Abstract

In recent years, coherent diffractive imaging has been considered as a promising alternative for information retrieval instead of conventional interference methods. Coherent diffractive imaging using the X-ray light source has opened up a new research perspective for the measurement of non-crystalline and biological specimens, and can achieve unprecedentedly high resolutions. In this paper, we show how a three-dimensional (3D) particle-like distribution and coherent diffractive imaging can be applied for a study of optical cryptography. An optical multiple-random-phase-mask encoding approach is used, and the plaintext is considered as a series of particles distributed in a 3D space. A topology concept is also introduced into the proposed optical cryptosystem. During image decryption, a retrieval algorithm is developed to extract the plaintext from the ciphertexts. In addition, security and advantages of the proposed optical cryptography topology are also analyzed.

© 2011 OSA

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2010 (6)

2009 (7)

2008 (8)

H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008).
[CrossRef]

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100(15), 155503 (2008).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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

P. Bao, F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval using multiple illumination wavelengths,” Opt. Lett. 33(4), 309–311 (2008).
[CrossRef] [PubMed]

T. J. Naughton, B. M. Hennelly, and T. Dowling, “Introducing secure modes of operation for optical encryption,” J. Opt. Soc. Am. A 25(10), 2608–2617 (2008).
[CrossRef]

Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008).
[CrossRef] [PubMed]

2007 (6)

P. Almoro, G. Pedrini, and W. Osten, “Aperture synthesis in phase retrieval using a volume-speckle field,” Opt. Lett. 32(7), 733–735 (2007).
[CrossRef] [PubMed]

A. Anand, G. Pedrini, W. Osten, and P. Almoro, “Wavefront sensing with random amplitude mask and phase retrieval,” Opt. Lett. 32(11), 1584–1586 (2007).
[CrossRef] [PubMed]

Y. Shi, G. Situ, and J. Zhang, “Multiple-image hiding in the Fresnel domain,” Opt. Lett. 32(13), 1914–1916 (2007).
[CrossRef] [PubMed]

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75(4), 043805 (2007).
[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] [PubMed]

V. Elser, I. Rankenburg, and P. Thibault, “Searching with iterated maps,” Proc. Natl. Acad. Sci. U.S.A. 104(2), 418–423 (2007).
[CrossRef] [PubMed]

2006 (6)

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006).
[CrossRef] [PubMed]

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97(2), 025506 (2006).
[CrossRef] [PubMed]

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused X-ray fields,” Nat. Phys. 2(2), 101–104 (2006).
[CrossRef]

J. Miao, C. C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T. K. Lee, and S. H. Risbud, “Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy,” Phys. Rev. Lett. 97(21), 215503 (2006).
[CrossRef] [PubMed]

X. Peng, P. Zhang, H. Wei, and B. Yu, “Known-plaintext attack on optical encryption based on double random phase keys,” Opt. Lett. 31(8), 1044–1046 (2006).
[CrossRef] [PubMed]

X. Peng, H. Wei, and P. Zhang, “Chosen-plaintext attack on lensless double-random phase encoding in the Fresnel domain,” Opt. Lett. 31(22), 3261–3263 (2006).
[CrossRef] [PubMed]

2005 (2)

G. Situ and J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30(11), 1306–1308 (2005).
[CrossRef] [PubMed]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102(43), 15343–15346 (2005).
[CrossRef] [PubMed]

2004 (5)

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

W. McBride, N. L. O’Leary, and L. J. Allen, “Retrieval of a complex-valued object from its diffraction pattern,” Phys. Rev. Lett. 93(23), 233902 (2004).
[CrossRef] [PubMed]

G. Situ and J. Zhang, “A lensless optical security system based on computer-generated phase only masks,” Opt. Commun. 232(1-6), 115–122 (2004).
[CrossRef]

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29(14), 1584–1586 (2004).
[CrossRef] [PubMed]

2003 (2)

V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20(1), 40–55 (2003).
[CrossRef]

J. Miao, K. O. Hodgson, T. Ishikawa, C. A. Larabell, M. A. LeGros, and Y. Nishino, “Imaging whole Escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. U.S.A. 100(1), 110–112 (2003).
[CrossRef] [PubMed]

2002 (4)

X. Peng, L. Yu, and L. Cai, “Double-lock for image encryption with virtual optical wavelength,” Opt. Express 10(1), 41–45 (2002).
[PubMed]

