Abstract

Information security with optical means, such as double random phase encoding, has been investigated by various researchers. It has been demonstrated that optical technology possesses several unique characteristics for securing information compared with its electronic counterpart, such as many degrees of freedom. In this paper, we present a review of optical technologies for information security. Optical security systems are reviewed, and theoretical principles and implementation examples are presented to illustrate each optical security system. In addition, advantages and potential weaknesses of each optical security system are analyzed and discussed. It is expected that this review not only will provide a clear picture about current developments in optical security systems but also may shed some light on future developments.

© 2014 Optical Society of America

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2014 (4)

S. Liu, C. Guo, and J. T. Sheridan, “A review of optical image encryption techniques,” Opt. Laser Technol. 57, 327–342 (2014).
[CrossRef]

A. Markman and B. Javidi, “Full-phase photon-counting double-random-phase encryption,” J. Opt. Soc. Am. A 31, 394–403 (2014).
[CrossRef]

W. Chen and X. Chen, “Double random phase encoding using phase reservation and compression,” J. Opt. 16, 025402 (2014).
[CrossRef]

A. Markman, B. Javidi, and M. Tehranipoor, “Photon-counting security tagging and verification using optically encoded QR codes,” IEEE Photon. J. 6, 6800609 (2014).
[CrossRef]

2013 (12)

W. Chen and X. 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]

I. Mehra, S. K. Rajput, and N. K. Nishchal, “Collision in Fresnel domain asymmetric cryptosystem using phase truncation and authentication verification,” Opt. Eng. 52, 028202 (2013).
[CrossRef]

W. Chen, G. Situ, and X. Chen, “High-flexibility optical encryption via aperture movement,” Opt. Express 21, 24680–24691 (2013).
[CrossRef]

J. M. Vilardy, M. S. Millán, and E. Pérez-Cabré, “Improved decryption quality and security of a joint transform correlator-based encryption system,” J. Opt. 15, 025401 (2013).
[CrossRef]

M. Cho and B. Javidi, “Three-dimensional photon counting double-random-phase encryption,” Opt. Lett. 38, 3198–3201 (2013).
[CrossRef]

W. Chen, X. Chen, A. Anand, and B. Javidi, “Optical encryption using multiple intensity samplings in the axial domain,” J. Opt. Soc. Am. A 30, 806–812 (2013).
[CrossRef]

Y. Shi, T. Li, Y. Wang, Q. Gao, S. Zhang, and H. Li, “Optical image encryption via ptychography,” Opt. Lett. 38, 1425–1427 (2013).
[CrossRef]

W. Chen, X. Chen, A. Stern, and B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photon. J. 5, 6900113 (2013).
[CrossRef]

W. Chen and X. Chen, “Optical image encryption based on multiple-region plaintext and phase retrieval in three-dimensional space,” Opt. Lasers Eng. 51, 128–133 (2013).
[CrossRef]

W. Chen and X. Chen, “Optical cryptography network topology based on 2D-to-3D conversion and phase-mask extraction,” Opt. Lasers Eng. 51, 410–416 (2013).
[CrossRef]

W. Liu, Z. Liu, and S. Liu, “Asymmetric cryptosystem using random binary phase modulation based on mixture retrieval type of Yang-Gu algorithm,” Opt. Lett. 38, 1651–1653 (2013).
[CrossRef]

2012 (6)

W. Chen, X. Chen, and C. J. R. Sheppard, “Optical image encryption based on phase retrieval combined with three-dimensional particle-like distribution,” J. Opt. 14, 075402 (2012).
[CrossRef]

S. S. Gorthi and E. Schonbrun, “Phase imaging flow cytometry using a focus-stack collecting microscope,” Opt. Lett. 37, 707–709 (2012).
[CrossRef]

W. Chen, X. Chen, and C. J. R. Sheppard, “Optical image encryption based on coherent diffractive imaging using multiple wavelengths,” Opt. Commun. 285, 225–228 (2012).
[CrossRef]

E. Pérez-Cabré, H. C. Abril, M. S. Millan, and B. Javidi, “Photon-counting double-random-phase encoding for secure image verification and retrieval,” J. Opt. 14, 094001 (2012).
[CrossRef]

E. Rueda, C. Ríos, J. F. Barrera, and R. Torroba, “Master key generation to avoid the use of an external reference wave in an experimental JTC encrypting architecture,” Appl. Opt. 51, 1822–1827 (2012).
[CrossRef]

