## High visibility two-photon interference with classical light |

Optics Express, Vol. 21, Issue 12, pp. 14056-14065 (2013)

http://dx.doi.org/10.1364/OE.21.014056

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### Abstract

Two-photon interference with independent classical sources, in which superposition of two indistinguishable two-photon paths plays a key role, is of limited visibility with a maximum value of 50%. By using a random-phase grating to modulate the wavefront of a coherent light, we introduce superposition of multiple indistinguishable two-photon paths, which enhances the two-photon interference effect with a signature of visibility exceeding 50%. The result shows the importance of phase control in the control of high-order coherence of classical light.

© 2013 OSA

## 1. Introduction

*“Each photon interferes only with itself. Interference between different photons never occurs”*[1].

6. L. Mandel, “Photon interference and correlation effects produced by independent quantum sources,” Phys. Rev. A **28**(2), 929–943 (1983) [CrossRef] .

6. L. Mandel, “Photon interference and correlation effects produced by independent quantum sources,” Phys. Rev. A **28**(2), 929–943 (1983) [CrossRef] .

7. H. Paul, “Interference between independent photons,” Rev. Mod. Phys. **58**(1), 209–231 (1986) [CrossRef] .

8. Z. Ou, “Quantum theory of fourth-order interference,” Phys. Rev. A **37**(5), 1607–1619 (1988) [CrossRef] [PubMed] .

9. D. Klyshko, “Quantum optics: quantum, classical, and metaphysical aspects,” Phys. Usp. **37**(11), 1097–1123 (1994) [CrossRef] .

10. 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**(18), 3600–3603 (1995) [CrossRef] [PubMed] .

12. K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. **89**(21), 213601 (2002) [CrossRef] [PubMed] .

13. G. Scarcelli, A. Valencia, and Y. Shih, “Two-photon interference with thermal light,” Europhys. Lett. **68**(5), 618–624 (2004) [CrossRef] .

15. Yan-Hua Zhai, Xi-Hao Chen, Da Zhang, and Ling-An Wu, “Two-photon interference with true thermal light,” Phys. Rev. A **72**(4),043805 (2005) [CrossRef] .

16. I. Agafonov, M. Chekhova, T. Iskhakov, and A. Penin, “High-visibility multiphoton interference of Hanbury Brown-Twiss type for classical light,” Phys. Rev. A **77**(5), 053801 (2008) [CrossRef] .

19. Y. Zhou, J. Simon, J. Liu, and Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A **81**(4), 043831 (2010) [CrossRef] .

## 2. Theoretical model and results

### 2.1. Random-phase grating

*N*-slit mask with specially designed random-phase structure shown in the inset of Fig. 2(a), in which

*b*is the transmission slit width and

*d*is the distance between neighboring slits, respectively. The phase encoded on the

*nth*transmission slit of the grating is designed to be Φ(

*x*,

_{s}*t*) = rect((

*x*−

_{s}*nd*)/

*b*)(

*n*− 1)

*ϕ*(

*t*), where rect(

*x*) is the one-dimensional rectangular function,

_{s}*x*is the position on the grating plane with

_{s}*x*=

_{s}*nd*being the center of the

*nth*slit of the grating,

*n*is a positive integer and the elementary phase

*ϕ*(

*t*) is a temporally random phase (In the following, we will use

*ϕ*to represent

*ϕ*(

*t*) for simplicity but without causing confusion). In this way, a random phase (

*n*− 1)

*ϕ*will be encoded on the light wave transmitting through the

*nth*slit. Such a random-phase grating can be realized through a spatial light modulator (SLM) in practice, as we will demonstrate experimentally in Sec. 3. In the following, we will consider the single-photon and two-photon interference effects when a collimated coherent light transmits through the random-phase grating, as shown in Fig. 2(b).

### 2.2. Theoretical results

20. R. Glauber, “The quantum theory of optical coherence,” Phys. Rev. **130**(6), 2529–2539 (1963) [CrossRef] .

*â*is the annihilation operator,

*β*=

_{j}*k*sin

*θ*and tan

_{j}*θ*=

_{j}*x*/

_{j}*f*(

*j*= 1, 2) with

*f*,

*k*and

*θ*being the focal length of the lens, the wave vector and the diffraction angle of the light wave, respectively. Here (

_{j}*n*− 1)

*ϕ*is the random phase encoded by the random-phase grating, while −(

*n*− 1)

*β*is the propagation phase difference between the optical paths from the first slit and the

_{j}d*nth*slit of the grating to the

*jth*detector.

