## Interference effects induced by non-local spatial filtering |

Optics Express, Vol. 19, Issue 18, pp. 17308-17317 (2011)

http://dx.doi.org/10.1364/OE.19.017308

Acrobat PDF (822 KB)

### Abstract

The spatial correlation between down-converted photons allows for non-local spatial filtering when two-photon coincidences are registered. This allows one to non-locally control the visibility of interference fringes, to observe ghost images and interference patterns, and to “retrieve” a coherent quantum image from an incoherent field distribution. We show theoretically that non-local spatial filtering can lead to counter-intuitive effects when the pump beam is no longer given by a Gaussian profile. Namely, increased non-local filtering can actually decrease the visibility of interference fringes, contrary to what has been observed so far. We explain this behavior through the transverse spatial parity entanglement of the down-converted photons.

© 2011 OSA

## 1. Introduction

1. S. P. Walborn, C. H. Monken, S. Pádua, and P. H. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. **495**, 87–139 (2010). [CrossRef]

2. D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. **74**, 3600–3603 (1995). [CrossRef] [PubMed]

4. R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. **92**, 033601 (2004). [CrossRef] [PubMed]

5. C. H. Monken, P. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A. **57**, 3123–3126 (1998). [CrossRef]

6. E. Fonseca, C. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. **82**, 2868–2871 (1999). [CrossRef]

7. W. A. T. Nogueira, S. P. Walborn, S. Pádua, and C. H. Monken, “Experimental observation of spatial antibunching of photons,” Phys. Rev. Lett. **86**, 4009–4012 (2001). [CrossRef] [PubMed]

8. A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Entangled-photon imaging of a pure phase object,” Phys. Rev. Lett. **93**, 213903 (2004). [CrossRef] [PubMed]

9. P. H. S. Ribeiro, S. Pádua, J. C. M. da Silva, and G. A. Barbosa, “Controlling the degree of visibility of young’s fringes with photon coincidence measurements,” Phys. Rev. A **49**, 4176–4179 (1994). [CrossRef] [PubMed]

10. G. A. Barbosa, “Quantum images in double-slit experiments with spontaneous down-conversion light,” Phys. Rev. A **54**, 4473–4478 (1996). [CrossRef] [PubMed]

11. D. N. Klyshko, “A simple method of preparing pure states of an optical field, of implementing the Einstein-Podolsky-Rosen experiment, and of demonstrating the complementarity principle,” Sov. Phys. Usp. **31**, 74–85 (1988). [CrossRef]

13. P. H. Souto Ribeiro and G. A. Barbosa, “Direct and ghost interference in double-slit experiments with coincidence measurements,” Phys. Rev. A **54**, 3489–3492 (1996). [CrossRef] [PubMed]

14. H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A **68**, 012323 (2003). [CrossRef]

15. A. N. de Oliveira, S. P. Walborn, and C. H. Monken, “Implementation of the Deutsch algorithm using polarization and transverse modes,” J. Opt. B: Quantum Semiclassical Opt. **7**, 288–292 (2005). [CrossRef]

17. T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Synthesis and analysis of entangled photonic qubits in spatial-parity space,” Phys. Rev. Lett. **99**, 250502 (2007). [CrossRef]

18. S. P. Walborn, S. Pádua, and C. H. Monken, “Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion,” Phys. Rev. A **71**, 053812 (2005). [CrossRef]

19. S. P. Walborn and C. H. Monken, “Transverse spatial entanglement in parametric down-conversion,” Phys. Rev. A **76**, 062305 (2007). [CrossRef]

## 2. Asymmetric interferometer

*D*

_{2}, while the other photon propagates directly to detector

*D*

_{1}. In our choice of coordinate system, the fields propagate along the

*z*-direction, as shown in Fig. 1. Due to the reflections at the 50-50 non-polarizing beam splitters (BS) and the mirrors, a field propagating through the AMZ suffers an even number of reflections when passing through one arm and an odd number of reflections when passing through the other. Under an odd number of reflections, the horizontal (with respect to the laboratory table) coordinate suffers a sign change, that is, in momentum space:

