Abstract

For performing phase retrieval in the extreme ultraviolet (EUV) regime more efficiently, developing polychromatic ptychography is desirable. As an alternative to the existing ptychographic information multiplexing (PIM) method, we present an another scheme where all monochromatic exit waves are expressed in terms of the amplitude of the transmission function and the thickness function of the object. Our proposed algorithm is a gradient based method and its validity is studied numerically. In addition, the sampling issue which appears in the polychromatic ptychography scheme and its influence to the reconstruction quality are discussed.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-ångström resolution,” Nature 559(7714), 343–349 (2018).
[Crossref]

2017 (4)

S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle bragg ptychography,” Nat. Mater. 16(2), 244–251 (2017).
[Crossref]

M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543(7645), 402–406 (2017).
[Crossref]

D. F. Gardner, M. Tanksalvala, E. R. Shanblatt, X. Zhang, B. R. Galloway, C. L. Porter, R. K. Jr, C. Bevis, D. E. Adams, H. C. Kapteyn, M. M. Murnane, and G. F. Mancini, “Subwavelength coherent imaging of periodic samples using a 13.5 nm tabletop high-harmonic light source,” Nat. Photonics 11(4), 259–263 (2017).
[Crossref]

A. Maiden, D. Johnson, and P. Li, “Further improvements to the ptychographical iterative engine,” Optica 4(7), 736–745 (2017).
[Crossref]

2016 (4)

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2(3), 310–322 (2016).
[Crossref]

S. Marchesini, H. Krishnan, B. J. Daurer, D. A. Shapiro, T. Perciano, J. A. Sethian, and F. R. N. C. Maia, “SHARP: a distributed GPU-based ptychographic solver,” J. Appl. Crystallogr. 49(4), 1245–1252 (2016).
[Crossref]

B. Zhang, D. F. Gardner, M. H. Seaberg, E. R. Shanblatt, C. L. Porter, R. Karl, C. A. Mancuso, H. C. Kapteyn, M. M. Murnane, and D. E. Adams, “Ptychographic hyperspectral spectromicroscopy with an extreme ultraviolet high harmonic comb,” Opt. Express 24(16), 18745–18754 (2016).
[Crossref]

P. D. Baksh, M. Odstrčil, H.-S. Kim, S. A. Boden, J. G. Frey, and W. S. Brocklesby, “Wide-field broadband extreme ultraviolet transmission ptychography using a high-harmonic source,” Opt. Lett. 41(7), 1317–1320 (2016).
[Crossref]

2015 (3)

2014 (6)

S. Witte, V. T. Tenner, D. W. Noom, and K. S. Eikema, “Lensless diffractive imaging with ultra-broadband table-top sources: from infrared to extreme-ultraviolet wavelengths,” Light: Sci. Appl. 3(3), e163 (2014).
[Crossref]

Y. S. G. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 22(26), 32082–32097 (2014).
[Crossref]

X. Huang, H. Yan, R. Harder, Y. Hwu, I. K. Robinson, and Y. S. Chu, “Optimization of overlap uniformness for ptychography,” Opt. Express 22(10), 12634–12644 (2014).
[Crossref]

M. D. Seaberg, B. Zhang, D. F. Gardner, E. R. Shanblatt, M. M. Murnane, H. C. Kapteyn, and D. E. Adams, “Tabletop nanometer extreme ultraviolet imaging in an extended reflection mode using coherent fresnel ptychography,” Optica 1(1), 39–44 (2014).
[Crossref]

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104(17), 171104 (2014).
[Crossref]

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
[Crossref]

2013 (2)

2012 (3)

M. Humphry, B. Kraus, A. Hurst, A. Maiden, and J. Rodenburg, “Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging,” Nat. Commun. 3(1), 730 (2012).
[Crossref]

A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. A 29(8), 1606–1614 (2012).
[Crossref]

J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3(1), 993 (2012).
[Crossref]

2010 (2)

K. A. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys. 59(1), 1–99 (2010).
[Crossref]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467(7314), 436–439 (2010).
[Crossref]

2009 (4)

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

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref]

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, “Diffractive imaging using partially coherent x rays,” Phys. Rev. Lett. 103(24), 243902 (2009).
[Crossref]

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 79(2), 023809 (2009).
[Crossref]

2008 (3)

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108(5), 481–487 (2008).
[Crossref]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref]

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16(10), 7264–7278 (2008).
[Crossref]

2007 (1)

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref]

2005 (1)

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21(1), 37–50 (2005).
[Crossref]

2004 (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]

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

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

2003 (1)

1997 (1)

1969 (1)

L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17(2), 86–92 (1969).
[Crossref]

Abbey, B.