*First-order spatial correlation function*— The first-order spatial correlation function in the detection plane can be expressed as [20

20. R. Glauber, “The quantum theory of optical coherence,” Phys. Rev. **130**(6), 2529–2539 (1963) [CrossRef] .

21. R. Glauber, “Coherent and incoherent state of radiation field,” Phys. Rev. **131**(6), 2766–2788 (1963) [CrossRef] .

*E*(

*x*) is the eigenvalue of the field operator

_{j}*Ê*

^{(+)}(

*x*) on the state of source (coherent state), and 〈⋯〉 represents the ensemble average. By substituting Eq. (1) into Eq. (2) and taking the condition 〈

_{j}*e*〉 = 0, one gets Here the term sin

^{iϕ}^{2}(

*β*/2)/(

_{j}b*β*/2)

_{j}b^{2}represents the diffraction from a single slit. It is evident that the intensity distribution in the focal plane is a sum of the diffraction intensities from

*N*different slits. There is no stationary first-order interference among these slits of the random-phase grating. However, the case will be totally different when one considers the two-photon interference.

*Second-order spatial correlation function*— The second-order spatial correlation function at the detection plane can be expressed as [20

20. R. Glauber, “The quantum theory of optical coherence,” Phys. Rev. **130**(6), 2529–2539 (1963) [CrossRef] .

21. R. Glauber, “Coherent and incoherent state of radiation field,” Phys. Rev. **131**(6), 2766–2788 (1963) [CrossRef] .

*e*〉 = 0, the second-order spatial correlation function can be deduced as (see Appendix A)

^{iϕ}*e*

^{−i(n−1)(β1d−ϕ)}

*e*

^{−i(m−1)(β2d−ϕ)}and

*e*

^{−i(n−1)(β2d−ϕ)}

*e*

^{−i(m−1)(β1d−ϕ)}correspond to the twin two-photon paths transmitting through a pair of slits (

*m*,

*n*) respectively, which are as those shown in Fig. 1: (1) one photon transmitting through the

*mth*slit goes to the detector D1, while the other transmitting through the

*nth*slit goes to the detector D2; and (2) one photon transmitting through the

*mth*slit goes to the detector D2, while the other transmitting through the

*nth*slit goes to the detector D1. Here we introduce the delta function

*δ*(

*m*−

*n*) to show that there is only one path when the two photons transmit through the same slit to trigger a coincidence count. It can also be found that a random phase (

*m*+

*n*− 2)

*ϕ*will be encoded on the amplitudes of the twin two-photon paths as represented in the first line of Eq. (5). For a

*N*-slit random-phase grating as shown in Fig. 2, there could be many such twin two-photon paths originated from different pairs of slits (

*m*,

*n*), and the amplitudes of those twin paths with equal (

*m*+

*n*) will contain the same random phase (

*m*+

*n*− 2)

*ϕ*. These twin two-photon paths are indistinguishable in principle. In this way, multiple different but indistinguishable two-photon paths are introduced through the

*N*-slit random-phase grating. As shown in the second line of Eq. (5), the amplitudes of all different but indistinguishable two-photon paths with the same random phase

*lϕ*(

*l*= 0, 1,⋯ , 2

*N*− 2) are superposed to calculate their contributions to the coincidence probability, and then the coincidence probability contributions from those with different random phases

*lϕ*are added to get the total coincidence probability. Next, we will show that such a superposition of multiple two-photon amplitudes would enhance the two-photon interference, leading to high-visibility two-photon interference for classical light.

^{2}((

*l′*+ 1)(

*β*

_{1}−

*β*

_{2})

*d*/2) / sin

^{2}((

*β*

_{1}−

*β*

_{2})

*d*/2) are of the similar formula as the multiple-slit single-photon interference function [22], and therefore can be called as multiple-slit two-photon interference function. It is seen that

*g*

^{(2)}(

*x*

_{1},

*x*

_{2}) in Eq. (6) is a sum of (2

*N*−1) multiple-slit two-photon interference functions sin

^{2}((

*l′*+1)(

*β*

_{1}−

*β*

_{2})

*d*/2) / sin

^{2}((

*β*

_{1}−

*β*

_{2})

*d*/2) introduced by the random-phase grating, each one is associated with a group of different but indistinguishable two-photon paths which are characterized by the same random phase

*lϕ*(

*l*= 0, 1,⋯ , 2

*N*− 2) in Eq. (5). These multiple-slit two-photon interference functions are periodical functions of the position difference (

*x*

_{1}−

*x*

_{2}) with the same period Λ =

*λf*/

*d*in the paraxial approximation, which is exactly the same as that of the multiple-slit single-photon interference pattern of a normal grating with respect to the position

*x*on the detection plane [22]. Therefore, two-photon interference fringes can be observed on the detection plane.

