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Abstract
A single-shot phase retrieval algorithm based on a random aperture and partially saturated diffraction pattern is proposed. The diffraction pattern in the saturated area could be retrieved during the iterative process, which circumvents the problem of limited dynamic range of the detector. Besides, the random aperture is easier to be manufactured and if the accuracy of the random aperture is high enough, the design value could be used directly for iterations. It has the potential to be adapted for different wavelengths without additional transmission measurement of the wave modulator. The validity has been demonstrated by simulations and experiment.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Coherent diffractive imaging (CDI) [1–4] is a widely used lensless imaging technique to retrieve phase information from recorded diffraction patterns using an iterative engine. As it does not require a high-quality imaging lens and in theory provides a diffraction-limited resolution, this technique plays an important role in a variety of fields, such as biology [5–7] and material science [8, 9], where the use of X-rays and electrons creates challenges for imaging lens manufacturing. In order to guarantee the convergence speed during the iterative process in CDI, different strategies are employed. The first strategy relies on information redundancy introduced by multiple diffraction patterns, such as the widely used ptychographic iterative engine (PIE) [10,11] providing an unlimited field of view and high convergence speed with a series of diffraction patterns recorded while translating the specimen parallel to the detector with a step size smaller than the probe diameter. However, the accumulated radiation damage [12,13] from X-rays or electrons is still inevitable in PIE, as well as fly-scan PIE [14], when imaging biological samples, such as frozen-hydrated biological specimens [15]; For XFEL (X-ray free-electron lasers) which providing ultra-bright X-ray pulses, single-shot ‘diffraction before destruction’ experiments are common [16]. Therefore, a single-shot phase retrieval algorithm is still desired. Based on the fact that the Fourier transform of their diffraction pattern could be divided distinctly into autocorrelation and cross-correlations for two sufficiently separated objects, single-shot phase retrieval with the diffraction of separated object is available [16]. For the continuous object, single-shot phase retrieval could be available in theory by employing another strategy, pre-characterized wave modulation [17], such as coherent modulation imaging (CMI) [18,19] which applies a pre-characterized random phase plate with a binary phase delay, 0 or π, to modulate the incident wave; this has been successfully used for phase imaging with X-rays [20]. However, although the convergence speed is high, the signal-to-noise ratio (SNR) of the final reconstruction is limited by not detecting high-order diffraction, which is strongly scattered by the phase plate and too weak to be recorded by the detector. As a result, the practical application of this approach is restricted to a certain extent compared with other algorithms.
As an alternative to a phase plate, a random aperture, which is insensitive to the wavelength and easy to manufacture with high precision, as compared with a phase plate, can be used as a wave modulator for single-shot phase retrieval. In fact, a random aperture has been used to improve the final reconstruction in CDI [21,22], and a set of aperture masks generated by a digital micro-mirror device (DMD) [23] or programmable spatial light modulator (SLM) [24,25] have been applied for phase retrieval. However, the demand for single-shot operation is not satisfied, and DMDs or SLMs are not applicable for X-rays and electrons. In addition, a sparsity constraint based on compressive sensing [26,27] could be used for single-shot phase imaging based on a random aperture; however, according to the results, including the multi-mask phase retrieval algorithms mentioned above, the final resolution is limited. As a result, a practical single-shot phase retrieval algorithm based on a random aperture is still absent.
In this paper, a single-shot phase retrieval algorithm, coherent amplitude modulation imaging (CAMI), based on a random aperture is proposed. Owing to the partially saturated diffraction pattern, space-limiting and band-limiting constraints, the competitive performance for phase retrieval is available compared with CMI under the condition of appropriate amounts of saturated pixels. The validity of the approach is demonstrated through numerical simulations and experiment. This can become a practical algorithm to satisfy the demands of single-shot phase imaging with high resolution in a variety of fields.