*q*→ −

_{y}*q*[20

_{y}20. S. P. Walborn, A. N. de Oliveira, S. Pádua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. **90**, 143601 (2003). [CrossRef] [PubMed]

21. S. P. Walborn, W. A. T. Nogueira, A. N. de Oliveira, S. Pádua, and C. H. Monken, “Multimode Hong-Ou-Mandel interferometry,” Mod. Phys. Lett. B **19**, 1–19 (2005). [CrossRef]

*ϕ*is the relative phase difference between optical paths 1 and 2, and

**q**=

*q*

_{x}**x̂**+

*q*

_{y}**ŷ**is the transverse component of the wave vector.

*l*= 1, 2 and

**r**

*is the detection position for the down-converted modes*

_{l}*l*= 1, 2 and |

*ψ*〉 is a two-photon state.

*k*is the magnitude of the wave vector

**k**and

*ρ*=

*x*

**x̂**+

*y*

**ŷ**is the transverse component of the position vector

**r**= (

*x*,

*y*,

*z*). The operator a

*(*

_{l}**q**) annihilates a photon in mode

*l*with transverse wave vector

**q**.

*x*

_{1},

*y*

_{1}), (

*x*

_{2},

*y*

_{2}) can be calculated: where

*ρ*′ =

*x*

**x̂**−

*y*

**ŷ**due to the odd number of reflections in one arm.

*P*(

*ρ*

_{1},

*ρ*

_{2}) is given by Eq. (2). The last two terms in Eq. (6) are responsible for the interference fringes.

*λ*= 2

_{c}*λ*, where

_{p}*λ*is the pump beam wavelength. We will also assume the paraxial and thin-crystal approximations, and consider that the polarization state of each down-converted photon pair is well defined. Under these approximations, the two-photon detection amplitude is given by [1

_{p}1. S. P. Walborn, C. H. Monken, S. Pádua, and P. H. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. **495**, 87–139 (2010). [CrossRef]

5. C. H. Monken, P. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A. **57**, 3123–3126 (1998). [CrossRef]

*U*is the transverse profile of the pump beam propagated to the detection region. We have assumed that the distance from the crystal to each detector is the same:

*z*

_{1}=

*z*

_{2}=

*Z*. For notational simplicity, the

*z*-dependence of

*U*has been omitted.

*U*(

*ρ*) is accurately described by a Hermite-Gaussian (HG) mode [23

23. B. E. A. Saleh and M. C. Teich, *Fundamental Photonics* (Wiley, 1991). [CrossRef]

*U*(

*ρ*) ≡

*U*(

_{x}*x*)

*U*(

_{y}*y*). In this case, the detection probability in Eq. (6) can be factorized as where

*P*(

_{d}*x*

_{1},

*x*

_{2}) ∝ |

*U*(

_{x}*x*

_{1}/2 +

*x*

_{2}/2)|

^{2}and where

*ξ*(

*y*

_{1},

*y*

_{2}) =

*ky*

_{1}

*y*

_{2}/2

*R*(

*Z*) comes from the quadratic phase of the Hermite-Gaussian pump beam profile

*U*[23

23. B. E. A. Saleh and M. C. Teich, *Fundamental Photonics* (Wiley, 1991). [CrossRef]

*R*(

*Z*) is the radius of curvature and

*k*= 2

*π*/

*λ*the wave number of the pump beam. We assume that the interferometer is perfectly aligned. Let us suppose that detectors

_{p}*D*

_{1}and

*D*

_{2}are equipped with square apertures of width 2

*a*

_{1}and 2

*a*

_{2}, centered at

*ρ*

_{1}=

*ρ*

_{2}= 0, respectively. The coincidence count rate is obtained by integrating over the square detection apertures. From Eqs. (8) and (9) it is evident that the transverse spatial dependence of the interference curves depends only on the

*y*coordinates of the detectors, and we notice that the

*dx*

_{1}and

*dx*

_{2}integrals will not affect the visibility of the interference fringes. With these considerations, we omit the

*x*-dependence of the coincidence count rate, and write where

*A*

_{1}and

*A*

_{2}are functions describing the apertures in the

*y*direction.

