## Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics

Optics Express, Vol. 14, Issue 4, pp. 1476-1486 (2006)

http://dx.doi.org/10.1364/OE.14.001476

Acrobat PDF (1872 KB)

### Abstract

A new method is proposed in this paper for the synthesis and encryption of information with digital holography technique and virtual optics. By using a three-step phase-shifting interferometry, the fused or subtracted digital hologram can be calculated from different interference patterns. To protect the digital data that can be transmitted through communication channel, an encryption approach based on virtual optics is also proposed. The encryption method proposed is based on extended fractional Fourier transforms. Both the encryption and decryption processes are performed in all-digital manner. The encrypted data and the synthesized data reconstructed numerically also can be stored and transmitted in the conventional communication channel. Numerical simulation results are given to verify the proposed idea.

© 2006 Optical Society of America

## 1. Introduction

1. D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, “Optical image synthesis (complex amplitude addition and subtraction) by holographic Fourier transformation,” Phys. Lett. **18**, 116–118 (1965). [CrossRef]

9. E. Tajahuerce, O. Matoba, S. C. Verrall, and B. Javidi, “Optoelectronic information encryption with phase-shifting interferometry,” Appl. Opt. **39**, 2313–2320 (2000). [CrossRef]

10. B. Javidi and T. Nomural, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. **25**, 887–889 (2000). [CrossRef]

15. N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. **235**, 253–259 (2004). [CrossRef]

19. L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. **28**, 1808–1810 (2003). [CrossRef] [PubMed]

20. J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A **14**, 3316–3322 (1997). [CrossRef]

## 2. Principle of information synthesis with digital holography technique

*U*

_{1}(

*x,y*) in CCD plane after Fresnel diffraction can be written as

*x*

_{0},

*y*

_{0}and

*x, y*are the coordinates of object plane and CCD plane, respectively. The function

*U*

_{0}(

*x*

_{0},

*y*

_{0}) represents the original object complex wave in object plane. Let the on-axis reference wave in CCD plane at the

*j*th (

*j*= 1, 2, 3) exposure be expressed as

_{j}is the phase shift introduced by phase retarders at the

*j*th exposure, and

*A*

_{r}

*, φ*

_{r}are the constant amplitude and the phase of reference wave, respectively.

*U*

_{1}(

*x,y*) =

*A*

_{0}(

*x,y*)exp[

*φ*

_{0}(

*x,y*)]. The digital hologram of the first image obtained from the three interference patterns can be expressed as [19

19. L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. **28**, 1808–1810 (2003). [CrossRef] [PubMed]

*A*

_{r}

*, φ*

_{r}are replaced with 1 and 0, respectively. In the same way, the corresponding digital hologram of the second image can be expressed as

*I′*

_{1},

*I′*

_{2},

*I′*

_{3}are the intensity distributions of the 1st, 2nd , 3rd interferograms respectively. Calculated from Eq. (6) and Eq. (7), a fused digital hologram can be written as

*f*means the data is obtained during fusion. Similarly, a subtracted digital hologram can also be obtained from Eq. (6) and Eq. (7), which is given by

*S*means the data is obtained during subtraction process. To obtain the synthesized image, a Fresnel inverse transform of the digital hologram has to be performed digitally. With digital holography technique, one can realize image synthesis by recording digital holograms

*U*

^{f}(

*x, y*) and

*U*

^{s}(

*x, y*) . When a half-wave plate and a quarter-wave plate are placed in the reference beam arm to generate phase shifts of 0 ,

*π*[9

9. E. Tajahuerce, O. Matoba, S. C. Verrall, and B. Javidi, “Optoelectronic information encryption with phase-shifting interferometry,” Appl. Opt. **39**, 2313–2320 (2000). [CrossRef]

8. G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double random phase encoding in the fractional Fourier domain,” Opt. Lett. **25**, 887–889 (2000). [CrossRef]

16. H. Kim, D. H. Kim, and Y. H. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Exp. **12**, 4912–4921 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4912. [CrossRef]

## 3. Encryption of digital hologram based on the concept of “virtual optics”

8. G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double random phase encoding in the fractional Fourier domain,” Opt. Lett. **25**, 887–889 (2000). [CrossRef]

14. N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-encrypted memory using cascaded extended fractional Fourier transform,” Opt. Lasers Eng. **42**, 141–151 (2004). [CrossRef]

18. X. Wang, D. Zhao, and L. Chen, “Image encryption based on extended fractional Fourier transform and digital holography technique,” Opt. Commun. (in press). [PubMed]

*n*segments {

*S*

_{1},

*S*

_{2},

*S*

_{3},⋯,

*S*

_{n}} digitally and each segment

*S*

_{k}(

*k*= 1,2,3,⋯

*n*) is extended fractional Fourier transformed two times with a random phase mask placed at the output plane of the first extended FRT. The set of

*n*encoded segments {

*k*th (

*k*= 1,2,3 ⋯

*n*) segment is used to illustrate this method and a one-dimensional representation is followed in this section. As shown as in Fig. 2(b), let

