## Experimental observation of coincidence fractional Fourier transform with a partially coherent beam

Optics Express, Vol. 14, Issue 16, pp. 6999-7004 (2006)

http://dx.doi.org/10.1364/OE.14.006999

Acrobat PDF (532 KB)

### 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

## 1. Introduction

^{1}

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

^{2}

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

^{3}–

^{5}

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

^{6}–

^{8}

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]

^{9}

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}

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}

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}

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}

13. X. Xue, H.Q. Wei, and A. G. Kirk, “Beam analysis by fractional Fourier transform,” Opt. Lett. **26**, 1746–1748 (2001). [CrossRef]

^{14}

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}

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]

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]

## 2. Experimental setup

*d*

_{0}≈ 10

*mm*) generated by a He-Ne laser (

*λ*= 632.8

*nm*) is used to illuminate a rotating ground-glass disk, and the transmitted light can be considered as a partially coherent light source. By controlling the rotating speed of the ground-glass disk, we can control the coherence of the transmitted light. After passing through the ground-glass disk, the light is split by a 50:50 beam splitter (BS) into two distinct optical paths. The transmitted beam going through path 1 (see Fig. 1) will arrive at single-photon detector D

_{1}located at

*u*

_{1}= 0 . Along this path, an object (double slits with slit distance

*d*= 195

*μm*and slit width

*a*= 85

*μm*) is placed between the beam splitter and D

_{1}, and a lens with a focal length of

*f*

_{1}is placed between the object and D

_{1}, and both the distances from the object to the lens and from the lens to D

_{1}are

*f*

_{1}. The reflected beam going through path 2 (see Fig. 1) arrives at single-photon detector D

_{2}connected with a single mode optical fiber whose tip (as a fiber probe) is scanning on the transverse plane. Along this path, a lens with a focal length of

*f*is placed between the beam splitter and the scanning fiber tip, and the distances from the light source to the lens and from the lens to the scanning fiber tip are

*l*

_{1}and

*l*

_{2}, respectively. Finally, the output signals from the two single-photon counting detectors are sent to an electronic coincidence circuit to measure the coincident counting rate.

## 3. Theoretical analysis

_{1}and D

_{2}

^{14}–

^{16}

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]

*h*

_{1}(

*x*

_{1},

*H*

_{1}) and

*h*

_{2}(

*x*

_{2},

*H*

_{2}) are the response functions associated with the two optical paths. <

*I*(

*u*

_{i}) > is the second order correlation function (i.e., the intensity at the

*i*-th detector) at point

*u*

_{i}, and depends only on the

*i*-th optical path,

*i*=1, 2. Γ(

*u*

_{1},

*u*

_{2}) is the second order cross correlation function at two different detecting points, and ⟨

*E*

_{s}(

*x*

_{1})

*x*

_{2})⟩ is the second order cross correlation in the source plane. Eqs. (2) and (3) are valid for an optical system with the invariance of linear translation in the paraxial regime.

*I*

_{0}can be expressed as:

*h*

_{1}[

*x*

_{1},

*u*

_{1}) and

*h*

_{2}(

*x*

_{2},

*u*

_{2}) into Eqs. (1)–(3), we obtain

^{14}

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}

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]

*H*(

*v*) is the transmission function of the object. For an object of double slits,

*H*(

*v*) is 1 for -

*d*/2-

*a*/2<

*v*<-

*d*/2 +

*a*/2 or

*d*/2-

*a*/2<v<

*d*/2 +

*a*/2, and is 0 elsewhere. Here we have assumed that14–15

**86**, 021112 (2005). [CrossRef]

*f*

_{e}is called the “standard focal length”,

^{3}

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

*pπ*/2 with

*p*the desired fractional order of the FRT. Eq. (6) gives the FRT expression of the object.14–15

**86**, 021112 (2005). [CrossRef]

*f*

_{e}and

*p*. The visibility of the coincidence FRT pattern is defined as

*V*= ∣Γ(

*u*

_{1}=0,

*u*

_{2})

*G*

^{(2)}(

*u*

_{1}=0,

*u*

_{2}). Since both <

*I*(

*u*

_{1}= 0) ><

*I*(

*u*

_{2})> and Γ(

*u*

_{1}=0,

*u*

_{2}) contribute to the coincident counting rate, the visibility of the FRT pattern of an object is practically zero for an incoherent light due to <

*I*(

*u*

_{1}= 0) ><

*I*(

*u*

_{2}) >= ∞ (cf. Eq. (5)).

