Compensation for the setup instability in ptychographic imaging
aa r X i v : . [ phy s i c s . i n s - d e t ] M a r Compensation for the setup instability inptychographic imaging Y UDONG Y AO , C HENG L IU , AND J IANQIANG Z HU Key Laboratory of High Power Laser and Physics, Shanghai Institute of Optics and Fine Mechanics,Chinese Academy of Sciences, Shanghai 201800, China University of Chinese Academy of Sciences, Beijing 100049, China * [email protected] Abstract:
The high-frequency vibration of the imaging system degrades the quality of thereconstruction of ptychography by acting as a low-pass filter on ideal diffraction patterns. Inthis study, we demonstrate that by subtracting the deliberately blurred diffraction patterns fromthe recorded patterns and adding the properly amplified subtraction to the original data, thehigh-frequency components lost by the vibration of the setup can be recovered, and thus theimage quality can be distinctly improved. Because no prior knowledge regarding the vibratingproperties of the imaging system is needed, the proposed method is general and simple and hasapplications in several research fields. © 2018 Optical Society of America
OCIS codes: (100.5070) Phase retrieval; (050.1970) Diffractive optics; (120.5050) Phase measurement; (070.0070)Fourier optics and signal processing
References and links
1. R. W. Gerchberg, "A practical algorithm for the determination of phase from image and diffraction plane pictures,"Optik , 237 (1972).2. J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. , 2758–2769 (1982).3. J. Miao, P. Charalambous, J. Kirz, and D. Sayre, "Extending the methodology of x-ray crystallography to allowimaging of micrometre-sized non-crystalline specimens," Nature , 342–344 (1999).4. I. K. Robinson, I. A. Vartanyants, G. Williams, M. Pfeifer, and J. Pitney, "Reconstruction of the shapes of goldnanocrystals using coherent x-ray diffraction," Phys. Rev. Lett. , 195505 (2001).5. D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman ,"Biological imaging by soft x-ray diffraction microscopy," Proceedings of the National Academy of Science ,15343–15346 (2005).6. J. Zuo, I. Vartanyants, M. Gao, R. Zhang, and L. Nagahara, "Atomic resolution imaging of a carbon nanotube fromdiffraction intensities," Science , 1419–1421 (2003).7. J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, "Beyond crystallography: Diffractive imaging usingcoherent x-ray light sources,"Science , 530–535 (2015).8. P. Bao, F. Zhang, G. Pedrini, and W. Osten, "Phase retrieval using multiple illumination wavelengths," Opt. Lett. , 309–311(2008).9. V. Y. Ivanov, M. Vorontsov, and V. Sivokon, "Phase retrieval from a set of intensity measurements: theory andexperiment," J. Opt. Soc. Am. A , 1515–1524 (1992).10. F. Zhang and J. Rodenburg, "Phase retrieval based on wave-front relay and modulation," Phys. Rev. B , 121104(2010).11. H. Tao, S. P. Veetil, X. Pan, C. Liu, and J. Zhu, "Lens-free coherent modulation imaging with collimated illumination,"Chinese Optics Letters , 071203 (2016).12. H. Faulkner and J. Rodenburg, "Movable aperture lensless transmission microscopy: a novel phase retrieval algo-rithm," Phys. Rev. Lett. , 023903 (2004).13. A. M. Maiden and