## 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

Interference is an essentially important topic in optical physics, resulting in many interesting phenomena and important applications. The key physics lying behind interference is the superposition principle. After the birth of quantum physics, it was realized that superposition of multiple single-photon paths plays a key role in the traditional optical interference phenomenon, which is usually known as Dirac’s famous statement: *“Each photon interferes only with itself. Interference between different photons never occurs”*[1].

In 1956, Hanbury Brown and Twiss (HBT) introduced the second-order correlation measurement, and reported a new type of interference effect between independent photons, i.e., the bunching effect of thermal light [2, 3]. Soon after, it was realized that superposition of two indistinguishable two-photon paths plays a key role in the HBT interferometer [4]. To explain it briefly, as shown in Fig. 1, for every pair of independent photons, there are two indistinguishable paths for the pair of photons to trigger a coincidence count. The phase of the complex amplitude of a two-photon path is composed of two components: the initial random phase component *ϕ*_{s1} + *ϕ*_{s2} with *ϕ _{si}* being the initial random phase of the point source emitting photon

*si*(

*i*= 1, 2), and the propagation phase component related to the optical paths for the photons propagating from the sources to the respective detectors [5]. Since the amplitudes of the two paths are always of the same initial random phase

*ϕ*

_{s1}+

*ϕ*

_{s2}, their interference term will survive in the ensemble average, leading to the constructive or destructive two-photon interference.

Later, Mandel gave a detailed theoretical analysis of HBT-type two-photon interference between two independent sources with random phases, and predicted that the visibility of two-photon interference fringes has a maximum value of 50% for classical light [6]. In general, for the case of two-photon interference with independent light sources, the coincidence counts consist of two parts: (i) The self-correlation part with the pair of photons from the same source which contributes a constant to the correlation function. (ii) The cross-correlation part with the pair of photons from different sources, which is dominated by the superposition of the two indistinguishable paths as depicted in Fig. 1, resulting in the two-photon interference fringes. It is the existence of part (i) that will low down the visibility of two-photon interference fringes. For a quantum source such as the single-photon state, the self-correlation contribution could be eliminated, and therefore giving rise to a 100%-visibility two-photon interference [6]. However, for a classical light, the self-correlation always contributes, which makes the visibility of two-photon interference fringes not exceeding 50%. This property of two-photon interference with independent random-phase sources was further discussed by Paul [7], Ou [8] and Klyshko [9], and confirmed experimentally for both the quantum light [10–12] and classical light [13–15]. Nevertheless, when one considers the multi-photon interference, the visibility could be higher than 50% for classical lights based on the third- and higher-order coherence [16–19].

In this paper, without the introduction of third- and higher-order correlation measurement, we explore another way to achieve high visibility two-photon interference with classical light. Instead of reducing or even removing the self-correlation contribution as in the case of quantum sources, we increase the cross-correlation contribution by introducing the superposition of multiple indistinguishable two-photon paths (path number >2) to enhance the two-photon interference effect of a classical light. This could be practically realized by introducing a random-phase grating to modulate the wavefront of a coherent light.

The paper is organized as follows. In Sec. 2, we described the structure of the random-phase grating, which was followed by a detailed theoretical study on both the first-order and the second-order spatial correlations of a coherent light transmitting through the random-phase grating. The experimental verification on the theoretical predictions was given in Sec. 3. And finally, we summarized the paper in Sec. 4.

## 2. Theoretical model and results

#### 2.1. Random-phase grating

The random-phase grating is shown schematically in Fig. 2(a). It is a transmission *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

To clearly illustrate the single-photon and two-photon interference effect of the light field transmitting through the random-phase grating, we will calculate the first-order and second-order correlation functions of the transmitting field in the Fraunhofer zone, i.e., in the focal plane of a lens put behind the random-phase grating, as shown in Fig. 2(b). For simplicity, we assume that the coherent light incident normally onto the random-phase grating is a plane wave and a single-mode one, in one-dimensional case, the field operator on the detection plane is expressed as [20–22]

*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, 21]

*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

^{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, 21]

*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.

Thus, the normalized second-order spatial correlation function can be calculated as (see Appendix B)

^{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.

