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

doi:10.1364/OE.14.006999

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

Fei Wang, Yangjian Cai, and Sailing He »View Author Affiliations

Optics Express, Vol. 14, Issue 16, pp. 6999-7004 (2006)
http://dx.doi.org/10.1364/OE.14.006999

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▸ Article Outline
– Abstract
1. Introduction
2. Experimental setup
3. Theoretical analysis
4. Experimental results
5. Conclusion
– References and links

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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.

1. Introduction

The concept of the fractional Fourier transform (FRT) proposed by Namias is a generalization of the conventional Fourier transform, and can be used as a mathematical tool for solving some theoretical physical problems. ¹

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

McBride and Kerr have introduced a rigorous definition for FRT. ²

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

Since Ozaktas, Mendlovic and Lohmann implemented FRT with some optical systems in 1993, ³–⁵

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

FRT has attracted more attention and much work ⁶–⁸

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]

has been done on its properties, optical implementation, and applications in e.g. signal processing, 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]

optical image encryption, ¹¹

B. Zhu, S. Liu, and Q. Ran, “Optical image encryption based on multi-fractional Fourier transforms,” Opt. Lett. 25, 1159–1161 (2000). [CrossRef]

beam shaping 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]

and beam analysis. ¹³

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

More recently, we have introduced the concept of coincidence FRT, and designed an optical system to implement it with some incoherent 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]

or partially coherent light. 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]

Coincidence FRT is a method to obtain the FRT pattern of an object by measuring the coincidence counting rate (i.e., the fourth order correlation) of two detected signals going through two different light paths (the object is located in one light path). Such an FRT system is based on the fourth order correlation of the light, and is quite different from the conventional FRT, which is based on a deterministic field or the second order correlation of the light [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]

]. In the present setup for the coincidence FRT, the information about the FRT of the object must be obtained from both light paths (instead of a single light path as in most of the conventional FRT setup), and thus have the advantages of e.g. improving the security against eavesdropping. In this paper, we analyze the visibility and quality of the coincidence FRT pattern of an object quantitatively, and we report the experimental observation of the coincidence FRT with some partially coherent light and analyze the results.

2. Experimental setup

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

The schematic diagram for the experimental setup is shown in Fig. 1. A laser beam (with a beam diameter of d ₀ ≈ 10mm) generated by a He-Ne laser (λ = 632.8nm) 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 D₁ located at u ₁ = 0 . Along this path, an object (double slits with slit distance d = 195μm and slit width a = 85μm) is placed between the beam splitter and D₁, and a lens with a focal length of f ₁ is placed between the object and D₁, and both the distances from the object to the lens and from the lens to D₁ are f ₁. The reflected beam going through path 2 (see Fig. 1) arrives at single-photon detector D₂ 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

From the optical coherence theory, we can obtain the following fourth order correlation function between D₁ and D₂ ¹⁴–¹⁶

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]

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},

(1)

where

< I (u_{i}) > = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} 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,

(2)

Γ (u_{1}, u_{2}) = \int_{- \infty}^{\infty} \int_{- \infty}^{\infty} 〈 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},

(3)

and where 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)

E_{s}^{*}

(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.

The second order correlation function for an incoherent light source with uniform intensity distribution I ₀ can be expressed as:

〈 E_{s} (x_{1}) {E_{s}}^{*} (x_{2}) 〉 = I_{0} δ (x_{1} - x_{2}) .

(4)

Substituting Eq. (4) and expressions for h ₁ [x ₁,u ₁) and h ₂ (x ₂,u ₂) into Eqs. (1)–(3), we obtain 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]

< I (u_{1} = 0) > < I (u_{2}) > = \infty,

(5)

Γ (u_{1} = 0, u_{2}) = \frac{I_{0}}{\sqrt{λ^{2} f_{1} f_{e}} \sin ϕ} \int_{- \infty}^{\infty} H (v_{1}) \exp [\frac{iπ}{λ f_{e} \tan ϕ} (v_{1}^{2} + v_{2}^{2}) - \frac{2 iπ}{λ f_{e} \sin ϕ} v_{1} v_{2}] d v_{1},

(6)

where 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

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]

l_{1} = f_{e} \tan \frac{ϕ}{2} + z, I_{2} = f_{e} \tan \frac{ϕ}{2}, f = \frac{f_{e}}{\sin ϕ},

(7)

where f_e is called the “standard focal length”, ³

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

and ϕ is defined by 0 = pπ/2 with p the desired fractional order of the FRT. Eq. (6) gives the FRT expression of the object.14–15

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]

The parameters for the optical system can be determined from Eq. (7) for the desired f_e and p. The visibility of the coincidence FRT pattern is defined as V = ∣Γ(u ₁=0,u ₂)

∣_{max}^{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)).