^{15}

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}

*g*(

*x*

_{1}-

*x*

_{2}) = exp[-(

*x*

_{1}-

*x*

_{2})

^{2}/2

*σ*

_{1}and

*σ*

_{g}are transverse spot size and transverse coherence width of the beam, respectively. Note that smaller

*σ*

_{g}corresponds to lower coherence of the beam. By substituting Eq. (8) and expressions for

*h*

_{1}(

*x*

_{1},

*u*

_{1}) and

*h*

_{2}(

*x*

_{2},

*u*

_{2}) into Eqs. (1)–(3) and applying Eq. (7), we can numerically calculate the coincidence FRT pattern of the object.

^{15}

**22**, 1798-1804 (2005) [CrossRef]

**22**, 1798-1804 (2005) [CrossRef]

^{15}

**22**, 1798-1804 (2005) [CrossRef]

*σ*

_{g}= 0 , σ

_{1}= ∞)

*D*corresponds to a low deviation (i.e., better quality of the FRT pattern). We calculate and show in Fig. 2 the dependence of the deviation factor and visibility of the coincidence FRT pattern for an object of double slits on the transverse coherence width of a partially coherent source. Here we choose

*f*

_{e}= 63cm,

*z*= 15cm,

*f*

_{1}= 25cm,

*a*= 85

*μm*,

*d*= 195

*μm*,

*λ*= 632.8

*μm*and

*σ*

_{I}= 2

*mm*. One sees from Fig. 2 that we can observe a coincidence FRT pattern with good visibility and reasonably good quality (i.e., smaller D) by choosing an appropriate non-zero value for the coherence width of the light source [too large

*σ*

_{g}is not good as the deviation (error) will be too large]. Our numerical results have also shown (not presented to save space) that the values of D and V also depend on the order p of the coincidence FRT. However, the dependence of D or V on the order p is much smaller than that on parameter σ

*(see e.g. Figs. 5 and 6).*

_{g}## 4. Experimental results

*σ*

_{g}of a partially coherent light beam. Here we use an experimental setup shown in Fig. 3 to measure

*σ*

_{g}of the partially coherent light beam used for our FRT experiment. From Eq. (8), we can find the following spectral degree of coherence

*σ*

_{g}of the partial coherent light beam can be obtained by measuring

*G*

^{(2)}(

*x*

_{1},

*x*

_{2}<)and <

*I*(

*x*

_{1})

*I*(

*x*

_{2}) of the beam in the setup of Fig. 3. Here we used a 2f-imaging system. The intensity distribution of the partially coherent beam is measured directly by a CCD. Figure 4 shows the square of the spectral degree of coherence for a partially coherent beam used in our FRT experiment. The width of function

*g*

^{2}(

*x*

_{1}-

*x*

_{2}) shown in Fig. 4 gives coherence width

*σ*

_{g}(about 15

*μm*here).

*p*for an object of double slits with a partially coherent beam (

*σ*

_{g}= 15

*μm*). The other parameters used in our experiment are

*f*

_{e}= 63cm,

*z*= 15cm,

*f*

_{1}= 25

*cm*. The visibility (mainly determined by

*σ*

_{g}for Fig. 5 (a)–(c) is nearly the same (about 0.069). Fig. 6 shows the experimental results of the coincidence FRT pattern (with fractional order

