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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 5 — Mar. 11, 2013
  • pp: 6162–6168
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Ptycholographic iterative engine with self-positioned scanning illumination

Xinchen Pan, Cheng Liu, Qiang Lin, and Jianqiang Zhu  »View Author Affiliations


Optics Express, Vol. 21, Issue 5, pp. 6162-6168 (2013)
http://dx.doi.org/10.1364/OE.21.006162


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Abstract

A special optical alignment is adopted and corresponding reconstruction algorithm is developed to reduce the reconstruction error induced by the hysteresis or backlash error of the translation stage in Ptychographical Iterative Engine (PIE) imaging with weak scattering specimen. In this suggested method, the positions of the scanning probe are determined directly from the recorded diffraction patterns rather than from the readout of the stage meter. This method not only remarkably improves the reconstruction quality, but also completely lowers the dependency of PIE on the device accuracy and accordingly enhances its feasibility for many applications with weak scattering specimen.

© 2013 OSA

1. Introduction

The ptychographical iterative engine (PIE) algorithm for solving the diffraction phase problem in electron or X-ray microscopy was proposed several years ago [1

1. J. M. Rodenburg, “Ptychography and related diffractive imaging methods,” Adv. Imaging Electron Phys. 150, 87–184 (2008). [CrossRef]

3

3. H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93(2), 023903 (2004). [CrossRef] [PubMed]

]. This approach combines the ideas of traditional ptychography and iterative phase retrieval [4

4. C. Liu, T. Walther, and J. M. Rodenburg, “Influence of thick crystal effects on ptychographic image reconstruction with moveable illumination,” Ultramicroscopy 109(10), 1263–1275 (2009). [CrossRef] [PubMed]

, 5

5. A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett. 35(15), 2585–2587 (2010). [CrossRef] [PubMed]

] to produce a powerful new technique to reconstruct the scattered wave front under situations where the illumination can be shifted laterally relative to the specimen. Using diffraction patterns measured from different positions of the incident beam, the complex transmission function of the specimen can be recovered in amplitude and phase, and thus the structural information of the specimen studied is extended. Compared to the other Coherent Diffraction Imaging (CDI) methods, PIE has wider field of view, faster computing convergence, and higher stability. Due to these advantages PIE method has attracted much attention in recent years and has been successfully realized with visible light, hard X-rays, and electron beam. Like the other CDI techniques the uniqueness of PIE lies in its capability to realize accurate phase measurement without using the interferometry, and thus it disposes off the requirements of high-quality optics and strictly stable environment and accordingly makes the phase measurement much easier. In PIE technique, the specimen needs to be stepped through a localized coherent “probe” wave front to generate a series of diffraction patterns at the plane of a detector, and during this scanning process the illuminated area of the specimen at each position should overlap with its neighbors. In practical experiments, about one hundred or more diffraction patterns are needed for the faithful reconstruction and rapid convergence in the iterative computationand thus the PIE technique heavily depends on the accuracy of the scanning mechanism. Generally the accuracy of the motorized stage is about several micrometers due to the backlash error, and the accuracy of the piezoelectric stage is about several nanometers [6

6. S. B. Jung and S. W. Kim, “Improvement of scanning accuracy of PZT piezoelectric actuators by feed-forward model-reference control,” Precis. Eng. 16(1), 49–55 (1994). [CrossRef]

,7

7. C. H. Ru and L. N. Sun, “Improving positioning accuracy of piezoelectric actuators by feedforward hysteresis compensation based on a new mathematical model,” Rev. Sci. Instrum. 76(9), 095111 (2005). [CrossRef]

] because of the hysteresis effect. So the exact specimen-probe position in practical experiments is essentially unknown, and accordingly the spatial resolution achieved is always much lower than that determined by the numerical aperture of the system. Limited by the current techniques, it is not easy to fully meet the requirement of PIE on the positioning precision, and the inaccuracy of the specimen-probe position has become the main error source for most of PIE researches.

