Abstract

In this paper, we propose and experimentally demonstrate phase-only authentication based on single-pixel optical imaging through scattering media. The propagating wave is sequentially modulated by using a series of random amplitude-only patterns embedded in a spatial light modulator (SLM), and then a series of one-dimensional (1D) intensity values is recorded by the single-pixel (bucket) detector. Subsequently, an intensity pattern just before the SLM is retrieved by using a correlation algorithm and then further propagates back to the object plane in which the object phase pattern is recovered to serve as reference. Then some single-pixel intensity values are randomly selected from the recorded data, and 1-bit compression is applied to the randomly selected data in order to generate 1D binary signals as ciphertext. A series of random amplitude-only patterns corresponding to the randomly selected single-pixel intensity values serve as principal keys. In a scattering environment, the proposed method is able to carry out phase-only authentication without visually rendering the plaintext, which has not been previously studied. It is found that phase-only authentication is sensitive to security keys, and the proposed method possesses high security. In addition, the proposed method is highly robust to noise contamination and data-loss contamination. Optical experimental results demonstrate the feasibility and effectiveness of the proposed method.

© 2021 Optical Society of America

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References

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2020 (2)

2019 (4)

W. Chen, “Single-shot in-line holographic authentication using phase and amplitude modulation,” Opt. Laser Eng. 121, 473–478 (2019).
[Crossref]

H. Hai, S. X. Pan, M. H. Liao, D. J. Lu, W. Q. He, and X. Peng, “Cryptanalysis of random-phase-encoding-based optical cryptosystem via deep learning,” Opt. Express 27, 21204–21213 (2019).
[Crossref]

Y. Xiao, L. N. Zhou, and W. Chen, “Direct single-step measurement of Hadamard spectrum using single-pixel optical detection,” IEEE Photon. Tech. Lett. 31, 845–848 (2019).
[Crossref]

Y. Xiao, L. N. Zhou, and W. Chen, “Experimental demonstration of ghost-imaging-based authentication in scattering media,” Opt. Express 27, 20558–20566 (2019).
[Crossref]

2016 (1)

2015 (6)

J. J. Wu, W. Liu, Z. J. Liu, and S. T. Liu, “Correlated-imaging-based chosen plaintext attack on general cryptosystems composed of linear canonical transforms and phase encodings,” Opt. Commun. 338, 164–167 (2015).
[Crossref]

Z. B. Zhang, X. Ma, and J. G. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6, 6225 (2015).
[Crossref]

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[Crossref]

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[Crossref]

M. Takeda, K. Nakano, H. Suzuki, and M. Yamaguchi, “Encrypted sensing based on digital holography for fingerprint images,” Opt. Photon. J. 5, 6–14 (2015).
[Crossref]

M. T. Shiu, Y. K. Chew, H. T. Chan, X. Y. Wong, and C. C. Chang, “Three-dimensional information encryption and anticounterfeiting using digital holography,” Appl. Opt. 54, A84–A88 (2015).
[Crossref]

2014 (2)

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photon. 6, 120–155 (2014).
[Crossref]

W. Chen, X. G. Wang, and X. D. Chen, “Security-enhanced phase encryption assisted by nonlinear optical correlation via sparse phase,” J. Opt. 17, 035702 (2014).
[Crossref]

2013 (4)

W. Chen and X. D. Chen, “Object authentication in computational ghost imaging with the realizations less than 5% of Nyquist limit,” Opt. Lett. 38, 546–548 (2013).
[Crossref]

W. Chen and X. Chen, “Ghost imaging for three-dimensional optical security,” Appl. Phys. Lett. 103, 221106 (2013).
[Crossref]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

S. K. Rajut and N. K. Nishchal, “Known-plaintext attack-based optical cryptosystem using phase-truncated Fresnel transform,” Appl. Opt. 52, 871–878 (2013).
[Crossref]

2012 (1)

M. Tanha, R. Kheradmand, and S. Ahmadi-Kandjani, “Gray-scale and color optical encryption based on computational ghost imaging,” Appl. Phys. Lett. 101, 101108 (2012).
[Crossref]

2011 (2)

2010 (3)

2009 (2)

O. Matoba, T. Nomura, E. Péréz-Cabré, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[Crossref]

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[Crossref]

2008 (4)

2006 (1)

2005 (2)

2004 (1)

T. J. Naughton and B. Javidi, “Compression of encrypted three-dimensional objects using digital holography,” Opt. Eng. 43, 2233–2238 (2004).
[Crossref]

1995 (1)

1989 (1)

1981 (1)

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[Crossref]

Agafonov, I. N.