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 79(2), 023809 (2009).
[Crossref]

Adams, D. E.

Aeppli, G.

M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543(7645), 402–406 (2017).
[Crossref]

Allain, M.

S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle bragg ptychography,” Nat. Mater. 16(2), 244–251 (2017).
[Crossref]

Baksh, P. D.

Balaur, E.

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, “Diffractive imaging using partially coherent x rays,” Phys. Rev. Lett. 103(24), 243902 (2009).
[Crossref]

Batey, D. J.

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
[Crossref]

Bean, R.

Berenguer, F.

Bevis, C.

D. F. Gardner, M. Tanksalvala, E. R. Shanblatt, X. Zhang, B. R. Galloway, C. L. Porter, R. K. Jr, C. Bevis, D. E. Adams, H. C. Kapteyn, M. M. Murnane, and G. F. Mancini, “Subwavelength coherent imaging of periodic samples using a 13.5 nm tabletop high-harmonic light source,” Nat. Photonics 11(4), 259–263 (2017).
[Crossref]

Boden, S. A.

Brocklesby, W. S.

Bunk, O.

M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543(7645), 402–406 (2017).
[Crossref]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467(7314), 436–439 (2010).
[Crossref]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108(5), 481–487 (2008).
[Crossref]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref]

Burdet, N.

C. Maia, F. R. N.

S. Marchesini, H. Krishnan, B. J. Daurer, D. A. Shapiro, T. Perciano, J. A. Sethian, and F. R. N. C. Maia, “SHARP: a distributed GPU-based ptychographic solver,” J. Appl. Crystallogr. 49(4), 1245–1252 (2016).
[Crossref]

Campbell, S. I.

Z. Dong, Y.-L. L. Fang, X. Huang, H. Yan, S. Ha, W. Xu, Y. S. Chu, S. I. Campbell, and M. Lin, “High-performance multi-mode ptychography reconstruction on distributed GPUs,” in 2018 New York Scientific Data Summit (NYSDS), (IEEE, 2018).

Chamard, V.

S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle bragg ptychography,” Nat. Mater. 16(2), 244–251 (2017).
[Crossref]

Chen, B.

F. Zhang, I. Peterson, J. Vila-Comamala, A. Diaz, F. Berenguer, R. Bean, B. Chen, A. Menzel, I. K. Robinson, and J. M. Rodenburg, “Translation position determination in ptychographic coherent diffraction imaging,” Opt. Express 21(11), 13592–13606 (2013).
[Crossref]

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 79(2), 023809 (2009).
[Crossref]

Chen, Z.

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-ångström resolution,” Nature 559(7714), 343–349 (2018).
[Crossref]

Chu, Y. S.

X. Huang, K. Lauer, J. N. Clark, W. Xu, E. Nazaretski, R. Harder, I. K. Robinson, and Y. S. Chu, “Fly-scan ptychography,” Sci. Rep. 5(1), 9074 (2015).
[Crossref]

X. Huang, H. Yan, R. Harder, Y. Hwu, I. K. Robinson, and Y. S. Chu, “Optimization of overlap uniformness for ptychography,” Opt. Express 22(10), 12634–12644 (2014).
[Crossref]

Z. Dong, Y.-L. L. Fang, X. Huang, H. Yan, S. Ha, W. Xu, Y. S. Chu, S. I. Campbell, and M. Lin, “High-performance multi-mode ptychography reconstruction on distributed GPUs,” in 2018 New York Scientific Data Summit (NYSDS), (IEEE, 2018).

Clark, J.

J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3(1), 993 (2012).
[Crossref]

Clark, J. N.

N. Burdet, X. Shi, D. Parks, J. N. Clark, X. Huang, S. D. Kevan, and I. K. Robinson, “Evaluation of partial coherence correction in x-ray ptychography,” Opt. Express 23(5), 5452–5467 (2015).
[Crossref]

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B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104(17), 171104 (2014).
[Crossref]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467(7314), 436–439 (2010).
[Crossref]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108(5), 481–487 (2008).
[Crossref]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref]

Porter, C. L.

