## Abstract

In this paper, we report an interference experiment in which a spatially incoherent light source illuminates two spatially separated apertures, whose superposition at the same place forms a double-slit. The experimental result exhibits a well-defined interference fringe solely through intensity measurements, in agreement with the theoretical analysis by means of the first-order spatial interference of the incoherent light. Consequently, the nonlocal double-slit interference with thermal light should be attributed to the first-order spatial correlation of incoherent field.

© 2009 OSA

## 1. Introduction

In classical optics, interference is implemented through intensity observation in a detection plane where two or more light beams are superposed. This is regarded as the first-order or one-photon interference. Recently, it has been shown that interference phenomena may occur through intensity correlation measurements or two-photon coincidence measurements even if the first-order interference disappears, such as “ghost” imaging, “ghost” interference, subwavelength interference and nonlocal double-slit interference [1–5]. These effects, regarded as the second-order or two-photon interference, manifest nonlocal features since an entangled two-photon source is involved. As an example, in the nonlocal double-slit interference, a pair of signal and idler photons generated by spontaneous parametric down conversion are scattered by two spatially separated apertures: none of them is a double-slit but their superposition at the same place forms a double-slit [5]. Though each intensity profile of the signal and idler beams does not exhibit any fringe, an interference pattern can be observed in the two-photon coincidence measurements of the two beams. These effects were attributed to the nonlocal character of the quantum entanglement. Recent theoretical and experimental results demonstrated that the similar second-order interference effects to the above can be performed by using a thermal light source [6–24]. Gao et al. [24] demonstrated that the interference from a nonlocal double-slit can be performed through the intensity correlation measurements of thermal light. The similarity between the two-photon entangled source and the thermal light source arouses different physical explanations, such as quantum vs. classical interpretation, to the nature of the second-order correlation of thermal light [25, 26]. There is still no consensus so far, and hence new experimental results would be helpful to reach the proper understanding.

## 2. Experimental results

In contrast to common knowledge that irregular phase distribution of an incoherent source shall degrade interference pattern in intensity observation, Zhang et al. first demonstrated that a spatially incoherent light source is capable of obtaining the coherence information through just the intensity distribution itself [27]. Here we report such an experiment in which the nonlocal double-slit interference can also be implemented through one-photon process. As sketched in Fig. 1
, the experimental setup is very similar to the recent proposal [27] on the first-order interference effect in an unbalanced interferometer using spatially incoherent light source, but with a nonlocal double-slit replacing the real double-slit. The source field is divided into two parts by the beamsplitter BS_{1}: one illuminates aperture A_{1} in path 1 and the other illuminates aperture A_{2} in path 2. BS_{2} is a $50/50$ beamsplitter, where the interference of the two fields may occur. Aperture A_{1}, a wire with a diameter of ${L}_{1}=0.2$ mm and aperture A_{2}, a slit with a width of ${L}_{2}=0.4$ mm, are placed at the equal distance ${z}_{0}=1$ cm from BS_{1}, and their superposition at the same position forms a double-slit of slit width $b=0.1$ mm and spacing $d=0.3$ mm. The distance between the aperture and charge-coupled device (CCD) camera is ${z}_{1}=38$ cm, and the two arms of the interferometer have the same optical path. A lens of focal length $f=19$ cm is set in the middle of path 2, so aperture A_{2} and the CCD screen are located in the two focal planes of lens L. The equal-optical-path condition can assure that one photon interferes with itself after passing through the two arms. However, there are different diffraction configurations in the two arms and it is the key in the present scheme since balanced interferometer washes out the information of the object [27].