2. Theory
As illustrated in Fig. 1(a), for a random aperture providing binary amplitude modulation for the incident wave, strong zero-order diffraction is recorded while the weak high-order diffraction is absent due to the limited dynamic range of the detector. This is universal for a weak scattering object, which has a narrow spectral width, as illustrated in Fig. 1(b). However, the high-order diffraction contains the key information on the aperture-induced modulation and is responsible for high convergence. As a result, the absence of information is the main obstacle for phase retrieval with weak wave modulation from random aperture. If the low frequency components of the random aperture are blocked, for example, by setting them to zero as shown in Fig. 1(c), the weak modulation information becomes detectable. As a result, using a beam stop, which is common in X-ray CDI [28,29], or just keeping the detector saturated, is essential. Although the zero-order diffraction is not recorded, it is not absolutely lost and can be retrieved during the iterative process considering that the diffracted wave is the convolution between the spectrum of the incident wave and random aperture. The validity to retrieve lost information for phase retrieval has been demonstrated by previous research work [20,30,31] despite that the capability is not unlimited if the saturation area was bigger than appropriate value. As a result, it is possible to realize single-shot phase retrieval based on weak wave modulation in theory, such as the random aperture. Besides, due to the weak high-order diffraction is strengthened, the tolerance to noise, as well as the SNR of final reconstructions, should be improved at the same time.
Fig. 1 (a) Illustration of the scattering properties of a random aperture; (b) complete power spectrum of the random aperture; (c) power spectrum of the random aperture after setting the low-frequency components to zero.
3. Simulations
According to the concept described above, a representative single-shot phase retrieval scheme based on a random aperture is shown in Fig. 2. The collimated incident beam passes through the focusing lens, forming the illumination probe with soft edge on the specimen located 10 mm downstream. The diffraction pattern of the specimen exit wave is recorded by the detector after the modulation by the random aperture. The distances between the specimen, random aperture, and detector are T and L, respectively, and the detector is located in the focal plane of the lens. First, numerical simulations are performed to investigate the validity of the proposed approach. The wavelength of the incident wave is 632.8 nm, and the illumination of the specimen with a radius of 1.5 mm is illustrated by the up-right inset of Fig. 2(b) and Fig. 2(c). A matrix containing pseudorandom values drawn from the standard uniform distribution in the interval (0, 1) is generated, and the random aperture H, as illustrated in the inset of Fig. 1(a), is obtained by setting the elements in the matrix with values greater than 0.5 to 1 while setting the remaining elements to 0 at the same time. The side length of each basic cell in H is S = 4 μm. The phase of H is constant and therefore neglected. The diffraction pattern I with dimensions of 3072×3072 pixels is recorded by a 12-bit detector with a pixel size of 9 μm × 9 μm. T = 101.2 mm, L = 49.7 mm, and f = T + L = 150.9 mm. The amplitude of the exit wave of the specimen is shown in Fig. 2(b) and the phase of the specimen ranging from −0.25π to 0.25π is illustrated by Fig. 2(c). Supposing when the simulative and relative exposure time is t0 = 1, the diffraction pattern I0 is exactly unsaturated, and the maximum intensity value is 4095. When the exposure time goes up and becomes t = nt0, n ≥ 1, the corresponding diffraction pattern becomes I = nI0, and the saturated diffraction pattern I′ is obtained by setting the intensity values greater than 4095 to 4095 after rounding to the nearest integer numbers. After making a random guess for the specimen exit wave ϕ0, the k-th step of the iterative process is described as follows.
- Propagate the k-th guess for the specimen exit wave ϕk to the random aperture for distance T and obtain φk = FFT{ϕk, T} = φ′k +φk ·(1−H), where FFT{} indicates the forward-propagation process, and φ′k = φk · H is the exit wave of the random aperture.
- Back-propagate Ẽk to the random aperture plane and obtain the revised exit wave of the random aperture by φ̃′k = FFT−1{Ẽk, L}, where FFT−1{} indicates the back-propagation process.
- Obtain the revised illumination of the random aperture through φ̃k = φ̃′k · H + φk · (1 − H).
- Forward-propagate φ̃k to the detector plane and obtain the focus by Fk = FFT{φ̃k, L}. Apply the circle constraint Pk with increasing radius and obtain the revised focused wave F̃k = Fk · Pk.
- Back-propagate F̃k to the specimen plane and obtain the revised exit wave of the specimen by ϕ̃k = FFT−1{F̃k, L + T}·M (rm), where M(rm) is a circle function with a radius of rm = 2.1 mm, which is larger than the actual radius of the illumination on specimen and illustrated by the up-right inset of Fig. 2(b). ϕ̃k is regarded as the new initial guess ϕk+1 for the (k + 1)-th iteration.
