## Ptycholographic iterative engine with self-positioned scanning illumination |

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

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

## 2. Basic principle of PIE imaging

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. 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]

**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.

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

_{0}(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]

^{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 q

_{n}(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

**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

*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

*um*

^{2}. 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 , 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 mm

^{2}

## 5. Conclusion

## Acknowledgments

## References and links

1. | J. M. Rodenburg, “Ptychography and related diffractive imaging methods,” Adv. Imaging Electron Phys. |

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. | H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. |

4. | C. Liu, T. Walther, and J. M. Rodenburg, “Influence of thick crystal effects on ptychographic image reconstruction with moveable illumination,” Ultramicroscopy |

5. | A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett. |

6. | S. B. Jung and S. W. Kim, “Improvement of scanning accuracy of PZT piezoelectric actuators by feed-forward model-reference control,” Precis. Eng. |

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. |

8. | A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy |

9. | J. M. Rodenburg and H. M. L. Faulkner, “phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. |

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 |

11. | J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. |

12. | J. R. Fienup and C. C. Wackerman, “Phase retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A |

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. |

14. | J. C. H. Spence, U. Weierstall, and M. Howells, “Coherence and sampling requirements for diffractive imaging,” Ultramicroscopy |

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 |

**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

- J. M. Rodenburg, “Ptychography and related diffractive imaging methods,” Adv. Imaging Electron Phys.150, 87–184 (2008). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy109(10), 1256–1262 (2009). [CrossRef] [PubMed]
- J. M. Rodenburg and H. M. L. Faulkner, “phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett.85(20), 4795–4797 (2004). [CrossRef]
- 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]
- J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt.21(15), 2758–2769 (1982). [CrossRef] [PubMed]
- J. R. Fienup and C. C. Wackerman, “Phase retrieval stagnation problems and solutions,” J. Opt. Soc. Am. A3(11), 1897–1907 (1986). [CrossRef]
- 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]
- J. C. H. Spence, U. Weierstall, and M. Howells, “Coherence and sampling requirements for diffractive imaging,” Ultramicroscopy101(2-4), 149–152 (2004). [CrossRef] [PubMed]
- 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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