Abstract

We propose and demonstrate a fractional Fourier domain encrypted holographic memory using an anamorphic optical system. The encryption is done by use of two statistically independent random-phase codes in the fractional Fourier domain. If the two random-phase codes are statistically independent white sequences, the encrypted data are stationary white noise. We exploit the capability of an optical system to process information in two dimensions by using two different sets of parameters along the two orthogonal axes to encode the data. The fractional Fourier transform parameters along with the random-phase codes constitute the key to the encrypted data. The knowledge of the key is essential to the successful decryption of data. The decoding of the encoded data is done by use of phase conjugation. We present a few experimental results.

© 2001 Optical Society of America

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References

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    [CrossRef] [PubMed]
  2. G. Unnikrishnan, J. Joseph, K. Singh, “Optical encryption system that uses phase conjugation in a photorefractive crystal,” Appl. Opt. 37, 8181–8185 (1998).
    [CrossRef]
  3. P. C. Mogensen, J. Glückstad, “A phase-based optical encryption system with polarisation encoding,” Opt. Commun. 173, 177–183 (2000).
    [CrossRef]
  4. B. Javidi, T. Nomura, “Securing information by use of digital holography,” Opt. Lett. 25, 28–30 (2000).
    [CrossRef]
  5. P. C. Mogensen, J. Glückstad, “Phase-only optical encryption,” Opt. Lett. 25, 566–568 (2000).
    [CrossRef]
  6. O. Matoba, B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24, 762–764 (1999).
    [CrossRef]
  7. O. Matoba, B. Javidi, “Encrypted optical storage with wavelength-key and random phase codes,” Appl. Opt. 38, 6785–6790 (1999).
    [CrossRef]
  8. O. Matoba, B. Javidi, “Encrypted optical storage with angular multiplexing,” Appl. Opt. 38, 7288–7293 (1999).
    [CrossRef]
  9. G. Unnikrishnan, J. Joseph, K. Singh, “Optical encryption using double random phase encoding in the fractional Fourier domain,” Opt. Lett. 25, 887–889 (2000).
    [CrossRef]
  10. H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes ed. (Academic, New York, 1999), Vol. 106, Chap. 4, pp. 239–291.
    [CrossRef]
  11. A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transforms in optics,” in Progress in Optics (Elsevier, Amsterdam, 1998), Vol. 38, Chap. 4, pp. 263–342.
    [CrossRef]
  12. D. Mendlovic, Z. Zalevsky, H. M. Ozaktas, “Applications of the fractional Fourier transforms to optical pattern recognition,” in Optical Pattern Recognition, F. T. S. Yu, S. Jutamulia, eds. (Cambridge U. Press, Cambridge, UK, 1998), Chap. 4, pp. 89–125.
  13. A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
    [CrossRef]
  14. M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
    [CrossRef]

2000

1999

1998

1997

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

1995

Erden, M. F.

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

Glückstad, J.

P. C. Mogensen, J. Glückstad, “A phase-based optical encryption system with polarisation encoding,” Opt. Commun. 173, 177–183 (2000).
[CrossRef]

P. C. Mogensen, J. Glückstad, “Phase-only optical encryption,” Opt. Lett. 25, 566–568 (2000).
[CrossRef]

Javidi, B.

Joseph, J.

Kutay, M. A.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes ed. (Academic, New York, 1999), Vol. 106, Chap. 4, pp. 239–291.
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transforms in optics,” in Progress in Optics (Elsevier, Amsterdam, 1998), Vol. 38, Chap. 4, pp. 263–342.
[CrossRef]

Matoba, O.

Mendlovic, D.

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

D. Mendlovic, Z. Zalevsky, H. M. Ozaktas, “Applications of the fractional Fourier transforms to optical pattern recognition,” in Optical Pattern Recognition, F. T. S. Yu, S. Jutamulia, eds. (Cambridge U. Press, Cambridge, UK, 1998), Chap. 4, pp. 89–125.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transforms in optics,” in Progress in Optics (Elsevier, Amsterdam, 1998), Vol. 38, Chap. 4, pp. 263–342.
[CrossRef]

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes ed. (Academic, New York, 1999), Vol. 106, Chap. 4, pp. 239–291.
[CrossRef]

Mogensen, P. C.