J. M. Rodenburg, "An improved ptychographical phase retrieval algorithm for diffractive imaging,"Ultramicroscopy ,1256–1262 (2009).14. H. Faulkner and J. Rodenburg, "Error tolerance of an iterative phase retrieval algorithm for moveable illuminationmicroscopy," Ultramicroscopy , 153–164 (2005).15. K. Stachnik, I. Mohacsi, I. Vartiainen, N. Stuebe, J. Meyer, M. Warmer, C. David, and A. Meents, "Influence offinite spatial coherence on ptychographic reconstruction," Appl. Phys. Lett. , 011105 (2015).16. N. Burdet, X. Shi, D. Parks, J. N. Clark, X. Huang, S. D. Kevan, and I. K. Robinson, "Evaluation of partial coherencecorrection in x-ray ptychography," Opt. Lett. , 5452–5467 (2015).17. L. Whitehead, G. Williams, H. Quiney, D. Vine, R. Dilanian, S. Flewett, K. Nugent, A. G. Peele, E. Balaur, andI. McNulty, "Diffractive imaging using partially coherent x rays," Phys. Rev. Lett. , 243902 (2009).8. J. N. Clark and A. G. Peele, "Simultaneous sample and spatial coherence characterisation using diffractive imaging,"Appl. Phys. Lett. , 154103 (2011).19. B. Chen, R. A. Dilanian, S. Teichmann, B. Abbey, A. G. Peele, G. J. Williams, P. Hannaford, L. Van Dao, H. M.Quiney, and K. A. Nugent, "Multiple wavelength diffractive imaging," Phys. Rev. A , 023809 (2009).20. P. Thibault and A. Menzel, "Reconstructing state mixtures from diffraction measurements," Nature , 68–71(2013).21. D. J. Batey, D. Claus, and J. M. Rodenburg, "Information multiplexing in ptychography," Ultramicroscopy ,13–21 (2014).22. W. Yu, S. Wang, S. Veetil, S. Gao, C. Liu, and J. Zhu, "High-quality image reconstruction method for ptychographywith partially coherent illumination," Phys. Rev. B , 241105 (2016).23. S. Dong, P. Nanda, K. Guo, J. Liao, and G. Zheng, "Incoherent fourier ptychographic photography using structuredlight," Photonics Research , 19–23 (2015).24. K. Guo, S. Dong, and G. Zheng, "Fourier ptychography for brightfield, phase, darkfield, reflective, multi-slice, andfluorescence imaging," IEEE Journal of Selected Topics in Quantum Electronics , 77–88 (2016).25. F. Wei, J.-Y. Choi, and S. Rah, "Experiences of the long term stability at sls," Proc. AIP , 38–41(2007).26. V. Schlott, M. Boge, B. Keil, P. Pollet, and T. Schilcher, "Fast orbit feedback and beam stability at the Swiss LightSource," Proc. AIP , 174–1812004.27. F. Zhang, I. Peterson, J. Vila-Comamala, A. Diaz, F. Berenguer, R. Bean, B. Chen, A. Menzel, I. K. Robinson, andJ. M. Rodenburg, "Translation position determination in ptychographic coherent diffraction imaging," Opt. Express , 13592–13606 (2013).28. A. Maiden, M. Humphry, M. Sarahan, B. Kraus, and J. Rodenburg, "An annealing algorithm to correct positioningerrors in ptychography," Ultramicroscopy , 64–72 (2012).29. M. Guizar-Sicairos and J. R. Fienup, "Phase retrieval with transverse translation diversity: a nonlinear optimizationapproach," Opt. Express , 7264–7278 (2008).30. M. Odstrcil, P. Baksh, S. Boden, R. Card, J. Chad, J. Frey, and W. Brocklesby, "Ptychographic coherent diffractiveimaging with orthogonal probe relaxation," Opt. Express , 8360–8369 (2016).31. J. N. Clark, X. Huang, R. J. Harder, and I. K. Robinson, "Dynamic imaging using ptychography," Phys. Rev. Lett. , 113901 (2014).
