Abstract

We present an optical encryption method for handling time-varying information by means of properly designing a four-dimensional mutual intensity function distribution. We present the theory and validate the basic concept with numerical simulations and experimental results.

© 2004 Optical Society of America

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References

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  1. H. Delfs, H. Knebl, Introduction to Cryptography (Springer-Verlag, Berlin, 2002), pp. 1–56.
    [CrossRef]
  2. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), pp. 491–528.
  3. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 170–226.
  4. D. Mendlovic, G. Shabtay, A. W. Lohmann, “Synthesis of spatial coherence,” Opt. Lett. 24, 361–363 (1999).
    [CrossRef]
  5. M. Fatih, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
    [CrossRef]
  6. Z. Zalevsky, D. Mendlovic, H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. A: Pure Appl. Opt. 2, 83–87 (2000).
    [CrossRef]
  7. Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
    [CrossRef] [PubMed]
  8. M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), pp. 265–266.
  9. S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [CrossRef] [PubMed]
  10. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGray-Hill, New York, 1996), pp. 101–119.

2000 (1)

Z. Zalevsky, D. Mendlovic, H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. A: Pure Appl. Opt. 2, 83–87 (2000).
[CrossRef]

1999 (1)

1996 (2)

M. Fatih, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef] [PubMed]

1983 (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Born, M.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), pp. 491–528.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), pp. 265–266.

Delfs, H.

H. Delfs, H. Knebl, Introduction to Cryptography (Springer-Verlag, Berlin, 2002), pp. 1–56.
[CrossRef]

Dorsch, R. G.

Fatih, M.

M. Fatih, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGray-Hill, New York, 1996), pp. 101–119.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 170–226.

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Knebl, H.

H. Delfs, H. Knebl, Introduction to Cryptography (Springer-Verlag, Berlin, 2002), pp. 1–56.
[CrossRef]

Lohmann, A. W.

Mendlovic, D.

Z. Zalevsky, D. Mendlovic, H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. A: Pure Appl. Opt. 2, 83–87 (2000).
[CrossRef]

D. Mendlovic, G. Shabtay, A. W. Lohmann, “Synthesis of spatial coherence,” Opt. Lett. 24, 361–363 (1999).
[CrossRef]

M. Fatih, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef] [PubMed]

Ozaktas, H. M.

Z. Zalevsky, D. Mendlovic, H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. A: Pure Appl. Opt. 2, 83–87 (2000).
[CrossRef]

M. Fatih, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

Shabtay, G.

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), pp. 491–528.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), pp. 265–266.

Zalevsky, Z.

Z. Zalevsky, D. Mendlovic, H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. A: Pure Appl. Opt. 2, 83–87 (2000).
[CrossRef]

Z. Zalevsky, D. Mendlovic, R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842–844 (1996).
[CrossRef] [PubMed]

J. Opt. A: Pure Appl. Opt. (1)

Z. Zalevsky, D. Mendlovic, H. M. Ozaktas, “Energetic efficient synthesis of general mutual intensity distribution,” J. Opt. A: Pure Appl. Opt. 2, 83–87 (2000).
[CrossRef]

Opt. Commun. (1)

M. Fatih, H. M. Ozaktas, D. Mendlovic, “Synthesis of mutual intensity distributions using the fractional Fourier transform,” Opt. Commun. 125, 288–301 (1996).
[CrossRef]

Opt. Lett. (2)

Science (1)

S. Kirkpatrick, C. D. Gelatt, M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[CrossRef] [PubMed]

Other (5)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGray-Hill, New York, 1996), pp. 101–119.

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), pp. 265–266.

H. Delfs, H. Knebl, Introduction to Cryptography (Springer-Verlag, Berlin, 2002), pp. 1–56.
[CrossRef]

M. Born, E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), pp. 491–528.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), pp. 170–226.

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Figures (12)

Fig. 1
Fig. 1

One-dimensional system.

Fig. 2
Fig. 2

(a) Opaque-transparent mask M(x, y) of the first setup; (b) phase-modulated mask of the first setup.

Fig. 3
Fig. 3

Basic setup of the 2-D system that provides at its output two correlated points.

Fig. 4
Fig. 4

Overall 2-D system.

Fig. 5
Fig. 5

Input MIF of 32 × 32 pixels used in first simulations and in the experiment.

Fig. 6
Fig. 6

(a) Desired mutual intensity; (b) output MIF from using FRT filtering and analytical design of the phase filter; (c) output MIF FT filtering and analytical design of the phase filter; (d) output MIF from using FT filtering and the iterative design of a phase filter.

Fig. 7
Fig. 7

(a) Input MIF of simulations from using an SA algorithm; (b) desired MIF of simulations from using an SA algorithm (top view); (c) obtained output MIF from using an initial filter calculated by the calculus of the variations method (top view); (d) output intensity distribution from using an initial filter calculated by the calculus of the variations method.

Fig. 8
Fig. 8

(a) Output MIF obtained from using the SA algorithm (top view); (b) output intensity distribution from using the SA algorithm.

Fig. 9
Fig. 9

(a) Top view of the standard deviation graph P(k, l); (b) diagonal section of P(k, l) (in radians) crossing indices (40, 90) and (90, 40).