P. C. Mogensen, J. Glückstad, “Phase-only optical encryption,” Opt. Lett. 25, 566–568 (2000).
[CrossRef]

P. C. Mogensen, J. Glückstad, “A phase-based optical encryption system with polarisation encoding,” Opt. Commun. 173, 177–183 (2000).
[CrossRef]

Nomura, T.

Ozaktas, H. M.

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

D. Mendlovic, Z. Zalevsky, H. M. Ozaktas, “Applications of the fractional Fourier transforms to optical pattern recognition,” in Optical Pattern Recognition, F. T. S. Yu, S. Jutamulia, eds. (Cambridge U. Press, Cambridge, UK, 1998), Chap. 4, pp. 89–125.

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes ed. (Academic, New York, 1999), Vol. 106, Chap. 4, pp. 239–291.
[CrossRef]

Refregier, P.

Sahin, A.

A. Sahin, H. M. Ozaktas, D. Mendlovic, “Optical implementations of two-dimensional fractional Fourier transforms and linear canonical transforms with arbitrary parameters,” Appl. Opt. 37, 2130–2141 (1998).
[CrossRef]

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

Singh, K.

Unnikrishnan, G.

Zalevsky, Z.

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transforms in optics,” in Progress in Optics (Elsevier, Amsterdam, 1998), Vol. 38, Chap. 4, pp. 263–342.
[CrossRef]

D. Mendlovic, Z. Zalevsky, H. M. Ozaktas, “Applications of the fractional Fourier transforms to optical pattern recognition,” in Optical Pattern Recognition, F. T. S. Yu, S. Jutamulia, eds. (Cambridge U. Press, Cambridge, UK, 1998), Chap. 4, pp. 89–125.

Appl. Opt.

Opt. Commun.

M. F. Erden, H. M. Ozaktas, A. Sahin, D. Mendlovic, “Design of dynamically adjustable anamorphic fractional Fourier transformer,” Opt. Commun. 136, 52–60 (1997).
[CrossRef]

P. C. Mogensen, J. Glückstad, “A phase-based optical encryption system with polarisation encoding,” Opt. Commun. 173, 177–183 (2000).
[CrossRef]

Opt. Lett.

Other

H. M. Ozaktas, M. A. Kutay, D. Mendlovic, “Introduction to fractional Fourier transform and its applications,” in Advances in Imaging and Electron Physics, P. W. Hawkes ed. (Academic, New York, 1999), Vol. 106, Chap. 4, pp. 239–291.
[CrossRef]

A. W. Lohmann, D. Mendlovic, Z. Zalevsky, “Fractional transforms in optics,” in Progress in Optics (Elsevier, Amsterdam, 1998), Vol. 38, Chap. 4, pp. 263–342.
[CrossRef]

D. Mendlovic, Z. Zalevsky, H. M. Ozaktas, “Applications of the fractional Fourier transforms to optical pattern recognition,” in Optical Pattern Recognition, F. T. S. Yu, S. Jutamulia, eds. (Cambridge U. Press, Cambridge, UK, 1998), Chap. 4, pp. 89–125.

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Figures (9)

Fig. 1
Fig. 1

Schematic of the optical setup used for encryption.

Fig. 2
Fig. 2

Fractional Fourier transformers used for encryption.

Fig. 3
Fig. 3

Block diagram of the proposed method for encryption. FRT, fractional Fourier transform.

Fig. 4
Fig. 4

Fractional Fourier transformers used for decryption.

Fig. 5
Fig. 5

Schematic of the optical setup used for decryption.

Fig. 6
Fig. 6

Schematic of an encrypted memory setup. PR, photorefractive.

Fig. 7
Fig. 7

Experimental setup: BS, beam splitter; BE, beam expander; M, mirror; R1, R2, random-phase masks; L1, L2, L3, L4, cylindrical lenses; L5 imaging lens; PRC, photorefractive crystal; O, object.

Fig. 8
Fig. 8

(a)–(c) Images to be encrypted; (d)–(f) encrypted images.

Fig. 9
Fig. 9

(a)–(c) Decrypted images by use of the correct key; (d)–(i) decrypted images by use of a wrong key.