### 2.1. Gaussian pump profile

*Z*= 0. In this case, the radius of curvature

*R*(

*Z*) is infinite, and

*ξ*(

*y*

_{1},

*y*

_{2}) = 0. Let us also assume that the pump beam is given by the usual Gaussian profile:

*ϕ*and this term is always nonnegative,

*C*

_{max}occurs when

*ϕ*= 0 and

*C*

_{min}occurs when

*ϕ*=

*π*. The visibility of fringes in the coincidence counts, or fourth-order visibility, is then

*D*

_{2}. This can be easily done by calculating the single count rate

*C*

_{2}(

*a*

_{2}), obtained by letting the width of the aperture

*A*

_{1}tend to infinity. Taking the limit,

*C*

_{2}(

*a*

_{2}) is

*C*

_{2}(

*a*

_{2}) has a maximum and minimum at

*ϕ*= 0 and

*ϕ*=

*π*, respectively. Using Eq. (14), the visibility for the interference fringes in the single counts, or second-order visibility, is Note that in the limiting case

*a*

_{2}→ ∞,

*V*

_{2}→ 0. Eq. (15) gives the visibility as a function of the size of the detection apertures and the beam width

*w*. Figure 2 shows a plot of the fourth-order visibility given by Eq. (15) as a function of

*a*

_{2}/

*w*for various values of

*a*

_{1}/

*w*. We have also plotted the second-order visibility

*V*

_{2}in Fig. 2. Note that the fourth-order visibility

*V*tends to the second-order visibility

*V*

_{2}for large values of

*a*

_{1}/

*w*.

### 2.2. Hermite-Gaussian HG_{01} pump profile

20. S. P. Walborn, A. N. de Oliveira, S. Pádua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. **90**, 143601 (2003). [CrossRef] [PubMed]

26. S. P. Walborn, W. A. T. Nogueira, S. Pádua, and C. H. Monken, “Optical Bell-state analysis in the coincidence basis,” Europhys. Lett. **62**, 161–167 (2003). [CrossRef]

_{01}is defined as Using this pump profile, the coincidence count rate Eq. (10) becomes

_{01}pump beam As in section 2.1, the second-order visibility

*V*

_{2}(

*a*

_{2}) can be found by calculating the single count rate

*C*

_{2}(

*a*

_{2}) at detector

*D*

_{2}as the limit of the coincidence counts when

*a*

_{1}tends to infinity. In this case, we obtain Again,

*C*

_{2}(

*a*

_{2}) has a maximum and minimum at

*ϕ*= 0 and

*ϕ*=

*π*, respectively. Using Eq. (14), the second-order visibility is

*a*

_{2}/

*w*as a function of

*a*

_{1}/

*w*. We note that the visibility is symmetric, that is

*V*(

*a*

_{1},

*a*

_{2}) =

*V*(

*a*

_{2},

*a*

_{1}). A striking feature can be observed: for fixed

*a*

_{2}(

*a*

_{1}), the fourth-order visibility can also increase as

*a*

_{1}(

*a*

_{2}) increases. In other words, increased spatial filtering of the non-interfering photon can decrease the visibility of the fourth-order interference, in contrast to what is observed for the Gaussian pump profile. Another interesting point of Eq. (22) is that

*V*(

*a*,

*a*) = 0 for all values of

*a*, as shown in Fig. 3. Therefore, there is always an aperture size for the detector of the non-interfering photon for which no interference is observed. These effects can be seen directly in the visibility in Eq. (22), which is given by the difference of two terms. When