D. F. Gardner, M. Tanksalvala, E. R. Shanblatt, X. Zhang, B. R. Galloway, C. L. Porter, R. K. Jr, C. Bevis, D. E. Adams, H. C. Kapteyn, M. M. Murnane, and G. F. Mancini, “Subwavelength coherent imaging of periodic samples using a 13.5 nm tabletop high-harmonic light source,” Nat. Photonics 11(4), 259–263 (2017).
[Crossref]

B. Zhang, D. F. Gardner, M. H. Seaberg, E. R. Shanblatt, C. L. Porter, R. Karl, C. A. Mancuso, H. C. Kapteyn, M. M. Murnane, and D. E. Adams, “Ptychographic hyperspectral spectromicroscopy with an extreme ultraviolet high harmonic comb,” Opt. Express 24(16), 18745–18754 (2016).
[Crossref]

Purohit, P.

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-ångström resolution,” Nature 559(7714), 343–349 (2018).
[Crossref]

Quiney, H. M.

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, “Diffractive imaging using partially coherent x rays,” Phys. Rev. Lett. 103(24), 243902 (2009).
[Crossref]

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 79(2), 023809 (2009).
[Crossref]

Raabe, J.

M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543(7645), 402–406 (2017).
[Crossref]

Rabiner, L.

L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17(2), 86–92 (1969).
[Crossref]

Rader, C.

L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17(2), 86–92 (1969).
[Crossref]

Robinson, I.

J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3(1), 993 (2012).
[Crossref]

Robinson, I. K.

Rodenburg, J.

M. Humphry, B. Kraus, A. Hurst, A. Maiden, and J. Rodenburg, “Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging,” Nat. Commun. 3(1), 730 (2012).
[Crossref]

Rodenburg, J. M.

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
[Crossref]

F. Zhang, I. Peterson, J. Vila-Comamala, A. Diaz, F. Berenguer, R. Bean, B. Chen, A. Menzel, I. K. Robinson, and J. M. Rodenburg, “Translation position determination in ptychographic coherent diffraction imaging,” Opt. Express 21(11), 13592–13606 (2013).
[Crossref]

A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. A 29(8), 1606–1614 (2012).
[Crossref]

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

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[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]

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

Ross, R.

Ruder, S.

S. Ruder, “An overview of gradient descent optimization algorithms,” arXiv:1609.04747 (2016).

Schafer, R.

L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17(2), 86–92 (1969).
[Crossref]

Schneider, P.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467(7314), 436–439 (2010).
[Crossref]

Seaberg, M. D.

Seaberg, M. H.

Sethian, J. A.

S. Marchesini, H. Krishnan, B. J. Daurer, D. A. Shapiro, T. Perciano, J. A. Sethian, and F. R. N. C. Maia, “SHARP: a distributed GPU-based ptychographic solver,” J. Appl. Crystallogr. 49(4), 1245–1252 (2016).
[Crossref]

Shanblatt, E. R.

Shapiro, D. A.

S. Marchesini, H. Krishnan, B. J. Daurer, D. A. Shapiro, T. Perciano, J. A. Sethian, and F. R. N. C. Maia, “SHARP: a distributed GPU-based ptychographic solver,” J. Appl. Crystallogr. 49(4), 1245–1252 (2016).
[Crossref]

Shi, X.

Spence, J.

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

Stockmar, M.

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104(17), 171104 (2014).
[Crossref]

Tanksalvala, M.

D. F. Gardner, M. Tanksalvala, E. R. Shanblatt, X. Zhang, B. R. Galloway, C. L. Porter, R. K. Jr, C. Bevis, D. E. Adams, H. C. Kapteyn, M. M. Murnane, and G. F. Mancini, “Subwavelength coherent imaging of periodic samples using a 13.5 nm tabletop high-harmonic light source,” Nat. Photonics 11(4), 259–263 (2017).
[Crossref]

Tate, M. W.

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-ångström resolution,” Nature 559(7714), 343–349 (2018).
[Crossref]

Teichmann, S.

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 79(2), 023809 (2009).
[Crossref]

Tenner, V. T.

S. Witte, V. T. Tenner, D. W. Noom, and K. S. Eikema, “Lensless diffractive imaging with ultra-broadband table-top sources: from infrared to extreme-ultraviolet wavelengths,” Light: Sci. Appl. 3(3), e163 (2014).
[Crossref]

Thibault, P.

