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Abstract
Ghost imaging based on the high-order correlation of optical field has developed rapidly and has been extended to the x-ray region. However, the limited flux leads to severe image deterioration. Here, an approach of Fourier-transform ghost imaging with super-Rayleigh speckles is proposed to realize high quality ghost imaging at low photon flux level. The super-Rayleigh speckles are designed by optimizing binary modulating screens based on the direct binary search algorithm. The experimental results show that the speckle contrast can be greatly enhanced and high visibility Fourier-transform diffraction pattern of the sample can be obtained. The sample’s image in spatial domain is successfully achieved even if the detected photon level decreases to 0.1 photons/pixel. This method is of great importance for high-resolution imaging in the photon-limited scenarios, especially for laboratory x-ray systems.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Ghost imaging(GI) has attracted much attention since its first demonstration in the quantum regime with entangled light generated by spontaneous parametric down-conversion [1]. In 2004, it was theoretically proved that ghost imaging can be also realized with classical thermal light [2,3], and later the experimental validation was performed with a pseudo-thermal source [4,5]. Since then, ghost imaging technology has developed rapidly and many schemes have been proposed, such as differential GI, compressive GI and correspondence GI [6–10]. Nowadays, ghost imaging has been widely applied in remote sensing, super-resolution imaging, spectral imaging, three-dimensional imaging, etc [11–14]. In 2016, ghost imaging was extended to the x-ray region and experimentally demonstrated in both real space and reciprocal space [15–22]. The real-space x-ray GI experiments have been carried out using synchrotron and laboratory x-ray sources [15–18]. In the reciprocal space, x-ray Fourier-transform ghost imaging(FGI) has been demonstrated using synchrotron x-rays [21,22], which provides the possibility of realizing high-resolution x-ray exploration under the framework of ghost imaging. Considering that the access to synchrotron is limited, small x-ray sources will be more suitable for laboratory and factory applications. However, the image quality is usually related to the photon flux. In the case of laboratory x-ray system, the brightness of the source is limited, which may leads to severe image deterioration. Moreover, in order to avoid radiation damage, the x-ray flux illuminating sample needs to be restricted, especially for biological samples [23–25]. To improve the image quality in ghost imaging, compressive sensing [26–28], superbunching pseudothermal light source [29,30] and speckle manipulation are useful strategies [31–35]. Compressive sensing is more suitable for the cases of insufficient samplings. Superbunching pseudothermal light source is hard to realize in x-ray imaging for the lack of the corresponding modulators. Here we provide a solution of realizing x-ray FGI under the condition of ultra-low photon flux based on speckle manipulation.
Typical speckle patterns used in ghost imaging include Hadamard basis pattern, orthogonal sinusoidal pattern, multi-scale pattern, etc [36–38]. They can enhance the measurement efficiency or the image resolution, but do not serve to increase the speckle contrast which plays a dominant role in ghost imaging under limited photon flux. The contrast of a speckle pattern is defined as $C=\sqrt {\left \langle {{I}^{2}} \right \rangle _s /{{\left \langle I \right \rangle }_s^{2}}-1}$, where $I$ is the intensity of the speckle pattern and $\left \langle \cdots \right \rangle _s$ denotes the spatial averaging over the speckle plane [39]. In conventional GI, the speckle generated by a modulating screen is Rayleigh speckle with the contrast of $C=1$ [34]. By optimizing the modulating screen, the super-Rayleigh speckle whose contrast is higher than 1 can be achieved. It may improve the noise immunity and realize ghost imaging with less signal photons. For visible light, the super-Rayleigh speckle is produced by using a SLM as the modulator and loading continuous phase distribution onto the SLM. The phase distribution is obtained through the inverse propagation of high-order Rayleigh speckle [39,40]. However, due to the limited fabrication technique of x-ray optics, a binary x-ray modulator is more realistic than the continuous one. The inverse propagation method is not applicable in this situation.