*a*

_{1}=

*a*

_{2}these two terms are equal, and

*V*= 0. Depending on whether

*a*

_{1}>

*a*

_{2}or

*a*

_{2}>

*a*

_{1}, the term inside the absolute value is positive or negative. Since

*V*is defined as the absolute value, this accounts for the increase in visibility when

*a*

_{1}and

*a*

_{2}are different. The dashed line in Fig. 3 represents the second order visibility

*V*

_{2}as a function of

*a*

_{2}/

*w*. Notice that

*V*

_{2}is zero only for large values of

*a*

_{2}/

*w*.

*V*

_{2}, the transverse coherence length in the case of the HG

_{01}pump beam is

*a*

_{2coh}≈ 0.5

*w*. However, for the fourth-order interference, it is not possible to define an effective coherence length as was done when the pump was Gaussian. It would be necessary to define a higher-order coherence “length” or coherence area, depending on the parameters of pump, signal and idler fields. In usual second-order coherence theory, the complex degree of coherence between points

**r**

_{1}and

**r**

_{2}generally decreases as a function of |

**r**

_{1}−

**r**

_{2}| [24]. The coherence area is then defined as the region in which the complex degree of coherence is above some cut-off value, typically 1/2 or 1/

*e*. For a circularly symmetric coherence function, for example, the coherence area is

*r*is the radius for which the coherence is above the cut-off value. In the case of the effective fourth-order coherence length for the HG

_{c}_{01}pump profile, it is not possible to make this simple distinction, and a more elaborate definition is necessary. We leave the careful derivation of this definition to future work.

*ξ*(

*y*

_{1},

*y*

_{2}) = 0 in the detection probability given by Eq. (9). The same qualitative effects will appear even when the waist of the pump laser is not at the detection plane. The propagation of the pump beam in this context appears in two forms. First, due to the divergence of the pump beam, the beam width is no longer given by the waist

*w*but by the

*z*-dependent width

*w*(

*Z*). Thus, replacing

*w*with

*w*(

*Z*) in all of the results obtained above accounts for the divergence of the pump beam. Secondly, since

*ξ*(

*y*

_{1},

*y*

_{2}) =

*ky*

_{1}

*y*

_{2}/2

*R*(

*Z*), the term cos[

*ϕ*+

*ξ*(

*y*

_{1},

*y*

_{2})] introduces a phase that depends on the spatial coordinates which appears only in the interference term in Eq. (9). We note first that this phase is identical for both the Gaussian and the HG

_{01}pump beam. The overall effect is a reduction in the visibilities obtained in both cases, but the form of the curves in Figs. 2 and 3 do not change. For

*Z*that is much less than the Rayleigh range of the pump beam,

*ξ*(

*y*

_{1},

*y*

_{2}) can be ignored. For example, for

*w*= 1 mm,

*λ*= 400 nm and

_{p}*Z*= 100 mm, (

*k*/2

*R*(

*Z*)) ≈ 0.01/

*w*

^{2}. Hence, the phase variation is slow compared to the variation of the amplitude of the fourth-order field.

## 3. Parity Correlations

*HG*

_{01}mode by decomposing the two-photon state in terms of Hermite-Gaussian modes [18

18. S. P. Walborn, S. Pádua, and C. H. Monken, “Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion,” Phys. Rev. A **71**, 053812 (2005). [CrossRef]

_{01}pump beam [16

16. T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Experimental violation of Bell’s inequality in spatial-parity space,” Phys. Rev. Lett. **99**, 170408 (2007). [CrossRef] [PubMed]

19. S. P. Walborn and C. H. Monken, “Transverse spatial entanglement in parametric down-conversion,” Phys. Rev. A **76**, 062305 (2007). [CrossRef]

*EO*〉

_{12}is the (normalized) state of photon pairs such that photon 1 and 2 have even and odd