- Repeat steps 1)–7) until the error ER(k) is reduced to the specified level.
Fig. 2 (a) Basic scheme for single-shot phase retrieval based on a random aperture; (b) amplitude of the specimen exit wave and the amplitude of the incident wave on the specimen is illustrated by the up-right inset; (c) phase of the specimen while the phase of the incident wave is illustrated by the up-right inset. (d) and (e) are the retrieved amplitude of exit wave and phase map of specimen respectively without considering any noise and saturations. The red dashed circle in (d) indicates the constraints in the specimen plane. Scale bar in (b), 1 mm and (b)(d) share the same scale bar. The color bars of (c) and (e) indicate phase in radians.
First of all, the simulation without any errors, such as the errors of quantization and saturation resulting from the limited dynamic range of detector, is considered. The perfect diffraction pattern is used for the iterations as is described above. After 200 iterations, the retrieved amplitude of exit wave and phase map of the specimen are shown in Fig. 2(d)(e) respectively. The red dashed lines in Fig. 2(d) illustrate the constraint boundary of M(rm). The final error ER is 0.002 and it means the incident wave of the random aperture could be retrieved very well. However, the recorded diffraction pattern is affected inevitably by different errors in practice. As a result, despite of saturation error, three additional errors, including the Poisson noise, quantization noise and uniformly distributed random background noise ranging from 0 to 50, are considered in the subsequent simulations. The practical diffraction pattern recorded by the detector is described by
where Poisson() means adding the Poisson noise, rand means the uniformly distributed random noise ranging from 0 to 1, and round() means rounding the intensity to the nearest integers. S and S′ indicate the unsaturated and saturated area respectively after considering the Poisson and random noise.Based on the scheme described by Eq. 3, four diffraction patterns with different relative exposure time t ranging from 1 to 30E3 are obtained in the simulations and the center parts of the diffraction patterns are illustrated in Fig. 3(a1)–(d1) respectively. It should be emphasized that the exposure time is relative and based on the hypothesis that the intensity of light source is constant. The practical exposure time could be reduced by enhancing the incident light intensity. When t = 1, as shown in Fig. 3(a1), the diffraction pattern is assumed to be fully unsaturated; when t goes up, the saturation area marked by red color appears and expands which could be obviously observed. The insets in the top-right corner of Fig. 3(a1)–(d1) show zoomed-in images of the center part. The phase retrieval process is repeated for each diffraction pattern with 200 iterations, with the reconstructed specimen exit waves shown in Fig. 3(a2)–(d2). The red dashed lines in Fig. 3(a2)–(d2) illustrate the constraint boundary of M(rm). The corresponding retrieved phase maps of the specimen under different relative exposure time are shown in Fig. 3(a3)–(d3) respectively. The phase maps are obtained by calculating the phase difference between the reconstructed exit wave and original illumination of the specimen.
Fig. 3 Results of the numerical simulations. (a)–(d) are the results of CAMI while the results of CMI are shown in (e)–(h). (a1)–(d1) and (e1)–(h1) are the diffraction patterns recorded for CAMI and CMI respectively with different exposure time. The red sections indicate the saturated area and the amplified views are illustrated by the corresponding up-right insets. The number of saturated pixels, indicated by N, for each column is approximately the same for fair comparisons. The random phase plate, providing binary phase delay 0 or π for CMI, has the same basic size S with the random aperture for CAMI. After 200 iterations, the retrieved exit waves and phase maps of the specimen with CAMI and CMI are shown by (a2)–(d2), (a3)–(d3) and (e2)–(h2), (e3)–(h3) respectively. The red dashed circles in (a2)–(d2) and (e2)–(h2) indicate the constraints in the specimen plane. (a1)–(d1) and (e1)–(h1) share the same scale bar. (a2)–(d2), (a3)–(d3), (e2)–(h2) and (e3)–(h3) share the same scale bar. The color bars of the third and sixth rows indicate phase in radians.