H. T. Chang, W. C. Lu, and C. J. Kuo, “Multiple-phase retrieval for optical security systems by use of random-phase encoding,” Appl. Opt. 41(23), 4825–4834 (2002).
[CrossRef] [PubMed]

Y. Zhang, C. H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4-6), 277–285 (2002).
[CrossRef]

X. Peng, Z. Cui, and T. Tan, “Information encryption with virtual-optics imaging system,” Opt. Commun. 212(4-6), 235–245 (2002).
[CrossRef]

2001 (1)

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87(19), 195505 (2001).
[CrossRef] [PubMed]

2000 (1)

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
[CrossRef]

1997 (1)

B. Javidi, “Securing information with optical technologies,” Phys. Today 50(3), 27–32 (1997).
[CrossRef]

1995 (1)

1994 (1)

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994).
[CrossRef] [PubMed]

1983 (1)

1982 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Abbey, B.

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

Ahmad, M. A.

Allen, L. J.

W. McBride, N. L. O’Leary, and L. J. Allen, “Retrieval of a complex-valued object from its diffraction pattern,” Phys. Rev. Lett. 93(23), 233902 (2004).
[CrossRef] [PubMed]

Almoro, P.

Amirbekian, B.

H. Jiang, D. Ramunno-Johnson, C. Song, B. Amirbekian, Y. Kohmura, Y. Nishino, Y. Takahashi, T. Ishikawa, and J. Miao, “Nanoscale imaging of mineral crystals inside biological composite materials using X-ray diffraction microscopy,” Phys. Rev. Lett. 100(3), 038103 (2008).
[CrossRef] [PubMed]

Anand, A.

Bao, P.

Beck, M.

M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994).
[CrossRef] [PubMed]

Beetz, T.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102(43), 15343–15346 (2005).
[CrossRef] [PubMed]

Bunk, O.

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. V. König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” N. J. Phys. 12(3), 035017 (2010).
[CrossRef]

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100(15), 155503 (2008).
[CrossRef] [PubMed]

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] [PubMed]

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] [PubMed]

Cai, L.

Cai, Z.

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused X-ray fields,” Nat. Phys. 2(2), 101–104 (2006).
[CrossRef]

Cao, L.

Chang, H. T.

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
[CrossRef]

Chen, C. C.

J. Miao, C. C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T. K. Lee, and S. H. Risbud, “Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy,” Phys. Rev. Lett. 97(21), 215503 (2006).
[CrossRef] [PubMed]

Chen, W.

Chen, X.

Clark, J. N.

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

Cui, Z.

X. Peng, Z. Cui, and T. Tan, “Information encryption with virtual-optics imaging system,” Opt. Commun. 212(4-6), 235–245 (2002).
[CrossRef]

Cullis, A. G.

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] [PubMed]

David, C.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100(15), 155503 (2008).
[CrossRef] [PubMed]

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J. Miao, C. C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T. K. Lee, and S. H. Risbud, “Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy,” Phys. Rev. Lett. 97(21), 215503 (2006).
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J. Miao, C. C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T. K. Lee, and S. H. Risbud, “Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy,” Phys. Rev. Lett. 97(21), 215503 (2006).
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J. Miao, K. O. Hodgson, T. Ishikawa, C. A. Larabell, M. A. LeGros, and Y. Nishino, “Imaging whole Escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. U.S.A. 100(1), 110–112 (2003).
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B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4(5), 394–398 (2008).
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M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. V. König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” N. J. Phys. 12(3), 035017 (2010).
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[CrossRef] [PubMed]

Miao, H.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102(43), 15343–15346 (2005).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

J. Miao, C. C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T. K. Lee, and S. H. Risbud, “Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy,” Phys. Rev. Lett. 97(21), 215503 (2006).
[CrossRef] [PubMed]

J. Miao, K. O. Hodgson, T. Ishikawa, C. A. Larabell, M. A. LeGros, and Y. Nishino, “Imaging whole Escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. U.S.A. 100(1), 110–112 (2003).
[CrossRef] [PubMed]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
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H. Jiang, D. Ramunno-Johnson, C. Song, B. Amirbekian, Y. Kohmura, Y. Nishino, Y. Takahashi, T. Ishikawa, and J. Miao, “Nanoscale imaging of mineral crystals inside biological composite materials using X-ray diffraction microscopy,” Phys. Rev. Lett. 100(3), 038103 (2008).
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J. Miao, C. C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T. K. Lee, and S. H. Risbud, “Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy,” Phys. Rev. Lett. 97(21), 215503 (2006).
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J. Miao, K. O. Hodgson, T. Ishikawa, C. A. Larabell, M. A. LeGros, and Y. Nishino, “Imaging whole Escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. U.S.A. 100(1), 110–112 (2003).
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O. Matoba, T. Nomura, E. Perez-Cabre, M. Í. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97(6), 1128–1148 (2009).
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B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4(5), 394–398 (2008).
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H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused X-ray fields,” Nat. Phys. 2(2), 101–104 (2006).
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W. McBride, N. L. O’Leary, and L. J. Allen, “Retrieval of a complex-valued object from its diffraction pattern,” Phys. Rev. Lett. 93(23), 233902 (2004).
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Paterson, D.