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104(17), 171104 (2014).
[Crossref]

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494(7435), 68–71 (2013).
[Crossref]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467(7314), 436–439 (2010).
[Crossref]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref]

Tian, L.

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2(3), 310–322 (2016).
[Crossref]

Tsai, E. H. R.

M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543(7645), 402–406 (2017).
[Crossref]

Van Dao, L.

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 79(2), 023809 (2009).
[Crossref]

Varma, P.

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2(3), 310–322 (2016).
[Crossref]

Vila-Comamala, J.

Vine, D. J.

Y. S. G. Nashed, D. J. Vine, T. Peterka, J. Deng, R. Ross, and C. Jacobsen, “Parallel ptychographic reconstruction,” Opt. Express 22(26), 32082–32097 (2014).
[Crossref]

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, “Diffractive imaging using partially coherent x rays,” Phys. Rev. Lett. 103(24), 243902 (2009).
[Crossref]

Waller, L.

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2(3), 310–322 (2016).
[Crossref]

Weierstall, U.

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

Wepf, R.

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467(7314), 436–439 (2010).
[Crossref]

Whitehead, L. W.

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, “Diffractive imaging using partially coherent x rays,” Phys. Rev. Lett. 103(24), 243902 (2009).
[Crossref]

Williams, G. J.

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, “Diffractive imaging using partially coherent x rays,” Phys. Rev. Lett. 103(24), 243902 (2009).
[Crossref]

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 79(2), 023809 (2009).
[Crossref]

Witte, S.

S. Witte, V. T. Tenner, D. W. Noom, and K. S. Eikema, “Lensless diffractive imaging with ultra-broadband table-top sources: from infrared to extreme-ultraviolet wavelengths,” Light: Sci. Appl. 3(3), e163 (2014).
[Crossref]

Wright, M. H.

W. Murray, M. H. Wright, and P. E. Gill, Practical Optimization (Emerald Publishing Limited, 1982).

Xie, S.

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-ångström resolution,” Nature 559(7714), 343–349 (2018).
[Crossref]

Xu, W.

X. Huang, K. Lauer, J. N. Clark, W. Xu, E. Nazaretski, R. Harder, I. K. Robinson, and Y. S. Chu, “Fly-scan ptychography,” Sci. Rep. 5(1), 9074 (2015).
[Crossref]

Z. Dong, Y.-L. L. Fang, X. Huang, H. Yan, S. Ha, W. Xu, Y. S. Chu, S. I. Campbell, and M. Lin, “High-performance multi-mode ptychography reconstruction on distributed GPUs,” in 2018 New York Scientific Data Summit (NYSDS), (IEEE, 2018).

Yan, H.

X. Huang, H. Yan, R. Harder, Y. Hwu, I. K. Robinson, and Y. S. Chu, “Optimization of overlap uniformness for ptychography,” Opt. Express 22(10), 12634–12644 (2014).
[Crossref]

Z. Dong, Y.-L. L. Fang, X. Huang, H. Yan, S. Ha, W. Xu, Y. S. Chu, S. I. Campbell, and M. Lin, “High-performance multi-mode ptychography reconstruction on distributed GPUs,” in 2018 New York Scientific Data Summit (NYSDS), (IEEE, 2018).

Zhang, B.

Zhang, F.

Zhang, X.

D. F. Gardner, M. Tanksalvala, E. R. Shanblatt, X. Zhang, B. R. Galloway, C. L. Porter, R. K. Jr, C. Bevis, D. E. Adams, H. C. Kapteyn, M. M. Murnane, and G. F. Mancini, “Subwavelength coherent imaging of periodic samples using a 13.5 nm tabletop high-harmonic light source,” Nat. Photonics 11(4), 259–263 (2017).
[Crossref]

Zhong, J.

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2(3), 310–322 (2016).
[Crossref]

Adv. Phys. (1)

K. A. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys. 59(1), 1–99 (2010).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

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

B. Enders, M. Dierolf, P. Cloetens, M. Stockmar, F. Pfeiffer, and P. Thibault, “Ptychography with broad-bandwidth radiation,” Appl. Phys. Lett. 104(17), 171104 (2014).
[Crossref]

IEEE Trans. Audio Electroacoust. (1)

L. Rabiner, R. Schafer, and C. Rader, “The chirp z-transform algorithm,” IEEE Trans. Audio Electroacoust. 17(2), 86–92 (1969).
[Crossref]

IEEE Trans. Comput. Imaging (1)