Due to the limited dynamic range of the detector and the quantization process, the high-order diffraction information is lost, and, as illustrated in Fig. 3(a2), the reconstruction is failed for t = 1. However, when t goes up, the reconstruction could be visibly improved as seen in Fig. 3(b2)–(d2)(b3)–(d3), and the partial saturation in the diffraction pattern is not an obstacle for reconstruction. In addition, when the random aperture was replaced by a random phase plate, which providing random and binary phase delay 0 or π with the same basic dimension 4 μm × 4 μm, the exactly unsaturated diffraction recorded is illustrated by Fig. 3(e1) when the relative exposure time is t = 813. The reconstructed amplitude of exit wave and the phase map of the specimen are illustrated by Fig. 3(e2)(e3) respectively after 200 iterations with CMI. The constraints and noise considered is the same with the simulations with random aperture. In order to fairly compare the difference between CAMI and CMI, the number of saturated pixels of diffraction patterns used for CMI are also controlled by choosing an appropriate relative exposure time. The update strategy in the detector plane is the same as is described by Eq. 2, where the diffraction wave in the saturated area is reserved. Besides, the constraint in the specimen plane, as is shown by the red dashed circles in Fig. 3(e2)–(h2), is also the same with CAMI. The results with different saturated area are illustrated by Fig. 3(f)–(h) respectively. The number of saturated pixels in each column of Fig. 3 is approximately the same. For example, the saturated pixels for CAMI when t=18E3 is N =27440 while the saturated pixels for CMI when t=5.28E3 is N =27461. In order to provide a numerical comparison, the error ER vs. iteration number curves for different saturated pixels for CAMI and CMI are shown in Fig. 4 with different colors. In all cases, the reconstruction becomes stable rapidly within 120 iterations; however, the final errors ER differ. Although longer exposure time means losing more diffraction information and slowing down the convergence speed, the lost information could be retrieved during the iterations and it can significantly improve the final ER, as well as the reconstruction quality. This tendency is applicative both for CAMI and CMI according to Fig. 4(a) and introducing the partially saturated diffraction pattern is a practical way to overcome the limitations of the detector. However, according to the final ER illustrated by Fig. 4(b), the improvements are different. When there is no saturation, the final ER of CMI is 0.409 while CAMI fails, however, when N goes up, the ER of CAMI approaches the ER of CMI gradually. When N ≈ 27440, the ER s of CAMI and CMI are comparative and when N goes up continually, such as when N ≈ 81573, the ER of CMI is 0.145 while the ER of CAMI goes down to 0.141, which is slightly lower than CMI. As a result, the concept of partial saturation is critical for CAMI, and by choosing an appropriate exposure time or incident wave intensity, the performance of CAMI is competitive compared with CMI. In general, CAMI could be an efficient and practical way for phase imaging and wave measurements with single-shot measurement.
Fig. 4 Variation of the error ER of the simulations shown in Fig. 3. The solid lines and dashed lines represent the results of CAMI and CMI respectively. (a) The variation of the error with iteration number and the curves with same color has approximate amounts of saturated pixels in the diffraction patterns. (b) The comparisons of final ER between CAMI and CMI after 200 iterations.
When the basic side length S of the random aperture increases to 6 μm, 8 μm and 12 μm, the reconstruction results under different exposure time are illustrated by Fig. 5 after 200 iterations. The partially saturated diffraction patterns are illustrated by Fig. 5(a)(d)(g) while the corresponding random apertures are shown in the up-right insets respectively. The reconstructed amplitude of exit wave and the phase map of the specimen are shown in the second and third column of Fig. 5. When S = 6μm and 8μm, the ER is around 0.175, but when S = 12μm, the ER goes up to 0.22 and the noise becomes more distinct according to Fig. 5(h), but it is still better than the results of CMI shown in Fig. 3(j). For specific S, different exposure time t is chosen and the best results are illustrated. As a result, the exposure time changes for different S. According to the simulations shown in Fig. 4, the longer exposure time could improve the final reconstructions, but when the basic size of random aperture goes up, the capability to retrieve the lost information is weakened at the same time. As a result, the appropriate exposure time decreases with bigger basic size S, which could be observed from the simulations. Besides, the quality of the final reconstructions deteriorates with bigger S. Considering the capability of manufacturing random aperture, the basic side length and the exposure time should be balanced.
