different wavelengths, the SGSP-60YAW rotating rail from SIGMAKOKI [11] was selected, which has a rotating accuracy of 0.005 deg/pulse. Fig. 3 shows the schematic of the X-ray quasi-monochromator for the X-ray pinhole camera. This quasi-monochromator is mounted on a rail that can be moved in and out of the optical path of the X-ray pinhole camera.

Figure 8:Full-screen View

(Color online) Extracted X-ray spectrum in the X-ray diagnostic beam line at SSRF.

Figure 3:Full-screen View

(Color online) X-ray quasi-monochromator for X-ray pinhole camera at SSRF.

The former X-ray pinhole camera system is shown in Fig 1. The addition of a single crystal to the light path requires consideration of the position of the existing YAG:Ce target or the position of the camera. Because of the glancing angle, the reflected X-ray will deviate from the original imaging point. In order to keep the X-ray within the range of the YAG:Ce scintillator screen, the monochromator crystal was placed close to the imaging system. The relationship between the distance from the monochromatic crystal to the YAG:Ce target and the deviation of the reflected light from the YAG:Ce center are shown in Fig. 4. In this case, the distance between the center of the crystal and the YAG:Ce target is 8 cm. The layout of the new X-ray pinhole camera is shown in Fig. 5.

Figure 4:Full-screen View

(Color online) Distance between the monochromatic crystal 'L’ and the YAG:Ce target and the deviation of the reflected light from the YAG:Ce center 'delta’.

Figure 5:Full-screen View

(Color online) Schematic of the X-ray pinhole camera combined with a quasi-monochromator.

A study on the PSF of the new X-ray pinhole camera is presented in the next subsection.

2.3 Point spread function study of the new X-ray pinhole camera

The image formed on the camera is the convolution of several independent contributions, including the beam size, pinhole, and the imaging system. We assume the source and the PSFs to be Gaussian. The root mean square (RMS) beam size can be expressed as follows [12]:

where C_mag is the magnification factor of the X-ray pinhole system, σ_img is the acquired image size, σ_aper is the geometric width of the pinhole PSFs, σ_diff is the diffraction width, and σ_scr is the imaging system width, which contains the effects of the YAG:Ce scintillator screen, lens, and CCD camera.

The diffraction width of the pinhole PSFs for a monochromatic photon beam is given by [12]

where A is the aperture size of the pinhole, D is the distance from the pinhole to the YAG:Ce target, and λ is the wavelength of the X-ray. The difficulty in this experiment involves determining the wavelength. For the former system, the method integrates this expression over the spectrum [7]. For the new pinhole camera, the wavelength can be calculated from the glancing angle. However, it is very hard to measure this angle directly. Therefore, a beam-based experimental method was employed to measure the glancing angle, which will be presented in the next section.

The geometric width of pinhole PSFs is given by [12]

where d is the distance from the source point to the pinhole.

However the PSF of the imaging system is hard to calculate. The imaging system is shown in Fig. 5. The CCD camera used in this work is Point Grey FL2-08S2M, which has a 12-bit ADC and has a resolution of 1031×776 pixels at 30 FPS with 4.65 μ m/pixel. Introducing a lens doubles the magnification of the imaging system. An experiment has been conducted at SSRF to measure the PSF of the imaging system [13].

3.1 Verifying the presence of quasi-monochromatic X-ray

The monochromator is based on Bragg reflection, and monochromatic light is reflected from the crystal surface. If the light that reaches the target is reflected light, and the spot position moves with the crystal. In addition, the monochromatic beam energy will also be lower than the energy of the full synchrotron spectrum. The idea of this experiment is to move the crystal and observe the resulting images.

Since the crystal surface was perpendicular to the YAG:Ce scintillator screen. In order to conveniently observe the reflected light on the CCD, the crystal was rotated 0.5 degrees before the experiment. Then the crystal was moved slowly into the light path. The process of verification is shown below.

First, the entire image of the photon at the source point was found at the CCD camera. When the edge of the spot in the observed image was obscured, it seems that the edge of the crystal begins obstructing the light path. Next, the crystal was moved further into the beam, and the spot on the CCD camera was also moved, resulting in an intensity decrease. This experiment was carried out using 500 electron beam bunches in the storage ring, and the current was approximately 200 mA. A 2 mm Cu filter was inserted in the light path, and the size of the rectangle pinhole was 50×20 μM. The exposure time of the CCD was 0.01 s, and the gain was 16 dB. Images in this process are shown in Fig. 6.

Figure 6:Full-screen View

(Color online) Images and profiles as the crystal was moved into the beam. (a) Image and profile before the crystal cuts the X-ray. (b) Only the edge of the crystal cuts the X-ray. (c) X-rays reflected from the crystal.

Figure 6 shows three typical results of the experiment presented above, and the profile was obtained using direct projection. Those figures show that during the last step of the experiment presented above, the intensity decreased as the spot moved. According to the verification ideas described above, the last image shows how X-rays are reflected from the crystal. In other words, this can be considered as quasi-monochromatic X-ray imaging. However, the intensity was very low, the Cu filter was been moved out before the next experiment. The image obtained at the camera can be seen in Fig. 7.

Figure 7:Full-screen View

(Color online) Image without Cu filter.

The images show horizontal compression due to the inherent deformation of the crystal surface in the horizontal direction and the angle of the YAG:Ce scintillator screen. This paper focuses on the vertical direction, and the vertical beam size measurement will be presented in the following sections.

