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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 16 — Aug. 3, 2009
  • pp: 13418–13424
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Optical image encryption based on interference of polarized light

Nan Zhu, Yongtian Wang, Juan Liu, Jinghui Xie, and Hao Zhang  »View Author Affiliations


Optics Express, Vol. 17, Issue 16, pp. 13418-13424 (2009)
http://dx.doi.org/10.1364/OE.17.013418


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Abstract

We proposed a novel architecture for optical image encryption based on interference between two polarized wavefronts. A polarization-selective diffractive optical element is employed to generate the desired polarized wavefronts by modulating the incident polarized light beam. The encryption algorithm for this new method is simple and does not need iterative encoding. Numerical simulation is performed to demonstrate the validity of this new proposed method.

© 2009 Optical Society of America

1. Introduction

In recent years, encryption techniques based on optical theories have been proposed and developed in the areas of data and information securities. Compared with conventional mathematical encryption methods optical encryption provides more information volume, free-degree as well as its ability of multi-dimension and parallel processing. Since Refregier and Javidi firstly reported their pioneer work about optical encryption based on double random phase encoding[1

1. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995). [CrossRef] [PubMed]

], several optical encryption techniques have been proposed in recent years[2

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

7

7. S. Liu, Q. Mi, and B. Zhu, “Optical image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26, 1242–1244 (2001). [CrossRef]

]. Very recently, Zhang et al. have proposed a novel method for optical image encryption based on interference[2

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

]. In their paper two phase-only masks are used to modulate the wavefronts of the incident light beams, in which two modulated light beams interfere with each other after combination with a half-mirror (HM), and finally generate an optical image in the output plane. Since all the information of an encryption image is stored in two separated phase-only masks, they must be presented together to decode the original image, which enhance the information security level. However, it is difficult to collimate two beams into collinear (co-axis) in an actual optical experiment since high precision is needed during the optical image decryption process.

Diffractive optical elements (DOEs) constructed as phase-only optical elements are widely used in a variety of applications where arbitrary wavefront generation is desired such as aberration correction, optical testing, optical communication and image processing. Especially in applications where volume, weight, and design flexibility are important issues, diffractive optical elements have proven to be useful alternatives for refractive optics. However, conventional diffractive optical elements are made of isotropic materials and, therefore, are largely insensitive to light polarization. The polarization-selective DOEs use polarization as another degree of freedom in optical and photonic system design and hence enable applications such as the polarization multiplexing and demultiplexing in the data transmission system, optical isolating and pickup for compact disc system, image processing and optical sensing, etc. In 1993 Ford et al. proposed an original technique to fabricate computer-generated polarization-selective DOEs with independent phase profiles for the two orthogonally linear polarizations[9

9. J. E. Ford, F. Xu, K. Urquhart, and Y. Fainman, “Polarization-selective computer-generated holograms,” Opt. Lett. 18, 456–458 (1993). [CrossRef] [PubMed]

]. In 1996 Xu et al. proposed another method, with experimental demonstration, to fabricate a polarization-selective DOE upon a single substrate of birefringent[10

10. F. Xu, R.-C. Tyan, Y. Fainman, and J. E. Ford, “Single-substrate birefringent computer-generated holograms,” Opt. Lett. 21, 516–518 (1996). [CrossRef] [PubMed]

].

In this paper, a novel method is proposed to encode an image in a single polarization-selective diffractive optical element. In the process of optical image decryption, this polarization-selective DOE that has encoded information for the target image is employed to generate two desired polarized wavefronts by modulating the incident polarized light beam. These two wavefronts then interfere and generate the encryption image. The encoding process is simple and does not need the iterative algorithm. The realization of optical image decryption also has the advantages of easier installation and collimation since all the optical elements are in a same optical axis.

2. General concept of polarization-selective DOEs

Conventional phase-only DOEs are fabricated by means of mapping an optical phase function into an etched surface-relief pattern on an isotropic substrate. Polarization-selective DOE is a phase-only diffractive optical element fabricated on a birefringent substrate where ordinary-and extraordinary-polarized light will have different refractive indexes. A polarization-selective DOE can effectively separate the incident light by polarization, control on incident beam according to its polarization status independently and simultaneously, act independently on each path, and recombine the beams, as shown in Fig. 1.