Fig. 5 (a), 5(d), 5(g) The partially saturated diffraction patterns when S = 6μm, 8 μm and 12 μm respectively. The center part of the random apertures are shown in the corresponding up-right insets. The corresponding amplitude of exit wave and the phase map of specimen are illustrated by (b)–(c), (e)–(f) and (h)–(i) respectively. The color bars of (c) (f) (i) indicate phase in radians.
4. Experiment
In addition, an experiment was performed with the same setup parameters as shown in Fig. 2(a), except the bit depth of the diffraction pattern changed to 14-bit, to demonstrate the practical validity of the proposed approach. A pumpkin stem section was used as the specimen, and a chrome silica plate with basic dimensions of 4 μm × 4 μm, as shown in Fig. 6(a), was employed as the random aperture. Considering that the precision of 0.2 μm is possible for the chrome plate with current manufacturing capabilities, and neglecting the phase distortion within several millimeters introduced by the silica plate, the designed pattern illustrated by the red map of Fig. 6(b) was used as the practical transmission function H. The micro-graph is also illustrated by the gray scale map of Fig. 6(a). The rotation angle between the modulator and detector is 0.0127 degrees which could be calculated by the angle registration between the design value of aperture and partial measurement with ptychography. The diffraction pattern with partially saturated area which is shown in Fig. 6(c) was recorded by a charge-coupled device (CCD, AVT F110B), with the red color indicating the saturated area. In order to revise the angle between the ideal and practical positions of the aperture relative to the detector, the diffraction pattern has been rotated by 0.0127 degrees before iterations. The top-right inset in Fig. 6(c) shows a magnified section marked by the white dashed box. The retrieved amplitude and phase of the exit wave in the specimen plane obtained after 600 iterations are shown in Fig. 6(d) and the top-right inset, respectively. In order to obtain the phase map of the specimen, another diffraction pattern without specimen was recorded, and the illumination on the specimen could be retrieved. As a result, the phase of specimen could be calculated by subtracting the phase of illumination from the exit wave of specimen and the result is illustrated in Fig. 6(e). The practical validity of the scheme proposed above could be demonstrated as well.
Fig. 6 (a) The photo of a chrome silica plate used as a random aperture in the experiment. (b) The comparison between the micro-graph of (a) and the design value, the grayscale map is the micro-graph and the red map above the micro-graph is the corresponding design value. The sections without chrome for red map is transparent. (c) Diffraction pattern recorded by the CCD. (d) Reconstructed amplitude of the specimen exit wave, the top-right inset shows the phase distribution. (e) The phase map the specimen after subtract the phase of illumination, which is retrieved from the diffraction pattern without specimen. The red dashed circles are the constraints in the specimen plane. The color bars of (d) and (e) indicate phase in radians.
Based on the properties of the proposed concept, it could be termed coherent amplitude modulation imaging (CAMI). Besides the SNR improvement compared with CMI, a random aperture is easier to manufacture with smaller basic dimensions than a binary random phase plate. If the accuracy of the manufactured random aperture is high enough, the designed value could be used directly for phase retrieval, which is demonstrated by the experiment. It means once the spatial position between the random aperture and detector is determined, it is adaptable for different coherent wavelengths. Although CMI could be used for different wavelengths, but the random phase plate should be pre-characterized for each wavelength individually and for the wavelengths with large difference, the phase plate should be re-manufactured to guarantee strong phase modulation. As a result, CAMI has a wider practical and potential applications in theory for phase imaging and wave diagnostics with single shot measurement, such as color imaging, multi-wavelength phase mapping and the measurement of M2 factor of different laser pulses.
Besides, the concept proposed above could be extended to X-ray imaging to improve the final SNR of CMI in theory. A focus lens in Fig. 2(a) is not required as long as the strong zero-order diffraction is restricted within a relative small section compared with the whole diffraction pattern. However, due to the strong penetrability of X-rays, especially hard X-rays, a pure random aperture is not possible in practice. The actual random wave modulator generated with commonly used metal, such as gold and tungsten, provides hybrid modulations on amplitude and phase. For example, at 6.2 KeV, the intensity transmission of 1.25 μm gold is 37.6% while the phase delay is π. As a result, the total transmission of the random phase plate is around 69%. A similar phenomenon has been described by Zhang [20] for CMI imaging with X-rays. However, considering the concept proposed above, a weakly scattering modulator with less phase delay, rather than π, is still practical in theory with higher SNR. Besides, the transmission could be enhanced at the same time. For example, the intensity transmission of 0.6 μm gold is 81.3% while the phase delay is 1.5 rad. It could be a potential technique to improve X-ray phase imaging based on wave modulation.