1. Introduction
Coherent diffraction imaging (CDI) is a promising technology for obtaining the complex trans-mission function of a specimen from the recorded diffraction intensity. As a lens-free technique,CDI can bypass the resolution limits imposed by the poor focusing optics available at shortwavelengths [1, 2] and can theoretically reach the diffraction-limited resolution. With X-ray andhigh-energy electrons, a resolution of nanometers or angstroms can be achieved; thus, CDI isbecoming an important tool in material and biological sciences [3–7]. Because the performanceof traditional CDI algorithms is not very satisfying with regard to convergence, accuracy, andreliability, several improved CDI methods have been proposed [8–11]. The ptychographic itera-tive engine (PIE) [12] is a scanning version of the CDI technique where the specimen is scannedthrough a localized illumination beam to a grid of positions and the resulting diffraction patternsare recorded. Using an iterative scheme with a proper overlapping ratio between two adjacentscanning positions, the modulus and phase of the transmission functions of the specimen andthe illumination beam can be reconstructed accurately and rapidly [13, 14]. In theory, PIE caneasily generate images with a resolution only limited by the numerical aperture of the detector;however, in practice, the resolution is affected by the flaws of the imaging system, especially forimaging with X-ray and electron beams. The coherence of synchrotron radiation sources is notas good as that of a laser beam and has been proven to be the largest barrier for PIE to reach thetheoretical diffraction-limited resolution [15, 16]. Several algorithms have been investigated toimprove the image quality of PIE with partially coherent illumination [17–22]. Although thesemethods require complete or partial prior knowledge of the properties of illumination or time-consuming computation and may slightly compromise the spatial resolution, the quality of thereconstruction can be distinctively improved in many cases. Furthermore, the recently developedtechnique of Fourier ptychography has potential to avoid the influence of the incoherence forachieving high-quality reconstruction [23, 24]. Another factor that limits the practical resolu-tion of PIE is the instability of the imaging system, including the vibration of the mechanicalcanning system, the tiny pointing-direction change, and the transverse shifting of the radiationbeam [25, 26]. Because the wavelengths of the X-ray and the electron beams are much smallerthan 1 nm, the tiny departure of the illumination beam from the right position and direction cangenerate obvious errors in the final reconstruction. Numerous methods have been proposed toefficiently correct the low-frequency vibration with a period far longer than the exposure timeof the detector [27–30]. However, there is no ideal method for effectively handling the high-frequency vibration of the experimental system, which makes the recorded diffraction intensitya summation of many diffraction intensities formed by the changing illumination during the ex-posure of the detector [31]. Although the influence of the high-frequency vibration can be treatedas a type of incoherence of the illumination, the aforementioned methods [17–22, 31] requirecomplete or partial prior knowledge of the properties of vibration-including the frequencies andamplitude-or massive data processing. However, in practice, the parameters of the vibration ofthe whole imaging system are difficult to measure in real time. To deal with the image-qualitydegradation induced by the setup instability, we must examine its influence on the recorded dataand the final reconstruction and then develop a method to circumvent this problem.In this study, we mathematically analyze how the recorded diffraction patterns and the finalreconstruction are blurred by the instability of the imaging system and then propose a simplenumerical method for enhancing the lost high-frequency components. The proposed method doesnot require any prior knowledge regarding the characteristics of the high-frequency vibration ofthe imaging system and can be extended to other CDI methods using X-ray and electron beamsto solve the problems related to imaging-system instability.
2. Principle of the method
Fig. 1. Flowchart of the reconstruction process using the conventional PIE method and theproposed method.