*β*

_{1}−

*β*

_{2})

*d*= ±2

*nπ*(

*n*= 0, 1, 2, ⋯) due to the constructive interference effect, i.e., when the phase difference among different but indistinguishable two-photon paths are an integer multiple of 2

*π*. The constructive interference peak for each multiple-slit two-photon interference function is (

*l′*+ 1)

^{2}, and therefore, one can get the interference peak of

*g*

^{(2)}(

*x*

_{1},

*x*

_{2}) to be (2

*N*

^{2}+ 1)/(3

*N*), according to Eq. (6). On the other hand, the minimum of

*g*

^{(2)}(

*x*

_{1},

*x*

_{2}) is achieved at the condition (

*β*

_{1}−

*β*

_{2})

*d*= ±(2

*n*+ 1)

*π*(

*n*= 0, 1, 2, ⋯) due to the destructive interference effect among multiple two-photon paths. However, the minimum of

*g*

^{(2)}(

*x*

_{1},

*x*

_{2}) is not zero but calculated to be 1/

*N*due to the existence of the cases when the two photons transmit through the same slit of the grating. Therefore, the visibility of the two-photon interference fringes is found to be

*V*= (

*N*

^{2}− 1)/(

*N*

^{2}+ 2), which grows quickly with the increase of slit number

*N*and exceeds 50% when

*N*> 2, as shown in Fig. 3.

## 3. Experimental demonstration and discussions

*Experimental setup*— Figure 4 shows the experimental setup that we used to measure the two-photon interference effect of the light field scattering from the random-phase grating. In our experiments, a single mode, continuous-wave laser with a wavelength of 780 nm was introduced as the light source, which was expanded and collimated through a beam expander to obtain a plane wave. The expanded and collimated light beam was then reflected by a beam splitter BS and incident normally onto a random-phase grating. Here the random-phase grating was composed of a

*N*-slit amplitude mask (

*b*= 72

*μ*m and

*d*= 400

*μ*m) and a reflection-type phase-only SLM (HEO 1080P from HOLOEYE Photonics AG, Germany) put just behind the mask. The light first transmitted through the

*N*-slit amplitude mask, and then was reflected back from the SLM and finally re-transmitted through the

*N*-slit amplitude mask again. Here we put the SLM as close as possible to the mask, ensuring that the light goes in and out of the same slit of the mask. The SLM provided the desired phase structure on the

*N*-slit mask as shown in the inset of Fig. 2(a). At last, the light waves scattered from the random-phase grating were collected by a lens L with a focal length

*f*= 80 cm. Both the intensity and the second-order spatial correlation measurements were performed on the focal plane of the lens L by using a charge coupled device (CCD) camera with a frame acquisition time of 0.79 ms.

*Results for traditional grating*— When there is no electric signal loaded on the SLM, our experimental configuration is essentially the same as a typical setup to measure the single-photon interference of a traditional

*N*-slit grating. In the experiment, we measured the stationary single-photon interference patterns of the

*N*-slit gratings (

*N*= 2, 3, 4 and 5, respectively). The results are shown in the first column of Fig. 5. As expected, stationary single-photon interference fringes described by the multiple-slit single-photon interference function sin

^{2}(

*Nβd*/2)/sin

^{2}(

*βd*/2) [22] were observed. The period between the neighboring principal intensity peaks was measured to be 1.57 mm on the detection plane, and (

*N*− 2) sub-peaks appear between the two neighboring principal peaks of the stationary single-photon interference fringes. Note that the normalized second-order spatial correlation function

*g*

^{(2)}(

*x*

_{1},

*x*

_{2}) in this case was confirmed to be a unity (not shown in Fig. 5).

*g*

^{(2)}(

*x*

_{1},

*x*

_{2}), which is calculated through a formula

*g*

^{(2)}(

*x*

_{1},

*x*

_{2}) = 〈

*I*(

*x*

_{1})

*I*(

*x*

_{2})〉/(〈

*I*(

*x*

_{1})〉〈

*I*(

*x*

_{2})〉) [18

18. X. Chen, I. Agafonov, K. Luo, Q. Liu, R. Xian, M. Chekhova, and L. Wu, “High-visibility, high-order lensless ghost imaging with thermal light,” Opt. Lett. **35**(8), 1166–1168 (2010) [CrossRef] [PubMed] .

23. Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nature Photonics **4**, 721–726 (2010) [CrossRef] .

*x*

_{1}−

*x*

_{2}= ±2

*nπf*/ (

*kd*) and minimized at the position differences

*x*

_{1}−

*x*

_{2}= ±(2

*n*+ 1)

*πf*/ (

*kd*) (

*n*= 0, 1, 2, ⋯), respectively. The period of the two-photon interference fringes Λ was measured to be 1.57 mm, in good agreement with the prediction of Eq. (6). On the other hand, sub-peaks typical for the single-photon interference fringes shown in the first column of Fig. 5 were not observed in the two-photon interference fringes in the third column of Fig. 5. This is due to the fact that

*g*

^{(2)}(

*x*

_{1},

*x*

_{2}) is a sum of (2

*N*− 1) different multiple-slit two-photon interference functions (see Eq. (6)), and these different multiple-slit two-photon interference functions are always in phase at their principal peaks but out of phase at the sub-peaks. Moreover, the visibility of the two-photon interference fringes was measured to be 44.9%, 59.1%, 62.3% and 71.9% for the

*N*-slit random-phase gratings with

*N*= 2, 3, 4 and 5, respectively. As predicted by Eq. (6), the visibility of the two-photon interference fringes increases with the increase of the slit number

*N*of the random-phase gratings and surpasses 50% when

*N*> 2.

## 4. Summary

*N*-slit random-phase grating reaches (

*N*

^{2}−1)/(

*N*

^{2}+2). Experimentally, the visibility of the two-photon interference fringes with a

*N*-slit random-phase grating (

*N*= 2, 3, 4 and 5) was measured to be 44.9%, 59.1%, 62.3% and 71.9%, respectively. The results show the possibility to control the high-order coherence of light through optical phase.

## Appendix A

*â*, is incident normally onto the random-phase grating as shown in Fig. 2(b), one arrives at by substituting Eq. (1) into Eq. (4). After taking the square of mould in Eq. (7), one needs to do the ensemble average for the terms with

*n′*,

*m′*,

*n*,

*m*being integer within [1,

*N*]. It is not difficult to find out that the ensemble average will survive for the two-photon case only if

*n′*+

*m′*equals

*n*+

*m*. Therefore, by removing the ensemble average operator 〈⋯〉 in Eq. (7), one arrives at Eq. (5).

## Appendix B

## Acknowledgments

## References and links

1. | P. Dirac, |

2. | R. Brown and R. Twiss, “Correlation between photons in two coherent beams of light,” Nature |

3. | R. Brown and R. Twiss, “A test of new type of stellar interferometer on sirius,” Nature |

4. | U. Fano, “Quantum theory of interference effects in the mixing of light from phase-independent sources,” Am. J. Phys. |

5. | J. Liu and G. Zhang, “Unified interpretation for second-order subwavelength interference based on Feynmans path-integral theory,” Phys. Rev. A |

6. | L. Mandel, “Photon interference and correlation effects produced by independent quantum sources,” Phys. Rev. A |

7. | H. Paul, “Interference between independent photons,” Rev. Mod. Phys. |

8. | Z. Ou, “Quantum theory of fourth-order interference,” Phys. Rev. A |

9. | D. Klyshko, “Quantum optics: quantum, classical, and metaphysical aspects,” Phys. Usp. |

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

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

12. | K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett. |

13. | G. Scarcelli, A. Valencia, and Y. Shih, “Two-photon interference with thermal light,” Europhys. Lett. |

14. | J. Xiong, D. Cao, F. Huang, H. Li, X. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. |

15. | Yan-Hua Zhai, Xi-Hao Chen, Da Zhang, and Ling-An Wu, “Two-photon interference with true thermal light,” Phys. Rev. A |

16. | I. Agafonov, M. Chekhova, T. Iskhakov, and A. Penin, “High-visibility multiphoton interference of Hanbury Brown-Twiss type for classical light,” Phys. Rev. A |

17. | D. Cao, J. Xiong, S. Zhang, L. Lin, L. Gao, and K. Wang, “Enhancing visibility and resolution in Nth-order intensity correlation of thermal light,” Appl. Phys. Lett. |

18. | X. Chen, I. Agafonov, K. Luo, Q. Liu, R. Xian, M. Chekhova, and L. Wu, “High-visibility, high-order lensless ghost imaging with thermal light,” Opt. Lett. |

19. | Y. Zhou, J. Simon, J. Liu, and Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A |

