## Abstract

The spatial second-order interference of two independent pseudothermal light beams in a Hong-Ou-Mandel interferometer is studied experimentally and theoretically. The similar cosine modulation in the second-order coherence function as the one with entangled-photon pairs in a Hong-Ou-Mandel interferometer is observed. Two-photon interference based on Feynman’s path integral theory is employed to interpret the results. The experimental results and theoretical simulations agree with each other very well.

© 2013 osa

## 1. Introduction

Two-photon interference is extensively studied in quantum optics and quantum information, among which, “Hong-Ou-Mandel (HOM) dip” or “Shih-Alley dip” is one of the most famous two-photon interference phenomena [1–3]. Most of the reported experiments with HOM interferometers are in the temporal regime with entangled-photon pairs [4–9] (and references therein) or classical light [10–13]. Recently, Kim *et al.* observed cosine modulation in the spatial second-order coherence function in a HOM interferometer with entangled-photon pairs [14]. In their experiments, they observed two-photon anticorrelation when two detectors are in the symmetrical positions. Based on the conclusions demonstrated by Saleh *et al.* that a number of spatial properties of entangled-photon pairs are analogous to those of ordinary photons generated by incoherent sources [15], two-photon anticorrelation should also be observed with thermal light in a HOM interferometer. However, it is predicted by Olivares *et al.* and experimentally verified by the same group that there is no correlation when two thermal light beams are incident to a HOM interferometer and these two detectors are in the symmetrical positions [16,17]. Hence it is important to understand what is the difference between the spatial second-order interference with thermal light and entangled-photon pairs in a HOM interferometer. In this paper, we will report the spatial second-order interference pattern with two independent pseudothermal light beams in a HOM interferometer and discuss the difference between the results with thermal light and the ones with entangled-photon pairs. We also employed our results to interpret some reported experiments with HOM interferometers.

## 2. Experiments

Our experimental setup is shown in Fig. 1. Two independent pseudothermal light sources are created by dividing a laser beam into two and sending them to two independent rotating ground glasses (RG), respectively [18]. The rotating frequencies of RG1 and RG2 are 34.29Hz and 32.21Hz, respectively. The temporal coherence lengths of these two pseudothermal light beams are 83.9 *μ*s and 90.8 *μ*s, respectively. The laser employed in our experiment is single-mode continuous wave laser with central wavelength 780 nm and frequency bandwidth 200 kHz (Newport, SWL-7513). The two-photon coincidence counts are measured by single-photon detectors (PerkinElmer, SPCM-AQRH-14-FC) and two-photon coincidence counting system (Becker & Hickl GmbH, SPC630), which are not shown in the figure. Two identical single-mode fibers with 5 *μ*m diameters are employed to couple the photons into the two single-photon detectors, respectively. The two-photon coincidence count time window is 61.6 ns. A half wave plate is used to control the polarization of one beam so that we can measure the second-order coherence functions when the polarizations of these two beams are orthogonal and parallel, respectively. Both focal lengths of the lens L_{1} and L_{2} are 50 mm. The distance between L*j* and RG*j* is 80 mm (*j* = 1, and 2). All the distances between the source planes and the detection planes are 910 mm.

The experimental results are shown in Fig. 2. The measurement interval for each dot is 200 seconds and these four different sets of data are measured with the same experimental parameters except the polarization of one beam is changed in the last measurement. The polarizations of these two beams are orthogonal in the first three measurements (Figs. 2(a), 2(b) and 2(c)) and then they are changed to be parallel for the fourth measurement (Fig. 2(d)) by rotating HP in the experimental setup. Figure 2(a) is the normalized second-order coherence function of the field emitted by S_{1} when S_{2} is blocked. Figure 2(b) is the normalized second-order coherence function of the field emitted by S_{2} when S_{1} is blocked. Both the spatial coherence lengths at the detection planes of these two sources are calculated to be 1.15 mm. Thus both the sizes of these two thermal sources are 0.6172 mm. Figures 2(c) and 2(d) are measured when the polarizations of these two light beams are orthogonal and parallel, respectively. The red lines in Figs. 2(c) and 2(d) are theoretical simulations by employing following derived Eqs. (8) and (9), respectively. The measured value of *g*^{(2)}(0) in Fig. 2(c) is 1.39 ± 0.07, which is less than the theoretical maximum value 1.5. The measured value of *g*^{(2)}(0) in Fig. 2(d) is 1.00 ± 0.05. The distance between the middle points of S_{1} and S′_{2} is calculated to be 1.5337 mm. Even if considering the lengths of S_{1} and S_{2} may not be exactly the same in the experiments, the theoretical simulations agree with the experimental results very well.