J. Zhong, L. Tian, P. Varma, and L. Waller, “Nonlinear optimization algorithm for partially coherent phase retrieval and source recovery,” IEEE Trans. Comput. Imaging 2(3), 310–322 (2016).
[Crossref]

Inverse Probl. (1)

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21(1), 37–50 (2005).
[Crossref]

J. Appl. Crystallogr. (1)

S. Marchesini, H. Krishnan, B. J. Daurer, D. A. Shapiro, T. Perciano, J. A. Sethian, and F. R. N. C. Maia, “SHARP: a distributed GPU-based ptychographic solver,” J. Appl. Crystallogr. 49(4), 1245–1252 (2016).
[Crossref]

J. Opt. Soc. Am. A (2)

Light: Sci. Appl. (1)

S. Witte, V. T. Tenner, D. W. Noom, and K. S. Eikema, “Lensless diffractive imaging with ultra-broadband table-top sources: from infrared to extreme-ultraviolet wavelengths,” Light: Sci. Appl. 3(3), e163 (2014).
[Crossref]

Nat. Commun. (2)

J. Clark, X. Huang, R. Harder, and I. Robinson, “High-resolution three-dimensional partially coherent diffraction imaging,” Nat. Commun. 3(1), 993 (2012).
[Crossref]

M. Humphry, B. Kraus, A. Hurst, A. Maiden, and J. Rodenburg, “Ptychographic electron microscopy using high-angle dark-field scattering for sub-nanometre resolution imaging,” Nat. Commun. 3(1), 730 (2012).
[Crossref]

Nat. Mater. (1)

S. O. Hruszkewycz, M. Allain, M. V. Holt, C. E. Murray, J. R. Holt, P. H. Fuoss, and V. Chamard, “High-resolution three-dimensional structural microscopy by single-angle bragg ptychography,” Nat. Mater. 16(2), 244–251 (2017).
[Crossref]

Nat. Photonics (1)

D. F. Gardner, M. Tanksalvala, E. R. Shanblatt, X. Zhang, B. R. Galloway, C. L. Porter, R. K. Jr, C. Bevis, D. E. Adams, H. C. Kapteyn, M. M. Murnane, and G. F. Mancini, “Subwavelength coherent imaging of periodic samples using a 13.5 nm tabletop high-harmonic light source,” Nat. Photonics 11(4), 259–263 (2017).
[Crossref]

Nature (4)

Y. Jiang, Z. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-ångström resolution,” Nature 559(7714), 343–349 (2018).
[Crossref]

M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543(7645), 402–406 (2017).
[Crossref]

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494(7435), 68–71 (2013).
[Crossref]

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467(7314), 436–439 (2010).
[Crossref]

Opt. Express (7)

Opt. Lett. (1)

Optica (2)

Phys. Rev. A (1)

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 79(2), 023809 (2009).
[Crossref]

Phys. Rev. Lett. (3)

L. W. Whitehead, G. J. Williams, H. M. Quiney, D. J. Vine, R. A. Dilanian, S. Flewett, K. A. Nugent, A. G. Peele, E. Balaur, and I. McNulty, “Diffractive imaging using partially coherent x rays,” Phys. Rev. Lett. 103(24), 243902 (2009).
[Crossref]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[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]

Sci. Rep. (1)

X. Huang, K. Lauer, J. N. Clark, W. Xu, E. Nazaretski, R. Harder, I. K. Robinson, and Y. S. Chu, “Fly-scan ptychography,” Sci. Rep. 5(1), 9074 (2015).
[Crossref]

Science (1)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref]

Ultramicroscopy (5)

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

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref]

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
[Crossref]

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

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108(5), 481–487 (2008).
[Crossref]

Other (6)

W. Murray, M. H. Wright, and P. E. Gill, Practical Optimization (Emerald Publishing Limited, 1982).

S. Ruder, “An overview of gradient descent optimization algorithms,” arXiv:1609.04747 (2016).

G. D. Martin, “Chirp z-transform spectral zoom optimization with MATLAB,” Tech. rep. (2005).

A. Fannjiang, “Raster grid pathology and the cure,” arXiv:1810.00852v3 (2018).

Z. Dong, Y.-L. L. Fang, X. Huang, H. Yan, S. Ha, W. Xu, Y. S. Chu, S. I. Campbell, and M. Lin, “High-performance multi-mode ptychography reconstruction on distributed GPUs,” in 2018 New York Scientific Data Summit (NYSDS), (IEEE, 2018).