20. | R. Glauber, “The quantum theory of optical coherence,” Phys. Rev. |

21. | R. Glauber, “Coherent and incoherent state of radiation field,” Phys. Rev. |

22. | G. Brooker, |

23. | Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nature Photonics |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(270.1670) Quantum optics : Coherent optical effects

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: May 2, 2013

Revised Manuscript: May 26, 2013

Manuscript Accepted: May 28, 2013

Published: June 5, 2013

**Citation**

Peilong Hong, Lei Xu, Zhaohui Zhai, and Guoquan Zhang, "High visibility two-photon interference with classical light," Opt. Express **21**, 14056-14065 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-12-14056

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### References

- P. Dirac, The Principles of Quantum Mechanics, 2nd edition (Oxford University, 1935).
- R. Brown and R. Twiss, “Correlation between photons in two coherent beams of light,” Nature177(4497), 27–29 (1956). [CrossRef]
- R. Brown and R. Twiss, “A test of new type of stellar interferometer on sirius,” Nature178(4541), 1046–1048 (1956). [CrossRef]
- U. Fano, “Quantum theory of interference effects in the mixing of light from phase-independent sources,” Am. J. Phys.29(8), 539–545 (1961). [CrossRef]
- J. Liu and G. Zhang, “Unified interpretation for second-order subwavelength interference based on Feynmans path-integral theory,” Phys. Rev. A82(1), 013822 (2010). [CrossRef]
- L. Mandel, “Photon interference and correlation effects produced by independent quantum sources,” Phys. Rev. A28(2), 929–943 (1983). [CrossRef]
- H. Paul, “Interference between independent photons,” Rev. Mod. Phys.58(1), 209–231 (1986). [CrossRef]
- Z. Ou, “Quantum theory of fourth-order interference,” Phys. Rev. A37(5), 1607–1619 (1988). [CrossRef] [PubMed]
- D. Klyshko, “Quantum optics: quantum, classical, and metaphysical aspects,” Phys. Usp.37(11), 1097–1123 (1994). [CrossRef]
- 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(18), 3600–3603 (1995). [CrossRef] [PubMed]
- E. J. S. Fonseca, C. H. Monken, and S. Pádua, “Measurement of the de Broglie wavelength of a multiphoton wave packet,” Phys. Rev. Lett.82(14), 2868–2871 (1999). [CrossRef]
- K. Edamatsu, R. Shimizu, and T. Itoh, “Measurement of the photonic de Broglie wavelength of entangled photon pairs generated by spontaneous parametric down-conversion,” Phys. Rev. Lett.89(21), 213601 (2002). [CrossRef] [PubMed]
- G. Scarcelli, A. Valencia, and Y. Shih, “Two-photon interference with thermal light,” Europhys. Lett.68(5), 618–624 (2004). [CrossRef]
- J. Xiong, D. Cao, F. Huang, H. Li, X. Sun, and K. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett.94(17), 173601 (2005). [CrossRef] [PubMed]
- Yan-Hua Zhai, Xi-Hao Chen, Da Zhang, and Ling-An Wu, “Two-photon interference with true thermal light,” Phys. Rev. A72(4),043805 (2005). [CrossRef]
- I. Agafonov, M. Chekhova, T. Iskhakov, and A. Penin, “High-visibility multiphoton interference of Hanbury Brown-Twiss type for classical light,” Phys. Rev. A77(5), 053801 (2008). [CrossRef]
- D. Cao, J. Xiong, S. Zhang, L. Lin, L. Gao, and K. Wang, “Enhancing visibility and resolution in Nth-order intensity correlation of thermal light,” Appl. Phys. Lett.92(20), 201102 (2008). [CrossRef]
- X. Chen, I. Agafonov, K. Luo, Q. Liu, R. Xian, M. Chekhova, and L. Wu, “High-visibility, high-order lensless ghost imaging with thermal light,” Opt. Lett.35(8), 1166–1168 (2010). [CrossRef] [PubMed]
- Y. Zhou, J. Simon, J. Liu, and Y. Shih, “Third-order correlation function and ghost imaging of chaotic thermal light in the photon counting regime,” Phys. Rev. A81(4), 043831 (2010). [CrossRef]
- R. Glauber, “The quantum theory of optical coherence,” Phys. Rev.130(6), 2529–2539 (1963). [CrossRef]
- R. Glauber, “Coherent and incoherent state of radiation field,” Phys. Rev.131(6), 2766–2788 (1963). [CrossRef]
- G. Brooker, Modern Classical Optics (Oxford University, 2003).
- Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nature Photonics4, 721–726 (2010). [CrossRef]

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