Ahmadi-Kandjani, S.

M. Tanha, R. Kheradmand, and S. Ahmadi-Kandjani, “Gray-scale and color optical encryption based on computational ghost imaging,” Appl. Phys. Lett. 101, 101108 (2012).
[Crossref]

Arcos, S.

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Bowman, A.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

Bowman, R.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

Bromberg, Y.

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[Crossref]

Carnicer, A.

Chan, H. T.

Chang, C. C.

Charrière, F.

Chekhova, M. V.

Chen, L. F.

Chen, W.

Y. Xiao, L. N. Zhou, and W. Chen, “Secured single-pixel ghost holography,” Opt. Laser Eng. 128, 106045 (2020).
[Crossref]

Y. Xiao, L. N. Zhou, and W. Chen, “Direct single-step measurement of Hadamard spectrum using single-pixel optical detection,” IEEE Photon. Tech. Lett. 31, 845–848 (2019).
[Crossref]

Y. Xiao, L. N. Zhou, and W. Chen, “Experimental demonstration of ghost-imaging-based authentication in scattering media,” Opt. Express 27, 20558–20566 (2019).
[Crossref]

W. Chen, “Single-shot in-line holographic authentication using phase and amplitude modulation,” Opt. Laser Eng. 121, 473–478 (2019).
[Crossref]

W. Chen and X. Chen, “Optical authentication via photon-synthesized ghost imaging using optical nonlinear correlation,” Opt. Laser Eng. 73, 123–127 (2015).
[Crossref]

W. Chen and X. D. Chen, “Grayscale object authentication based on ghost imaging using binary signals,” Europhys. Lett. 110, 44002 (2015).
[Crossref]

W. Chen, X. G. Wang, and X. D. Chen, “Security-enhanced phase encryption assisted by nonlinear optical correlation via sparse phase,” J. Opt. 17, 035702 (2014).
[Crossref]

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photon. 6, 120–155 (2014).
[Crossref]

W. Chen and X. D. Chen, “Object authentication in computational ghost imaging with the realizations less than 5% of Nyquist limit,” Opt. Lett. 38, 546–548 (2013).
[Crossref]

W. Chen and X. Chen, “Ghost imaging for three-dimensional optical security,” Appl. Phys. Lett. 103, 221106 (2013).
[Crossref]

Chen, X.

W. Chen and X. Chen, “Optical authentication via photon-synthesized ghost imaging using optical nonlinear correlation,” Opt. Laser Eng. 73, 123–127 (2015).
[Crossref]

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photon. 6, 120–155 (2014).
[Crossref]

W. Chen and X. Chen, “Ghost imaging for three-dimensional optical security,” Appl. Phys. Lett. 103, 221106 (2013).
[Crossref]

Chen, X. D.

W. Chen and X. D. Chen, “Grayscale object authentication based on ghost imaging using binary signals,” Europhys. Lett. 110, 44002 (2015).
[Crossref]

W. Chen, X. G. Wang, and X. D. Chen, “Security-enhanced phase encryption assisted by nonlinear optical correlation via sparse phase,” J. Opt. 17, 035702 (2014).
[Crossref]

W. Chen and X. D. Chen, “Object authentication in computational ghost imaging with the realizations less than 5% of Nyquist limit,” Opt. Lett. 38, 546–548 (2013).
[Crossref]

X. D. Chen, Computational Methods for Electromagnetic Inverse Scattering (Wiley-IEEE, 2018).

Chen, X. H.

Chew, Y. K.

Cho, M.

Clemente, P.

Colomb, T.

Cuche, E.

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Depeursinge, C.

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Durán, V.

Edgar, M. P.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

Emery, Y.

Fu, L.

Gan, W. W.

Gong, W. L.

P. L. Zhang, W. L. Gong, X. Shen, and S. S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

Guo, C. L.

Guo, Q. C.

Hai, H.

Han, S. S.

P. L. Zhang, W. L. Gong, X. Shen, and S. S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

He, W. Q.