In the PIE method, the specimen O ( r ) is fixed on a two-dimensional (2D) translation stageand illuminated by a localized probe P ( r ) . Assuming that the specimen is sufficiently thin, theexiting wave from the specimen is ϕ ( r , R ) = O ( r − R ) P ( r ) , and the recorded diffraction intensityis I ( k , R ) ∝ |ℑ[ ϕ ( r , R )]| in most experiments with short-wavelength sources, where k is thereciprocal coordinate with respect to the real space coordinate r in the specimen plane, and R enotes the position of the specimen during the raster scanning. With the recorded diffractionpatterns, the complex amplitudes of the specimen and the probe can be reconstructed. Theflowchart of the reconstruction process of PIE is shown in Fig. 1, and a detailed description isfound in the literature [12, 13]. In the PIE algorithm, the illumination probe is assumed to be absolutely static during the dataacquisition; however, its direction and position change continually within a small range withrespect to the specimen and detector owing to the instability of the imaging system. In mostcases, the direction and position of the illumination beam change simultaneously during theexposure of the detector, but for simplicity, we analyze them separately to determine how therecorded data are influenced mathematically and then consider their combined effects.According to the principles of Fourier optics, any illumination beam can be decomposed intoa series of spherical waves of different strengths; thus, we can assume a spherical illuminatingprobe without a loss of generality. The spherical illumination is expressed in Fourier form as e P ( k ) = W ( k ) exp (− j πλ zk ) , where λ is the wavelength of the probe, and z is the distancebetween the focal spot of the probe and the specimen, and W ( k ) is determined by the numericalaperture of the illuminating optics. For a short wavelength or in the far field, the diffractiondistribution in the detector plane is the Fourier transform of the wave exiting the specimen. Thecomplex amplitude of the diffraction pattern in the detector plane can be expressed as Φ ( k ) = e P ( k ) ⊗ e O ( k ) = Õ n e P ( k − k n ) e O ( k n ) . (1)where e O ( k ) is the Fourier transform of the transmission function of the specimen. The recordedintensity with static illumination is I ( k ) = | Õ n e P ( k − k n ) e O ( k n )| = I ( k ) + Õ m , n A m ( k ) A n ( k ) cos [ θ ( k )] . (2)The first item I ( k ) is the intensity summation of all diffracted beams, and the second item Í m , n A m ( k ) A n ( k ) cos [ θ ( k )] represents the interference between different spatial-frequency com-ponents. I ( k ) = Í n | e P ( k − k n ) e O ( k n )| A m ( k ) = | e P ( k − k m ) e O ( k m )| A n ( k ) = | e P ( k − k n ) e O ( k n )| θ ( k ) = πλ z [ k ( k m − k n ) − ( k m − k n )] + φ mn . (3)Here, k m − k n is the frequency of the interference fringe formed by the m th and n th diffractionorders, and φ m , n is the additional phase introduced by the specimen. Considering the coordinatetransformation r c = λ Lk on the CCD plane, where L is the distance between the specimen andthe CCD, the diffraction pattern intensity can be described in the frequency domain as e I ( u ) = e I ( u ) + Õ m , n e A m ( u ) ⊗ e A n ( u )| u = z ( k m − k n )/ L . (4)where u is the reciprocal coordinate with respect to the real space coordinate r c in the CCDplane.hen considering the pointing instability of the imaging system, the illumination beam tiltedan angle of α can be expressed as, P ′ ( r ) = P ( r ) exp (− j π r sin αλ ) . (5)So the Fourier transform of the probe is e P ′ ( k ) = e P ( k − ∆ k ) , where ∆ k = sin α / λ . The recordedintensity with tilted illumination becomes I ′ ( k ) = | Õ n e P ′ ( k − k n ) e O ( k n )| = I ( k − ∆ k ) . (6)Assuming that the vibration in the pointing direction of the illumination beam follows the normaldistribution H ( ∆ k ) = exp (− ∆ k / K ) , where K is a constant related to the vibrating propertiesof the imaging system. The recorded intensity with high-frequency vibration in the pointingdirection of the illuminating beam can be interpreted as a summation of the diffraction patternsof all possible illuminations with different pointing directions. Thus, it can be expressed as I v ( k ) = ∫ + ∞−∞ I ( k − ∆ k ) H ( ∆ k ) d ∆ k ∫ + ∞−∞ H ( ∆ k ) d ∆ k = √ π K I ( k ) ⊗ exp (− k K ) . (7)Considering the coordinate transform r c = λ Lk in the detector plane, the recorded intensity canbe expressed in the frequency domain as e I v ( u ) = e I ( u ) exp [−( πλ LK ) u ] . (8)Thus, the influence of the vibration in the pointing direction of the illuminating beam acts as alow-pass filter on the ideal diffraction patterns.On the other hand, when the illumination probe suffers from transverse positioning vibration,the recorded diffraction pattern can be expressed as a summation of the diffraction intensitiesformed by all possible illumination beams with diverse transverse shifts. The probe with atransverse shift of δ is expressed as P ′′ ( r ) = P ( r + δ ) . And its Fourier transform is e P ′′ ( k ) = e P ( k ) exp ( j π k δ ) . The corresponding diffraction-pattern intensity is I ′′ ( k ) = | Õ n e P ′′ ( k − k n ) e O ( k n )| = I ( k ) + Õ m , n A m ( k ) A n ( k ) cos [ θ ( k + ∆ )] . (9)Assume that the transverse shifting of the illuminating beam has a normal distribution H ( ∆ ) = exp (− ∆ / D ) , where D is determined by the standard deviation of ∆ = δ / λ z . The recordedintensity with high-frequency positioning vibration is a summation of different intensities fordifferent illumination shifts: I v ( k ) = ∫ + ∞−∞ I ′′ ( k ) H ( ∆ ) d ∆ ∫ + ∞−∞ H ( ∆ ) d ∆ = I ( k ) + (10) Õ m , n A m ( k ) A n ( k ) cos [ θ ( k )] exp [−( πλ zD ) ( k m − k n ) ] . he Fourier transform of the recorded diffraction intensity becomes e I v ( u ) = e I ( u ) + exp [−( πλ LD ) u ]× Õ m , n e A m ( u ) ⊗ e A n ( u )| u = z ( k m − k n )/ L . (11)Compared with Eq. (4), the position vibration degrades the contrast of the interference fringe inthe recorded intensity, acting as a low-pass filter.In practice, the pointing and transverse vibration of the illumination beam can occur simulta-neously; thus, the Fourier transform of the recorded diffraction intensity is e I v ( u ) = { e I ( u ) + exp [−( πλ LD ) u ]× Õ m , n e A m ( u ) ⊗ e A n ( u )} exp [−( πλ LK ) u ] . (12)Because e I ( u ) has a narrow frequency band for spherical illumination, Eq.(12) can be approxi-mated as e I v ( u ) ≈ e I ( u ) + exp (− C u ) Õ m , n e A m ( u ) ⊗ e A n ( u ) . (13)where C = πλ L √ K + D . Clearly, the instability of the illumination beam during the exposureof the detector leads to the loss of high-frequency components of the diffraction intensity, andthe contrast of the interference fringes is differently suppressed depending on their frequency.As reported in the literature [22], change in the contrast of the diffraction patterns leads tomathematical ambiguity and generates errors in the final reconstruction. At first glance, it appears that the Fourier component of the ideal diffraction pattern can be ob-tained by dividing exp (− C u ) to the e I v ( u ) by each possible C until a satisfactory reconstructionis achieved. However, because this operation can seriously amplify the errors or noise, and theexpired resolution improvement can be submerged by them, it cannot be adopted in practice.In the proposed method, the recorded intensities e I v ( k ) are modified before the conventionalPIE process, as shown in Fig.1. The recorded diffraction intensities are deliberately blurred viaconvolution with the Gaussian function exp (− k / K v ) , where K v is a constant chosen to slightlyblur the recorded diffraction pattern. The deliberately blurred intensity pattern is expressed inthe frequency domain as e I ′ v ( u ) = e I ( u ) + exp (− C u ) exp (− C v u )× Õ m , n e A m ( u ) ⊗ e A n ( u ) . (14)where C v = πλ LK v . We subtract the blurred pattern e I ′ v ( k ) from the original recorded diffraction e I v ( k ) and then add the subtracted pattern to e I v ( k ) after multiplying it by a constant β : I c ( k ) = I v ( k ) + β [ I v ( k ) − I ′ v ( k )] . (15)The Fourier transform of the modified intensity pattern is e I c ( u ) = e I ( u ) + [ + β − β exp (− C v u )] exp (− C u ) Õ m , n e A m ( u ) ⊗ e A n ( u ) . (16)learly, when [ + β − β exp (− C v u )] exp (− C u ) in Eq.(16) is wider than exp (− C u ) , e I c ( u ) is close to the Fourier transform of the vibration-free diffraction pattern, and then the influenceof the imaging system instability on the reconstructed image can be remarkably suppressed.