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97(2), 025506 (2006).
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Peele, A. G.

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H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused X-ray fields,” Nat. Phys. 2(2), 101–104 (2006).
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Perez-Cabre, E.

O. Matoba, T. Nomura, E. Perez-Cabre, M. Í. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97(6), 1128–1148 (2009).
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B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4(5), 394–398 (2008).
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M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006).
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M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. V. König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” N. J. Phys. 12(3), 035017 (2010).
[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] [PubMed]

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[CrossRef] [PubMed]

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I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87(19), 195505 (2001).
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[CrossRef] [PubMed]

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused X-ray fields,” Nat. Phys. 2(2), 101–104 (2006).
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H. Jiang, D. Ramunno-Johnson, C. Song, B. Amirbekian, Y. Kohmura, Y. Nishino, Y. Takahashi, T. Ishikawa, and J. Miao, “Nanoscale imaging of mineral crystals inside biological composite materials using X-ray diffraction microscopy,” Phys. Rev. Lett. 100(3), 038103 (2008).
[CrossRef] [PubMed]

J. Miao, C. C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T. K. Lee, and S. H. Risbud, “Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy,” Phys. Rev. Lett. 97(21), 215503 (2006).
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V. Elser, I. Rankenburg, and P. Thibault, “Searching with iterated maps,” Proc. Natl. Acad. Sci. U.S.A. 104(2), 418–423 (2007).
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M. G. Raymer, M. Beck, and D. F. McAlister, “Complex wave-field reconstruction using phase-space tomography,” Phys. Rev. Lett. 72(8), 1137–1140 (1994).
[CrossRef] [PubMed]

Refregier, P.

Renker, D.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100(15), 155503 (2008).
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Risbud, S. H.

J. Miao, C. C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T. K. Lee, and S. H. Risbud, “Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy,” Phys. Rev. Lett. 97(21), 215503 (2006).
[CrossRef] [PubMed]

Robinson, I. K.

R. Harder, M. Liang, Y. Sun, Y. Xia, and I. K. Robinson, “Imaging of complex density in silver nanocubes by coherent x-ray diffraction,” N. J. Phys. 12(3), 035019 (2010).
[CrossRef]

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006).
[CrossRef] [PubMed]

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87(19), 195505 (2001).
[CrossRef] [PubMed]

Rodenburg, J. M.

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] [PubMed]

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

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R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Sayre, D.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102(43), 15343–15346 (2005).
[CrossRef] [PubMed]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
[CrossRef]

Schlichting, I.

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. V. König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” N. J. Phys. 12(3), 035017 (2010).
[CrossRef]

Shapiro, D.

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102(43), 15343–15346 (2005).
[CrossRef] [PubMed]

Sheng, Y.

Sheppard, C. J. R.

Shi, Y.

Shimobaba, T.

Singh, K.

Situ, G.

Song, C.

H. Jiang, D. Ramunno-Johnson, C. Song, B. Amirbekian, Y. Kohmura, Y. Nishino, Y. Takahashi, T. Ishikawa, and J. Miao, “Nanoscale imaging of mineral crystals inside biological composite materials using X-ray diffraction microscopy,” Phys. Rev. Lett. 100(3), 038103 (2008).
[CrossRef] [PubMed]

J. Miao, C. C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T. K. Lee, and S. H. Risbud, “Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy,” Phys. Rev. Lett. 97(21), 215503 (2006).
[CrossRef] [PubMed]

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R. Harder, M. Liang, Y. Sun, Y. Xia, and I. K. Robinson, “Imaging of complex density in silver nanocubes by coherent x-ray diffraction,” N. J. Phys. 12(3), 035019 (2010).
[CrossRef]

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H. Jiang, D. Ramunno-Johnson, C. Song, B. Amirbekian, Y. Kohmura, Y. Nishino, Y. Takahashi, T. Ishikawa, and J. Miao, “Nanoscale imaging of mineral crystals inside biological composite materials using X-ray diffraction microscopy,” Phys. Rev. Lett. 100(3), 038103 (2008).
[CrossRef] [PubMed]

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Tan, T.