To solve this problem, this study selects a special optical alignment for the data recording and develops a corresponding algorithm for the reconstruction. In our method, the pinhole is mounted on a translation stage to generate an illumination probe moveable relative to the fixed weak scattering specimen, and the CCD camera is placed on the image plan of the pinhole, then when the pinhole is translated laterally, the specimen will be illuminated by a moving ‘probe’ wave front. When the specimen is weak scattering, the real positions of the pinhole can be determined from the recorded diffraction patterns directly. In other words, while the common PIE technique uses the readout of the stage meter to determine the probe position, our suggested PIE method uses the ‘readout’ of the diffraction patterns instead. Thus the positioning error included in the readout of the stage meter will not affect the resolution of the reconstructed image, and accordingly the dependency of PIE on the accuracy of the scanning mechanism is remarkably lowered for many applications with weak scattering specimen.

2. Basic principle of PIE imaging

We briefly outline here the PIE algorithm, and details can be found elsewhere [8

8. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109(10), 1256–1262 (2009). [CrossRef] [PubMed]

10

10. F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82(12), 121415 (2010). [CrossRef]

]. The optical setup for the PIE technique is schematically shown in Fig. 1
Fig. 1 The principle of standard PIE method
, where the specimen with a transmission function of q(r) is fixed on a translation stage and illuminated by a probe with distribution P(r), where r is the coordinate of the object plane. When CCD is in the far field of the specimen the recorded diffraction pattern is proportional to the absolute square of the Fourier transform of the scattered wave function. In this case, the recorded intensity can be written as I(k), here I(k)∝|FFT[Ψ(r)]|. The momentum transfer k is the reciprocal coordinate of the direct space coordinate r. The far-field intensities are recorded for different sample-to-probe positions shifted by a vector R. The relationship between the exit wave and the illumination probe is Ψ(r, R) = q(r)P(r-R), where R is the shift of the probe to the object.

Like other CDI techniques [11

11. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982). [CrossRef] [PubMed]

15

15. M. C. Scott, C. C. Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius, U. Dahmen, B. C. Regan, and J. W. Miao, “Electron tomography at 2.4-angstrom resolution,” Nature 483(7390), 444–447 (2012). [CrossRef] [PubMed]

], PIE makes the reconstruction iteratively by forward propagating and backward propagating the wave between the object and the recording planes, and the phase retrieval is started with a random guess for the transmission function q0(r). The iteration reconstruction procedure is [8

8. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109(10), 1256–1262 (2009). [CrossRef] [PubMed]

] as follows:
  • a. The exit field is calculated from the known illumination and the current guess of the object qn(r):
    ψn,i(r,Ri)=qn(r)P(rRi).
    (1)
  • b. Amplitude and phase in reciprocal space are calculated from the Fourier transform of the exit field:
    ψ˜n,i(k)=FFT[ψn,i(r,Ri)]=An,iexp[iαn,i(k)].
    (2)
  • c. The amplitude is replaced by the square root of the measured values:
    ψ˜n,i(k)=Ii(k)exp[iαn,i(k)].
    (3)
  • d. The updated guess of the exit field is obtained by inverse Fourier transform:
    ψnew(k)=FFT1[Ii(k)exp[iαn,i(k)].
    (4)
  • e. The n + 1 guess of the object is updated at the position of the current illumination:
    qn+1(r)=qn(r)+U(r)[ψnew(r)ψn,i(r)],
    (5)
where the update function U(r) is U(r) = |P(r)|/max(|P(r)|)P*(r)/[|P(r)|2 + γ], where γ is an appropriately chosen constant to suppress the noise, in a way identical to a Wiener filter in conventional deconvolution methods. With the repetition of the above five steps, the reconstructed transmission function qn(r) will converge to the experimental distribution. The novelty of PIE, which makes it different from the other similar techniques, is that, it calculates the complex object function at a set of specimen positions, where the illuminated area partially overlaps its neighbor. This makes PIE quicker in convergence and wider in the field of view, however this also lets PIE heavily rely on the accuracy of the scanning mechanism.