In the experiment we first use a pseudo-thermal light source as incoherent source, which is formed by passing a He-Ne laser beam of wavelength 632.8 nm through a slowly rotating (0.005 Hz) ground glass disk G. A huge number of speckles illuminating the object have the average size about 0.01 mm, which is much smaller than that of the apertures. Since the source is spatially incoherent, it is clear that the average intensity distribution of the diffraction field in each arm is homogeneous. We now observe the interference patterns at the two outgoing ports of the beamsplitter BS_{2}, which are recorded by either of two CCD cameras. As the ground glass rotates slowly, the interference patterns in the detection plane fluctuate randomly. However, if we average over a number of frames, a well-defined interference pattern emerges [27]. Experimental results are shown in Fig. 2
, where (a) and (b) are the average intensity patterns $\u3008{I}_{1}\u3009$ and $\u3008{I}_{2}\u3009$ registered by CCD_{1} and CCD_{2}, respectively. We will show in the next section that the fringes fit well the Fourier spectrum of a double-slit, modulated by a quadratic phase factor. Since the interference terms at the two outgoing ports have a phase shift *π* due to the reflection and transmission of the beamsplitter, the two fringes are complementary. For a $50/50$ beamsplitter, the difference and sum of the two patterns gives the net interference pattern and the intensity background, as shown in Figs. 2(c) and 2(d), respectively. As a matter of fact, the homogeneous intensity background in Fig. 2(d) verifies the incoherence of the source.

To further demonstrate above effect, a true thermal light source should be taken into account. We replace the pseudo-thermal light source in Fig. 1 by a Na lamp of wavelength 589.3 nm with an illumination area of $10\times 10$ mm^{2}. In this case, the coherence time of the Na lamp is much shorter than the responses time of the CCD camera, so the interference patterns can appear directly on the CCD screen, as shown in Fig. 3
. The patterns are similar to that in Fig. 2, except for the slightly different spacings, owing to the different wavelengths of the two sources.

To confirm whether the interference patterns above are related to the spatial incoherence, we may compare them with the results obtained in the same interferometer using coherent light. We simply remove the ground glass in Fig. 1. The experimental results are shown in Fig. 4 , where the interference patterns are stationary and completely different from that in Figs. 2 and 3.

## 3. Theoretical analysis

We now present the theoretical explanation for the experimental results. Let ${E}_{s}(x)$and ${E}_{j}(x)$ be the source field at BS_{1} and the field of path $j(=1,2)$ at the recording plane, respectively, and*x*, the transverse position across the beam. The field diffraction in path *j* is described as

*k*is the wavenumber of the beam; ${A}_{1}(x)=1-\mathrm{rect}(x/{L}_{1})$ and ${A}_{2}(x)=\mathrm{rect}(x/{L}_{2})$are designated as the transmission functions of the two apertures, respectively, and their product $D(x)$ is a double-slit function with slit width $b=({L}_{2}-{L}_{1})/2$ and spacing $d=({L}_{2}+{L}_{1})/2$, i.e., $D(x)={A}_{1}(x){A}_{2}(x)=\mathrm{rect}[(x-d/2)/b]+\mathrm{rect}[(x+d/2)/b]$. The propagation of the mutual coherence in the interferometer is given by

*j*and the interference term, respectively.

For a spatially incoherent light source, the first-order field correlation function satisfies$\u3008{E}_{s}^{\ast}({x}_{0}){E}_{s}({x}_{0}^{\prime})\u3009={I}_{s}\delta ({x}_{0}-{x}_{0}^{\prime})$, and the interference term is written as

_{2}in the interferometer [27]. However, the two intensity distributions, $\u3008{E}_{1}^{\ast}(x){E}_{1}(x)\u3009$and$\u3008{E}_{2}^{\ast}(x){E}_{2}(x)\u3009$, are homogeneous. Figure 5 shows the numerical simulation of Eq. (5), fitting well with the experimental results in Fig. 2, apart from minor asymmetry. Any misalignment in the optical system may cause the asymmetry in the interference pattern.