J. Goodman, Introduction to Fourier Optics, McGraw-Hill physical and quantum electronics series (W. H. Freeman, 2005).

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Figures (12)

Fig. 1.
Fig. 1. Polychromatic ptychography configuration with plane-wave illumination.
Fig. 2.
Fig. 2. Simulation results for validating Algorithm 1. The illumination includes polychromatic plane-waves. (a1)-(a8) are the simulation results when the illumination is monochromatic, reconstructed by employing the PIE algorithm. (b1)-(b8), (c1)-(c8) and (d1)-(d8) show the reconstructions with implementing Algorithm 1, for the case where 2, 5 and 20 spectral components are included in the probe’s spectrum, respectively. A schematic demonstration regarding how we generate the temporal spectrum can be found on the left hand column. Note that the distances in frequency between adjacent frequencies are identical, and all the wavelengths share the same number of photons.
Fig. 3.
Fig. 3. The final values of the NER and the NEF after our simulation converged. In (a) we show the calculated NERs with 1-20 spectral components in the probe’s spectrum, and with the TPN varies from $10^{5}$ to $10^{7}$. The NEFs for the same settings are depicted in (b). Note that the signal-to-noise ratio is so large that the noise has negligible influence when the TPN equals $10^{7}$, therefore in this plot the blue dots and red ones are almost overlap.
Fig. 4.
Fig. 4. Graphical description of the two different ways of computing the polychromatic diffraction intensity pattern in our simulation. (a) illustrates the calculation process in Eq. (4), which includes the wavelength-dependency of the wavefield propagation. (b) describes the wavefield propagation model which is without the wavelength-dependency. The red frame represents the boundary of an imaginary detector $\textbf {1}_{D}(\textbf {r}')$.
Fig. 5.
Fig. 5. The numerical experiment results corresponding to the situation where the wavefield propagation is wavelength-independent, as in Fig. 4(b). In these plots same choices have been made for the number of wavelengths and noise as in Fig. 3.
Fig. 6.
Fig. 6. A comparison simulation result between Algorithm 1 and the PIM method, with noise-free measurements. The NEF values in the right figure are identical for the two methods.
Fig. 7.
Fig. 7. The reconstructed object functions for the situation where the probe has 20 spectral components. Noise-free measurements are used in this simulation and the propagation model follows the scheme in Eq. (4).
Fig. 8.
Fig. 8. Simulation results with probe reconstruction following the scheme in Eq. (9). On the left the probe’s temporal spectrum is shown. In (a1)-(a4) are the reconstructed probe function when the initial probe has a complex profile. The difference between the original complex profile probe and the reconstructed probe are illustrated in (a5)-(a6). (b1)-(b4) are the results with a polychromatic plane-wave initial probe function.
Fig. 9.
Fig. 9. Graphical description of the two different ways of computing the polychromatic diffraction intensity pattern. The red frame represents the boundary of an imaginary detector $\textbf {1}_{D}(\textbf {r}')$. (a) illustrates the situation where the detector can records incomplete data. (b) describes the situation where the detector is able to measure the maximum spatial frequency component for the largest wavelength $\lambda _{N}$, hence also for the wavelengths shorter than $\lambda _{N}$ each monochromatic diffraction intensities are zero-padded in accordance with Eq. (4).
Fig. 10.
Fig. 10. A comparison between simulation results of the two sampling situations shown in Figs. 9(a) and 9(b), with noise-free measurements. The orange dots are related to the propagation model which is illustrated in Fig. 9(a), while the blue dots corresponds to the sampling scheme which is shown in Fig. 9(b).
Fig. 11.
Fig. 11. The evolution of NEF and NER in our simulation described in Subsection 3.1.1. Each curve in (a) and (b) is related to one situation in Fig. 3.
Fig. 12.
Fig. 12. Simulation results about how the overlap ratio affects the performance of polychromatic ptychography. In this plot the diffraction intensity measurements are noise free. (a) shows the reconstructed object when the probe only contains 5 wavelengths, and when the overlap ratio equals 0%, 80% and 99%, respectively. In (b) we demonstrate the NERs for a series of wavelengths and for overlap ratios ranging from 0% to 99%.