Fig. 1. Effective function of a polarization-selective DOE: PBS’s, polarizing beam splitters; M’s, mirrors.

A polarization-selective DOE consists of a two-dimensional array of pixels, where each pixel has a phase delay corresponding to the etched surface-relief depth of the birefringent substrate. For a same surface-relief depth d, ordinary- and extraordinary-polarized light will have different phase delay Φo and Φe respectively since they have different refractive indexes. Consider a polarization-selective DOE fabricated in a birefringent substrate with the optical axis parallel to the surface of the substrate, using geometrical optics, we can find the corresponding phase delays caused by the surface-relief depth compared with those of an unetched pixel for the ordinary- and extraordinary polarized waves

Φo=2π(no1)dλ,
(1)
Φe=2π(ne1)dλ,
(2)

whereλ is the wavelength of the incident wave in vacuum, no and ne are the refractive indexes of birefringent substrate for ordinary- and extraordinary-polarized light, respectively. Here we consider that air is the material surrounding the substrate. It is well known that, for diffractive optical elements the absolute phase delay Φo and Φe are normally not important. We can always express Φo and Φe as

Φo=φo+2pπ,
(3)
Φe=φe+2,
(4)

where p and q are integers and φo and φe are phase values in the region of 0 to 2π, as

φo+2pπ=2π(no1)dλ,
(5)
φe+2qπ=2π(ne1)dλ.
(6)

For diffractive optical elements the values that make sense are φo and φe.

By carefully designing the surface-relief depth d, it can be achieved that the optical path through each pixel have the desired phase delay φo for ordinary-polarized light and φe for extraordinary-polarized light simultaneously and independently. In general, if φo and φe are arbitrary values, we cannot find an accurate common solution d for both Equations 5 and 6. However if we introduce small errors δp and δq, such as

φo+2pπ+δp=2π(no1)dλ,
(7)
φe+2qπ+δq=2π(ne1)dλ,
(8)

there exists an approximate solution for d when the values of integers p and q are arbitrarily large, as

d=λ[φo+2pπ+δp]2π(no1)=λ[φe+2qπ++δq]2π(ne1).
(9)

The introduction of δp and δq will slightly affect the function of the polarization-selective DOE. The set of quantities of δp and δq is a tradeoff between the accurate function of polarization-selective DOE and its fabrication cost. When δp and δq are very small values the integers p and q can have correspondingly large values which means the surface-relief of DOE can have very deep etched depth and causes high fabrication cost. Economically δp and δq are set to controllably appropriate values meanwhile the function of DOE can still be maintained at an acceptable range and meet the design requirement.

By using the same method, as is described above, to each pixel independently on the polarization-selective DOE, a surface relief pattern d(m,n) can be obtained, where (m,n) refers to the discrete coordinates of pixel array on a polarization-selective DOE.

3. Optical encryption algorithm

The encryption problem is to separate an image into two phase functions φo and φe, then encoding these two phase functions in a single polarization-selective diffractive optical element. If the intensity distribution of the encrypted image is o(m,n), the object function o′(m,n) can be constructed as

o=mn=omnexp[i2πrandmn],
(10)

where rand(m,n) generates a random distribution between 0 and 1. This constructed field distribution can be expressed as the interference of optical fields generated by phase distributions φo and φe

omn={exp(iφo)}+{exp(iφe)},
(11)

where 𝔽{…} expresses the Fourier transform. If we define Q=𝔽-1{o′(m,n)},where 𝔽-1{…} expresses the inverse Fourier transform, we can have

exp(iφo)=Qexp(iφe).
(12)

Since φo and φe consist of phase-only values, we can have

Qexp(iφe)2=Qexp(iφe)Q
exp(iφe)*=1,
(13)

Finally we obtain two phase distributions as

φo=arg(Q)±arccos[abs(Q)2];
(14)
φo=arg[Qexp(iφo)];
(15)

where arg(Q) and abs(Q) return the phase and the amplitude, respectively.

Once the phase distributions φo and φe are obtained we encode them into the surface-relief pattern of the polarization-selective DOE according to the algorithm described in part 2 of this paper.