The visibility of two-photon interference fringes can be calculated through a formula
$V=\left({g}_{\mathit{max}}^{\left(2\right)}-{g}_{\mathit{min}}^{\left(2\right)}\right)/\left({g}_{\mathit{max}}^{\left(2\right)}+{g}_{\mathit{min}}^{\left(2\right)}\right)$, where
${g}_{\mathit{max}}^{\left(2\right)}$ and
${g}_{\mathit{min}}^{\left(2\right)}$ are the peak and valley, respectively, of the interference fringes described by Eq. (6). It is not hard to find out that these multiple-slit two-photon interference functions are peaked at the same position differences satisfying (*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.

In the following Sec. 3, we will give an experimental verification on the high-visibility two-photon interference fringes described by Eq. (6) for a coherent light transmitting through the random-phase gratings.

## 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.

Figure 5 shows the measured single-photon and two-photon interference patterns on the detection plane (i.e., the focal plane of the lens L) at different conditions, in which the empty circles are the experimental results while the red curves are the theoretical fits, respectively.

*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).

*Results for random-phase grating* — When the SLM was loaded with the random phases, the random-phase grating was constructed. In this case, there should be no stationary single-photon interference fringes since the phase difference between every two slits changes randomly with time. The second column of Fig. 5 shows the experimental results, in which each one is an intensity average over 10000 frames of the intensity distribution measured by the CCD camera, corresponding to 10000 realization of the random elementary phase *ϕ* which is uniformly distributed within [0, 2*π*]. Note that the elementary phase *ϕ* was kept to be a fixed value with a duration time of 500 ms for each intensity measurement, but it changed randomly from one measurement to the other. It is seen that the single-photon interference fringes were almost erased, leaving an intensity distribution enveloped by a diffraction profile (see the respective red curves) described by Eq. (3). One notes that there are still some residual intensity fluctuations which deviate from the intensity envelop predicted by Eq. (3). This is mainly due to the unavoidable phase flicker of the SLM during each intensity measurement.

Although the single-photon interference fringes disappear with the random-phase grating, two-photon interference fringes appear as predicted by Eq. (6). The third column of Fig. 5 shows the measured second-order spatial correlation function *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})〉) [18,23] by using the same 10000 frames of the measured intensity distributions as those used in the second column of Fig. 5. Here the red curves are the theoretical fits using Eq. (6). It is seen that the second-order correlation function exhibits itself in the form of high quality interference fringes, in good agreement with the theoretical prediction by Eq. (6).

It is seen from the experimental results shown in the third column of Fig. 5 that, the two-photon interference fringes are peaked at the position differences *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.

*Further discussions* — One may note that, except for the *N* = 2 case, the random phases encoded on the slits of the random-phase grating are not fully independent but indeed correlated with respect to each other. Therefore, our case is different from the case discussed by Mandel [6], where classical lights with fully independent random phases are considered, and which in fact corresponds to the *N* = 2 case in our configuration. For classical lights with fully independent random phases, the visibility of two-photon interference fringes cannot exceed 50%, as also confirmed by the *N* = 2 case in our configuration (*V* = 44.9%, see the two-photon interference fringes in the top first one of the third column in Fig. 5). More importantly, our results show that, by appropriately controlling the random phase structure encoded on a coherent light field, one could achieve two-photon interference fringes with the visibility exceeding 50%. It is known that controlling optical phase plays a key role in the single-photon interference effect [22], our results show that it may also play an important role in controlling the high-order coherence of light.

## 4. Summary

In summary, we have designed a kind of two-photon grating with a special random-phase structure, through which the single-photon interference is smeared out but the two-photon interference appears. With such a random-phase grating, superposition of multiple indistinguishable two-photon paths is introduced, which leads to high-visibility two-photon interference fringes of classical light. Theoretically, the visibility of the two-photon interference fringes for a coherent light transmitting through a *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

For the case when a coherent light, which is the eigenstate of the annihilation operator *â*, is incident normally onto the random-phase grating as shown in Fig. 2(b), one arrives at

*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

Eq. (6) can be calculated from Eq. (5) as

## Acknowledgments

This work was supported by the 973 program ( 2013CB328702), the CNKBRSF ( 2011CB922003), the NSFC ( 11174153, 90922030 and 10904077), the 111 project ( B07013), and the Fundamental Research Funds for the Central Universities.

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