3. Results
Fig. 2. Diffraction patterns obtained (a) with stable illumination, (b) with unstable illumina-tion, and (c) by modification via the proposed method; reconstructed modulus obtained (d)with stable illumination, (e) with unstable illumination, and (f) using corrected intensities;reconstructed phase obtained (g) with stable illumination, (h) with unstable illumination,and (i) using corrected intensities. The upper insets in (d)-(i) show the reconstructed illumi-nation fields.
A numerical simulation is performed using the proposed algorithm to check its feasibility.A divergent spherical wave with a wavelength of 632.8 nm is used for the illumination. Twopictures are used as the modulus and phase, respectively, of the specimen. The diameter of theilluminated area on the specimen is 0.74 mm, and 10 ×
10 diffraction intensities are recordedwith a step size of 0.185 mm. The charge-coupled device (CCD) camera has a resolution of256 ×
256 pixels and a pixel size of 7.4 µ m. For comparison, the diffractive intensities withstable illumination are calculated, one of which is shown in Fig.2(a). When the unstable imagingsystem results in diffraction pattern vibration with variance of 46 µ m in the detector plane,the intensity distributions are calculated by adding diffraction intensities under many slightlydifferent illuminations with varying incident angles and transverse shifts. The diffraction patternscorresponding to the same illuminating position of Fig.2(a) is shown in Fig.2(b), where theintensity is obviously blurred. This coincides with Eq.(13). The reconstructed modulus andphase of the specimen with stable illumination are very clear as shown in Figs.2(d) and (g),respectively. Figs.2(e) and (h) show the reconstructed complex transmission of the specimenwith the unstable illumination, where the resolution is apparently decreased compared withFigs.2(d) and (g). Corrected diffraction patterns are obtained from the raw recorded data usingthe proposed method. A Gaussian function is used to slightly blur the recorded intensities with K v = . mm − . And the obtained blurred diffraction patterns are subtracted from the rawecorded diffraction patterns. Then the modified intensity patterns are obtained by adding theamplified subtractions to the raw data with β = .
5. As shown in Fig.2(c), the contrast ofthe corrected diffraction pattern is obviously improved compared with the raw data and thedistribution is similar to that of Fig.2(a). The reconstructed modulus and phase of the specimenusing the corrected intensities are shown in Figs.2(f) and (i), respectively, where the resolutionis remarkably improved compared with Figs.2(e) and (h). The insets in Figs.2(d)-(f) and (g)-(i)show the reconstructed modulus and phase, respectively, of the illumination field, indicating thatthe quality of the retrieved illumination is also improved using the proposed method.To quantifythe performance of the proposed method, the normalized root-mean-square error metric [13]is calculated, as shown in Fig. 3. The accuracy of the reconstruction are obviously improvedwhen the proposed method is used to correct the recorded diffraction patterns before the iterativecomputation.
Fig. 3. Progress of the error of the conventional PIE method for a static imaging system(blue dashed line),an an unstable system (green dashed line); the proposed method for anunstable system(red line).
Fig. 4. Schema of the experimental setup.
The proof-of-principle experiment is conducted with visible light as shown in Fig. 4, wherehe divergent laser beam from a He-Ne laser is slightly diffused by a rotating plastic diffuserbefore irradiating on the sample to simulate the instability of the imaging system. A cross sectionof a monocotyledon placed on a 2D translation stage is used as the sample. The illuminationarea on the specimen is ∽ ×
10 diffraction patterns are recorded by the CCDcamera while the sample is scanned relative to the instable beam with a step size of 0.37 mm.