J. F. Barrera, M. Tebaldi, C. Ríos, E. Rueda, N. Bolognini, and R. Torroba, “Experimental multiplexing of encrypted movies using a JTC architecture,” Opt. Express 20, 3388–3393 (2012).
[CrossRef]

2011 (13)

F. Mosso, J. F. Barrera, M. Tebaldi, N. Bolognini, and R. Torroba, “All-optical encrypted movie,” Opt. Express 19, 5706–5712 (2011).
[CrossRef]

B. Yang, Z. Liu, B. Wang, Y. Zhang, and S. Liu, “Optical stream-cipher-like system for image encryption based on Michelson interferometer,” Opt. Express 19, 2634–2642 (2011).
[CrossRef]

P. Kumar, J. Joseph, and K. Singh, “Optical image encryption using a jigsaw transform for silhouette removal in interference-based methods and decryption with a single spatial light modulator,” Appl. Opt. 50, 1805–1811 (2011).
[CrossRef]

W. Chen and X. Chen, “Optical multiple-image encryption based on multiplane phase retrieval and interference,” J. Opt. 13, 115401 (2011).
[CrossRef]

H. T. Chang, H. E. Hwang, C. L. Lee, and M. T. Lee, “Wavelength multiplexing multiple-image encryption using cascaded phase-only masks in the Fresnel transform domain,” Appl. Opt. 50, 710–716 (2011).
[CrossRef]

X. Wang and D. Zhao, “Multiple-image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain,” Opt. Commun. 284, 148–152 (2011).
[CrossRef]

A. Alfalou, C. Brosseau, N. Abdallah, and M. Jridi, “Simultaneous fusion, compression, and encryption of multiple images,” Opt. Express 19, 24023–24029 (2011).
[CrossRef]

E. Pérez-Cabré, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36, 22–24 (2011).
[CrossRef]

W. Chen and X. Chen, “Optical cryptography topology based on a three-dimensional particle-like distribution and diffractive imaging,” Opt. Express 19, 9008–9019 (2011).
[CrossRef]

W. Chen and X. Chen, “Optical image encryption using multilevel Arnold transform and noninterferometric imaging,” Opt. Eng. 50, 117001 (2011).
[CrossRef]

N. K. Nishchal and T. J. Naughton, “Flexible optical encryption with multiple users and multiple security levels,” Opt. Commun. 284, 735–739 (2011).
[CrossRef]

W. Chen, X. Chen, and C. J. R. Sheppard, “Optical double-image cryptography based on diffractive imaging with a laterally-translated phase grating,” Appl. Opt. 50, 5750–5757 (2011).
[CrossRef]

W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13, 075404 (2011).
[CrossRef]

2010 (9)

2009 (8)

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A Pure Appl. Opt. 11, 125406 (2009).
[CrossRef]

N. Savage, “Digital spatial light modulators,” Nat. Photonics 3, 170–172 (2009).
[CrossRef]

H. E. Hwang, H. T. Chang, and W. N. Lie, “Multiple-image encryption and multiplexing using a modified Gerchberg–Saxton algorithm and phase modulation in Fresnel-transform domain,” Opt. Lett. 34, 3917–3919 (2009).
[CrossRef]

P. Kumar, A. Kumar, J. Joseph, and K. Singh, “Impulse attack free double-random-phase encryption scheme with randomized lens-phase functions,” Opt. Lett. 34, 331–333 (2009).
[CrossRef]

W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282, 3680–3685 (2009).
[CrossRef]

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photon. 1, 589–636 (2009).
[CrossRef]

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

M. He, Q. Tan, L. Cao, Q. He, and G. Jin, “Security enhanced optical encryption system by random phase key and permutation key,” Opt. Express 17, 22462–22473 (2009).
[CrossRef]

2008 (3)

2007 (2)

2006 (10)

L. Chen and D. Zhao, “Optical image encryption with Hartley transforms,” Opt. Lett. 31, 3438–3440 (2006).
[CrossRef]

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[CrossRef]

2005 (1)

2004 (1)

2003 (4)

2001 (2)

2000 (6)

1999 (4)

1998 (1)

1997 (3)

B. Javidi and A. Sergent, “Fully phase encoded key and biometrics for security verification,” Opt. Eng. 36, 935–942 (1997).
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I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997).
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1996 (2)

B. L. Volodin, B. Kippelen, K. Meerholz, B. Javidi, and N. Peyghambarian, “A polymeric optical pattern-recognition system for security verification,” Nature 383, 58–60 (1996).
[CrossRef]

R. K. Wang, I. A. Watson, and C. Chatwin, “Random phase encoding for optical security,” Opt. Eng. 35, 2464–2469 (1996).
[CrossRef]

1995 (1)

1994 (1)

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
[CrossRef]

1989 (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 35, 237–246 (1972).