X. Peng, Z. Cui, and T. Tan, “Information encryption with virtual-optics imaging system,” Opt. Commun. 212(4-6), 235–245 (2002).
[CrossRef]

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Y. Zhang, C. H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4-6), 277–285 (2002).
[CrossRef]

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W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009).
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M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. V. König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” N. J. Phys. 12(3), 035017 (2010).
[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] [PubMed]

V. Elser, I. Rankenburg, and P. Thibault, “Searching with iterated maps,” Proc. Natl. Acad. Sci. U.S.A. 104(2), 418–423 (2007).
[CrossRef] [PubMed]

D. Shapiro, P. Thibault, T. Beetz, V. Elser, M. Howells, C. Jacobsen, J. Kirz, E. Lima, H. Miao, A. M. Neiman, and D. Sayre, “Biological imaging by soft x-ray diffraction microscopy,” Proc. Natl. Acad. Sci. U.S.A. 102(43), 15343–15346 (2005).
[CrossRef] [PubMed]

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G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97(2), 025506 (2006).
[CrossRef] [PubMed]

Unnikrishnan, G.

Vartanyants, I. A.

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006).
[CrossRef] [PubMed]

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87(19), 195505 (2001).
[CrossRef] [PubMed]

Wang, B.

Wang, Y.

H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008).
[CrossRef]

Wei, H.

Williams, G. J.

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

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97(2), 025506 (2006).
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M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006).
[CrossRef] [PubMed]

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87(19), 195505 (2001).
[CrossRef] [PubMed]

Xia, Y.

R. Harder, M. Liang, Y. Sun, Y. Xia, and I. K. Robinson, “Imaging of complex density in silver nanocubes by coherent x-ray diffraction,” N. J. Phys. 12(3), 035019 (2010).
[CrossRef]

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Xu, L.

Yao-Yao, C.

Yong-Liang, X.

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[CrossRef] [PubMed]

Y. Zhang, C. H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4-6), 277–285 (2002).
[CrossRef]

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Y. Zhang, C. H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4-6), 277–285 (2002).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (2)

W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95(20), 201103 (2009).
[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. Opt. Soc. Am. (1)

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

N. J. Phys. (2)

R. Harder, M. Liang, Y. Sun, Y. Xia, and I. K. Robinson, “Imaging of complex density in silver nanocubes by coherent x-ray diffraction,” N. J. Phys. 12(3), 035019 (2010).
[CrossRef]

M. Dierolf, P. Thibault, A. Menzel, C. M. Kewish, K. Jefimovs, I. Schlichting, K. V. König, O. Bunk, and F. Pfeiffer, “Ptychographic coherent diffractive imaging of weakly scattering specimens,” N. J. Phys. 12(3), 035017 (2010).
[CrossRef]

Nat. Phys. (2)

H. M. Quiney, A. G. Peele, Z. Cai, D. Paterson, and K. A. Nugent, “Diffractive imaging of highly focused X-ray fields,” Nat. Phys. 2(2), 101–104 (2006).
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[CrossRef]

Nature (2)

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442(7098), 63–66 (2006).
[CrossRef] [PubMed]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400(6742), 342–344 (1999).
[CrossRef]

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H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008).
[CrossRef]

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Y. Zhang, C. H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4-6), 277–285 (2002).
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Optik (Stuttg.) (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttg.) 35, 237–246 (1972).

Phys. Rev. A (1)

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75(4), 043805 (2007).
[CrossRef]

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W. McBride, N. L. O’Leary, and L. J. Allen, “Retrieval of a complex-valued object from its diffraction pattern,” Phys. Rev. Lett. 93(23), 233902 (2004).
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[CrossRef] [PubMed]

J. Miao, C. C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. Ramunno-Johnson, T. K. Lee, and S. H. Risbud, “Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy,” Phys. Rev. Lett. 97(21), 215503 (2006).
[CrossRef] [PubMed]

H. Jiang, D. Ramunno-Johnson, C. Song, B. Amirbekian, Y. Kohmura, Y. Nishino, Y. Takahashi, T. Ishikawa, and J. Miao, “Nanoscale imaging of mineral crystals inside biological composite materials using X-ray diffraction microscopy,” Phys. Rev. Lett. 100(3), 038103 (2008).
[CrossRef] [PubMed]