In this paper, an approach of FGI with super-Rayleigh speckles has been proposed to realize high quality x-ray ghost imaging at low photon flux level. The theoretical model of this approach has been established, and the super-Rayleigh speckle is obtained by optimizing the binary modulating screen based on the direct binary search(DBS) algorithm. Relevant experimental results show that the quality of the Fourier-transform diffraction pattern of the sample obtained under low photon flux can be significantly improved, and the sample’s image in real space is successfully recovered.
2. Theory and method
2.1 Principle of FGI with super-Rayleigh speckles
The scheme of our method is shown in Fig. 1. The light emitted from the source is divided into two beams after passing through a modulating screen. In the reference beam, the light propagates freely and is recorded by a panel detector. In the test beam, a sample is inserted and the optical field is detected by a point detector. The distance from the modulating screen to the sample is $d_1$, and the distance from the sample to the detector is $d_2$. The distance from the modulating screen to the panel detector is $d_1+d_2$. Different from conventional FGI scheme, the modulating screen in our scheme is specifically designed to generate super-Rayleigh speckles.
The correlation between the detected intensity fluctuations of the reference beam and the test beam can be expressed as [3]
Suppose the area of the detector is large enough, the source can be regarded as a phase-conjugation mirror, then the relationship between ${{E}_{t}}\left ( {{x}_{t}} \right )$ and ${{E}_{r}}\left ( {{x}_{r}} \right )$ can be described as [41]
Here, $t(x)$ is the transmittance of the sample, $\lambda$ is the wavelength.
The super-Rayleigh speckle field in the reference beam can be obtained by adding an exponential factor $n$ to a conventional Rayleigh speckle field $E_{Ray}(x_{r})$ [39], which is
The optical field with Rayleigh distribution can be regarded as a Gaussian random process with zero mean. According to the Gaussian moment theorem, the fourth-order correlation of the optical field in the reference beam can be obtained as following [40]
Similarly, the second-order correlation of the optical field in the reference beam is
Assuming that the spatial coherence width of the Rayleigh speckle field is small enough, the second-order correlation of the Rayleigh speckle field is proportional to a delta function, which is [35]
Substituting Eq. (2)–Eq. (6) into Eq. (1), after some calculations, we have
The speckles generated with different $n$ may influence the visibility of the ghost imaging system. The visibility of ghost imaging can be evaluated by the normalized second-order intensity correlation of the speckle field [42], which is defined as [43]
Similar to the previous derivation, the peak value of ${{g}^{\left ( 2 \right )}}$ can be obtained, which is
For Rayleigh speckles, $n=1$ and $g_{\max }^{\left ( 2 \right )}=2$. When it comes to super-Rayleigh speckles with $n>1$, the value of $g_{\max }^{\left ( 2 \right )}$ will be higher than 2. Therefore, the imaging visibility can be improved by exploiting super-Rayleigh speckles. This is of great significance for achieving high-quality ghost imaging under photon-limited condition, especially in the presence of noise.
2.2 Super-Rayleigh speckle generation
Considering the fabrication difficulty of x-ray modulator, we choose a binary modulating model to generate the super-Rayleigh speckle. As a start, a Rayleigh speckle is generated with a random binary modulating screen through the Fresnel diffraction calculation. Then the target super-Rayleigh speckle is obtained by adding an exponential factor $n$ to the Rayleigh speckle. Finally, considering the optimization target is a binary modulating screen, an optimization process based on the direct binary search(DBS) algorithm [44] is carried out to find the matching binary modulating screen for this target super-Rayleigh speckle.
The binary modulating screen can be treated as a matrix composed of binary transmittance elements, which is
Figure 2 shows the flow chart of the optimization process. In the beginning, a random binary matrix ${{T}_{b}}\left ( k,l \right )$ is initialized, and its corresponding Rayleigh speckle pattern is obtained by the Fresnel diffraction calculation. The objective function is the mean square error (MSE) between the obtained speckle pattern and the target speckle pattern, which is
After switching, we update the matrix and the corresponding speckle pattern, and calculate the new MSE and compare it with the previous one. If the MSE decreases, the updated matrix and speckle pattern are retained, otherwise the previous matrix and speckle pattern are retained. Now, if the MSE is less than the preset value, the optimization is successful and the matching matrix for the target super-Rayleigh speckle is achieved. If not, another pixel is selected and the optimization process is executed again.