Tables (2)

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Algorithm 1. polychromatic ptychography algorithm with plane-wave illumination

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Table 1. The maximum measurable spatial frequency for 30 n m , 40 n m and 50 n m wavelength, with a detector which contains a 320 × 320 array of 15 μ m pixels. The propagation distance is assumed to be 1 c m . The diameter of the circular aperture which lies inside the probe function is 10 μ m . Hence the Fresnel number is 1 / 3 for 30 n m wavelength.

Equations (45)

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Ψ j ( r , λ k ) = P ( r R j ) A ( r ) exp [ i λ 1 λ k ϕ ( r ) ] ,
P ( r ) = { P ( r ) , | r | r 0 , 0 , | r | > r 0 .
E j ( A , ϕ ) = [ I M , j ( r ) I E , j ( r ) ] 2 d r ,
I E , j ( r ) = 1 D ( r ) k | F { Ψ j ( r , λ k ) } ( r λ k z ) | 2 ,
1 D ( r ) = { 1 , | x | x D , | y | y D , 0 , | x | > x D , | y | > y D ,
r = x e ^ x + y e ^ y ,
{ A n + 1 ( r ) = A n ( r ) + δ A k { [ P j ( r ) ] exp [ i λ 1 λ k ϕ n ( r ) ] Δ Ψ j , n ( r , λ k ) } , ϕ n + 1 ( r ) = ϕ n ( r ) + δ ϕ k { [ P j ( r ) ] i λ 1 λ k A n ( r ) exp [ i λ 1 λ k ϕ n ( r ) ] Δ Ψ j , n ( r , λ k ) } ,
Δ Ψ j , n ( r , λ k ) = F 1 { 1 D ( r ) ( I M , j ( r ) I E , j , n ( r ) + α 1 ) F [ Ψ j , n ( r , λ k ) ] ( r λ k z ) } ( r , λ k ) ,
{ A n + 1 ( r ) = A n ( r ) + δ A k { [ P j , n ( r ) ] exp [ i λ 1 λ k ϕ n ( r ) ] Δ Ψ j , n ( r , λ k ) } , ϕ n + 1 ( r ) = ϕ n ( r ) + δ ϕ k { [ P j , n ( r ) ] i λ 1 λ k A n ( r ) exp [ i λ 1 λ k ϕ n ( r ) ] Δ Ψ j , n ( r , λ k ) } , P n + 1 ( r ) = P n ( r ) + δ P k A n ( r + R j ) exp [ i λ 1 λ k ϕ n ( r + R j ) ] Δ Ψ j , n ( r + R j , λ k ) ,
E F ( n ) = j r | I M , j ( r ) I E , j , n ( r ) | 2 j r | I M , j ( r ) | ,
E R ( n ) = r | O ( r ) γ O n ( r ) | 2 r | O ( r ) | 2 ,
{ O n ( r ) = A n ( r ) exp [ i ϕ n ( r ) ] , γ = r O ( r ) O n ( r ) r | O n ( r ) | 2 .
overlap ratio = 1 d L ,
Ψ j ( r ) = P ( r R j ) O ( r ) = P j ( r ) O ( r )
P ( r ) = { P ( r ) , | r | r 0 , 0 , | r | > r 0 ,
I E , j ( r ) = | F ( Ψ j ) ( r λ z ) | 2 = | Ψ j ( r ) exp ( i 2 π λ z r r ) d r | 2 ,
E = j [ I M , j ( r ) I E , j ( r ) ] 2 d r ,
E j ( P , O ) = [ I M , j ( r ) I E , j ( r ) ] 2 d r .
δ E j ( P , O ) ( δ O ) = 2 ( I M , j ( r ) I E , j ( r ) 1 ) [ F ( P j O ) ( r ) F ( P j δ O ) ( r ) ] d r = 2 [ ( I M , j ( r ) I E , j ( r ) 1 ) F ( P j O ) ( r ) F ( P j δ O ) ( r ) d r ] = 2 { F 1 [ ( I M , j ( r ) I E , j ( r ) 1 ) F ( P j O ) ( r ) ] P j ( r ) δ O ( r ) d r } = 2 [ Δ Ψ j ( r ) P j ( r ) δ O ( r ) d r ] ,
