Abstract

The coincidence Fractional Fourier transform (FRT) is implemented with a partially coherent light source experimentally. The visibility and quality of the coincidence FRT pattern of an object are investigated theoretically. The FRT pattern of an object is obtained by measuring the coincidence counting rate between the detected signals passing through two different optical paths. The experimental results are analyzed and found to be consistent with the theoretical results.

©2006 Optical Society of America

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References

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  1. V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980)
    [Crossref]
  2. A. C. McBride and F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. App. Math. 39, 159–175 (1987)
    [Crossref]
  3. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993)
    [Crossref]
  4. D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A 10, 1875–1881 (1993)
    [Crossref]
  5. H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A 10, 2522–2531 (1993)
    [Crossref]
  6. A. W. Lohmann, D. Medlovic, and Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics Vol. XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998).
    [Crossref]
  7. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, New York, 2001).
  8. A. Torre, “The fractional Fourier transform and some of its applications to optics,” in Progress in Optics Vol. XLIII, E. Wolf, ed. (Elsevier, Amsterdam, 2002).
    [Crossref]
  9. D. Mendlovic, Z. Zalevsky, R.G. Dorsch, Y. Bitran, A.W. Lohmann, and H. Ozaktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12, 2424–2431 (1995).
    [Crossref]
  10. S. C. Pei, M.H. Yeh, and T. L. Luo, “Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform,” IEEE Trans. Signal Process. 47, 2883–2888 (1999).
    [Crossref]
  11. B. Zhu, S. Liu, and Q. Ran, “Optical image encryption based on multi-fractional Fourier transforms,” Opt. Lett. 25, 1159–1161 (2000).
    [Crossref]
  12. Y. Zhang, B. Dong, B. Gu, and G. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A 15, 1114–1120 (1998).
    [Crossref]
  13. X. Xue, H.Q. Wei, and A. G. Kirk, “Beam analysis by fractional Fourier transform,” Opt. Lett. 26, 1746–1748 (2001).
    [Crossref]
  14. Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phy. Lett. 86, 021112 (2005).
    [Crossref]
  15. Y. Cai and S. Zhu, “Coincidence fractional Fourier transform with partially coherent light radiation,” J. Opt. Soc. Am. A 22, 1798-1804 (2005)
    [Crossref]
  16. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, New York, 1995)

2005 (2)

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phy. Lett. 86, 021112 (2005).
[Crossref]

Y. Cai and S. Zhu, “Coincidence fractional Fourier transform with partially coherent light radiation,” J. Opt. Soc. Am. A 22, 1798-1804 (2005)
[Crossref]

2001 (1)

2000 (1)

1999 (1)

S. C. Pei, M.H. Yeh, and T. L. Luo, “Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform,” IEEE Trans. Signal Process. 47, 2883–2888 (1999).
[Crossref]

1998 (1)

1995 (1)

1993 (3)

1987 (1)

A. C. McBride and F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. App. Math. 39, 159–175 (1987)
[Crossref]

1980 (1)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980)
[Crossref]

Bitran, Y.

Cai, Y.

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phy. Lett. 86, 021112 (2005).
[Crossref]

Y. Cai and S. Zhu, “Coincidence fractional Fourier transform with partially coherent light radiation,” J. Opt. Soc. Am. A 22, 1798-1804 (2005)
[Crossref]

Dong, B.

Dorsch, R.G.

Gu, B.

Kerr, F. H.

A. C. McBride and F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. App. Math. 39, 159–175 (1987)
[Crossref]

Kirk, A. G.

Kutay, M. A.

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

Lin, Q.

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phy. Lett. 86, 021112 (2005).
[Crossref]

Liu, S.

Lohmann, A. W.

A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993)
[Crossref]

A. W. Lohmann, D. Medlovic, and Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics Vol. XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998).
[Crossref]

Lohmann, A.W.

Luo, T. L.

S. C. Pei, M.H. Yeh, and T. L. Luo, “Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform,” IEEE Trans. Signal Process. 47, 2883–2888 (1999).
[Crossref]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, New York, 1995)

McBride, A. C.

A. C. McBride and F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. App. Math. 39, 159–175 (1987)
[Crossref]

Medlovic, D.

A. W. Lohmann, D. Medlovic, and Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics Vol. XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998).
[Crossref]

Mendlovic, D.

Namias, V.

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980)
[Crossref]

Ozaktas, H.

Ozaktas, H. M.

Pei, S. C.

S. C. Pei, M.H. Yeh, and T. L. Luo, “Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform,” IEEE Trans. Signal Process. 47, 2883–2888 (1999).
[Crossref]

Ran, Q.

Torre, A.

A. Torre, “The fractional Fourier transform and some of its applications to optics,” in Progress in Optics Vol. XLIII, E. Wolf, ed. (Elsevier, Amsterdam, 2002).
[Crossref]

Wei, H.Q.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, New York, 1995)

Xue, X.

Yang, G.

Yeh, M.H.

S. C. Pei, M.H. Yeh, and T. L. Luo, “Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform,” IEEE Trans. Signal Process. 47, 2883–2888 (1999).
[Crossref]

Zalevsky, Z.

D. Mendlovic, Z. Zalevsky, R.G. Dorsch, Y. Bitran, A.W. Lohmann, and H. Ozaktas, “New signal representation based on the fractional Fourier transform: definitions,” J. Opt. Soc. Am. A 12, 2424–2431 (1995).
[Crossref]

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

A. W. Lohmann, D. Medlovic, and Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics Vol. XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998).
[Crossref]

Zhang, Y.