3. The principle of the suggested method

Figure 2
Fig. 2 The diagram of the suggested method. The thin laser beam illuminates the object after transmitting a pinhole. A CCD camera is placed on the image plane of the pinhole to record the diffraction pattern behind the lens
shows the experimental setup of our suggested method, where the parallel light beam is incident on the pinhole mounted on a translation stage. When there is no specimen the pinhole can be clearly imaged by the lens onto the CCD target. When the pinhole is shifted by a distance of ∆d in the direction perpendicular to the optical axis, its image will move by a distance of ΔdSOSIon the CCD plane in Fig. 2, so that the real position of the pinhole can be measured independent of the accuracy of the translation stage.

When there is a specimen between the pinhole and the lens, the pinhole image will be distorted to some degree depending on the specimen property. For a strongly scattering specimen, the wavefront incident on CCD will be disturbed thoroughly and cannot form a pinhole image. In contrast, for the weakly scattering specimen whose transmission function can be described with the first order Born approximation, the un-diffracted beam dominates the exit wave of the specimen and still can form a roughly clear pinhole image except for some modulations formed by the diffracted beams. Fortunately, most of the specimens investigated in PIE are weak scattering, and the pinhole image at least its outline can be identified in the diffraction patterns recorded with Fig. 2, and accordingly the pinhole position can be determined in the way described above.

In Fig. 2 the ‘probe’ P(r) illuminated on the specimen can be calculated with the Fresnel formula using the parameters of pinhole shape, the distance l, and the laser wavelength λ. If the complex transmission function of the object is assumed as O(r), then its exit wave will be the product of P(r)O(r). Practically this exit wave will propagate from the left to the right along the axis, but mathematically we can conjugate it to P*(r)O*(r) and then numerically backward propagate it to the pinhole plane, generating a diffraction pattern I'(r) on the pinhole plate. In other words, I'(r) is the diffraction pattern formed by the illumination of P*(r) incident on the object of O*(r). Since the pinhole and the CCD are on two object-image conjugate planes in our setup, the distribution of I'(r) can be calculated by shrinking or magnifying the recorded diffraction pattern I(r). Thus the transmission function of O*(r) and O(r) can be reconstructed using the above described PIE (or ePIE) algorithm from the recorded diffraction patterns without using the readout of stage meter, and thus the influence of the inaccurate scanning can be reduced entirely.

4. Experimental results

To illustrate the feasibility of the suggested method, a proof-of-principle experiment is done with visible light at 632.8 nm. The optics is aligned according to Fig. 2. The diameter of the pinhole used is 5 mm. The pixel number of the CCD is 2048 × 2048, and the size of each pixel is 7.4 × 7.4 um2. The focal length of the lens used is 100 mm, and the distance between the pinhole and the lens is 181 mm, that is, the magnification of the imaging system is 1.23. The accuracy of the translation stage is about several micrometers. Diffraction patterns are recorded at 11 × 11 positions during the raster scanning of the pinhole at a step of about 500 µm. Two of these diffraction patterns are shown in Fig. 3
Fig. 3 ‘Reading’ the position of pinhole from diffraction patterns. The two diffraction patterns reproduce the shape of the pinhole very well, and their distance can be determined by binarizeing them and doing the cross correlation.
, where the images of the pinhole can be clearly identified, and it is easy to determine their distance at an accuracy of one of tenth pixel by using a proper threshold to binarize these two images and calculating their cross correlation. The distance calculated between the two pinhole images in Fig. 3 is 4.736 mm in x-direction and 5.587 mm in y-direction. The sample studied is a fixed herbage transverse section of about 1.5 × 1.5 mm2

With the diffraction patterns recorded and their distances determined, iterative reconstruction can be done with the standard PIE algorithm described above. Figure 4
Fig. 4 The reconstruction of the sample using the method proposed: (a) amplitude, (b) phase of the reconstruction. Details can be derived from them and structure of the sample is shown clearly. The scale bar is 444 μm and the phase passes from -π to-π.
shows the reconstructed modules and phase images, where the cell structure can be clearly identified at spatial resolution of about 8 μm.