3. Results and discussion
Experimental validation was performed with visible light, and a spatial light modulator (SLM) was adopted in the experiment. SLM is usually amplitude-only type or phase-only type, and modulators with complex transmittance cannot be directly simulated by SLM. However, binary modulators with complex transmittance are more practical in the x-ray region. In order to simulate this feature, we adopted the Fourier transform holographic method in the experiment.
The experiment setup is presented in Fig. 3. The diagram for the Fourier transform holographic method is shown in the blue dotted box. When the hologram of an object located on the front focal plane of a lens is illuminated, the object appears on the back focal plane and can be extracted by an aperture. In our experiment, the hologram of the designed modulating screen was loaded onto an amplitude-only SLM positioned at the front focus of the lens, and the pattern of the modulating screen with complex transmittance was obtained at the back focal plane of the lens.
Fig. 3. Experimental setup. The binary modulating screen is generated based on the Fourier transform holographic method.
The experimental layout to generate the pattern of the binary modulating screen is displayed in the green dotted box. The 532 nm laser light passes through an adjustable attenuation plane (AAP), a polarizer P1, a 10$\times$ beam expander, a beam splitter, and illuminates the amplitude-only SLM with a pixel size of $8\;\mathrm{\mu}{\rm m}$. The light modulated by the SLM is reflected and goes through a polarizer P2. The polarization directions of P1 and P2 are in accordance with the long axis and short axis direction of the SLM, respectively. The lens is placed behind P2, and the optical distance from the SLM to the lens is equal to the focal length. The aperture A1 is positioned at the back focal plane of the lens. After loading the hologram onto the SLM, the pattern of the binary modulating screen can be taken out with A1.
The aperture A2 is placed in the beam to define an illumination area of $1\, {\rm mm} \times 1 \, {\rm mm}$ on the sample plane. When the sample is inserted into the beam close to A2, the optical path serves as the test beam. When the sample is moved out, the optical path serves as the reference beam. At the end of the optical path is a camera with a pixel size of $6.5 \;\mathrm{\mu}\text {m}\times 6.5\;\mathrm{\mu}\text {m}$. The distance from A1 to A2 was 11 cm, and the distance from A2 to the detector was 19 cm.
In the experiment, the DBS algorithm was implemented to generate the super-Rayleigh speckle. The size of the binary modulating screen was $2.7\text { } {\rm mm}$, and the dimension of the corresponding binary matrix describing the screen was $100\times 100$. The transmission difference between the two possible states of the modulating screen was $11{\% }$, and the phase difference between them was $\pi$. The matrix was randomly initialized and the corresponding Rayleigh speckle was obtained through Fresnel diffraction simulation. The target super-Rayleigh speckle was produced by adding an exponential factor $n=3$ to the Rayleigh speckle. The preset MSE value in the optimization was set to $6.8\times 10^{-6}$. When a higher value of $n$ is adopted, the convergence rate of the algorithm will slow down.