The reason why we first measure the spatial Hanbury Brown and Twiss (HBT) bunching peak of individual thermal source is to assure the symmetrical positions of these two detectors. It is well-known that the spatial two-photon bunching peak of thermal light is observed when the two detectors are in the symmetric positions [19]. Once one of the detector’s position is fixed, the symmetric position of the other detector is uniquely determined, no matter where the thermal source be (Figs. 2(a) or 2(b)) or more than one independent thermal sources are employed (Figs. 2(c) and 2(d)). In our experiments, we scan D_{1} to observe the second-order coherence functions in all four different conditions when the position of D_{2} is fixed.

## 3. Theory

Both classical and quantum theories can be employed to interpret the experimental results [20, 21]. In order to comparing the results between thermal light and entangled-photon pairs, we will employ two-photon interference based on Feynman’s path integral theory to interpret our experimental results [3, 22, 23]. We assume the intensity of these two thermal sources are the same. There are eight different ways for two photons to trigger a two-photon coincidence count in our experimental setup. For instance, there are two different ways when both photons are emitted by S_{2}. The corresponding Feynman’s two-photon propagators are *A*_{a21,b22} and *A*_{a22,b21}, respectively, where *A*_{a21,b22} is the Feynman’s two-photon propagator that photon *a* goes through S_{2} and detected by D_{1} and photon *b* goes through S_{2} and detected by D_{2}. All the eight different propagators are *A*_{a21,b22}, *A*_{a22,b21}, *A*_{a11,b12}, *A*_{a12,b11}, *A*_{a22,b11}, *A*_{a21,b12}, *A*_{a11,b22}, and *A*_{a12,b21}. If these eight different ways are indistinguishable, the second-order coherence function in the experimental setup is [22]

**r**

*,*

_{β}*t*) is the space-time coordinate of photon detection event by D

_{β}*(*

_{β}*β*= 1, and 2) and 〈...〉 is ensemble average.

*A*=

_{aim,bjn}*e*

^{iφaim,bjn}

*A′*, where the extra phase,

_{aim,bjn}*φ*, is the sum of the initial phases and the phase changes of photons

_{aim,bjn}*a*and

*b*due to BS

_{2}.

*A′*is two-photon propagation function. Supposing the phase changes of photons transmitted and reflected by BS are 0 and

_{aim,bjn}*π*/2, respectively [24], the extra phases of these eight Feynman’s two-photon propagators are shown in Table 1. Where

*φ*is the initial phase of photon

_{a}_{β}*a*emitted by S

*(*

_{β}*β*= 1, and 2), other symbols are defined similarly. The phase changes due to BS

_{1}do not affect the results, for the rotating ground glasses give random phases to the scattered photons [13,18].

Substituting the extra phases into Eq. (1) and with straightforward derivation, it is easy to get

*π*phase difference between these two pathes, respectively [1–3]. In general, there are 64 terms after finishing the modulus square. However, when S

_{1}and S

_{2}are two independent thermal light sources, all the terms including the phase

*φ*,

_{αj}*φ′*,

_{αj}*φ″*, and

_{αj}*φ′″*(

_{αj}*α*=

*a*, and

*b; j*= 1, and 2) will disappear, for the ensemble average of these terms equal zero, respectively. Eq. (2) can be simplified as

*A′*is the Feynman’s one-photon propagator corresponding to photon

_{bjn}*b*goes to D

*via S*

_{n}*[25, 26],*

_{j}**k**is the wave vector of photon emitted by S

_{jn}*goes to D*

_{j}*and*

_{n}**r**is the position vector from S

_{jn}*to D*

_{j}*.*

_{n}*ω*is the central frequency of the laser and

*t*is the photon detection time of D

_{n}*. In our experiments, the bandwidth of the laser is 200 kHz and we only measure spatial correlation, thus we will drop the temporal part in the following calculations.*

_{n}Substituting Eqs. (4) and (5) into Eq. (3) and considering both sources have finite lengthes, the second-order coherence function can be calculated. The first and second terms on the right sides of Eq. (3) are ordinary two-photon spatial bunching peaks of thermal light, which is calculated in many textbooks of quantum optics (for instance, see [23, 24]). The last two terms of Eq. (3) can be calculated with the same method by taking the equation,

*x*

_{1}and

*x*

_{2}are the transverse spatial coordinates of D

_{1}and D

_{2}, respectively. Other symbols are defined in Fig. 1. The first and second terms on the right of Eq.(7) correspond to typical HBT spatial bunching of the fields emitted by S

_{1}and S

_{2}, respectively. The third term corresponds to the two-photon interference when one photon comes from each source, respectively.