For easy comparison we also do the reconstruction by using the readout of the stage meter as the positions of the scanning pinhole, the reconstructed image is shown in Fig. 5
Fig. 5 The reconstructed images of (a) modules and (b) phase using the readout of the stage driver. Compared these two images to that in Fig. 4, remarkable distortion can be found.
, which is much blurred and obviously distorted in comparison with the Fig. 4. This remarkable difference comes from the low accuracy of the motorized translation stage used, which is only about 5 μm in both x and y directions. This experimental result matches the above theoretical analysis very well.

For clarity, we show the difference between the pinhole positions determined with the suggested method and the readout of the translation stage meter in Fig. 6
Fig. 6 The comparison of positions between the readout from translation stage driver and calculated using the suggested method. The plus sign indicates the position of the pinhole determined from the recorded data sets, and the tiny circle indicates the readout of the stage driver.
, where the plus sign indicates the position of the pinhole determined from the recorded data sets, and the tiny circle indicates the readout of the stage driver. The obvious difference between them is the positioning error of common PIE method, which generates noise and distortions in the reconstructions as shown in Fig. 5.

5. Conclusion

In conclusion, a special optical setup is adopted to record the diffraction patterns and the corresponding reconstruction algorithm is developed to do PIE imaging to reduce the reconstruction distortion induced by the probe-specimen positioning error. The pinhole is mounted on a translation stage to generate a moveable probe to illuminate the fixed specimen, and a CCD camera on the image plane of the pinhole is used to record the diffraction patterns. When the specimen is a weak scattering object the shape of the pinhole can be clearly identified from the recorded diffraction patterns, and thus the relative positions between the probe and the specimen are directly determined from the recorded diffraction patterns rather than from the stage meter. While the reconstruction quality can be greatly improved with this method, the dependency of the PIE method on the device precision is remarkably lowered, enhancing the feasibility of PIE for practical applications with weak scattering specimen. It is also worthy to note that, the principle suggested can also be used for ePIE with electron microscope in theory. Though the use of the electromagnetic lens will generate some distortions in the reconstructed image, compared to the information obtained from the phase image this distortion is tolerable

Acknowledgments

This research is supported by the One Hundred Persons Project of Chinese Academy of Sciences.

References and links

1.

J. M. Rodenburg, “Ptychography and related diffractive imaging methods,” Adv. Imaging Electron Phys. 150, 87–184 (2008). [CrossRef]

2.

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,” Nature Commun. 3, 730 (2012), doi:. [CrossRef]

3.

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93(2), 023903 (2004). [CrossRef] [PubMed]

4.

C. Liu, T. Walther, and J. M. Rodenburg, “Influence of thick crystal effects on ptychographic image reconstruction with moveable illumination,” Ultramicroscopy 109(10), 1263–1275 (2009). [CrossRef] [PubMed]

5.

A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett. 35(15), 2585–2587 (2010). [CrossRef] [PubMed]

6.

S. B. Jung and S. W. Kim, “Improvement of scanning accuracy of PZT piezoelectric actuators by feed-forward model-reference control,” Precis. Eng. 16(1), 49–55 (1994). [CrossRef]

7.

C. H. Ru and L. N. Sun, “Improving positioning accuracy of piezoelectric actuators by feedforward hysteresis compensation based on a new mathematical model,” Rev. Sci. Instrum. 76(9), 095111 (2005). [CrossRef]

8.

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109(10), 1256–1262 (2009). [CrossRef] [PubMed]

9.