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E. Pérez-Cabré, H. C. Abril, M. S. Millan, and B. Javidi, “Photon-counting double-random-phase encoding for secure image verification and retrieval,” J. Opt. 14, 094001 (2012).
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W. Chen, G. Situ, and X. Chen, “High-flexibility optical encryption via aperture movement,” Opt. Express 21, 24680–24691 (2013).
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W. Chen, X. Chen, and C. J. R. Sheppard, “Optical image encryption based on phase retrieval combined with three-dimensional particle-like distribution,” J. Opt. 14, 075402 (2012).
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W. Chen and X. Chen, “Optical multiple-image encryption based on multiplane phase retrieval and interference,” J. Opt. 13, 115401 (2011).
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W. Chen, X. Chen, and C. J. R. Sheppard, “Optical double-image cryptography based on diffractive imaging with a laterally-translated phase grating,” Appl. Opt. 50, 5750–5757 (2011).
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W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13, 075404 (2011).
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W. Chen and X. Chen, “Optical image encryption using multilevel Arnold transform and noninterferometric imaging,” Opt. Eng. 50, 117001 (2011).
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W. Chen, X. Chen, and C. J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35, 3817–3819 (2010).
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W. Chen and X. Chen, “Space-based optical image encryption,” Opt. Express 18, 27095–27104 (2010).
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W. Chen and X. Chen, “Quantitative phase retrieval of a complex-valued object using variable function orders in the fractional Fourier domain,” Opt. Express 18, 13536–13541 (2010).
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W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282, 3680–3685 (2009).
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Chen, X.

W. Chen and X. Chen, “Double random phase encoding using phase reservation and compression,” J. Opt. 16, 025402 (2014).
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W. Chen and X. Chen, “Object authentication in computational ghost imaging with the realizations less than 5% of Nyquist limit,” Opt. Lett. 38, 546–548 (2013).
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W. Chen and X. Chen, “Ghost imaging for three-dimensional optical security,” Appl. Phys. Lett. 103, 221106 (2013).
[CrossRef]

W. Chen and X. Chen, “Optical cryptography network topology based on 2D-to-3D conversion and phase-mask extraction,” Opt. Lasers Eng. 51, 410–416 (2013).
[CrossRef]

W. Chen, G. Situ, and X. Chen, “High-flexibility optical encryption via aperture movement,” Opt. Express 21, 24680–24691 (2013).
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W. Chen, X. Chen, A. Anand, and B. Javidi, “Optical encryption using multiple intensity samplings in the axial domain,” J. Opt. Soc. Am. A 30, 806–812 (2013).
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W. Chen and X. Chen, “Optical image encryption based on multiple-region plaintext and phase retrieval in three-dimensional space,” Opt. Lasers Eng. 51, 128–133 (2013).
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W. Chen, X. Chen, A. Stern, and B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photon. J. 5, 6900113 (2013).
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W. Chen, X. Chen, and C. J. R. Sheppard, “Optical image encryption based on phase retrieval combined with three-dimensional particle-like distribution,” J. Opt. 14, 075402 (2012).
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W. Chen, X. Chen, and C. J. R. Sheppard, “Optical image encryption based on coherent diffractive imaging using multiple wavelengths,” Opt. Commun. 285, 225–228 (2012).
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W. Chen and X. Chen, “Optical cryptography topology based on a three-dimensional particle-like distribution and diffractive imaging,” Opt. Express 19, 9008–9019 (2011).
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W. Chen and X. Chen, “Optical multiple-image encryption based on multiplane phase retrieval and interference,” J. Opt. 13, 115401 (2011).
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W. Chen and X. Chen, “Optical image encryption using multilevel Arnold transform and noninterferometric imaging,” Opt. Eng. 50, 117001 (2011).
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W. Chen, X. Chen, and C. J. R. Sheppard, “Optical double-image cryptography based on diffractive imaging with a laterally-translated phase grating,” Appl. Opt. 50, 5750–5757 (2011).
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W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13, 075404 (2011).
[CrossRef]

W. Chen, X. Chen, and C. J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35, 3817–3819 (2010).
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W. Chen and X. Chen, “Space-based optical image encryption,” Opt. Express 18, 27095–27104 (2010).
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W. Chen and X. Chen, “Quantitative phase retrieval of a complex-valued object using variable function orders in the fractional Fourier domain,” Opt. Express 18, 13536–13541 (2010).
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J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data by using polarized light,” Opt. Commun. 260, 109–112 (2006).
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Hennelly, B.

