Abstract

A lensless optical system for implementing the coincidence fractional Fourier transform (FRT) is proposed. The conditions for the lensless optical system to implement the coincidence FRT with incoherent light and entangled photon pairs are discussed. The results offer a novel scheme for FRTs and thus suggest useful applications.

© 2006 Optical Society of America

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References

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  1. D. Mendlovic and H. M. Ozaktas, J. Opt. Soc. Am. A 10, 1875 (1993).
    [CrossRef]
  2. H. M. Ozaktas and D. Mendlovic, J. Opt. Soc. Am. A 10, 2522 (1993).
    [CrossRef]
  3. A. W. Lohmann, J. Opt. Soc. Am. A 10, 2181 (1993).
    [CrossRef]
  4. C. Candan, M. A. Kutay, and H. M. Ozaktas, IEEE Trans. Signal Process. 48, 1329 (2000).
    [CrossRef]
  5. A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, Opt. Lett. 21, 281 (1996).
    [CrossRef] [PubMed]
  6. T. Alieva, M. J. Bastiaans, and M. L. Calvo, EURASIP J. Appl. Signal Process 2005, 1498 (2005).
    [CrossRef]
  7. A. W. Lohmann, D. Medlovic, and Z. Zalevsky, in Progress in Optics Vol. XXXVIII, E.Wolf, ed. (Elsevier, 1998).
  8. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
  9. Y. Cai and S. Zhu, Appl. Phys. Lett. 86, 021112 (2005).
    [CrossRef]
  10. Y. Cai and S. Zhu, J. Opt. Soc. Am. A 22, 1798 (2005).
    [CrossRef]
  11. Y. Cai, Q. Lin, and S. Zhu, J. Opt. Soc. Am. A 23, 835 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
  13. J. Hua, L. Liu, and G. Li, Appl. Opt. 36, 512 (1997).
    [CrossRef] [PubMed]
  14. H. Hwang, Opt. Commun. 223, 47 (2003).
    [CrossRef]
  15. V. B. Gavrikov, V. P. Likhachev, J. D. T. Arruda-Neto, and A. L. Bonini, Phys. Rev. A 65, 022903 (2002).
    [CrossRef]

2006 (1)

2005 (3)

T. Alieva, M. J. Bastiaans, and M. L. Calvo, EURASIP J. Appl. Signal Process 2005, 1498 (2005).
[CrossRef]

Y. Cai and S. Zhu, Appl. Phys. Lett. 86, 021112 (2005).
[CrossRef]

Y. Cai and S. Zhu, J. Opt. Soc. Am. A 22, 1798 (2005).
[CrossRef]

2003 (1)

H. Hwang, Opt. Commun. 223, 47 (2003).
[CrossRef]

2002 (1)

V. B. Gavrikov, V. P. Likhachev, J. D. T. Arruda-Neto, and A. L. Bonini, Phys. Rev. A 65, 022903 (2002).
[CrossRef]

2000 (1)

C. Candan, M. A. Kutay, and H. M. Ozaktas, IEEE Trans. Signal Process. 48, 1329 (2000).
[CrossRef]

1997 (1)

1996 (1)

1995 (1)

1993 (3)

Alieva, T.

T. Alieva, M. J. Bastiaans, and M. L. Calvo, EURASIP J. Appl. Signal Process 2005, 1498 (2005).
[CrossRef]

Arruda-Neto, J. D. T.

V. B. Gavrikov, V. P. Likhachev, J. D. T. Arruda-Neto, and A. L. Bonini, Phys. Rev. A 65, 022903 (2002).
[CrossRef]

Bastiaans, M. J.

T. Alieva, M. J. Bastiaans, and M. L. Calvo, EURASIP J. Appl. Signal Process 2005, 1498 (2005).
[CrossRef]

Bonini, A. L.

V. B. Gavrikov, V. P. Likhachev, J. D. T. Arruda-Neto, and A. L. Bonini, Phys. Rev. A 65, 022903 (2002).
[CrossRef]

Cai, Y.

Calvo, M. L.