J. M. Rodenburg and H. M. L. Faulkner, “phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 4795–4797 (2004). [CrossRef]

10.

F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82(12), 121415 (2010). [CrossRef]

11.

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21(15), 2758–2769 (1982). [CrossRef] [PubMed]

12.

J. R. Fienup and C. C. Wackerman, “Phase retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A 3(11), 1897–1907 (1986). [CrossRef]

13.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. D. Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4(5), 394–398 (2008). [CrossRef]

14.

J. C. H. Spence, U. Weierstall, and M. Howells, “Coherence and sampling requirements for diffractive imaging,” Ultramicroscopy 101(2-4), 149–152 (2004). [CrossRef] [PubMed]

15.

M. C. Scott, C. C. Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius, U. Dahmen, B. C. Regan, and J. W. Miao, “Electron tomography at 2.4-angstrom resolution,” Nature 483(7390), 444–447 (2012). [CrossRef] [PubMed]

OCIS Codes
(100.5070) Image processing : Phase retrieval
(110.1650) Imaging systems : Coherence imaging
(120.5050) Instrumentation, measurement, and metrology : Phase measurement

ToC Category:
Image Processing

History
Original Manuscript: January 10, 2013
Revised Manuscript: February 9, 2013
Manuscript Accepted: February 10, 2013
Published: March 4, 2013

Citation
Xinchen Pan, Cheng Liu, Qiang Lin, and Jianqiang Zhu, "Ptycholographic iterative engine with self-positioned scanning illumination," Opt. Express 21, 6162-6168 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-5-6162


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References

  1. J. M. Rodenburg, “Ptychography and related diffractive imaging methods,” Adv. Imaging Electron Phys.150, 87–184 (2008). [CrossRef]
  2. 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,” Nature Commun.3, 730 (2012), doi:. [CrossRef]
  3. H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett.93(2), 023903 (2004). [CrossRef] [PubMed]
  4. C. Liu, T. Walther, and J. M. Rodenburg, “Influence of thick crystal effects on ptychographic image reconstruction with moveable illumination,” Ultramicroscopy109(10), 1263–1275 (2009). [CrossRef] [PubMed]
  5. A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett.35(15), 2585–2587 (2010). [CrossRef] [PubMed]
  6. S. B. Jung and S. W. Kim, “Improvement of scanning accuracy of PZT piezoelectric actuators by feed-forward model-reference control,” Precis. Eng.16(1), 49–55 (1994). [CrossRef]
  7. C. H. Ru and L. N. Sun, “Improving positioning accuracy of piezoelectric actuators by feedforward hysteresis compensation based on a new mathematical model,” Rev. Sci. Instrum.76(9), 095111 (2005). [CrossRef]
  8. A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy109(10), 1256–1262 (2009). [CrossRef] [PubMed]
  9. J. M. Rodenburg and H. M. L. Faulkner, “phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett.85(20), 4795–4797 (2004). [CrossRef]
  10. F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B82(12), 121415 (2010). [CrossRef]
  11. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.21(15), 2758–2769 (1982). [CrossRef] [PubMed]
  12. J. R. Fienup and C. C. Wackerman, “Phase retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A3(11), 1897–1907 (1986). [CrossRef]
  13. B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. D. Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys.4(5), 394–398 (2008). [CrossRef]
  14. J. C. H. Spence, U. Weierstall, and M. Howells, “Coherence and sampling requirements for diffractive imaging,” Ultramicroscopy101(2-4), 149–152 (2004). [CrossRef] [PubMed]
  15. M. C. Scott, C. C. Chen, M. Mecklenburg, C. Zhu, R. Xu, P. Ercius, U. Dahmen, B. C. Regan, and J. W. Miao, “Electron tomography at 2.4-angstrom resolution,” Nature483(7390), 444–447 (2012). [CrossRef] [PubMed]

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