3.2 Measurement of wavelength

After verifying that the X-ray is monochromatic, the next important thing is to determine the X-ray wavelength, which is very important for calculating the diffraction width of PSFs.

A spectrum is a very useful tool that can be used to characterize the intensity distribution at each wavelength of synchrotron light. If the intensity of the X-rays at each wavelength can be measured, the spectrogram can be obtained from experiments. The intensity of the images obtained from the CCD camera can be used to reflect the intensity of the X-rays. In this work, the sum of the intensities of the images is used to characterize the intensity of the X-rays. By adjusting the angle of the crystal, X-rays with different wavelengths or energies can be obtained, and the spectrum of the extracted X-ray photon beam can be scanned out. In order to obtain the wavelength value corresponding to each intensity, the theoretical spectrogram needs to be used as a reference. The theoretical spectrum can be calculated using the X-ray Oriented Programs (XOP) [14].

Figure 8 shows the theoretical photon beam spectra calculated by XOP after passing through a 1 mm Al window and a 2 mm Cu filter. These curves can be used to fit the spectrum obtained from the experiment, and it is then possible to obtain the wavelength. Fig. 9 shows the integrated intensity of images at each angle, where the glancing angle of the X-ray was changed from large to small in 0.30 deg increments.

Figure 9:Full-screen View

(Color online) Integrated intensity of images at each angle.

By fitting the experimental data in Fig. 9, the wavelength can be extracted. The energy of the peak intensity obtained at this experiment is 28.7 keV, the wavelength is 0.043 nm, and the glancing angle is 3.92 deg. According to Eq. 3, the PSF of the diffraction pattern is 5.5 μ m, while the calculated diffraction contribution of the white X-ray is 6.0 μM [7].

3.3 Beam-based calibration

The idea of the novel beam-based calibration technique requires adjusting the angle of the crystal and observing the image at the CCD camera. The PSF is then derived from these measurements.

Eq. 2 can be rewritten as follows:

The beam size σ_beam and the PSFs other than the diffraction contribution $({\sigma }_{\text{aper}}^2+{\sigma }_{\text{scr}}^2)$ are assumed to be constant. Using the diffraction width of PSFs, the size of the image, and the theoretical beam size, the complete PSF of the pinhole system can be derived.

The theoretical beam size can be described as follows [15]:

where S is the beam size, β and η are the betatron and dispersion functions, respectively, and ϵ and σ_ϵ are the emittance and the relative energy spread of the electron beam, respectively. In this case, β is 12.65 m, σ_ϵ is 0.99e-3, η is 0.0, and ϵ is 32.3 pm-rad in the vertical plane. According to Eq. (6), the beam size in the vertical plane is 20.2 μM. The diffraction width can be calculated with the wavelength obtained from the previous subsection. We assume the beam size of 20.2 μM is constant.

Grazing angles and values of ${\sigma }_{\text{diff}}^2$, ${\sigma }_{\text{img}}^2$, and ${\sigma }_{\text{img}}^2-{(C_{\text{mag}}{\sigma }_{\text{beam}})}^2$ are shown in Table 1.

TABLE. 1 shows the experiment data. According to Eq. 5, $({\sigma }_{\text{aper}}^2+{\sigma }_{\text{scr}}^2)$ is the only unknown variable. This parameter is determined using the measured image sizes and σ_diff. From the data shown in the Table 1, the data conforms to the trend represented by the formula, it is expected that the total PSF can be obtained. This is a multi-equation process of finding approximate solutions, which is more accurate than using a single equation. At the same time, this can effectively reduce the random error.

The combined value of the geometric and imaging system widths of the pinhole PSFs is determined to be 37.7 μM, and the calibrated complete PSF of the pinhole camera is 38.1 μM at a grazing angle of 3.92.

3.4 Results and Discussion

According to the theoretical study of pinhole PSFs presented in Sect. 2, the calculated PSF is 18.4 μM, while the imaging system width is estimated to 10 μM. This calculated PSF is much smaller than the calibrated PSF. One possible reason is that Eq. 4 is not completely suitable for calculating the geometric width, which is based on the assumption of a Gaussian PSF. According to this consideration, simulations have been performed to calculate the PSF.

SRW [16] is a wave optics simulation code that can take the actual wave front of the light emitted by a filament-like source in the bending magnet, propagate it through the pinhole, and project it onto the scintillator. In this work, SRW code was used to calculate the theoretical PSF on the scintillator. In the simulation process, the vertical beam size was set to 20.2 μM, the optical path is shown in Fig. 5, and the photon energy was set to 28.7 keV. The image size obtained at the scintillator is 45.07 μM. In other words, the PSF on the scintillator is 33.35 μM.

Compared with the value obtained from simulation, the calibrated PSF is larger, and there are two possible reasons. The first is that the thickness of the pinhole is also an important factor. However, the thickness of the pinhole was not taken into account in the simulation. The second is that the imaging system width of pinhole PSFs was not contained in the simulation results.

According to the comparison between the theoretical values and the calibrated PSF, the calibrated PSF was larger but more comprehensive. In summary, the new experimental method can derive the PSF of the X-ray pinhole camera. Combined with the X-ray quasi-monochromator, the PSF calibration can be implemented during user operation without any particular beam conditions. With this method, the resolution of the pinhole camera will be higher and the PSF calibration become more convenient.