Hennelly, B. M.

Horner, J. L.

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33, 1752–1756 (1994).
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Hwang, H. E.

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Javidi, B.

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A. Markman and B. Javidi, “Full-phase photon-counting double-random-phase encryption,” J. Opt. Soc. Am. A 31, 394–403 (2014).
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M. Cho and B. Javidi, “Three-dimensional photon counting double-random-phase encryption,” Opt. Lett. 38, 3198–3201 (2013).
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W. Chen, X. Chen, A. Stern, and B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photon. J. 5, 6900113 (2013).
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W. Chen, X. Chen, A. Anand, and B. Javidi, “Optical encryption using multiple intensity samplings in the axial domain,” J. Opt. Soc. Am. A 30, 806–812 (2013).
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E. Pérez-Cabré, H. C. Abril, M. S. Millan, and B. Javidi, “Photon-counting double-random-phase encoding for secure image verification and retrieval,” J. Opt. 14, 094001 (2012).
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M. S. Millán, E. Pérez-Cabré, and B. Javidi, “Multifactor authentication reinforces optical security,” Opt. Lett. 31, 721–723 (2006).
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O. Matoba and B. Javidi, “Secure holographic memory by double-random polarization encryption,” Appl. Opt. 43, 2915–2919 (2004).
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S. Kishk and B. Javidi, “Watermarking of three-dimensional objects by digital holography,” Opt. Lett. 28, 167–169 (2003).
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B. Javidi and T. Nomura, “Securing information by use of digital holography,” Opt. Lett. 25, 28–30 (2000).
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B. Javidi and A. Sergent, “Fully phase encoded key and biometrics for security verification,” Opt. Eng. 36, 935–942 (1997).
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B. Javidi, “Securing information with optical technologies,” Phys. Today 50(3), 27–32 (1997).
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B. L. Volodin, B. Kippelen, K. Meerholz, B. Javidi, and N. Peyghambarian, “A polymeric optical pattern-recognition system for security verification,” Nature 383, 58–60 (1996).
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P. Réfrégier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995).
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B. L. Volodin, B. Kippelen, K. Meerholz, B. Javidi, and N. Peyghambarian, “A polymeric optical pattern-recognition system for security verification,” Nature 383, 58–60 (1996).
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A. Markman, B. Javidi, and M. Tehranipoor, “Photon-counting security tagging and verification using optically encoded QR codes,” IEEE Photon. J. 6, 6800609 (2014).
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A. Markman and B. Javidi, “Full-phase photon-counting double-random-phase encryption,” J. Opt. Soc. Am. A 31, 394–403 (2014).
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O. Matoba, T. Nomura, E. Pérez-Cabré, M. S. Millán, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
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M. S. Millán, E. Pérez-Cabré, and B. Javidi, “Multifactor authentication reinforces optical security,” Opt. Lett. 31, 721–723 (2006).
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M. S. Millán, E. Pérez-Cabré, and B. Javidi, “Multifactor authentication reinforces optical security,” Opt. Lett. 31, 721–723 (2006).
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B. L. Volodin, B. Kippelen, K. Meerholz, B. Javidi, and N. Peyghambarian, “A polymeric optical pattern-recognition system for security verification,” Nature 383, 58–60 (1996).
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W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282, 3680–3685 (2009).
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B. Javidi and A. Sergent, “Fully phase encoded key and biometrics for security verification,” Opt. Eng. 36, 935–942 (1997).
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B. L. Volodin, B. Kippelen, K. Meerholz, B. Javidi, and N. Peyghambarian, “A polymeric optical pattern-recognition system for security verification,” Nature 383, 58–60 (1996).
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X. Wang and D. Zhao, “Multiple-image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain,” Opt. Commun. 284, 148–152 (2011).
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Figures (27)

Figure 1
Figure 1

Schematic setup for DRPE in the Fourier domain.