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]

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[CrossRef] [PubMed]

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V. Elser, I. Rankenburg, and P. Thibault, “Searching with iterated maps,” Proc. Natl. Acad. Sci. U.S.A. 104(2), 418–423 (2007).
[CrossRef] [PubMed]

J. Miao, K. O. Hodgson, T. Ishikawa, C. A. Larabell, M. A. LeGros, and Y. Nishino, “Imaging whole Escherichia coli bacteria by using single-particle x-ray diffraction,” Proc. Natl. Acad. Sci. U.S.A. 100(1), 110–112 (2003).
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[CrossRef] [PubMed]

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).
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Figures (7)

Fig. 1
Fig. 1

Tree cryptography topology structure based on the proposed diffractive imaging: BSC, Beam splitter cube; M, Phase-only mask.

Fig. 2
Fig. 2

Star cryptography topology structure based on the proposed diffractive imaging.

Fig. 3
Fig. 3

A schematic optical cryptography setup based on diffractive imaging using the lateral translation of a phase-only mask: M, Phase mask; CCD, Charge-coupled device.

Fig. 4
Fig. 4

Flow chart for the decryption based on the proposed retrieval algorithm: ⊗, A multiplication operation.

Fig. 5
Fig. 5

(a) The plaintext (a binary image); (b) the original phase-only mask M3; (c) the typical phase-only mask M3 after a lateral translation; (d) a typical diffraction intensity map (i.e., ciphertext); (e) a relationship between the number of iterations and the iterative errors; and (f) a decrypted image using correct security keys.

Fig. 6
Fig. 6

The decrypted images at one section when the distance d 1 of (a) 20 mm, (b) 50 mm, (c) 80 mm, (d) 160 mm, (e) 250 mm and (f) 300 mm is directly used.

Fig. 7
Fig. 7

The decrypted images using (a) a wrong phase-only mask (M4); (b) a wrong translation amount for the phase-only mask (M3); (c) an error of 4 mm in the distance d 4 ; (d) a FRFT function order error of 0.08. Decrypted images when the ciphertexts are contaminated by (e) random noise and (f) 25% occlusions.

Equations (9)

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O '     ( x 1 , y 1 ) = j λ + + O       ( ξ , η )       M 1 ( ξ , η ) exp [ j     k ( x 1 ξ ) 2 + ( y 1 η ) 2 + [ d 1 ( ξ , η ) ] 2 ] ( x 1 ξ ) 2 + ( y 1 η ) 2 + [ d 1 ( ξ , η ) ] 2         d ξ     d η ,
O       ( x 1 , y 1 ) = O     ' ( x 1 , y 1 )     .
O '     ( x 1 , y 1 ) = F T 1 ( { F T [ O       ( ξ , η )       M 1 ( ξ , η ) ] } T ( f ξ , f η ) ) = A S A [ O       ( ξ , η )       M 1 ( ξ , η ) ]       ,
T ( f ξ , f η ) = exp [ j 2 π     d 1 ( ξ , η ) λ × 1 ( λ     f ξ ) 2 ( λ     f η ) 2 ] .
FRF T a [ O ( x 2 ) M 3 ( x 2 ) ] = + O ( x 2 ) M 3 ( x 2 )       P a ( x 3 , x 2 )     d x 2 ,
P a ( x 3 , x 2 ) = { R       exp     { j π [ x 3 2 cot     ( a π / 2 ) + x 2 2 cot ( a π / 2 ) 2 x 3 x 2 csc ( a π / 2 ) ]     }                         if       a 2 m δ ( x 3 x 2 )                                                                                                                                                                                                 if       a = 4 m δ ( x 3 + x 2 )                                                                                                                                                                                                 if       a = 4 m ± 2 ,
I h ( μ , ν ) = | A S A d 4 { [ FRFT ( { A S A d 2 [ ( { A S A d 1 ( ξ , η ) [ O       ( ξ , η )       M 1 ( ξ , η ) ] } )                                                                                       M 2 ( x 1 , y 1 ) ] } M 3 h ( x 2 , y 2 ) ) ] M 4 ( x 3 , y 3 ) } | 2 ,
K n ' ( x 2 , y 2 ) = K n ( x 2 , y 2 )       M 3 1 ( x 2 , y 2 ) ;
CC = [ cov ( O , O D ) ] / ( σ O × σ O D ) ,

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