Figure 4(a)-(c) shows the patterns of the simulated Rayleigh speckle, the target super-Rayleigh speckle, and the super-Rayleigh speckle obtained in our experiment. It can be observed that the distribution of the experimental super-Rayleigh speckle is very similar to that of the target speckle. The contrast of the experimental super-Rayleigh speckle is 2.5, which is higher than the contrast of the Rayleigh speckle. The contrast of the Rayleigh speckle is 1.0. The normalized second-order intensity correlation functions of the experimental and target super-Rayleigh speckle were calculated according to Eq. (8) with 753 measurements. The ${g}^{(2)}$ values versus $\Delta x={{x}_{{{r}_{1}}}}-{{x}_{{{r}_{2}}}}$ are plotted in Fig. 4(d). The ${g}^{(2)}$ peak value of the experimental super-Rayleigh speckle is about 4.8, which is much higher than 2. This indicates that high visibility ghost imaging can be realized with the designed super-Rayleigh speckle. Due to the difference between the target speckle and the super-Rayleigh speckle generated by optimization, as well as the background noise in the experiment, the contrast of the experimental super-Rayleigh speckle is lower than expected. Hence, the ${g}^{(2)}$ peak value of the experimental super-Rayleigh speckle is lower than the peak of the target super-Rayleigh speckle. Figure 4(e) gives the convergence rate of the optimization process. As the number of the selected pixels increases, the curve finally drops to the preset MSE value.
Fig. 4. Generation of the super-Rayleigh speckle. (a) is the simulated Rayleigh speckle pattern, (b) is the target super-Rayleigh speckle pattern, and (c) is the super-Rayleigh speckle pattern obtained in the experiment. (d) is the ${g}^{(2)}$ curves calculated from the target and experimental super-Rayleigh speckle patterns. (e) is the convergence rate of the optimization process.
The intensity of the super-Rayleigh speckle is concentrated on several speckle grains, and the rest areas are dark. This facilitates the speckle detection at low photon flux level, and can improve the image quality. We adjusted the AAP to control the photon flux, and inserted a sample into the beam. The sample consists of seven square holes, six of which are arranged around the central hole in a regular hexagon, as shown in Fig. 5(c). The size of each square is $100\;\mathrm{\mu}\text {m}$, and the side length of the hexagon is $200\;\mathrm{\mu}\text {m}$. The theoretical Fourier-transform diffraction pattern of the sample is shown in Fig. 5(d). In the experiment, 753 pairs of speckle patterns were acquired, and a region of 700 $\times$ 700 pixels in each pattern was used to calculate the intensity correlation.
Fig. 5. FGI results under photon-limited conditions. (a) and (b) are the results of FGI with super-Rayleigh speckles and Rayleigh speckles under the condition of $PPP_t= 5$ and $PPP_t = 0.1$, respectively. (c) is the sample’s photograph. (d) is the theoretical Fourier-transform diffraction pattern of the sample. (e) is the visibility curves of the Fourier-transform diffraction patterns obtained with super-Rayleigh and Rayleigh speckles at different photon flux levels.
Figure 5 presents the results of FGI with super-Rayleigh speckles. For quantitative evaluation, two parameters related to the signal photon and noise level are introduced. The photon per pixel(PPP) is defined as $PPP={{N}_{ph}}/{N}$, where $N_{ph}$ is the number of signal photons measured when acquiring an image and $N$ is the total number of pixels of the image [45]. Considering that a series of speckle patterns are used in each imaging process, we use the average PPP of these patterns to describe the photon efficiency. The signal-to-noise ratio(SNR) is defined as $SNR={S}/{\sigma }$, where $S$ is the mean signal intensity of the image and $\sigma$ is the standard deviation of the background noise [46]. Similarly, we use the average SNR of the speckle patterns to reflect the noise level. In the experiment, the average PPP and average SNR were calculated from the speckle pattern obtained in the test beam, which can be described as
The background noise image was recorded by turning off the laser. $N_{ph}$ was calculated by subtracting the photon number of the noise image from the photon number of the speckle pattern. $S$ was obtained by subtracting the noise intensity from the speckle intensity, and $\sigma$ was the standard deviation of the intensity distribution of the noise image.