Tables (2)

Tables Icon

Table 1 Physical Parameters (Focal Lengths and Distances) used to Encrypt the Images

Tables Icon

Table 2 AFRT Parameters used to Encrypt the Images

Equations (29)

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fox1, y1= fix, yBx1, y1; x, ydx dy,
Bxo, yo; xi, yi=K expiπ xo2λRaxexpiπsaxxo2Max2cot ϕax-2xoxiMaxcsc ϕax+xi2 cot ϕax×expiπ yo2λRayexpiπsayyo2May2cot ϕay-2yoyiMaycsc ϕay+yi2 cot ϕay,
expiπx2λRax+y2λRayfax, ayxMaxsax, yMaysay,
fax,ayx, y= fx, yexpiπx2tan ϕax+xxsin ϕax+x2tan ϕaxexpiπy2tan ϕay+xxsin ϕay+x2tan ϕaydx dy,
expiπλx2Rax+y2Ray.
tan ϕax=λsax2d1+d2-d1d2/f1x1-d2/f1x;ϕax=axπ2; sax2=λf1x,
tan ϕay=λsay2d1+d2-d1d2/f1y1-d2/f1y;ϕay=ayπ2; say2=λf1y,
Max2=1-d2f1x2+λ2sax4d1+d2-d1d2/f1x2,
May2=1-d2f1y2+λ2say4d1+d2-d1d2/f1y2,
1Rax=λsax2tan ϕaxMax2-1f1x-d2,
1Ray=λsay2tan ϕayMay2-1f1y-d2.
gx1, y1= fx, yR1x, yB1x1, y1; x, ydxdy,
B1x1, y1; x, y=K1 expiπ x12λRaxexpiπsaxx12Max2cot ϕax-2x1xMaxcsc ϕax+x2 cot ϕax×expiπ y12λRayexpiπsayy12May2cot ϕay-2y1yMaycsc ϕay+y2 cot ϕay,
Ψx2, y2= gx1, y1R2x1, y1×B2x2, y2; x1, y1dx1dy1,
B2x2, y2; x1, y1=K2 expiπ x22λRbx×expiπsbxx22Mbx2cot ϕbx-2x2x1Mbx×csc ϕbx+x12 cot ϕbx×expiπ y22λRby×expiπsbyy22Mby2 cot ϕby-2y2y1Mby×csc ϕby+y12 cot ϕby.
Ψx2, y2=  fx, yR1x, yB1x1, y1; x, y×R2x1, y1B2x2, y2; x1, y1dxdydx1dy1.
EΨx2, y2Ψ*x˜2, y˜2= |fx, y|2dxdy×δx2-x˜2, y2-y˜2,
gdx3, y3= Ψ*x2, y2B3x3, y3; x2, y2dx2dy2,
B3x3, y3; x2, y2=K2 expiπ x22λRbxexpiπsbxx32 cot ϕbx-2x3x2Mbxcsc ϕbx+x22Mbx2cot ϕbx×expiπ y22λRbyexpiπsbyy32 cot ϕby-2y3y2Mbycsc ϕby+y22Mby2cot ϕby.
fdx4, y4= gdx3, y3R2x3, y3×B4x4, y4; x3, y3dx3dy3,
B4x4, y4; x3, y3=K4 expiπ x32λRaxexpiπsaxx42 cot ϕax-2x4x3Maxcsc ϕax+x32Max2cot ϕax×expiπ y32λRayexpiπsayy42 cot ϕay-2y4y3Maycsc ϕay+y32May2cot ϕay.
fdx4, y4=f*x4, y4R1*x4, y4.
ER1x, yR1*x, y=δx-x, y-y,
ER2x, yR2*x, y=δx-x, y-y,
Ψx2, y2= fx, yR1x, yB1x1, y1; x, y×R2x1, y1B2x2, y2; x1, y1dx dy dx1dy1,
EΨx2, y2Ψ*x˜2, y˜2=K  fx, yf*x˜, y˜×ER1x, yR1*x, yER2x, yR2*x, y×B1x1, y1; x, yB1*x˜1, y˜1; x˜, y˜×B2x2, y2; x1, y1×B2*x˜2, y˜2; x˜1, y˜1dxdy dx1dy1dx˜dy˜dx˜1dy˜1.
EΨx2, y2Ψ*x˜2, y˜2=K  |fx, y|2×B2x2, y2; x1, y1B2*x˜2, y˜2; x1, y1dxdx1dydy1
× B2x2, y2; x1, y1B2*x˜2, y˜2; x1, y1dx1dy1=Kδx2-x˜2, y2-y˜2,
EΨx2, y2Ψ*x˜2, y˜2= |fx, y|2dxdy.

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