Figure 2
Figure 2

Amplitude-only DRPE: (a) encoding process and (b) decoding process.

Figure 3
Figure 3

Fully phase DRPE: (a) encoding process and (b) decoding process.

Figure 4
Figure 4

Schematic setup for DRPE in the Fresnel domain.

Figure 5
Figure 5

DRPE in the Fresnel domain: (a) encoding process and (b) decoding process.

Figure 6
Figure 6

Schematic setup for optical encryption using multidimensional keys. Various degrees of freedom can be used to encrypt the data/image, and wavelength sensitivity can also be tested. Different from Sections 2.1 and 2.2, phase-only mask M1 is not bonded with the input image.

Figure 7
Figure 7

Schematic illustration of polarization encoding in an optical security system.

Figure 8
Figure 8

(a) Input image to be encrypted, (b) amplitude of the encrypted data, and (c) photon-counting encrypted data using the number of photons Np=103. Reprinted from [55].

Figure 9
Figure 9

(a) Decrypted image corresponding to Fig. 8(c), and (b) correlation output using a kth-law nonlinear correlator with k=0.30. Reprinted from [55].

Figure 10
Figure 10

Schematic setup for diffractive-imaging-based optical encoding.

Figure 11
Figure 11

Flow chart for image decryption in a diffractive-imaging-based optical security system.

Figure 12
Figure 12

Diffractive-imaging-based optical encoding: (a) an input image and (b)–(d) three diffraction intensity patterns (i.e., ciphertexts).

Figure 13
Figure 13

Diffractive-imaging-based optical encoding: (a) and (b) decryption using correct keys (22 iterations), (c) and (d) decryption using the wrong wavelength (600 iterations), and (e) and (f) decryption using the wrong phase-only mask (M2, 600 iterations).

Figure 14
Figure 14

Schematic setup for 2D-phase-retrieval-based optical encoding.

Figure 15
Figure 15

Extracted phase-only masks (a) M1 and (b) M2, (c) the iterative process by using a phase-retrieval algorithm for the encoding, and (d) decrypted image obtained by using correct security keys.

Figure 16
Figure 16

Decrypted images obtained by using (a) wrong phase-only mask M1, (b) wrong phase-only mask M2, (c) wrong distance d2, and (d) wrong wavelength.

Figure 17
Figure 17

Schematic setup for 3D-phase-retrieval-based optical encoding.

Figure 18
Figure 18

(a) Input image, (b) extracted phase-only mask M1, (c) phase-only mask M2, and (d) phase-only mask M3.

Figure 19
Figure 19

(a) Decrypted image obtained by using correct keys, and (b) and (c) decrypted images obtained at one sectional plane.

Figure 20
Figure 20

Schematic setup for non-iterative interference-based optical encoding.

Figure 21
Figure 21

Extracted phase-only masks (a) M1 and (b) M2, (c) decrypted image using wrong M1 and M2, (d) decrypted image using wrong FrFT function order, and (e) decrypted image using correct keys.

Figure 22
Figure 22

Schematic setup for phase-truncated optical encoding.

Figure 23
Figure 23

(a) Phase-only mask M1, (b) phase-only mask M2, (c) decryption key Pw(μ,ν), (d) decryption key Po(ξ,η), and (e) the ciphertext.

Figure 24
Figure 24

(a) Decrypted image obtained by using correct keys, (b) decrypted image obtained by using M1 and M2 for the decryption, and (c) decrypted image obtained by using wrong decryption key Pw(μ,ν).

Figure 25
Figure 25

Schematic setup for sparsity-driven optical information authentication.

Figure 26
Figure 26

(a) Decrypted image obtained by using correct keys, (b) authentication result corresponding to (a), (c) decrypted image obtained by using wrong wavelength (error of 10.0 nm), and (d) authentication result corresponding to (c).

Figure 27
Figure 27

Similar image case: the authentication result.