5. Conclusion
In conclusion, a single-shot phase retrieval algorithm CAMI based on the modulation of a binary random aperture and partially saturated diffraction pattern is proposed. The limited dynamic range of the detector could be circumvented by abandoning the strong zero-order diffractions. The lost information in the saturated area could be reconstructed during the iterative process. Compared with CMI based on random phase plate, the performance of CAMI could be competitive by choosing appropriate amounts of saturated pixels. For visible light, the design value could be used directly for iterations if the accuracy of the random aperture is high enough and the validity has been demonstrated by experiments. As a result, it is adaptable for different coherent wavelengths for phase imaging and wave diagnostics. For strong penetrability light sources, such as X-rays, the concept of CAMI is still practical in theory to improve the imaging quality of CMI and enhance the transmission of the modulator, which provides more alternative and flexible designs for wave modulator.
Funding
National Natural Science Foundation of China (61675215); Shanghai Sailing Program (17YF1428700); National Innovation Fund (CXJJ-17S059).
References
1. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef] [PubMed]
2. W. Hua, G. Zhou, Y. Wang, P. Zhou, S. Yang, C. Peng, F. Bian, X. Li, and J. Wang, “Measurement of the spatial coherence of hard synchrotron radiation using a pencil beam,” Chin. Opt. Lett. 15, 033401 (2017). [CrossRef]
3. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342 (1999). [CrossRef]
4. J. M. Rodenburg, Ptychography and Related Diffractive Imaging Methods (Elsevier, 2008), vol. Volume 150, pp. 87–184.
5. M. J. Humphry, B. Kraus, A. C. Hurst, A. M. Maiden, and J. M. Rodenburg, “Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging,” Nat Commun 3, 730 (2012). [CrossRef] [PubMed]
6. M. W. M. Jones, K. D. Elgass, M. D. Junker, M. D. de Jonge, and G. A. van Riessen, “Molar concentration from sequential 2-d water-window x-ray ptychography and x-ray fluorescence in hydrated cells,” Sci. Reports 6, 24280 (2016). [CrossRef]
7. L. Shemilt, E. Verbanis, J. Schwenke, A. K. Estandarte, G. Xiong, R. Harder, N. Parmar, M. Yusuf, F. Zhang, and I. K. Robinson, “Karyotyping human chromosomes by optical and x-ray ptychography methods,” Biophys. J. 108, 706–713 (2015). [CrossRef] [PubMed]
8. M. Beckers, T. Senkbeil, T. Gorniak, M. Reese, K. Giewekemeyer, S. C. Gleber, T. Salditt, and A. Rosenhahn, “Chemical contrast in soft x-ray ptychography,” Phys Rev Lett 107, 208101 (2011). [CrossRef] [PubMed]
9. C. Zhu, R. Harder, A. Diaz, V. Komanicky, A. Barbour, R. Xu, X. Huang, Y. Liu, M. S. Pierce, A. Menzel, and H. You, “Ptychographic x-ray imaging of surfaces on crystal truncation rod,” Appl. Phys. Lett. 106, 101604 (2015). [CrossRef]
10. H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: A novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004). [CrossRef] [PubMed]
11. J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004). [CrossRef]
12. S. Chen, J. Deng, Y. Yuan, C. Flachenecker, R. Mak, B. Hornberger, Q. Jin, D. Shu, B. Lai, and J. Maser, “The bionanoprobe: hard x-ray fluorescence nanoprobe with cryogenic capabilities,” J. Synchro. Radiat. 21, 66–75 (2014). [CrossRef]
13. R. M. Glaeser, “Invited review article: Methods for imaging weak-phase objects in electron microscopy,” Rev. Sci. Instruments 84, 312 (2013). [CrossRef]
14. X. Huang, K. Lauer, J. N. Clark, W. Xu, E. Nazaretski, R. Harder, I. K. Robinson, and Y. S. Chu, “Fly-scan ptychography,” Sci. Reports 5, 9074 (2015).