T. Alieva, M. J. Bastiaans, and M. L. Calvo, EURASIP J. Appl. Signal Process 2005, 1498 (2005).
[CrossRef]

Candan, C.

C. Candan, M. A. Kutay, and H. M. Ozaktas, IEEE Trans. Signal Process. 48, 1329 (2000).
[CrossRef]

Dorsch, R. G.

Gavrikov, V. B.

V. B. Gavrikov, V. P. Likhachev, J. D. T. Arruda-Neto, and A. L. Bonini, Phys. Rev. A 65, 022903 (2002).
[CrossRef]

Hua, J.

Hwang, H.

H. Hwang, Opt. Commun. 223, 47 (2003).
[CrossRef]

Konforti, N.

Kutay, M. A.

C. Candan, M. A. Kutay, and H. M. Ozaktas, IEEE Trans. Signal Process. 48, 1329 (2000).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Li, G.

Likhachev, V. P.

V. B. Gavrikov, V. P. Likhachev, J. D. T. Arruda-Neto, and A. L. Bonini, Phys. Rev. A 65, 022903 (2002).
[CrossRef]

Lin, Q.

Liu, L.

Lohmann, A. W.

Medlovic, D.

A. W. Lohmann, D. Medlovic, and Z. Zalevsky, in Progress in Optics Vol. XXXVIII, E.Wolf, ed. (Elsevier, 1998).

Mendlovic, D.

Ozaktas, H. M.

C. Candan, M. A. Kutay, and H. M. Ozaktas, IEEE Trans. Signal Process. 48, 1329 (2000).
[CrossRef]

D. Mendlovic and H. M. Ozaktas, J. Opt. Soc. Am. A 10, 1875 (1993).
[CrossRef]

H. M. Ozaktas and D. Mendlovic, J. Opt. Soc. Am. A 10, 2522 (1993).
[CrossRef]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

Zalevsky, Z.

A. W. Lohmann, D. Mendlovic, and Z. Zalevsky, Opt. Lett. 21, 281 (1996).
[CrossRef] [PubMed]

D. Mendlovic, Z. Zalevsky, N. Konforti, R. G. Dorsch, and A. W. Lohmann, Appl. Opt. 34, 7615 (1995).
[CrossRef] [PubMed]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

A. W. Lohmann, D. Medlovic, and Z. Zalevsky, in Progress in Optics Vol. XXXVIII, E.Wolf, ed. (Elsevier, 1998).

Zhu, S.

Appl. Opt. (2)

Appl. Phys. Lett. (1)

Y. Cai and S. Zhu, Appl. Phys. Lett. 86, 021112 (2005).
[CrossRef]

EURASIP J. Appl. Signal Process (1)

T. Alieva, M. J. Bastiaans, and M. L. Calvo, EURASIP J. Appl. Signal Process 2005, 1498 (2005).
[CrossRef]

IEEE Trans. Signal Process. (1)

C. Candan, M. A. Kutay, and H. M. Ozaktas, IEEE Trans. Signal Process. 48, 1329 (2000).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (1)

H. Hwang, Opt. Commun. 223, 47 (2003).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

V. B. Gavrikov, V. P. Likhachev, J. D. T. Arruda-Neto, and A. L. Bonini, Phys. Rev. A 65, 022903 (2002).
[CrossRef]

Other (2)

A. W. Lohmann, D. Medlovic, and Z. Zalevsky, in Progress in Optics Vol. XXXVIII, E.Wolf, ed. (Elsevier, 1998).

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

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Figures (2)

Fig. 1
Fig. 1

Lensless optical system for implementing the coincidence FRT with incoherent light.

Fig. 2
Fig. 2

Normalized FRT pattern of a slit with slit width h = 0.02 mm for various values of the incoherent source’s transverse size σ I .