Javidi, B.

Juvells, I.

Katz, O.

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[Crossref]

Kelly, K. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Kemper, B.

Kheradmand, R.

M. Tanha, R. Kheradmand, and S. Ahmadi-Kandjani, “Gray-scale and color optical encryption based on computational ghost imaging,” Appl. Phys. Lett. 101, 101108 (2012).
[Crossref]

Lancis, J.

Laska, J. N.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Liao, M. H.

Lim, J. S.

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[Crossref]

Liu, Q.

Liu, S. T.

J. J. Wu, W. Liu, Z. J. Liu, and S. T. Liu, “Correlated-imaging-based chosen plaintext attack on general cryptosystems composed of linear canonical transforms and phase encodings,” Opt. Commun. 338, 164–167 (2015).
[Crossref]

Liu, W.

J. J. Wu, W. Liu, Z. J. Liu, and S. T. Liu, “Correlated-imaging-based chosen plaintext attack on general cryptosystems composed of linear canonical transforms and phase encodings,” Opt. Commun. 338, 164–167 (2015).
[Crossref]

Liu, Y. Q.

Liu, Z. J.

J. J. Wu, W. Liu, Z. J. Liu, and S. T. Liu, “Correlated-imaging-based chosen plaintext attack on general cryptosystems composed of linear canonical transforms and phase encodings,” Opt. Commun. 338, 164–167 (2015).
[Crossref]

Lu, D. J.

Luo, K. H.

Ma, X.

Z. B. Zhang, X. Ma, and J. G. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6, 6225 (2015).
[Crossref]

Magistretti, P.

Magistretti, P. J.

Marquet, P.

Matoba, O.

O. Matoba, T. Nomura, E. Péréz-Cabré, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[Crossref]

Millan, M. S.

O. Matoba, T. Nomura, E. Péréz-Cabré, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[Crossref]

Montes-Usategui, M.

Muniraj, I.

Nakano, K.

M. Takeda, K. Nakano, H. Suzuki, and M. Yamaguchi, “Encrypted sensing based on digital holography for fingerprint images,” Opt. Photon. J. 5, 6–14 (2015).
[Crossref]

Naughton, T. J.

T. J. Naughton and B. Javidi, “Compression of encrypted three-dimensional objects using digital holography,” Opt. Eng. 43, 2233–2238 (2004).
[Crossref]

Nishchal, N. K.

Nomura, T.

O. Matoba, T. Nomura, E. Péréz-Cabré, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[Crossref]

Oppenheim, A. V.

A. V. Oppenheim and J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[Crossref]

Padgett, M. J.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

Pan, S. X.

Peng, B. Y.

Peng, X.

Pérez-Cabré, E.

Péréz-Cabré, E.

O. Matoba, T. Nomura, E. Péréz-Cabré, M. S. Millan, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[Crossref]

Rajut, S. K.

Rappaz, B.

Refregier, P.

Shapiro, J. H.

J. H. Shapiro, “Computational ghost imaging,” Phys. Rev. A 78, 061802 (2008).
[Crossref]

Shen, X.

P. L. Zhang, W. L. Gong, X. Shen, and S. S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
[Crossref]

Sheridan, J. T.

Shiu, M. T.

Silberberg, Y.

Y. Bromberg, O. Katz, and Y. Silberberg, “Ghost imaging with a single detector,” Phys. Rev. A 79, 053840 (2009).
[Crossref]

Sun, B.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

Sun, T.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Suzuki, H.

M. Takeda, K. Nakano, H. Suzuki, and M. Yamaguchi, “Encrypted sensing based on digital holography for fingerprint images,” Opt. Photon. J. 5, 6–14 (2015).
[Crossref]

Tajahuerce, E.

Takeda, M.

M. Takeda, K. Nakano, H. Suzuki, and M. Yamaguchi, “Encrypted sensing based on digital holography for fingerprint images,” Opt. Photon. J. 5, 6–14 (2015).
[Crossref]

Takhar, D.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008).
[Crossref]

Tanha, M.

M. Tanha, R. Kheradmand, and S. Ahmadi-Kandjani, “Gray-scale and color optical encryption based on computational ghost imaging,” Appl. Phys. Lett. 101, 101108 (2012).
[Crossref]

Tian, N.

Torres-Company, V.