*x*,

*x*

_{1}and

*x*

_{2}denote the coordinates of the input plane and the output plane of the first and second extended FRT, respectively. The complex distribution in output plane can be written as [8

8. G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double random phase encoding in the fractional Fourier domain,” Opt. Lett. **25**, 887–889 (2000). [CrossRef]

14. N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-encrypted memory using cascaded extended fractional Fourier transform,” Opt. Lasers Eng. **42**, 141–151 (2004). [CrossRef]

18. X. Wang, D. Zhao, and L. Chen, “Image encryption based on extended fractional Fourier transform and digital holography technique,” Opt. Commun. (in press). [PubMed]

## 4. Decryption of digital hologram

*d*means the signal is obtained during decryption and

*x*′

_{1}) denotes the distribution in the output plane of the first extended FRT during the decryption process that can be given by

*S′*

_{k}(

*x*′

_{2}) = [

*x*′

_{2})]

^{*}. By substituting

*x*

_{2}) from Eq. (12) into Eq. (18) and

*x*′

^{1}) from Eq. (18) into Eq. (17), one can show the decrypted function

*x*′) is equal to

*S*

_{k}(

*x*). With all the correct decrypted digital segments, the fused or subtracted digital hologram can be retrieved.

## 5. Numerical simulation results

*λ*

_{1}=520nm,

*f*

_{1}=17cm,

*d*

_{1}= 16cm,

*f*

_{2}= 21cm,

*d*

_{2}=22cm,

*d*

_{3}=24cm, and

*d*

_{4}=29cm. The parameters of the extended FRT encryption system as calculated from Eqs. (14)–(16) are

*a*

_{1}=35.4111 + 35.4111

*i*,

*b*

_{1}=15.8363-15.8363

*i*,

*φ*

_{1}=1.5708-0.1312

*i*,

*a*

_{2}=39.2154 ,

*b*

_{2}=24.0144 and

*φ*

_{2}=1.8063 . The real part and imaginary part of the encrypted results are as shown in Fig. 5(e) and Fig. 5(f), respectively. With all the correct keys, the real part and imaginary part of decrypted results of the digital hologram segment are shown in Fig. 5(g) and Fig. 5(h), respectively. Figure 5(i) and Fig. 5(j) show the real part and imaginary part of the decrypted results with correct random phase codes but incorrect parameters (

*a*

_{1}= 32.2990 + 32.2990

*i*,

*b*

_{1}=16.1495-16.1495

*i*,

*φ*

_{1}=1.5708-0.1174

*i*,

*a*

_{2}=36.7115 ,

*b*

_{2}= 27.3631 and

*φ*

_{2}=1.9316 corresponding to the case of

*λ*

_{1}=560nm,

*f*

_{1}= 17cm,

*d*

_{1}= 16cm,

*f*

_{2}= 19cm,

*d*

_{2}= 16cm,

*d*

_{3}=24cm, and

*d*

_{4}= 28cm). Figure 5(k) and Fig. 5(l) show the real part and imaginary part of the decrypted results with incorrect random phase codes but correct parameters. It can be shown that digital hologram segments cannot be retrieved without knowledge of the respective random phase codes and parameters.

*I*

_{1}(

*i,j*) and

*I*

_{2}(

*i,j*) denote the values of the original image and the decrypted image at the pixel (

*i, j*) , respectively. The MSE between the real part of decrypted digital hologram as shown in Fig. 5(a) and the real part of input digital hologram as shown in Fig. 6(a) is about 3.54×10

^{-12}. The corresponding MSE between the imaginary parts as shown in Fig. 5(b) and Fig. 6(b) is about 3.56×10

^{-12}.

## 6. Conclusions

## Acknowledgment

## References and links

1. | D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, “Optical image synthesis (complex amplitude addition and subtraction) by holographic Fourier transformation,” Phys. Lett. |

2. | J. F. Ebersole, “Optical image subtraction,” Opt. Eng. |

3. | F. T. S. Yu and A. Tai, “Image subtraction with an encoded extended incoherent source,” Appl. Opt. |

4. | A. E. Chiou and P. Yeh, “Parallel image subtraction using a phase-conjugate Michelson interferometer,” Opt. Lett. |

5. | S. Zhivkova and M. Miteva, “Image subtraction using fixed holograms in photorefractive Bi12TiO20 crystals,” Opt. Lett. |

6. | M. Y. Shih, A. Shishido, and I. C. Khoo, “All-optical image processing by means of a photosensitive nonlinear liquid-crystal film: edge enhancement and image addition subtraction,” Opt. Lett. |

7. | P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. |

8. | G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double random phase encoding in the fractional Fourier domain,” Opt. Lett. |

9. | E. Tajahuerce, O. Matoba, S. C. Verrall, and B. Javidi, “Optoelectronic information encryption with phase-shifting interferometry,” Appl. Opt. |

10. | B. Javidi and T. Nomural, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. |

11. | N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption using a localized fractional Fourier transform,” Opt. Eng. |