Equations (15)

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G ( 2 ) ( u 1 , u 2 ) = E ( u 1 ) E ( u 2 ) E * ( u 2 ) E * ( u 1 ) = h 1 ( u 1 , x 1 ) h 2 ( u 2 , x 2 ) h 2 * ( u 2 , x 3 ) h 1 * ( u 1 , x 4 ) × E ( x 1 ) E ( x 2 ) E * ( x 3 ) E * ( x 4 ) d x 1 d x 2 d x 3 d x 4 = I ( u 1 ) I ( u 2 ) + g 2 ( u 1 , u 2 ) ,
I ( u i ) = h i ( u i , x 1 ) h i * ( u i , x 2 ) E ( x 1 ) E * ( x 2 ) d x 1 d x 2 , i = 1 , 2 ,
g 2 ( u 1 , u 2 ) = Γ ( u 1 , u 2 ) 2 = E ( x 1 ) E * ( x 2 ) h 1 ( u 1 , x 1 ) h 2 * ( u 2 , x 2 ) d x 1 d x 2 2 .
h 1 ( u 1 , x 1 ) = ( 1 λ 2 z 1 z 2 ) 1 2 exp [ i π λ z 1 ( x 1 ν 1 ) 2 i π λ z 2 ( ν 1 u 1 ) 2 ] H ( ν 1 ) d ν 1 ,
h 2 ( u 2 , x 2 ) = ( i λ z ) 1 2 exp [ i π λ z ( x 2 u 2 ) 2 ] .
Γ ( x 1 , x 2 ) = E ( x 1 ) E * ( x 2 ) = exp [ ( x 1 2 + x 2 2 ) 4 σ I 2 ] δ ( x 1 x 2 ) ,
I ( u 1 ) = 2 σ I 2 π λ 2 z 1 z 2 H ( ν 1 ) H * ( ν 2 ) exp [ 2 σ I 2 π λ 2 z 1 2 ( ν 1 ν 2 ) 2 i π λ ( 1 z 1 + 1 z 2 ) ( ν 1 2 ν 2 2 ) + 2 i π λ z 2 ( ν 1 ν 2 ) u 1 ] d ν 1 d ν 2 ,
g 2 ( u 1 , u 2 ) = π λ 3 z 1 z 2 z A 1 H ( ν 1 ) exp [ π 2 A 1 λ 2 ( ν 1 z 1 u 2 z ) 2 i π ν 1 2 λ z 1 + i π u 2 2 λ z i π λ z 2 ( ν 1 u 1 ) 2 ] d ν 1 2 ,
z z 1 = f e tan ϕ , u 2 = u 2 e cos ϕ
g 2 ( u 2 e cos ϕ ) = g 2 ( u 1 , u 2 e cos ϕ ) d u 1 = 1 λ 2 z 1 z H ( ν 1 ) exp [ i π λ f e tan ϕ ( ν 1 2 + u 2 e 2 ) 2 i π λ f e sin ϕ ν 1 u 2 e ] d ν 1 2 .
G ( 2 ) ( u 1 , u 2 ) = Ψ E ( u 1 ) E ( u 2 ) E * ( u 2 ) E * ( u 1 ) Ψ = 0 , 0 E ( x 1 ) E ( x 2 ) Ψ 2 = ψ ( u 1 , u 2 ) 2 ,
ψ ( u 1 , u 2 ) = E p ( x ) h 1 ( u 1 , x ) h 2 ( u 2 , x ) d x .
G ( 2 ) ( u 1 , u 2 ) = π λ 3 z 1 z 2 z A 1 H ( ν ) exp [ π 2 λ 2 A 1 ( ν z 1 + u 2 z ) 2 i π λ z 1 ν 2 i π λ z u 2 2 i π λ z ( ν u 1 ) 2 ] d ν 2 ,
z + z 1 = f e tan ϕ , u 2 = u 2 e cos ϕ
G ( 2 ) ( u 2 e cos ϕ ) = G ( 2 ) ( u 1 , u 2 e cos ϕ ) d u 1 = 1 λ 2 z 1 z H ( ν 1 ) exp [ i π λ f e tan ϕ ( ν 1 2 + u 2 e 2 ) + 2 i π λ f e sin ϕ ν 1 u 2 e ] d ν 1 2 .

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