In a practical situation, to increase the visibility of the FRT pattern of an object, we can replace the incoherent light source with a partially coherent light beam (with a finite spot size). 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]

Here we consider a typical partially coherent light source, namely, a partially coherent Gaussian Schell-model (GSM) source. The second order correlation function in the GSM source plane has the following form: ¹⁶

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

〈 E_{s} (x_{1}) {E_{s}}^{*} (x_{2}) 〉 = Γ (x_{1}, x_{2}) = \sqrt{I (x_{1})} I (x_{2}) g (x_{1} - x_{2}) = \exp [- \frac{(x_{1}^{2} + x_{2}^{2})}{4 σ_{I}^{2}} - \frac{{(x_{1} - x_{2})}^{2}}{2 σ_{g}^{2}}],

(8)

where g(x ₁-x ₂) = exp[-(x ₁ -x ₂)²/2

σ_{g}^{2}

is the spectral degree of coherence, 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

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

As can be seen from Ref. [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]

], when the beam’s coherence increases (with a fixed spot size), the coincidence FRT pattern gradually disappears (i.e., the quality of the FRT pattern degrades) whereas the visibility of the coincidence FRT pattern increases. 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]

To analyze the quality of the FRT pattern of an object quantitatively, here we introduce the following deviation factor to denote the deviation of the coincidence FRT pattern for a partially coherent light from that for an incoherent light (i.e., σ_g = 0 , σ₁ = ∞)

D = \frac{[\int_{- \infty}^{\infty} ∣ \frac{{∣ Γ (u_{1} = 0, u_{2}) ∣}^{2}}{{∣ Γ (u_{1} = 0, u_{2}) ∣}_{max}^{2}} - \frac{{∣ Γ {(u_{1} = 0, u_{2})}_{σ_{g} = 0, σ_{I} = \infty} ∣}^{2}}{{∣ Γ {(u_{1} = 0, u_{2})}_{σ_{g} = 0, σ_{I} = \infty} ∣}_{max}^{2}} ∣ d u_{2}]}{\int_{- \infty}^{\infty} \frac{{∣ Γ {(u_{1} = 0, u_{2})}_{σ_{g} = 0, σ_{I} = \infty} ∣}^{2}}{{∣ Γ {(u_{1} = 0, u_{2})}_{σ_{g} = 0, σ_{I} = \infty} ∣}_{max}^{2}} d u_{2} .}

(9)

A smaller value of 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 = 85μm, d = 195μm, λ = 632.8μm and σ_I = 2mm. 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_g (see e.g. Figs. 5 and 6).

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.

4. Experimental results

It is difficult to measure directly the coherence width 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

g (x_{1} - x_{2}) = \frac{Γ (x_{1}, x_{2})}{\sqrt{I (x_{1}) I (x_{2})}} = \exp [\frac{- {(x_{1} - x_{2})}^{2}}{2 σ_{g}^{2}}] .

(10)

Using Eqs. (1) and (10), we can obtain the following relation

g^{2} (x_{1} - x_{2}) = \exp [\frac{- {(x_{1} - x_{2})}^{2}}{σ_{g}^{2}}] = \frac{G^{(2)} (x_{1}, x_{2})}{< I (x_{1}) > < I (x_{2}) > - 1} .

(11)

Thus, the transverse coherence width 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).

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

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

Figure 5 shows our experimental results of the coincidence FRT pattern for different values of the fractional order p for an object of double slits with a partially coherent beam (σ_g = 15μm). The other parameters used in our experiment are f_e = 63cm, z = 15cm, f ₁ = 25cm. 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

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

) are also shown in Figs. 5 and 6. From these two figures one sees that the experimental results agree reasonably well with the theoretical results.

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 (σ_g = 15μm).

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 σ_g .

5. Conclusion

In conclusion, we have analyzed quantitatively the visibility and quality of the coincidence FRT pattern of an object when a partially coherent light source is used. We have reported the corresponding experimental observation of the coincidence FRT. The experimental results are in a good agreement with the corresponding theoretical results. The physical nature of the coincidence FRT with an incoherent or partially coherent light is due to the Hanbury Brown–Twiss effect ¹⁴–¹⁵

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]

The FRT of an object has been obtained by measuring the fourth-order correlation function (coincidence counting rate) between the detected signals passing through two optical paths, which is quite different from the conventional FRT. Further investigations on the applications of this coincidence FRT to metrology, lithography, holography and signal processing, etc., can be carried out in future study.

Acknowledgment

This research is partially supported by the National Basic Research Program (973) of China (2004CB719800).

References and links

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)

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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