Fig. 5. Diffraction patterns (a) in the stable case, (b) in the unstable case, and (c) modifiedusing the proposed method; reconstructed modulus of the specimen (d) in the stable case,(e) in the unstable case, and (f) obtained using the proposed method; reconstructed phaseof the specimen (g) in the stable case, (h) in the unstable case, and (i) obtained using theproposed method.
Recorded intensities obtained with stable and unstable laser beams are shown in Figs.5(a) and(b), respectively. It is clear that the recorded intensities using unstable imaging system are totallyblurred. In order to overcome this limitation, the newly proposed method is applied to modifythe recorded patterns. The Gaussian function with K v = . mm − is adopted to slightly blurthe recorded diffraction intensities. According to the proposed method, the corrected diffractionpatterns are obtained with β =
5. As shown in Fig.5(c), the contrast of the corrected diffractionpattern is remarkably improved and similar to that shown in Fig.4(a). Figs.5(d) and (g) show thereconstructed modulus and phase, respectively, of the sample with stable illumination, where thefine structures of individual cells are clear. Figs.5(e) and (h) show the reconstructed image of thespecimen with the unstable imaging system, where the individual cell is hardly resolved. Withthe corrected diffraction patterns, the reconstructed image of the specimen shown in Figs.5(f)and (i) are generated. Here, the quality of the reconstructed images is distinctly improved, andthe images obtained are roughly identical to those for the stable illumination.To quantify the resolvability of the proposed method, a USAF1951 target is measured. Asshown in Fig.6, group 5, which is resolvable for the stable illumination, is obviously blurredfor the unstable system. After the proposed method is applied, the elements in Group 5 becomedistinguished, as shown in Fig.6(c) and (f). ig. 6. Reconstructed modulus distribution of the USAF 1951 resolution target for (a) stableillumination, (b) unstable illumination, and (c) the proposed method; normalized value ofthe red line in (a) for (d) stable illumination, (e) unstable illumination, and (f) the proposedmethod.
4. Discussion
Fig. 7. Filter window obtained with unstable illumination and by modification with differentvalues of β β To show the dependency of the resolution-improvement effect on the value of β , a numericalsimulation is performed with the same parameters that were used for the simulation in Fig.2. Fig.7shows the effect of the proposed method with various curves, where the blue dashed line indicatesthe low-pass filter related to the vibration of the imaging system, and the other curves show theincreased width of the low-pass filter with varying β . The width of the low-pass filter increasesremarkably with increasing β , and for β = .
5, the width of [ + β − β exp (− C v u )] exp (− C u ) is about twice that of the exp (− C u ) , and accordingly the resolution of the final reconstructionwith these generated diffraction patterns can be remarkably improved. However, when β = β should be carefully selectedto assure a satisfactory final reconstruction.Fig.8 shows the reconstruction results with different β values according to the aforementionedanalysis, where Fig.8 (a) and (b) are the reconstructed modulus and phase images, respectively,obtained using the raw data acquired with the unstable system. Fig.8(c) to (h) show the recon-structed modulus and phase images with β = .
5, 1, and 5. These results indicate that themodulus and phase reconstructed using the proposed method produce a better reconstructionquality than those obtained directly with unstable data. On the other hand, the effect of theproposed method depends on the properly chosen value of β , which is coincident with the curvesshown in Fig.7. In the aforementioned theoretical analysis, the vibration of the illumination is assumed to havea normal distribution, and another Gaussian function is used to convolve with the recordedintensities to slightly blur the patterns. But the characteristics of a real imaging system areunknown and may be more complex, so it appears that the proposed method is difficult toimplement for real experiments. However, the fundamental principle of the proposed method isto recover the high-frequency components lost because of the instability of the radiation source,and other functions that can realize this purpose can replace the Gaussian function in the analysis.For example, circular functions, triangular functions, and para-curves can be adopted to slightlyblur the diffraction patterns for improving the resolution without considering the exact propertiesof the vibration of the imaging system. This feature makes the proposed method more applicableto real experiments and is demonstrated by the following simulations and experiments.