Figure 5(a) is the results of FGI with super-Rayleigh speckles under the condition of $PPP_t = 5$ and $SNR_t = 0.57$. In our experiment, the standard deviation $\sigma$ = 10.9. The left side shows a pair of speckle patterns recorded in the reference beam(up) and the test beam(down), and the right side shows the Fourier transform diffraction pattern obtained by intensity correlation calculation. For comparison, the corresponding results of Rayleigh speckle are also presented. The total photon flux in the Rayleigh and super-Rayleigh speckle experiments remained unchanged. The diffraction pattern obtained from the super-Rayleigh speckles is better than that from the conventional Rayleigh speckles because of the darker background and the sharper diffraction peaks. The color bars next to the speckle patterns show the range of photon counts. The maximum photon counts of the speckle recorded in the test beam are almost the same for the Rayleigh and super-Rayleigh cases. While in the reference beam, the maximum photon count of the super-Rayleigh speckle is much higher than that of the Rayleigh speckle. It means that higher photon counts can be detected by using super-Rayleigh speckle at the same illumination flux. This provides the potential of using super Rayleigh speckle to enhance the noise immunity of FGI.
Figure 5(b) is the results under the condition of $PPP_t = 0.1$ and $SNR_t = 0.01$. The detected speckles are drowned in the noise for both the Rayleigh and the super-Rayleigh cases. The Fourier-transform diffraction pattern of the Rayleigh case deteriorates sharply, but the diffraction pattern of the super-Rayleigh case is still clear. Thus, high-quality Fourier-transform diffraction pattern can be achieved at low photon flux level with super-Rayleigh speckles.
The performance of FGI with super-Rayleigh speckles was further investigated by the visibility evaluation. The visibility of an image is defined as [42]
Figure 6 gives the recovered images of the sample in spatial domain. The widely used hybrid input-output algorithm was adopted in phase retrieval [47]. It can be seen that in the case of super-Rayleigh speckle, the shapes of the sample are restored well with clear edges, while the reconstructed images in the case of Rayleigh speckle are distorted. Especially when $PPP_t$ is reduced to 0.1, most of the sample information is lost. Therefore, FGI with super-Rayleigh speckles has better performance than traditional FGI with Rayleigh speckles, and can realize high-quality imaging at a limited photon flux level.
Fig. 6. Recovered images of the sample in spatial domain. (a) and (b) are the results reconstructed from the Fourier-transform diffraction patterns in Fig. 5(a) and Fig. 5(b), respectively.
In x-ray applications, no SLM-like devices are available, and porous membrane can be fabricated as the binary modulating screen. The thickness of the membrane is usually set to produce a phase difference of $\pi$ between the area with and without holes. For x-ray with a wavelength of 1 nm, a golden porous membrane with a thickness of about 340 nm provides a phase difference of $\pi$. Then the DBS algorithm can be used to design the hole distribution on the membrane, and the super-Rayleigh speckle will be obtained.
A proof-of-concept simulation of x-ray FGI with super-Rayleigh speckles was performed. In our simulation, the x-ray beam illuminating the modulating screen was partially coherent and the x-ray wavelength was 1 nm. The modulating screen was positioned at a distance of 0.6 m from a laboratory x-ray source with the source size of $20\;\mathrm{\mu}{\rm m}$. This configuration resulted in a coherence length of approximately $30\;\mathrm{\mu}{\rm m}$ on the plane of the modulating screen (calculated as $1\text { nm}\times 0.6\text { m}\;/\;\text {20}\;\mathrm{\mu}\text {m = 30 }\;\mathrm{\mu}\text {m}$). The pore size of the modulating screen was set to 150 nm. The sample used in our simulation is similar to a triangular DNA origami [48], which is shown in Fig. 7(a). Its corresponding theoretical Fourier-transform diffraction pattern is displayed in Fig. 7(b). The Fourier-transform diffraction patterns can be achieved by intensity correlation calculation, and the results are given in Fig. 7(f) and (g). Obviously, the pattern obtained from the super-Rayleigh speckles is clearer than the Rayleigh ones, and contains more structural information of the sample. It indicates the potential of realizing FGI with a tabletop x-ray source using super-Rayleigh speckles. The impact of the coherent length was also considered, and the results are shown in Fig. 7(c)-(e). The partial coherence is evaluated by the value of $R_c$, which is the ratio of the spatial coherence length on the plane of the modulating screen to the pore size of the modulating screen. Under the condition of ${{R}_{c}}\ge \text {10}$, the diffraction peaks in the diffraction pattern are visible and contain the Fourier information of the sample. As the value of $R_c$ increases, the RMSE value decreases, and the quality of the diffraction pattern will be enhanced, leading to better image quality.