12. | L. Yu and L. Cai, “Multidimensional data encryption with digital holography,” Opt. Commun. |

13. | X. Peng, L. Yu, and L. Cai, “Digital watermarking in three-dimensional space with a virtual-optics imaging modality,” Opt. Commun. |

14. | N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-encrypted memory using cascaded extended fractional Fourier transform,” Opt. Lasers Eng. |

15. | N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. |

16. | H. Kim, D. H. Kim, and Y. H. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Exp. |

17. | L. Chen and D. Zhao, “Optical image encryption based on fractional wavelet transform,” Opt. Commun. |

18. | X. Wang, D. Zhao, and L. Chen, “Image encryption based on extended fractional Fourier transform and digital holography technique,” Opt. Commun. (in press). [PubMed] |

19. | L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. |

20. | J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A |

**OCIS Codes**

(070.6020) Fourier optics and signal processing : Continuous optical signal processing

(090.0090) Holography : Holography

(100.2000) Image processing : Digital image processing

**ToC Category:**

Holography

**History**

Original Manuscript: November 15, 2005

Revised Manuscript: January 10, 2006

Manuscript Accepted: February 3, 2006

Published: February 20, 2006

**Citation**

Xiaogang Wang, Daomu Zhao, Feng Jing, and Xiaofeng Wei, "Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics," Opt. Express **14**, 1476-1486 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-4-1476

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### References

- D. Gabor, G. W. Stroke, R. Restrick, A. Funkhouser, and D. Brumm, "Optical image synthesis (complex amplitude addition and subtraction) by holographic Fourier transformation," Phys. Lett. 18, 116-118 (1965). [CrossRef]
- J. F. Ebersole, "Optical image subtraction," Opt. Eng. 14, 436-447 (1975).
- F. T. S. Yu and A. Tai, "Image subtraction with an encoded extended incoherent source," Appl. Opt. 20, 4082-4088 (1981). [CrossRef] [PubMed]
- A. E. Chiou and P. Yeh, "Parallel image subtraction using a phase-conjugate Michelson interferometer," Opt. Lett. 11, 306-308 (1986) [CrossRef] [PubMed]
- S. Zhivkova and M. Miteva, "Image subtraction using fixed holograms in photorefractive Bi12TiO20 crystals," Opt. Lett. 16, 750-751 (1991). [CrossRef] [PubMed]
- M. Y. Shih, A. Shishido, and I. C. Khoo, "All-optical image processing by means of a photosensitive nonlinear liquid-crystal film: edge enhancement and image addition subtraction," Opt. Lett. 26, 1140-1142 (2001). [CrossRef]
- 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]
- G. Unnikrishnan, J. Joseph, and K. Singh, "Optical encryption by double random phase encoding in the fractional Fourier domain," Opt. Lett. 25, 887-889 (2000). [CrossRef]
- E. Tajahuerce, O. Matoba, S. C. Verrall, and B. Javidi, "Optoelectronic information encryption with phase-shifting interferometry," Appl. Opt. 39, 2313-2320 (2000). [CrossRef]
- B. Javidi and T. Nomural, "Optical encryption by double-random phase encoding in the fractional Fourier domain," Opt. Lett. 25, 887-889 (2000). [CrossRef]
- N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, "Optical encryption using a localized fractional Fourier transform," Opt. Eng. 42, 3566-3571 (2003). [CrossRef]
- L. Yu and L. Cai, "Multidimensional data encryption with digital holography," Opt. Commun. 215, 271-284 (2003). [CrossRef]
- X. Peng, L. Yu, and L. Cai, "Digital watermarking in three-dimensional space with a virtual-optics imaging modality," Opt. Commun. 226, 155-165 (2003). [CrossRef]
- N. K. Nishchal, J. Joseph, and K. Singh, "Fully phase-encrypted memory using cascaded extended fractional Fourier transform," Opt. Lasers Eng. 42, 141-151 (2004). [CrossRef]
- N. K. Nishchal, J. Joseph, and K. Singh, "Securing information using fractional Fourier transform in digital holography," Opt. Commun. 235, 253-259 (2004). [CrossRef]
- H. Kim, D. H. Kim, and Y. H. Lee, "Encryption of digital hologram of 3-D object by virtual optics," Opt. Express 12, 4912-4921 (2004). [CrossRef]
- L. Chen and D. Zhao, "Optical image encryption based on fractional wavelet transform," Opt. Commun. 254, 361-367 (2005). [CrossRef]
- X. Wang, D. Zhao, and L. Chen, "Image encryption based on extended fractional Fourier transform and digital holography technique," Opt. Commun. (in press). [PubMed]
- L. Z. Cai, Q. Liu, and X. L. Yang, "Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps," Opt. Lett. 28, 1808-1810 (2003). [CrossRef] [PubMed]
- J. Hua, L. Liu, and G. Li, "Extended fractional Fourier transforms," J. Opt. Soc. Am. A 14, 3316-3322 (1997). [CrossRef]

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