As for the spatially coherent light, it has$\u3008{E}_{s}^{\ast}({x}_{0}){E}_{s}({x}_{0}^{\prime})\u3009={E}_{s}^{\ast}({x}_{0}){E}_{s}({x}_{0}^{\prime})$, and the first-order correlation function Eq. (3) is separable as

*σ*characterizes the spot size. Using Eq. (6) and Eq. (4), we calculate 1D interference patterns in Fig. 6 . As we expected, the net interference pattern in Fig. 6 (c) corresponds to the product of the diffraction fields of the two apertures, and the intensity background in Fig. 6 (d) coincides with the sum of the two diffraction patterns of A

_{1}and A

_{2}. No information about the double-slit can be obtained when A

_{1}and A

_{2}are illuminated coherently. The theoretical curves are in a good agreement with the experimental results for the coherent light case of Fig. 4. A slight mismatch of some side-peaks comes from our simple laser model in the theoretical simulation.

## 4. Summary

In summary, we have demonstrated that the interference of a nonlocal double-slit can be realized through one-photon process. Our theoretical analysis showed that two spatially separated apertures can be joined together in the first-order field correlation of spatially incoherent light. The previous experiment on the nonlocal double-slit interference with thermal light relies on intensity correlation measurements. Physically, the second-order field correlation of thermal light is decomposable into the first-order correlations, one of which, the modulus of the first-order cross field correlation at two positions, i.e., ${\left|\u3008{E}_{1}^{*}({x}_{1}){E}_{2}({x}_{2})\u3009\right|}^{2}$, records the interference. In essence, the nonlocal double-slit interference with thermal light should be attributed to the first-order field correlation of incoherent light. However, the second-order field correlation of entangled photon pair cannot be degraded into the first-order ones, and hence the corresponding nonlocal double-slit effect and other quantum imaging are based on the true second-order interference.

## Acknowledgment

This work was supported by the National Fundamental Research Program of China, Project No. 2006CB921404, and the National Natural Science Foundation of China, Project No. 10874019.

## References and links

**1. **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]

**2. **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 **52**(5), R3429–R3432 (1995). [CrossRef] [PubMed]

**3. **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]

**4. **A. F. Abouraddy, B. E. A. Saleh, A. V. Sergienko, and M. C. Teich, “Role of entanglement in two-photon imaging,” Phys. Rev. Lett. **87**(12), 123602 (2001). [CrossRef] [PubMed]

**5. **E. J. S. Fonseca, P. H. S. Ribeiro, S. Pádua, and C. H. Monken, “Quantum interference by a nonlocal double slit,” Phys. Rev. A **60**(2), 1530–1533 (1999). [CrossRef]

**6. **R. S. Bennink, S. J. Bentley, and R. W. Boyd, ““Two-Photon” coincidence imaging with a classical source,” Phys. Rev. Lett. **89**(11), 113601 (2002). [CrossRef] [PubMed]

**7. **R. S. Bennink, S. J. Bentley, R. W. Boyd, and J. C. Howell, “Quantum and classical coincidence imaging,” Phys. Rev. Lett. **92**(3), 033601 (2004). [CrossRef] [PubMed]

**8. **A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Correlated imaging, quantum and classical,” Phys. Rev. A **70**(1), 013802 (2004). [CrossRef]

**9. **A. Gatti, E. Brambilla, M. Bache, and L. A. Lugiato, “Ghost imaging with thermal light: comparing entanglement and classical correlation,” Phys. Rev. Lett. **93**(9), 093602 (2004). [CrossRef] [PubMed]

**10. **J. Cheng and S. S. Han, “Incoherent coincidence imaging and its applicability in X-ray diffraction,” Phys. Rev. Lett. **92**(9), 093903 (2004). [CrossRef] [PubMed]

**11. **K. G. Wang and D. Z. Cao, “Subwavelength coincidence interference with classical thermal light,” Phys. Rev. A **70**(4), 041801 (2004). [CrossRef]