Fig. 7. Simulation results for FGI with a laboratory x-ray source. (a) is the image of the sample and (b) is the theoretical Fourier-transform diffraction pattern of the sample. (c)-(e) show the impact of the coherent length. (c) and (d) are the Fourier transform diffraction patterns for different $R_c$. (e) is the relationship between RMSE and $R_c$. (f) and (g) are the FGI results with super-Rayleigh speckles and Rayleigh speckles, respectively. The Fourier-transform diffraction patterns are in logarithmic scale.
4. Conclusion
In summary, we present an approach of FGI with super-Rayleigh speckles to realize high quality x-ray ghost imaging under limited photon flux. It is proved theoretically that the Fourier-transform diffraction pattern of the sample can be extracted by intensity correlation calculation. The super-Rayleigh speckle and the corresponding binary modulating screen can be obtained using the DBS algorithm. Theoretical analysis and experimental results show that the peak value of ${{g}^{\left ( 2 \right )}}$ for super-Rayleigh speckle is greater than 2, which indicates that the designed super-Rayleigh speckle can enhance the imaging visibility. When the detected photon level decreases to 0.1 photons/pixel, the Fourier-transform diffraction pattern of the sample still has high visibility, and the sample’s image in spatial domain can be successfully recovered. This method has great potential for realizing high quality ghost imaging, especially for laboratory x-ray systems, and it can also be used in other photon limited scenarios.
Funding
National Natural Science Foundation of China (11627811); National Key Research and Development Program of China (2017YFB0503303).
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.
References
1. A. V. Belinskii and D. N. Klyshko, “Two-photon optics: diffraction, holography, and transformation of two-dimensional signals,” J. Exp. Theor. Phys. 78(3), 259–262 (1994).
2. A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. 93(9), 093602 (2004). [CrossRef]
3. J. Cheng and S. Han, “Incoherent coincidence imaging and its applicability in x-ray diffraction,” Phys. Rev. Lett. 92(9), 093903 (2004). [CrossRef]
4. F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. 94(18), 183602 (2005). [CrossRef]
5. A. Valencia, G. Scarcelli, M. D’Angelo, and Y. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. 94(6), 063601 (2005). [CrossRef]
6. F. Ferri, D. Magatti, L. Lugiato, and A. Gatti, “Differential ghost imaging,” Phys. Rev. Lett. 104(25), 253603 (2010). [CrossRef]
7. O. Katz, Y. Bromberg, and Y. Silberberg, “Compressive ghost imaging,” Appl. Phys. Lett. 95(13), 131110 (2009). [CrossRef]
8. K. H. Luo, B. Q. Huang, W. M. Zheng, and L. A. Wu, “Nonlocal imaging by conditional averaging of random reference measurements,” Chin. Phys. Lett. 29(7), 074216 (2012). [CrossRef]
9. K. W. C. Chan, M. N. O’Sullivan, and R. W. Boyd, “High-order thermal ghost imaging,” Opt. Lett. 34(21), 3343–3345 (2009). [CrossRef]
10. C. Hu, R. Zhu, H. Yu, and S. Han, “Correspondence fourier-transform ghost imaging,” Phys. Rev. A 103(4), 043717 (2021). [CrossRef]