Tables (1)

Tables Icon

Table 1. Brief Summary of Key Features and Possible Drawbacks of Various Optical Security Systemsa

Equations (31)

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H(μ,ν)=FT{P(x,y)exp[jϕ(x,y)]},
O(ξ,η)=IFT{H(μ,ν)exp[jφ(μ,ν)]},
H^(μ,ν)={FT[O^(ξ,η)]}{exp[jφ(μ,ν)]}*,
S(x,y)={IFT[H^(μ,ν)]}{exp[jϕ(x,y)]}*,
O(ξ,η)=FrTd2,λ{(FrTd1,λ{P(x,y)exp[jϕ(x,y)]})exp[jφ(μ,ν)]},
S(x,y)=[FrTd1,λ({FrTd2,λ[O^(ξ,η)]}{exp[jφ(μ,ν)]}*)]{exp[jϕ(x,y)]}*,
Pd(lj;λj)=[λj]ljeλjlj!,lj=0,1,2,
I(h)(ξ,η)=|FrTd2+Δd×(h1){(FrTd1{P(x,y)exp[jϕ(x,y)]})exp[jφ(μ,ν)]}|2,
O(n)(ξ,η)=FrTd2+Δd×(h1){(FrTd1{P(n)(x,y)exp[jϕ(x,y)]})exp[jφ(μ,ν)]},
O^(n)(ξ,η)=I(h)(ξ,η)[O(n)(ξ,η)/|O(n)(ξ,η)|].
P^(n)(x,y)=[FrTd1({FrT[d2+Δd×(h1)][O^(n)(ξ,η)]}{exp[jφ(μ,ν)]}*)]{exp[jϕ(x,y)]}*,
IE=x,y[|P^(n)(x,y)||P^(n1)(x,y)|]2.
O(n)(x,y)=FrTd2{(FrTd1{exp[jϕ(n)(μ,ν)]})exp[jφ(n)(ξ,η)]}.
O^(n)(x,y)=P(x,y)O(n)(x,y)/|O(n)(x,y)|,
exp[jφ^(n)(ξ,η)]=(FrTd2[O^(n)(x,y)]FrTd1{exp[jϕ(n)(μ,ν)]})/|FrTd2[O^(n)(x,y)]FrTd1{exp[jϕ(n)(μ,ν)]}|,
exp[jϕ^(n)(μ,ν)]=FrTd1({FrTd2[O^(n)(x,y)]}{exp[jφ^(n)(ξ,η)]}*)|FrTd1({FrTd2[O^(n)(x,y)]}{exp[jφ^(n)(ξ,η)]}*)|.
P^(x,y)=|FrTd2{(FrTd1{exp[jϕ^(n)(μ,ν)]})exp[jφ^(n)(ξ,η)]}|2,
O(i,n)(k,l)=FrFTbi{(FrFTa2{(FrFTa1{exp[jϕ(i,n)(x,y)]})exp[jφ(μ,ν)]})×exp[jR(ξ,η)]},
O^(i,n)(k,l)={[2Ω|P(i)(k,l)||O(i,n)(k,l)|]×O(i,n)(k,l)/|O(i,n)(k,l)|if(k,l)iO(i,n)(k,l)if(k,l)i.
O(i,n)(x,y)=FrFTa1([FrFTa2({FrFTbi[O^(i,n)(k,l)]}{exp[jR(ξ,η)]}*)]×{exp[jφ(μ,ν)]}*),
exp[jϕ^(i,n)(x,y)]=O(i,n)(x,y)/|O(i,n)(x,y)|,
P^(k,l)=Ii=1,2;(k,l)i|FrFTbi{(FrFTa2{(FrFTa1{exp[jϕ^(x,y)]})exp[jφ(μ,ν)]})×exp[jR(ξ,η)]}|2,
P(x,y)exp[jR(x,y)]=FrFTα{exp[jϕ(μ,ν)]}+FrFTα{exp[jφ(μ,ν)]},
ϕ(μ,ν)=ang(FrFTα{P(x,y)exp[jR(x,y)]})arccos{[abs(FrFTα{P(x,y)exp[jR(x,y)]})]/2},
φ(μ,ν)=ang(FrFTα{P(x,y)exp[jR(x,y)]}exp[jϕ(μ,ν)]),
W(μ,ν)=FT{P(x,y)exp[jϕ(x,y)]},
Aw(μ,ν)=|W(μ,ν)|,
Pw(μ,ν)=W(μ,ν)/|W(μ,ν)|.
O(ξ,η)=IFT{Aw(μ,ν)exp[jφ(μ,ν)]}.
P^(x,y)=|IFT{|FT[Ao(ξ,η)Po(ξ,η)]|Pw(μ,ν)}|,
A(x,y)=|IFT(|{FT[P^(x,y)]}*{FT[P(x,y)]}|t{FT[P^(x,y)]}*{FT[P(x,y)]}|{FT[P^(x,y)]}*{FT[P(x,y)]}|)|2,

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