**12. **Y. J. Cai and S. Y. Zhu, “Ghost interference with partially coherent radiation,” Opt. Lett. **29**(23), 2716–2718 (2004). [CrossRef] [PubMed]

**13. **F. Ferri, D. Magatti, A. Gatti, M. Bache, E. Brambilla, and L. A. Lugiato, “High-resolution ghost image and ghost diffraction experiments with thermal light,” Phys. Rev. Lett. **94**(18), 183602 (2005). [CrossRef] [PubMed]

**14. **D. Z. Cao, J. Xiong, and K. G. Wang, “Geometrical optics in correlated imaging systems,” Phys. Rev. A **71**(1), 013801 (2005). [CrossRef]

**15. **A. Valencia, G. Scarcelli, M. D’Angelo, and Y. H. Shih, “Two-photon imaging with thermal light,” Phys. Rev. Lett. **94**(6), 063601 (2005). [CrossRef] [PubMed]

**16. **J. Xiong, D. Z. Cao, F. Huang, H. G. Li, X. J. Sun, and K. G. Wang, “Experimental observation of classical subwavelength interference with a pseudothermal light source,” Phys. Rev. Lett. **94**(17), 173601 (2005). [CrossRef] [PubMed]

**17. **D. Zhang, Y. H. Zhai, L. A. Wu, and X. H. Chen, “Correlated two-photon imaging with true thermal light,” Opt. Lett. **30**(18), 2354–2356 (2005). [CrossRef] [PubMed]

**18. **Y. H. Zhai, X. H. Chen, D. Zhang, and L. A. Wu, “Two-photon interference with true thermal light,” Phys. Rev. A **72**(4), 043805 (2005). [CrossRef]

**19. **G. Scarcelli, V. Berardi, and Y. H. Shih, “Phase-conjugate mirror via two-photon thermal light imaging,” Appl. Phys. Lett. **88**(6), 061106 (2006). [CrossRef]

**20. **L. Basano and P. Ottonello, “Experiment in lensless ghost imaging with thermal light,” Appl. Phys. Lett. **89**(9), 091109 (2006). [CrossRef]

**21. **R. Borghi, F. Gori, and M. Santarsiero, “Phase and amplitude retrieval in ghost diffraction from field-correlation measurements,” Phys. Rev. Lett. **96**(18), 183901 (2006). [CrossRef] [PubMed]

**22. **M. Bache, D. Magatti, F. Ferri, A. Gatti, E. Brambilla, and L. A. Lugiato, “Coherent imaging of a pure phase object with classical incoherent light,” Phys. Rev. A **73**(5), 053802 (2006). [CrossRef]

**23. **A. Gatti, M. Bache, D. Magatti, E. Brambilla, F. Ferri, and L. A. Lugiato, “Coherent imaging with pseudo-thermal incoherent light,” J. Mod. Opt. **53**(5-6), 739–760 (2006). [CrossRef]

**24. **L. Gao, J. Xiong, L. F. Lin, W. Wang, S. H. Zhang, and K. G. Wang, “Interference from nonlocal double-slit with pseudo-thermal light,” Opt. Commun. **281**(10), 2838–2841 (2008). [CrossRef]

**25. **G. Scarcelli, V. Berardi, and Y. H. Shih, “Can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?” Phys. Rev. Lett. **96**(6), 063602 (2006). [CrossRef] [PubMed]

**26. **A. Gatti, M. Bondani, L. A. Lugiato, M. G. A. Paris, and C. Fabre, “Comment on ‘can two-photon correlation of chaotic light be considered as correlation of intensity fluctuations?’,” Phys. Rev. Lett. **98**(3), 039301, discussion 039302 (2007). [CrossRef] [PubMed]

**27. **S. H. Zhang, L. Gao, J. Xiong, L. J. Feng, D. Z. Cao, and K. G. Wang, “Spatial interference: from coherent to incoherent,” Phys. Rev. Lett. **102**(7), 073904 (2009). [CrossRef] [PubMed]