11. B. I. Erkmen, “Computational ghost imaging for remote sensing,” J. Opt. Soc. Am. A 29(5), 782–789 (2012). [CrossRef]
12. Z. Chen, J. Shi, Y. Li, Q. Li, and G. Zeng, “Super-resolution thermal ghost imaging based on deconvolution,” Eur. Phys. J. Appl. Phys. 67(1), 10501 (2014). [CrossRef]
13. Z. Liu, S. Tan, J. Wu, E. Li, X. Shen, and S. Han, “Spectral camera based on ghost imaging via sparsity constraints,” Sci. Rep. 6(1), 1–10 (2016). [CrossRef]
14. N. D. Hardy and J. H. Shapiro, “Computational ghost imaging versus imaging laser radar for three-dimensional imaging,” Phys. Rev. A 87(2), 023820 (2013). [CrossRef]
15. D. Pelliccia, A. Rack, M. Scheel, V. Cantelli, and D. M. Paganin, “Experimental x-ray ghost imaging,” Phys. Rev. Lett. 117(11), 113902 (2016). [CrossRef]
16. A. Schori and S. Shwartz, “X-ray ghost imaging with a laboratory source,” Opt. Express 25(13), 14822–14828 (2017). [CrossRef]
17. A.-X. Zhang, Y.-H. He, L.-A. Wu, L.-M. Chen, and B.-B. Wang, “Tabletop x-ray ghost imaging with ultra-low radiation,” Optica 5(4), 374–377 (2018). [CrossRef]
18. Y. Klein, A. Schori, I. P. Dolbnya, K. Sawhney, and S. Shwartz, “X-ray computational ghost imaging with single-pixel detector,” Opt. Express 27(3), 3284–3293 (2019). [CrossRef]
19. A. M. Kingston, D. Pelliccia, A. Rack, M. P. Olbinado, Y. Cheng, G. R. Myers, and D. M. Paganin, “Ghost tomography,” Optica 5(12), 1516–1520 (2018). [CrossRef]
20. Y. Y. Kim, L. Gelisio, G. Mercurio, et al., “Ghost imaging at an xuv free-electron laser,” Phys. Rev. A 101(1), 013820 (2020). [CrossRef]
21. H. Yu, R. Lu, S. Han, H. Xie, G. Du, T. Xiao, and D. Zhu, “Fourier-transform ghost imaging with hard x rays,” Phys. Rev. Lett. 117(11), 113901 (2016). [CrossRef]
22. Z. Tan, H. Yu, R. Zhu, R. Lu, S. Han, C. Xue, S. Yang, and Y. Wu, “Single-exposure Fourier-transform ghost imaging based on spatial correlation,” Phys. Rev. A 106(5), 053521 (2022). [CrossRef]
23. P. Villanueva-Perez, S. Bajt, and H. N. Chapman, “Dose efficient compton x-ray microscopy,” Optica 5(4), 450–457 (2018). [CrossRef]
24. M. Weik, R. B. Ravelli, G. Kryger, S. McSweeney, M. L. Raves, M. Harel, P. Gros, I. Silman, J. Kroon, and J. L. Sussman, “Specific chemical and structural damage to proteins produced by synchrotron radiation,” Proc. Natl. Acad. Sci. 97(2), 623–628 (2000). [CrossRef]
25. R. B. Ravelli and S. M. McSweeney, “The ’fingerprint’ that x-rays can leave on structures,” Structure 8(3), 315–328 (2000). [CrossRef]
26. V. Katkovnik and J. Astola, “Compressive sensing computational ghost imaging,” J. Opt. Soc. Am. A 29(8), 1556–1567 (2012). [CrossRef]
27. M. Aßmann and M. Bayer, “Compressive adaptive computational ghost imaging,” Sci. Rep. 3(1), 1545 (2013). [CrossRef]
28. S.-M. Zhao and P. Zhuang, “Correspondence normalized ghost imaging on compressive sensing,” Chin. Phys. B 23(5), 054203 (2014). [CrossRef]
29. Y. Zhou, F.-l. Li, B. Bai, H. Chen, J. Liu, Z. Xu, and H. Zheng, “Superbunching pseudothermal light,” Phys. Rev. A 95(5), 053809 (2017). [CrossRef]