Fig. 8. Reconstructed (a) modulus and (b) phase using unstable system; reconstructed (c)modulus and (d) phase using modified intensity patterns with β = .
5; reconstructed (e)modulus and (f) phase using modified intensity patterns with β =
1; reconstructed (g)modulus and (h) phase using modified intensity patterns with β = When the vibration of the imaging system follows a normal distribution, the parameters arethe same as that were used for the simulations in Fig.2. As previously discussed, when thediffraction patterns are deliberately blurred via convolution with another Gaussian function exp (− k / K v ) , the width of the low-pass filter shown by the red line in Fig.9(a) becomes muchwider than that for the dashed line related to the instability. The same results can be obtained by ig. 9. When the vibration follows normal distribution, filter window (a) modified withGaussian function ( K v = . mm − , β = . K v = . mm − , β = . K v = . mm − , β = K v = . mm − , β = . K v = . mm − , β = . K v = . mm − , β = . K v = . mm − , β = K v = . mm − , β = . K v = . mm − , β = . using the circular function circ ( k / K v ) (Fig.9(b)) and the triangular function Λ ( k / K v ) (Fig.9(c)).For the case where the vibration of the imaging system follows a random distribution with themaximum extent of 52 . µ m in the detector plane, the low-pass filter window of the recordedintensity pattern is shown by the black dashed line in Fig.9(d). The modified results obtained bythe deliberately blurring diffraction patterns via convolution with a Gaussian function, circularfunction, and triangular function are shown as the red lines in Fig.9(d) to (f), respectively.When the vibration of the imaging system has a triangular distribution and results in an extentof 36 . µ m in the detector plane, the low-pass filter window of the recorded intensity patternhas the profile of the black dashed line shown in Fig.9(g). The modified results obtained bythe deliberately blurring diffraction patterns via convolution with a Gaussian function, circularfunction, and triangular function are shown as the red lines in Fig.9(g) to (i), respectively.And an experiment is carried out to verify the practicality of the proposed method, and theparameters are the same with Fig.5. The reconstructed images shown in Fig.10(a) and (b) arethe reconstructions using the raw data recorded with instable illumination, which are seriouslyblurred. The reconstructed images shown in Fig.10 (c) and (d) are obtained by using the Gaussianfunction with K v = . mm − and β = K v = . mm − and β = K v = . mm − and = Fig. 10. Reconstructed (a) modulus and (b) phase of the sample with unstable illumination;reconstructed (c) amplitude and (d) phase using the intensity patterns modified with Gaus-sian function; reconstructed (e) modulus and (f) phase using the intensity patterns modifiedwith circular function; reconstructed (g) modulus and (h) phase using the intensity patternsmodified with triangular function.
5. Conclusion
In conclusion, a simple method is proposed to relax the requirement of ptychography for thestability of the imaging system. The influence of the instability of the imaging system canbe described as a low-pass filter acting on the ideal diffraction intensities; the high-frequencycomponents are lost in the recorded data, generating a blurred image in the final reconstruction.Using the proposed method, the recorded intensity patterns are corrected by subtracting thedeliberately blurred diffraction patterns from the original ones and then adding the amplifiedsubtracted patterns to the recorded data. The resolution of the final reconstruction can besignificantly improved by using the corrected diffraction patterns. The feasibility of the proposedmethod is demonstrated via both numerical simulations and experiments on an optical bench.Because the proposed method does not require any prior knowledge of the characteristics of theillumination beam or massive calculation during the iterative process, it is an easy approach foracquiring a high-quality reconstruction with an unstable imaging system. The method can beextended to other CDI techniques for imaging with short-wavelength irradiation, such as freeelectrons or soft X-ray lasers.