Vittert, L. E.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

von Bally, G.

Wang, A.

Wang, X. G.

W. Chen, X. G. Wang, and X. D. Chen, “Security-enhanced phase encryption assisted by nonlinear optical correlation via sparse phase,” J. Opt. 17, 035702 (2014).
[Crossref]

Wei, H. Z.

Welsh, S.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340, 844–847 (2013).
[Crossref]

Wong, X. Y.

Wu, J. J.

J. J. Wu, W. Liu, Z. J. Liu, and S. T. Liu, “Correlated-imaging-based chosen plaintext attack on general cryptosystems composed of linear canonical transforms and phase encodings,” Opt. Commun. 338, 164–167 (2015).
[Crossref]

Wu, L. A.

Xian, R.

Xiao, Y.

Y. Xiao, L. N. Zhou, and W. Chen, “Secured single-pixel ghost holography,” Opt. Laser Eng. 128, 106045 (2020).
[Crossref]

Y. Xiao, L. N. Zhou, and W. Chen, “Direct single-step measurement of Hadamard spectrum using single-pixel optical detection,” IEEE Photon. Tech. Lett. 31, 845–848 (2019).
[Crossref]

Y. Xiao, L. N. Zhou, and W. Chen, “Experimental demonstration of ghost-imaging-based authentication in scattering media,” Opt. Express 27, 20558–20566 (2019).
[Crossref]

Xu, D. L.

Yamaguchi, M.

M. Takeda, K. Nakano, H. Suzuki, and M. Yamaguchi, “Encrypted sensing based on digital holography for fingerprint images,” Opt. Photon. J. 5, 6–14 (2015).
[Crossref]

Yu, B.

Zhang, P.

Zhang, P. L.

P. L. Zhang, W. L. Gong, X. Shen, and S. S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
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Zhang, Z. B.

Z. B. Zhang, X. Ma, and J. G. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6, 6225 (2015).
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Zhong, J. G.

Z. B. Zhang, X. Ma, and J. G. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6, 6225 (2015).
[Crossref]

Zhou, L. N.

Y. Xiao, L. N. Zhou, and W. Chen, “Secured single-pixel ghost holography,” Opt. Laser Eng. 128, 106045 (2020).
[Crossref]

Y. Xiao, L. N. Zhou, and W. Chen, “Experimental demonstration of ghost-imaging-based authentication in scattering media,” Opt. Express 27, 20558–20566 (2019).
[Crossref]

Y. Xiao, L. N. Zhou, and W. Chen, “Direct single-step measurement of Hadamard spectrum using single-pixel optical detection,” IEEE Photon. Tech. Lett. 31, 845–848 (2019).
[Crossref]

Adv. Opt. Photon. (1)

Appl. Opt. (5)

Appl. Phys. Lett. (2)

W. Chen and X. Chen, “Ghost imaging for three-dimensional optical security,” Appl. Phys. Lett. 103, 221106 (2013).
[Crossref]

M. Tanha, R. Kheradmand, and S. Ahmadi-Kandjani, “Gray-scale and color optical encryption based on computational ghost imaging,” Appl. Phys. Lett. 101, 101108 (2012).
[Crossref]

Europhys. Lett. (1)

W. Chen and X. D. Chen, “Grayscale object authentication based on ghost imaging using binary signals,” Europhys. Lett. 110, 44002 (2015).
[Crossref]

IEEE Photon. Tech. Lett. (1)

Y. Xiao, L. N. Zhou, and W. Chen, “Direct single-step measurement of Hadamard spectrum using single-pixel optical detection,” IEEE Photon. Tech. Lett. 31, 845–848 (2019).
[Crossref]

IEEE Signal Process. Mag. (1)

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J. Opt. (1)

W. Chen, X. G. Wang, and X. D. Chen, “Security-enhanced phase encryption assisted by nonlinear optical correlation via sparse phase,” J. Opt. 17, 035702 (2014).
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Nat. Commun. (1)

Z. B. Zhang, X. Ma, and J. G. Zhong, “Single-pixel imaging by means of Fourier spectrum acquisition,” Nat. Commun. 6, 6225 (2015).
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Opt. Commun. (1)