30. J. Liu, R. Zhuang, X. Zhang, C. Wei, H. Zheng, Y. Zhou, H. Chen, Y. He, and Z. Xu, “Simple and efficient way to generate superbunching pseudothermal light,” Opt. Commun. 498, 127264 (2021). [CrossRef]
31. X.-F. Liu, M.-F. Li, X.-R. Yao, W.-K. Yu, G.-J. Zhai, and L.-A. Wu, “High-visibility ghost imaging from artificially generated non-gaussian intensity fluctuations,” AIP Adv. 3(5), 052121 (2013). [CrossRef]
32. E.-F. Zhang, W.-T. Liu, and P.-X. Chen, “Ghost imaging with non-negative exponential speckle patterns,” J. Opt. 17(8), 085602 (2015). [CrossRef]
33. Z. Sun, F. Tuitje, and C. Spielmann, “Toward high contrast and high-resolution microscopic ghost imaging,” Opt. Express 27(23), 33652–33661 (2019). [CrossRef]
34. K. Kuplicki and K. W. C. Chan, “High-order ghost imaging using non-rayleigh speckle sources,” Opt. Express 24(23), 26766–26776 (2016). [CrossRef]
35. S. Zhang, W. Wang, R. Yu, and X. Yang, “High-order correlation of non-rayleigh speckle fields and its application in super-resolution imaging,” Laser Phys. 26(5), 055007 (2016). [CrossRef]
36. M.-J. Sun, L.-T. Meng, M. P. Edgar, M. J. Padgett, and N. Radwell, “A russian dolls ordering of the hadamard basis for compressive single-pixel imaging,” Sci. Rep. 7(1), 1–7 (2017). [CrossRef]
37. S. M. Khamoushi, Y. Nosrati, and S. H. Tavassoli, “Sinusoidal ghost imaging,” Opt. Lett. 40(15), 3452–3455 (2015). [CrossRef]
38. S. Ma, C. Hu, C. Wang, Z. Liu, and S. Han, “Multi-scale ghost imaging lidar via sparsity constraints using push-broom scanning,” Opt. Commun. 448, 89–92 (2019). [CrossRef]
39. Y. Bromberg and H. Cao, “Generating non-rayleigh speckles with tailored intensity statistics,” Phys. Rev. Lett. 112(21), 213904 (2014). [CrossRef]
40. S. Liu, Z. Liu, C. Hu, E. Li, X. Shen, and S. Han, “Spectral ghost imaging camera with super-rayleigh modulator,” Opt. Commun. 472, 126017 (2020). [CrossRef]
41. R. Zhu, H. Yu, R. Lu, Z. Tan, and S. Han, “Spatial multiplexing reconstruction for fourier-transform ghost imaging via sparsity constraints,” Opt. Express 26(3), 2181–2190 (2018). [CrossRef]
42. W. Gong and S. Han, “A method to improve the visibility of ghost images obtained by thermal light,” Phys. Lett. A 374(8), 1005–1008 (2010). [CrossRef]
43. R. J. Glauber, “The quantum theory of optical coherence,” Phys. Rev. 130(6), 2529–2539 (1963). [CrossRef]
44. M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 26(14), 2788–2798 (1987). [CrossRef]
45. X. Liu, J. Shi, X. Wu, and G. Zeng, “Fast first-photon ghost imaging,” Sci. Rep. 8, 1–8 (2018). [CrossRef]
46. D. W. O’Connell, K. S. Morgan, G. Ruben, F. Schaff, L. C. Croton, G. A. Buckley, D. M. Paganin, K. Uesugi, and M. J. Kitchen, “Photon-counting, energy-resolving and super-resolution phase contrast x-ray imaging using an integrating detector,” Opt. Express 28(5), 7080–7094 (2020). [CrossRef]
47. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982). [CrossRef]
48. H. Jun, X. Wang, W. P. Bricker, and M. Bathe, “Automated sequence design of 2D wireframe DNA origami with honeycomb edges,” Nat. Commun. 10(1), 5419 (2019). [CrossRef]