Δ Ψ j ( r ) = F 1 [ ( I M , j ( r ) I E , j ( r ) 1 ) F ( Ψ j ) ( r ) ] ( r ) ,
minimize δ O δ E j ( δ O ) subject to δ O 2 = c o n s t ,
δ O 2 = δ O ( r ) δ O ( r ) d r .
L j ( δ O ) = δ E j ( δ O ) λ O δ O 2 ,
[ Δ Ψ j ( r ) P j ( r ) δ O ~ ( r ) d r ] = [ λ O δ O ( r ) δ O ~ ( r ) d r ] .
[ Δ Ψ j ( r ) P j ( r ) ] δ O ~ ( r ) d r = λ O [ δ O ( r ) ] δ O ~ ( r ) d r .
[ Δ Ψ j ( r ) P j ( r ) ] = λ O [ δ O ( r ) ] .
[ Δ Ψ j ( r ) P j ( r ) ] [ i δ O ~ ( r ) ] d r = λ O [ δ O ( r ) ] [ i δ O ~ ( r ) ] d r ,
[ Δ Ψ j ( r ) P j ( r ) ] = λ O [ δ O ( r ) ] ,
Δ Ψ j ( r ) P j ( r ) = λ O δ O ( r ) .
O n + 1 ( r ) = O n ( r ) β O P n ( r R j ) Δ Ψ j , n ( r ) ,
P n + 1 ( r ) = P n ( r ) β P O n ( r + R j ) Δ Ψ j , n ( r + R j ) ,
Ψ j ( r , λ k ) = P ( r R j ) A ( r ) exp [ i λ 1 λ k ϕ ( r ) ]
I E , j ( r ) = 1 D ( r ) λ k | F [ Ψ j ( r , λ k ) ] ( r λ k z ) | 2 = 1 D ( r ) λ k | Ψ j ( r , λ k ) exp ( i 2 π λ k z r r ) d r | 2 ,
1 D ( r ) = { 1 , | x | x D , | y | y D , 0 , | x | > x D , | y | > y D ,
r = x e ^ x + y e ^ y .
{ δ E j ( P , A , ϕ ) ( δ A ) = 2 λ k { Δ Ψ j ( r , λ k ) P j ( r ) exp [ i λ 1 λ k ϕ ( r ) ] } δ A ( r ) d r , δ E j ( P , A , ϕ ) ( δ ϕ ) = 2 λ k { i λ 1 λ k Δ Ψ j ( r , λ k ) P j ( r ) A ( r ) exp [ i λ 1 λ k ϕ ( r ) ] } δ ϕ ( r ) d r , δ E j ( P , A , ϕ ) ( δ P ) = λ k { Δ Ψ j ( r , λ k ) P j ( r ) A ( r ) exp [ i λ 1 λ k ϕ ( r ) ] δ P ( r ) d r } ,
Δ Ψ j , n ( r , λ k ) = F 1 { 1 D ( r ) ( I M , j ( r ) I E , j , n ( r ) 1 ) F [ Ψ j , n ( r , λ k ) ] ( r λ k z ) } ( r , λ k ) .
{ A n + 1 ( r ) = A n ( r ) + δ A k { [ P j , n ( r ) ] exp [ i λ 1 λ k ϕ n ( r ) ] Δ Ψ n ( r , λ k ) } , ϕ n + 1 ( r ) = ϕ n ( r ) + δ ϕ k { i λ 1 λ k [ P j , n ( r ) ] A n ( r ) exp [ i λ 1 λ k ϕ n ( r ) ] Δ Ψ j , n ( r , λ k ) } , P n + 1 ( r ) = P n ( r ) + δ P k A n ( r + R j ) exp [ i λ 1 λ k ϕ n ( r + R j ) ] Δ Ψ j , n ( r + R j , λ k ) ,
e N = TPN N k = 1 N h ν k ,
e N = TPN N h [ ν 1 + ν N + m = 1 N 2 m ν 1 + ( N m 1 ) ν N N 1 ] ,
e N = TPN 2 h ( ν 1 + ν N ) .
I E , j ( r ) = | F ( Ψ j ) ( r ) | 2 = | exp ( i α ) F ( Ψ j ) ( r ) | 2 = | F [ Ψ j exp ( i α ) ] ( r ) | 2 = | F ( Ψ j ) ( r ) | 2 ,
F ( Ψ j ) ( r ) = F ( Ψ j ) ( r ) .
Ψ j ( r ) = P ( r R j ) O ( r ) = P ( r R j ) C ( r ) O ( r ) 1 C ( r ) = P ( r R j ) O ( r ) ,
{ P ( r R j ) = P ( r R j ) C ( r ) , O ( r ) = O ( r ) 1 C ( r ) .

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