J. J. Wu, W. Liu, Z. J. Liu, and S. T. Liu, “Correlated-imaging-based chosen plaintext attack on general cryptosystems composed of linear canonical transforms and phase encodings,” Opt. Commun. 338, 164–167 (2015).
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Y. Xiao, L. N. Zhou, and W. Chen, “Secured single-pixel ghost holography,” Opt. Laser Eng. 128, 106045 (2020).
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Opt. Photon. J. (1)

M. Takeda, K. Nakano, H. Suzuki, and M. Yamaguchi, “Encrypted sensing based on digital holography for fingerprint images,” Opt. Photon. J. 5, 6–14 (2015).
[Crossref]

Phys. Rev. A (3)

P. L. Zhang, W. L. Gong, X. Shen, and S. S. Han, “Correlated imaging through atmospheric turbulence,” Phys. Rev. A 82, 033817 (2010).
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X. D. Chen, Computational Methods for Electromagnetic Inverse Scattering (Wiley-IEEE, 2018).

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Figures (13)

Fig. 1.
Fig. 1. Schematic experimental setup: SLM, spatial light modulator.
Fig. 2.
Fig. 2. Flowchart for the proposed method.
Fig. 3.
Fig. 3. (a) Two transmission objects: the region inside the dashed-line box represents object 1, and the region inside the solid-line box represents object 2; (b) region of a reflective object (indicated by the solid-line box) used as object 3; and (c) typical ciphertext (i.e., a series of binary signals).
Fig. 4.
Fig. 4. PCE values versus the number of binary signals (i.e.,ciphertext).
Fig. 5.
Fig. 5. (a)–(c) Three reference phase patterns respectively corresponding to three objects in Figs. 3(a) and 3(b); (d), (h), and (l) three decrypted phase patterns respectively corresponding to the three objects; (e), (j), and (o) nonlinear correlation maps generated between reference phase patterns and their correspondingly decrypted phase patterns when correct keys are used for the decoding; and (f), (g), (i), (k), (m), and (n) nonlinear correlation maps generated between incorrect references and the decrypted phase patterns.
Fig. 6.
Fig. 6. (a)–(c) Three reference phase patterns respectively corresponding to the three objects in Figs. 3(a) and 3(b); (d), (h), and (l) three decrypted phase patterns obtained by using wrong amplitude-only patterns; and (e)–(g), (i)–(k), and (m)–(o) nonlinear correlation maps generated between reference phase patterns and the decrypted phase patterns.
Fig. 7.
Fig. 7. PCE values versus axial propagation distances.
Fig. 8.
Fig. 8. (a)–(d) Decrypted phase patterns respectively using noise levels of 0.2, 0.4, 0.6, 0.8; (e) reference phase pattern corresponding to object 1; and (f)–(i) nonlinear correlation maps between reference phase and the decrypted phase patterns [i.e., (a)–(d)].
Fig. 9.
Fig. 9. PCE values versus noise levels.
Fig. 10.
Fig. 10. (a)–(d) Decrypted phase patterns respectively obtained by using ciphertext-loss levels of 20.0%, 40.0%, 60.0%, and 80.0%; (e) reference phase pattern corresponding to object 1; and (f)–(i) nonlinear correlation maps generated between the reference phase [i.e., (e)] and the decrypted phase [i.e., (a)–(d)].
Fig. 11.
Fig. 11. PCE values versus ciphertext-loss levels.
Fig. 12.
Fig. 12. (a) Two cascaded diffusers placed at the object wave path, (b) two cascaded diffusers placed at the reference wave path, and (c) multiple cascaded diffusers at both the object wave and reference wave paths.
Fig. 13.
Fig. 13. (a)–(c) Three reference phase patterns respectively corresponding to the three objects; (d), (h), and (l) three decrypted phase patterns respectively corresponding to the three objects by using correct security keys; (e), (j), and (o) nonlinear correlation maps generated between the reference phase patterns and their correspondingly decrypted phase patterns; and (f), (g), (i), (k), (m), and (n) nonlinear correlation maps generated between incorrect references and the decrypted phase patterns.

Equations (4)

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H r e f = 1 T i = 1 T ( B i B i ) ( P i P i ) ,
O r e f ( x , y ) = I F T { F T [ H r e f ( x , y ) ] S T ( μ , ν ) } ,
N C = | I F T ( | Φ | k × Φ | Φ | ) | 2 ,
P C E = max N C N C .

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