X-ray Fourier-transform ghost imaging (XFGI) is a new imaging technology that has emerged in recent years. Unlike traditional X-ray imaging, the resolution limit of this imaging method is theoretically limited only by the X-ray wavelength, which makes it possible to obtain higher spatial resolution than real-space imaging. Moreover, the phase information of the sample can also be obtained by this technique, which is of great significance for the imaging of weakly absorbing biological samples. However, the current FGI experiments mainly utilize transmissive pseudo-thermal light sources. The longitudinal size of the speckle generated by the transmissive source is too large, which limits the longitudinal resolution of imaging. Meanwhile, the luminous flux of transmissive pseudo-thermal light is low, which makes it difficult to carry out three-dimensional FGI. In contrast, the reflection method can provide more room for design and adjustment, allowing the longitudinal size of the speckle to be reduced. Furthermore, grazing incidence can be used to increase the reflected light flux and hence improve imaging quality. Therefore, the three-dimensional characteristics of the speckle field of the reflective pseudo-thermal light source are studied in hope of guiding and improving the light source system of three-dimensional FGI and thereby enhancing the longitudinal imaging resolution.

According to the Fresnel diffraction theory, the position of the pending sample is taken as the observation position, and the correlation function of the intensity fluctuation of the speckle field emitted by the scattering screen is derived. The speckle size is defined as the full width at half maximum of the speckle field. The longitudinal speckle size of speckle fields emitted by scattering screens at different spatial positions and inclination angles is discussed. Then, the numerical simulation based on statistical optics is carried out in the visible light region. Through the simulation, the light-field intensity distribution of the speckle field is obtained, and the normalized light intensity fluctuation correlation function between the observation position and its adjacent area under the ensemble average is calculated to verify the theoretical results. Moreover, various factors affecting the longitudinal speckle size at the observation position of the speckle field are analyzed, such as the size of the scattering screen, propagation distance, scattering angle, and azimuth angle. Finally, for high-resolution three-dimensional XFGI, the longitudinal speckle size of the grazing-incidence silicon scattering screen reflective pseudo-thermal light source is also considered.

From the point of view of statistical optics, the longitudinal speckle size of speckle fields emitted by scattering screens at different spatial positions and inclination angles is given, which is consistent with the numerical simulation results under the ensemble average (Fig. 3). In the visible light region, the influence of various factors on the longitudinal speckle size at the observation position is discussed and numerically simulated (Fig. 4). The longitudinal speckle size can be effectively reduced by the following four strategies, i.e., increasing the size of the scattering screen, increasing the azimuth angle of the scattering screen, shortening the propagation distance, and reducing the scattering angle. In the end, the longitudinal speckle size at the observation position is decreased to less than 100 nm under the rational design of the spatial position and inclination angles of the silicon scattering screen in the X-ray region. When the wavelength of the incident light wave is 0.1 nm, the size of the silicon scattering screen is 20 mm, the incident angle and the scattering angle are both 89.86°, the azimuth angle of the scattering screen is 20°, and the propagation distance is 20 mm, the longitudinal speckle size at the observation position can reach 79.31 nm, and the lateral speckle size can reach 28.94 nm (Fig. 5) .

In this paper, the second-order statistical characteristics of the speckle field generated by the reflective pseudo-thermal light source are studied, and the intensity fluctuation correlation function of the speckle field of the reflective pseudo-thermal light source is theoretically derived. In addition, the longitudinal speckle size of the scattering screen placed at different spatial positions and inclination angles is given. The numerical simulation results based on the statistical optics are consistent with the theoretical results. The results show that the size, spatial position, and tilt state of the scattering screen will affect the speckle size at the observation position. The size of the speckle decreases with both the incident light wave's wavelength and the propagation distance. The size of the projection scattering screen in the scattering direction can be increased, which will reduce the size of speckles under a larger scattering screen and a smaller scattering angle. Moreover, the longitudinal speckle size decreases with the increasing azimuth of the scattering screen and changes abruptly at small angles since the speckle size along the scattering path is significantly bigger than that in other directions. Finally, the longitudinal speckle size of the reflective pseudo-thermal light source under different parameters is given in the X-ray region. The longitudinal speckle size at the observation position can be effectively reduced to less than 100 nm under the rational design of the spatial position and inclination angle of the scattering screen. In this way, the longitudinal resolution of imaging can be improved, which is of great significance for the development of three-dimensional FGI.