4. Optical decryption setup

In the process of optical image decryption, a polarization-selective DOE that has encoded surface-relief pattern for the target image is employed to generate two desired orthogonal linear polarized wavefronts by modulating the incident polarized light beam. The schematic view of the optical decrypting setup is illuminated in Fig. 2. The polarization directions of both two polarization plates are 45° to the horizontal of the system, as is indicated by two green arrows in Fig. 2. The optical axis of the polarization-selective DOE birefringent crystal substrate is parallel to the surface of the substrate wafer. A collimated monochromatic light beam incidents into Polarization Plate 1 and generate a plane wave with 45° linearly polarized light. This polarized light then perpendicularly incidents into the polarization-selective DOE. The polarization-selective DOE modulates the incident polarized light to generate the desired wavefronts for ordinary- and extraordinary-polarized light, respectively. After Polarization Plate 2, these two wavefronts interfere and generate an interferogram (decryption image) on the focal plane of the Fourier lens.

Fig. 2. Schematic of the optical image decryption setup.

5. Numerical simulations

The computer simulations are implemented to show the validity of this optical encryption method. The parameters used in the simulations are as follows: (1) the original image to be encrypted is a 256 gray-level image of a panda, as shown in Fig. 3(a); (2) the birefringent crystal substrate of the polarization-selective DOE is chose as TiO2 and its refractive index are no=2.4317, ne=2.6827 for λ=632.8nm; (3) the size of the polarization-selective DOE is 100mm×100mm; (4) the pixel numbers of the polarization-selective DOE and the encrypted image are 256×256; (5) the wavelength of the illuminating light is λ=632.8nm; (6) the focal length of the Fourier lens is 1.58m. To evaluate the quality of the encrypted image relative to the original one, a Relative Error (RE) function is defined as

RE=Σm=1NΣn=1Nrmno'mn2Σm=1NΣn=1No'mn2,
(16)

where o′(m,n) and r (m, n) denote the amplitude values of the original and encrypted images, respectively. Figure 3(b) shows the reconstructed image with the matched polarization-selective DOE and system configuration. The corresponding RE of the encrypted image is 0.0047. The noise of the encryption image is mainly due to the limited surface-relief depth of the polarization-selective DOE. Figure 3(c) displays the surface-relief pattern of the polarization-selective DOE. The maximum etch depth is less than 36µm, which is less than one-tenth of the pixel size on the polarization-selective DOE. The selection of this maximum value for etch depth is to ease the fabrication of polarization-selective DOE since the fabrication of DOE with deep etch depth is a time-consuming process, meanwhile the quality of the decoded image is acceptable.

Fig. 3. (a). The original image for encryption, (b). decryption image with the matched polarization-selective DOE, and (c) the surface-relief pattern of the polarization-selective DOE.

To investigate the decryption sensitivity of the working wavelength and the surface-relief depth of the polarization-selective DOE, we calculate the REs as functions of the wavelength and the deviation ratio δd/d of surface-relief depth, respectively. The wavelength of the illuminating light λ can be used as encryption key. The dependences of RE on λ and δd/d are shown in Figs. 4(a) and Fig. 4(b), respectively.

Fig. 4. (a). Dependence of RE on wavelength, (b) Dependence of RE on deviation ratio of polarization-selective DOE surface-relief depth.

It is clear that only when λ and d are exact values does RE reach a very small value. When RE>0.2, there is a failure to distinguish the decoded image. In the case of minimal deviations, the RE increases exponentially and it is a failure to recognize the original image visually, as is it zoomed in the insets. The REs for the parameters around the accurate values of λ and d are highly sensitive. The wavelength sensitivity is around 0.8 nm and the surface-relief deviation of polarization-selective DOE is about ±0.3%. The high sensitivity will cause great difficulty in copying the encryption system and meantime the sensitivity is also in the region of state of the art of optical fabrication that the optical encryption process can be implemented conveniently.

6. Conclusions

In summary, a new system for optical image encryption based on interference of polarized light is proposed. The image is encoded into the etched surface-relief pattern of a polarization-selective DOE. The image encryption by the polarization-selective DOE will highly increase the information security and still maintain the parameter sensitivity in an acceptable region. The encoding process does not need the iterative algorithm hence the encryption can be simply processed. The simulation results demonstrate that the validity of optical image encryption can be achieved successfully by the polarization-selective DOE. The co-axis optical arrangement with the polarization-selective DOE will also ease the optical alignment, and it can be operated by professionally secure employee without any difficulties in the future.