*y*-indices: and |

*HG*〉 are single photon states in Hermite-Gaussian modes. The asymmetric interferometer is sensitive to the parity of the input field [14

_{n}14. H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A **68**, 012323 (2003). [CrossRef]

17. T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Synthesis and analysis of entangled photonic qubits in spatial-parity space,” Phys. Rev. Lett. **99**, 250502 (2007). [CrossRef]

*P*

_{1}+

*P*

_{2}in Fig. 4) is composed of two contributions (

*P*

_{1}and

*P*

_{2}) that are out of phase. The visibility depends on the difference in the overall weight of each contribution.

*C*(

_{EO}*a*

_{1},

*a*

_{2}) is the number of coincidence counts from the even-odd term, as a function of the width of the detection apertures. The two-photon state in this case is symmetric, and as a result

*C*(

_{EO}*a*,

*b*) =

*C*(

_{OE}*b*,

*a*). Thus, if the detectors are the same size,

*C*(

_{EO}*a*,

*a*) =

*C*(

_{OE}*a*,

*a*) and

*V*= 0, as is shown in Fig. 3. Depending on the size of the detectors, it is possible to have

_{hg}*C*(

_{EO}*a*

_{1},

*a*

_{2}) >

*C*(

_{OE}*a*

_{1},

*a*

_{2}) or

*C*(

_{EO}*a*

_{1},

*a*

_{2}) <

*C*(

_{OE}*a*

_{1},

*a*

_{2}). Both of these cases result in non-zero visibility. Thus, due to the parity correlations of the down-converted photons, the appearance of interference fringes depends non-trivially on the size of the detector apertures. The parity correlations produce a different effect in the case of the Gaussian pump beam. In this case, the two-photon state in Hermite-Gaussian expansion is [18

18. S. P. Walborn, S. Pádua, and C. H. Monken, “Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion,” Phys. Rev. A **71**, 053812 (2005). [CrossRef]

*α*>

*β*. In this case the down-converted photons are produced with the same parity. The visibility for a Gaussian pump beam can be written as When

*a*

_{1}or

*a*

_{2}is small,

*C*(

_{EE}*a*

_{1},

*a*

_{2}) ≫

*C*(

_{OO}*a*

_{1},

*a*

_{2}), and interference fringes with large visibility are observed. As

*a*

_{1}and

*a*

_{2}grow,

*C*(

_{EE}*a*

_{1},

*a*

_{2}) and

*C*(

_{OO}*a*

_{1},

*a*

_{2}) become more and more similar, and the visibility becomes reduced.

## 4. Discussion

27. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express **12**, 5448–5456 (2004). [CrossRef] [PubMed]

14. H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A **68**, 012323 (2003). [CrossRef]

28. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. **88**, 257901 (2002). [CrossRef] [PubMed]

*b*oth detectors. Though the results presented above were derived specifically for parametric down-conversion, they apply to any physical realization of the entangled state of the form Eq. (7).

## 5. Conclusion

9. P. H. S. Ribeiro, S. Pádua, J. C. M. da Silva, and G. A. Barbosa, “Controlling the degree of visibility of young’s fringes with photon coincidence measurements,” Phys. Rev. A **49**, 4176–4179 (1994). [CrossRef] [PubMed]

13. P. H. Souto Ribeiro and G. A. Barbosa, “Direct and ghost interference in double-slit experiments with coincidence measurements,” Phys. Rev. A **54**, 3489–3492 (1996). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | S. P. Walborn, C. H. Monken, S. Pádua, and P. H. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. |

2. | D. V. Strekalov, A. V. Sergienko, D. N. Klyshko, and Y. H. Shih, “Observation of two-photon ghost interference and diffraction,” Phys. Rev. Lett. |

3. | T. B. Pittman, Y. H. Shih, D. V. Strekalov, and A. V. Sergienko, “Optical imaging by means of two-photon quantum entanglement,” Phys. Rev. A |