Fig. 10
Fig. 10

Setup used in the experiment. The Fourier transform was used instead of the FRT.

Fig. 11
Fig. 11

Experimental results: (a) fringe pattern when correlated points in the output plane interfere and a diffuser is rotating; (b) cross section of the spectrum of (a); (c) fringe pattern when noncorrelated points in the output plane interfere and a diffuser is rotating; (d) cross section of the spectrum of (c); (e) fringe pattern when the diffuser is still; (f) cross section of the spectrum of (e).

Fig. 12
Fig. 12

Schematic illustration of the MIF through a FRT-transforming system.

Equations (34)

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Γ12τ=uP1, t+τu*P2, t,
Γ12τ=Γ120exp-2πiυ¯τJ12 exp-2πiυ¯τ,
J12outx1, x2= J12intp1, p2Kx1; p1×K*x2; p2dp1dp2,
J12x1, x2=1x1=x21x1=η1, x2=η21x1=η2, x2=η10otherwise,
up, q, t= ukp, q, t= ukp, qexpi2πν¯texpiφkt= Akrectp-pk/εprectqL×expi2πν¯texpiφkt Akδp-pkrectqL×expi2πν¯texpiφkt,
Ux, y, t=A1 exp-i 2πλ¯f xp1sin cyLλ¯f×expi2πν¯texpiϕ1t+A2 exp-i 2πλ¯f xp2sin cyLλ¯f×expi2πν¯texpiφ2t.
Dx, y, t=|Ux, y, t|2=|A1|2+|A2|2+2 ReA1A2 exp2πλ¯f xp2-p1expiφ1t-φ2tsin c2yLλ¯f=|A1|2+|A2|2+2A1A2 cosφ1t-φ2t+2πλ¯f xp2-p1sin c2yLλ¯f.
L=λ¯f|p2-p1|,
PSDt=L/4πsinφ1t-φ2t.
J12x1, x2=1x1=x21x1=η1, x2=η21x1=η2, x2=η10otherwise,
ũpx=- ux¯Bpx, x¯dx¯,Bpx, x¯=exp-iπϕˆ/4-ϕ/2|sinϕ|1/2×expiπx2+x¯2cotϕ-2xx¯sinϕ,
J12inputx1, x2=1iλ¯zexpiπλ¯zx12+y12-x22+y22Source Iξ, η×exp-2πiλ¯zx1-x2ξ+y1-y2ηdξdη =1iλ¯zexpiπλ¯zx12+y12-x22+y22×Iξ, η|fx=x1-x2/λ¯z, fy=y1-y2/λ¯z,
Hx, y=expiωx, y,
 |J12outputx1; x2-J12desiredx1; x2|2dx1dx2.
F=σphase difference+a×σIntensity,σphase difference=stdΔφn,σIntensity=stdIm,Im=n |uoutm; n|2,
Dx, y, t=|A1|2+|A2|2+2A1A2×cosφ1t-φ2t+2πλ¯f xp2-p1.
Jinputx1, x2=12πσ21/2exp-x1-x222σ2,
Jinputx1, x2=exp-iπλ¯dx12-x22×exp-x1-x222σ2,
Pk, l=stdΔφk,ln,
uoutx, y=1jλ¯f  uinη, ξ×exp-2πiλ¯fxη+yξdηdξ,
rA, B=1NmnAmn-A¯Bmn-B¯,N=mnAmn-A¯2×mnBmn-B¯21/2,
PSDt=1N-L/2L/2 xA21+cosφ1t-φ2t+2πL xsinc2yLλ¯fdx,N=-L/2L/2 A21+cosφ1t-φ2t+2πL xsinc2yLλ¯fdx.
PSDφ1t-φ2t=L/4πsinπ+[φ1t-φ2t.
M=-- |J1dx1, x2-J1x1, x2|2dx1dx2.
M=-- |J1dx1, x2-J1x1, x2|2dx1dx2.
J1x1, x2=-- B-p2x1, x1Bp2x2, x2×J1x, x2dxdx2,J0x1, x2=-- Bp1x1, x1B-p1x2, x2×J0x1, x2dx1dx2
J1x1, x2=J0x1, x2Hx1H*x2.
M=--|J1dx1, x2|2+|J0x1, x2|2-2 ReJ1d*x1, x2Hx1H*x2×J0x1, x2dx1dx2.
ωxωx+δωx,
expiωx+δωxexpiωx1+iδωx,
εM=2 - δωx2--ReJ1d*x2, x1J0x2, x1+J1d*x1, x2J0x1, x2sinωx1-ωx2+ImJ1d*x2, x1J0x2, x1-J1d*x1, x2×J0x1, x2cosωx1-ωx2dx1dx2.
- GRx1, x2sinωx1-ωx2+G1x1, x2cosωx1-ωx2dx1=0,
GRx1, x2=-ReJ1d*x2, x1J0x2, x1+J1d*x1, x2J0x1, x2,G1x1, x2=ImJ1d*x2, x1J0x2, x1-J1d*x1, x2J0x1, x2.
ωx2=tan-1- GRx1, x2sinωx1+G1x1, x2cosωx1dx1×- GRx1, x2cosωx1-G1x1, x2sinωx1dx1-1.

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