Zhu, B.

Zhu, S.

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phy. Lett. 86, 021112 (2005).
[Crossref]

Y. Cai and S. Zhu, “Coincidence fractional Fourier transform with partially coherent light radiation,” J. Opt. Soc. Am. A 22, 1798-1804 (2005)
[Crossref]

Appl. Phy. Lett. (1)

Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phy. Lett. 86, 021112 (2005).
[Crossref]

IEEE Trans. Signal Process. (1)

S. C. Pei, M.H. Yeh, and T. L. Luo, “Fractional Fourier series expansion for finite signals and dual extension to discrete-time fractional Fourier transform,” IEEE Trans. Signal Process. 47, 2883–2888 (1999).
[Crossref]

IMA J. App. Math. (1)

A. C. McBride and F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. App. Math. 39, 159–175 (1987)
[Crossref]

J. Inst. Math. Its Appl. (1)

V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Its Appl. 25, 241–265 (1980)
[Crossref]

J. Opt. Soc. Am. A (6)

Opt. Lett. (2)

Other (4)

A. W. Lohmann, D. Medlovic, and Z. Zalevsky, “Fractional transformations in optics,” in Progress in Optics Vol. XXXVIII, E. Wolf, ed. (Elsevier, Amsterdam, 1998).
[Crossref]

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

A. Torre, “The fractional Fourier transform and some of its applications to optics,” in Progress in Optics Vol. XLIII, E. Wolf, ed. (Elsevier, Amsterdam, 2002).
[Crossref]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, New York, 1995)

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Figures (6)

Fig. 1.
Fig. 1. Experimental setup for realizing a coincidence FRT with a partially coherent light.

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Fig. 2.
Fig. 2. Dependence of the deviation factor and visibility of the coincidence FRT pattern for an object of double slits on the transverse coherence width of the partially coherent light beam.

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Fig. 3.
Fig. 3. Experimental setup for measuring the coherence width of a partially coherent light source

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Fig. 4.
Fig. 4. Square of the spectral degree of coherence (along x 1 - x 2) for a partially coherent light source used in our experiment

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Fig. 5.
Fig. 5. Experimental results of the coincidence FRT pattern for different fractional order p for an object of double slits with a partially coherent beam (σg = 15μm).

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Fig. 6.
Fig. 6. Experimental results of the coincidence FRT pattern (with p=1) for an object of double slits with partially coherent beams of different coherence width σg .

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Equations (11)

Equations on this page are rendered with MathJax. Learn more.

G ( 2 ) u 1 u 2 = 〈 E ( u 1 ) E ( u 2 ) E * ( u 2 ) E * ( u 1 ) 〉 = < I ( u 1 ) > < I ( u 2 ) > + ∣ Γ u 1 u 2 ∣ 2 ,
< I ( u i ) > = ∫ − ∞ ∞ ∫ − ∞ ∞ h i x 1 u i h i * x 2 u i 〈 E s ( x 1 ) E s * ( x 2 ) 〉 d x 1 d x 2 i = 1,2 ,
Γ u 1 u 2 = ∫ − ∞ ∞ ∫ − ∞ ∞ 〈 E s ( x 1 ) E s * ( x 2 ) 〉 h 1 x 1 u 1 h 2 * x 2 u 2 d x 1 d x 2 ,
〈 E s ( x 1 ) E s * ( x 2 ) 〉 = I 0 δ ( x 1 − x 2 ) .
< I ( u 1 = 0 ) > < I ( u 2 ) > = ∞ ,
Γ ( u 1 = 0 , u 2 ) = I 0 λ 2 f 1 f e sin ⁡ ϕ ∫ − ∞ ∞ H ( v 1 ) exp [ iπ λ f e tan ⁡ ϕ ( v 1 2 + v 2 2 ) − 2 iπ λ f e sin ⁡ ϕ v 1 v 2 ] d v 1 ,
l 1 = f e tan ⁡ ϕ 2 + z , I 2 = f e tan ⁡ ϕ 2 , f = f e sin ⁡ ϕ ,
〈 E s ( x 1 ) E s * ( x 2 ) 〉 = Γ x 1 x 2 = I ( x 1 ) I ( x 2 ) g ( x 1 − x 2 ) = exp [ − ( x 1 2 + x 2 2 ) 4 σ I 2 − ( x 1 − x 2 ) 2 2 σ g 2 ] ,
D = [ ∫ − ∞ ∞ ∣ ∣ Γ ( u 1 = 0 , u 2 ) ∣ 2 ∣ Γ ( u 1 = 0 , u 2 ) ∣ max 2 − ∣ Γ ( u 1 = 0 , u 2 ) σ g = 0 , σ I = ∞ ∣ 2 ∣ Γ ( u 1 = 0 , u 2 ) σ g = 0 , σ I = ∞ ∣ max 2 ∣ d u 2 ] ∫ − ∞ ∞ ∣ Γ ( u 1 = 0 , u 2 ) σ g = 0 , σ I = ∞ ∣ 2 ∣ Γ ( u 1 = 0 , u 2 ) σ g = 0 , σ I = ∞ ∣ max 2 d u 2 .
g ( x 1 − x 2 ) = Γ x 1 x 2 I ( x 1 ) I ( x 2 ) = exp [ − ( x 1 − x 2 ) 2 2 σ g 2 ] .
g 2 ( x 1 − x 2 ) = exp [ − ( x 1 − x 2 ) 2 σ g 2 ] = G ( 2 ) x 1 x 2 < I ( x 1 ) > < I ( x 2 ) > − 1 .

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