15. J. Deng, D. J. Vine, S. Chen, Q. Jin, Y. S. G. Nashed, T. Peterka, S. Vogt, and C. Jacobsen, “X-ray ptychographic and fluorescence microscopy of frozen-hydrated cells using continuous scanning,” Sci. Reports 7, 445 (2017). [CrossRef]
16. B. Leshem, R. Xu, Y. Dallal, J. Miao, B. Nadler, D. Oron, N. Dudovich, and O. Raz, “Direct single-shot phase retrieval from the diffraction pattern of separated objects,” Nat. Commun. 7, 10820 (2016). [CrossRef] [PubMed]
17. I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008). [CrossRef] [PubMed]
18. H. Tao, S. P. Veetil, X. Pan, C. Liu, and J. Zhu, “Lens-free coherent modulation imaging with collimated illumination,” Chin. Opt. Lett. 14, 071203 (2016).
19. F. Zhang and J. M. Rodenburg, “Phase retrieval based on wave-front relay and modulation,” Phys. Rev. B 82, 121104 (2010). [CrossRef]
20. F. Zhang, B. Chen, G. R. Morrison, J. Vila-Comamala, M. Guizar-Sicairos, and I. K. Robinson, “Phase retrieval by coherent modulation imaging,” Nat. Commun. 7, 13367 (2016). [CrossRef] [PubMed]
21. A. Anand, G. Pedrini, W. Osten, and P. Almoro, “Wavefront sensing with random amplitude mask and phase retrieval,” Opt. Lett. 32, 1584–1586 (2007). [CrossRef] [PubMed]
22. M. H. Seaberg, A. d’Aspremont, and J. J. Turner, “Coherent diffractive imaging using randomly coded masks,” Appl. Phys. Lett. 107, 231103 (2015). [CrossRef]
23. E. Peters, P. Clemente, E. Salvador-Balaguer, E. Tajahuerce, P. Andrés, D. G. Pérez, and J. Lancis, “Real-time acquisition of complex optical fields by binary amplitude modulation,” Opt. Lett. 42, 2030–2033 (2017). [CrossRef] [PubMed]
24. Z.-J. Cheng, B.-Y. Wang, Y.-Y. Xie, Y.-J. Lu, Q.-Y. Yue, and C.-S. Guo, “Phase retrieval and diffractive imaging based on babinet’s principle and complementary random sampling,” Opt. Express 23, 28874–28882 (2015). [CrossRef] [PubMed]
25. B.-Y. Wang, L. Han, Y. Yang, Q.-Y. Yue, and C.-S. Guo, “Wavefront sensing based on a spatial light modulator and incremental binary random sampling,” Opt. Lett. 42, 603–606 (2017). [CrossRef] [PubMed]
26. R. Horisaki, R. Egami, and J. Tanida, “Experimental demonstration of single-shot phase imaging with a coded aperture,” Opt. Express 23, 28691–28697 (2015). [CrossRef] [PubMed]
27. R. Horisaki, Y. Ogura, M. Aino, and J. Tanida, “Single-shot phase imaging with a coded aperture,” Opt. Lett. 39, 6466–6469 (2014). [CrossRef] [PubMed]
28. A. Suzuki, K. Shimomura, M. Hirose, N. Burdet, and Y. Takahashi, “Dark-field x-ray ptychography: Towards high-resolution imaging of thick and unstained biological specimens,” Sci. Reports 6, 35060 (2016). [CrossRef]
29. Y. Takahashi, A. Suzuki, N. Zettsu, Y. Kohmura, Y. Senba, H. Ohashi, K. Yamauchi, and T. Ishikawa, “Towards high-resolution ptychographic x-ray diffraction microscopy,” Phys. Rev. B 83, 214109 (2011). [CrossRef]
30. A. M. Maiden, M. J. Humphry, F. Zhang, and J. M. Rodenburg, “Superresolution imaging via ptychography,” J. Opt. Soc. Am. A 28, 604–612 (2011). [CrossRef]
31. X. Pan, S. P. Veetil, B. Wang, C. Liu, and J. Zhu, “Ptychographical imaging with partially saturated diffraction patterns,” J. Mod. Opt. 621–8 (2015).