4. | R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. |

5. | C. H. Monken, P. S. Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A. |

6. | E. Fonseca, C. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett. |

7. | W. A. T. Nogueira, S. P. Walborn, S. Pádua, and C. H. Monken, “Experimental observation of spatial antibunching of photons,” Phys. Rev. Lett. |

8. | A. F. Abouraddy, P. R. Stone, A. V. Sergienko, B. E. A. Saleh, and M. C. Teich, “Entangled-photon imaging of a pure phase object,” Phys. Rev. Lett. |

9. | P. H. S. Ribeiro, S. Pádua, J. C. M. da Silva, and G. A. Barbosa, “Controlling the degree of visibility of young’s fringes with photon coincidence measurements,” Phys. Rev. A |

10. | G. A. Barbosa, “Quantum images in double-slit experiments with spontaneous down-conversion light,” Phys. Rev. A |

11. | D. N. Klyshko, “A simple method of preparing pure states of an optical field, of implementing the Einstein-Podolsky-Rosen experiment, and of demonstrating the complementarity principle,” Sov. Phys. Usp. |

12. | A. V. Belinskii and D. N. Klyshko, “Two-photon optics: diffraction, holography, and transformation of two-dimensional signals,” J. Exp. Theor. Phys. |

13. | P. H. Souto Ribeiro and G. A. Barbosa, “Direct and ghost interference in double-slit experiments with coincidence measurements,” Phys. Rev. A |

14. | H. Sasada and M. Okamoto, “Transverse-mode beam splitter of a light beam and its application to quantum cryptography,” Phys. Rev. A |

15. | A. N. de Oliveira, S. P. Walborn, and C. H. Monken, “Implementation of the Deutsch algorithm using polarization and transverse modes,” J. Opt. B: Quantum Semiclassical Opt. |

16. | T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Experimental violation of Bell’s inequality in spatial-parity space,” Phys. Rev. Lett. |

17. | T. Yarnall, A. F. Abouraddy, B. E. A. Saleh, and M. C. Teich, “Synthesis and analysis of entangled photonic qubits in spatial-parity space,” Phys. Rev. Lett. |

18. | S. P. Walborn, S. Pádua, and C. H. Monken, “Conservation and entanglement of Hermite-Gaussian modes in parametric down-conversion,” Phys. Rev. A |

19. | S. P. Walborn and C. H. Monken, “Transverse spatial entanglement in parametric down-conversion,” Phys. Rev. A |

20. | S. P. Walborn, A. N. de Oliveira, S. Pádua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. |

21. | S. P. Walborn, W. A. T. Nogueira, A. N. de Oliveira, S. Pádua, and C. H. Monken, “Multimode Hong-Ou-Mandel interferometry,” Mod. Phys. Lett. B |

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

23. | B. E. A. Saleh and M. C. Teich, |

24. | M. Born and E. Wolf, |

25. | P. H. Souto Ribeiro, S. Pádua, J. C. Machado da Silva, and G. A. Barbosa, “Control of Young’s fringes visibility by stimulated down-conversion,” Phys. Rev. A |

26. | S. P. Walborn, W. A. T. Nogueira, S. Pádua, and C. H. Monken, “Optical Bell-state analysis in the coincidence basis,” Europhys. Lett. |

27. | G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetsov, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express |

28. | J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: June 6, 2011

Revised Manuscript: August 3, 2011

Manuscript Accepted: August 8, 2011

Published: August 18, 2011

**Citation**

S. P. Walborn, P. H. Souto Ribeiro, and C. H. Monken, "Interference effects induced by non-local spatial filtering," Opt. Express **19**, 17308-17317 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-18-17308

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### References

- S. P. Walborn, C. H. Monken, S. Pádua, and P. H. S. Ribeiro, “Spatial correlations in parametric down-conversion,” Phys. Rep. 495, 87–139 (2010). [CrossRef]
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