Acknowledgement

This work was supported by the Innovation Team Development Program of the Chinese Ministry of Education (Grant No. IRT0606) and the National Basic Research Program of China (Grant No. 2006CB302901).

References and links

1.

P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995). [CrossRef] [PubMed]

2.

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

3.

J. F. Heanue, M. C. Bashaw, and L. Hesselink, “Encrypted holographic data storage based on orthogonal-phase-code multiplexing,” Appl. Opt. 34, 6012–6015 (1995). [CrossRef] [PubMed]

4.

N. Yoshikawa, M. Itoh, and T. Yatagai, “Binary computer-generated holograms for security applications from a synthetic double-exposure method by electron-beam lithography,” Opt. Lett. 23, 1483–1485 (1998). [CrossRef]

5.

O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24, 762–764 (1999). [CrossRef]

6.

B. Zhu, S. Liu, and Q. Ran, “Optical image encryption based on multifractional Fourier transforms,” Opt. Lett. 25, 1159–1161 (2000). [CrossRef]

7.

S. Liu, Q. Mi, and B. Zhu, “Optical image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26, 1242–1244 (2001). [CrossRef]

8.

O. Matoba and B. Javidi, “Optical retrieval of encrypted digital holograms for secure real-time display,” Opt. Lett. 27, 321–323 (2002). [CrossRef]

9.

J. E. Ford, F. Xu, K. Urquhart, and Y. Fainman, “Polarization-selective computer-generated holograms,” Opt. Lett. 18, 456–458 (1993). [CrossRef] [PubMed]

10.

F. Xu, R.-C. Tyan, Y. Fainman, and J. E. Ford, “Single-substrate birefringent computer-generated holograms,” Opt. Lett. 21, 516–518 (1996). [CrossRef] [PubMed]

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(100.2000) Image processing : Digital image processing

ToC Category:
Image Processing

History
Original Manuscript: April 22, 2009
Revised Manuscript: July 9, 2009
Manuscript Accepted: July 11, 2009
Published: July 20, 2009

Citation
Nan Zhu, Yong-tian Wang, Juan Liu, Jing-hui Xie, and Hao Zhang, "Optical image encryption based on interference of polarized light," Opt. Express 17, 13418-13424 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-13418


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References

  1. P. Refregier and B. Javidi, "Optical image encryption based on input plane and Fourier plane random encoding," Opt. Lett. 20, 767-769 (1995). [CrossRef] [PubMed]
  2. Y. Zhang and B. Wang, "Optical image encryption based on interference," Opt. Lett. 33, 2443-2445 (2008). [CrossRef] [PubMed]
  3. J. F. Heanue, M. C. Bashaw, and L. Hesselink, "Encrypted holographic data storage based on orthogonal-phase-code multiplexing," Appl. Opt. 34, 6012-6015 (1995). [CrossRef] [PubMed]
  4. N. Yoshikawa, M. Itoh, and T. Yatagai, "Binary computer-generated holograms for security applications from a synthetic double-exposure method by electron-beam lithography," Opt. Lett. 23, 1483-1485 (1998). [CrossRef]
  5. O. Matoba and B. Javidi, "Encrypted optical memory system using three-dimensional keys in the Fresnel domain," Opt. Lett. 24, 762-764 (1999). [CrossRef]
  6. B. Zhu, S. Liu, and Q. Ran, "Optical image encryption based on multifractional Fourier transforms," Opt. Lett. 25, 1159-1161 (2000). [CrossRef]
  7. S. Liu, Q. Mi, and B. Zhu, "Optical image encryption with multistage and multichannel fractional Fourier-domain filtering," Opt. Lett. 26, 1242-1244 (2001). [CrossRef]
  8. O. Matoba and B. Javidi, "Optical retrieval of encrypted digital holograms for secure real-time display," Opt. Lett. 27, 321-323 (2002). [CrossRef]
  9. J. E. Ford, F. Xu, K. Urquhart, and Y. Fainman, "Polarization-selective computer-generated holograms," Opt. Lett. 18, 456-458 (1993). [CrossRef] [PubMed]
  10. F. Xu, R.-C. Tyan, Y. Fainman, and J. E. Ford, "Single-substrate birefringent computer-generated holograms," Opt. Lett. 21, 516-518 (1996). [CrossRef] [PubMed]

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