*p*=1) for an object of double slits with partially coherent beams of different

*σ*

_{g}values (by changing the rotating speed of the ground glass). The visibilities for Fig. 6 (a)–(c) are 0.069, 0.15 and 0.31, respectively. Figure 6 clearly shows that as the beam’s coherence increases the coincidence FRT pattern disappears gradually whereas the visibility of the coincidence FRT pattern increases. For comparison, the corresponding theoretical results (calculated for partial coherent beams

^{15}

**22**, 1798-1804 (2005) [CrossRef]

## 5. Conclusion

^{14}–

^{15}

**86**, 021112 (2005). [CrossRef]

## Acknowledgment

## References and links

1. | V. Namias, “The fractional Fourier transform and its application in quantum mechanics,” J. Inst. Math. Its Appl. |

2. | A. C. McBride and F. H. Kerr, “On Namia’s fractional Fourier transforms,” IMA J. App. Math. |

3. | A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A |

4. | D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” J. Opt. Soc. Am. A |

5. | H. M. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation: II,” J. Opt. Soc. Am. A |

6. | A. W. Lohmann, D. Medlovic, and Z. Zalevsky, “Fractional transformations in optics,” in |

7. | H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, |

8. | A. Torre, “The fractional Fourier transform and some of its applications to optics,” in |

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 |

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. |

11. | B. Zhu, S. Liu, and Q. Ran, “Optical image encryption based on multi-fractional Fourier transforms,” Opt. Lett. |

12. | Y. Zhang, B. Dong, B. Gu, and G. Yang, “Beam shaping in the fractional Fourier transform domain,” J. Opt. Soc. Am. A |

13. | X. Xue, H.Q. Wei, and A. G. Kirk, “Beam analysis by fractional Fourier transform,” Opt. Lett. |

14. | Y. Cai, Q. Lin, and S. Zhu, “Coincidence fractional Fourier transform with entangled photon pairs and incoherent light,” Appl. Phy. Lett. |

15. | Y. Cai and S. Zhu, “Coincidence fractional Fourier transform with partially coherent light radiation,” J. Opt. Soc. Am. A |

16. | L. Mandel and E. Wolf, |

**OCIS Codes**

(070.2580) Fourier optics and signal processing : Paraxial wave optics

(070.2590) Fourier optics and signal processing : ABCD transforms

**ToC Category:**

Fourier Optics and Optical Signal Processing

**History**

Original Manuscript: March 27, 2006

Revised Manuscript: June 28, 2006

Manuscript Accepted: July 12, 2006

Published: August 7, 2006

**Citation**

Fei Wang, Yangjian Cai, and Sailing He, "Experimental observation of coincidence fractional Fourier transform with a partially coherent beam," Opt. Express **14**, 6999-7004 (2006)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-16-6999

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### References

- V. Namias, "The fractional Fourier transform and its application in quantum mechanics," J. Inst. Math. Its Appl. 25, 241-265 (1980) [CrossRef]
- A. C. McBride and F. H. Kerr, "On Namia’s fractional Fourier transforms," IMA J. App. Math. 39, 159-175 (1987) [CrossRef]
- A. W. Lohmann, "Image rotation, Wigner rotation, and the fractional Fourier transform," J. Opt. Soc. Am. A 10, 2181-2186 (1993) [CrossRef]
- D. Mendlovic and H. M. Ozaktas, "Fractional Fourier transforms and their optical implementation: I," J. Opt. Soc. Am. A 10, 1875-1881 (1993) [CrossRef]
- H. M. Ozaktas and D. Mendlovic, "Fractional Fourier transforms and their optical implementation: II," J. Opt. Soc. Am. A 10, 2522-2531 (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]
- 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]
- 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]
- 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]
- B. Zhu, S. Liu, and Q. Ran, "Optical image encryption based on multi-fractional Fourier transforms," Opt. Lett. 25, 1159-1161 (2000). [CrossRef]
- 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]
- X. Xue, H.Q. Wei, and A. G. Kirk, "Beam analysis by fractional Fourier transform," Opt. Lett. 26, 1746-1748 (2001). [CrossRef]
- 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]
- L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge, New York, 1995)

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