Equations (3) and (7) can be employed to interpret above experimental results. It is easy to see when the sizes of two pseudothermal sources are different, the second-order coherence functions will be different even if the relative position of the sources, *i.e.*, *d*, is fixed. In our experiments, we consider both the lengths of S_{1} and S_{2} equal *l* for simplicity, which is also our experimental condition.

When the polarizations of these two light beams are orthogonal, the last two terms in Eq. (3) should be changed into |*A′*_{a22,b11}|^{2} + |*A′*_{a21,b12}|^{2} and |*A′*_{a11,b22}|^{2} + |*A′*_{a12,b21}|^{2}, respectively. For these different ways to trigger a two-photon coincidence count are distinguishable. Probabilities instead of probability amplitudes should be added to get the final probability distribution [22]. The normalized second-order coherence function in this condition can be expressed as

_{1}, S

_{2}, and one photon comes from each source, respectively. While the coincidence counts in the bunching peak only consist of both two photons come from one source (either S

_{1}or S

_{2}). One photon comes from each source respectively does not contribute to the peak for these two photons are distinguishable.

When the polarizations of the two beams are parallel, the normalized second-order coherence function can be simplified as

*i.e.*,

*x*

_{1}=

*x*

_{2},

*g*

^{(2)}(0) always equals 1 for all different relative positions of these two sources. This is due to the constructive interference when both photons are from the same source cancels the destructive interference when one photon comes from each source, which correspond to the second and third terms on the right hand side of Eq. (9), respectively.

When the relative distance between these two sources, *i.e.*, *d*, does not equal zero, there is a cosine modulation in the spatial second-order coherence function. We also analyze how the cosine modulation changes with *d*. Figure 3 shows the simulated results of *g*^{(2)}(*x*_{1} − *x*_{2}) for different values of *d*. The parameters are the same as the ones in the experiments except the visibility is ideal in simulation. There is no correlation peak or dip when *d* = 0, for *g*^{(2)}(*x*_{1} −*x*_{2}) always equals 1. As *d* increases, the period of the modulation decreases. However, *g*^{(2)}(0) always equals 1 for different values of *d*, which is consistent with the conclusions given by Olivares *et. al.*[16, 17].

## 4. Discussions

In the above, we have successfully interpreted the observed spatial second-order interference patterns in our experiments. Comparing the observed second-order interference pattern shown in Fig. 2(d) in our experiments with Fig. 2(a) in Kim *et al.*’s paper with entangled photon pairs [14], it is easy to find the main difference between these two is there is a constant background in our experiments while there is no background in the entangled-photon pair experiment. Based on the calculations above, the background two-photon coincidence counts correspond to the first and second terms on the right hand side of Eq. (7), which are the coincidence counts when both photons are emitted by the same source (either S1 or S2). If there is a way to eliminate these background coincidence counts, the spatial second-order interference pattern with thermal light in a HOM interferometer would be the same as the one with entangled-photon pairs. In fact, the background coincidence counts in our experiments can be eliminated by employing two thermal sources with different central frequencies and detecting the two-photon coincidence counts with sum frequency generation [12] or two-photon absorption [27]. The two-photon detection system can be specially designed that it only responds to the coincidence count that two photons comes from different sources [28]. Hence both quantum and classical light sources can be employed to observe two-photon interference pattern without background. The only difference is that when quantum light source is employed, the entangled-photon pairs make sure there are only two photons and only the last two terms of Eq. (3) contribute to the final observed two-photon interference pattern [1–3]. While in classical light source case, the special detection systems make sure only one photon from each source can trigger a two-photon coincidence count, which also means only the last two terms of Eq. (3) contribute to the observed two-photon interference pattern [12,28]. Hence two-photon anticorrelation with thermal light beams in a HOM interferometer is observable when a special two-photon detection system as suggested above is employed. If normal two-photon detection systems just as the one in our experiments and the one in Brida *et al.*’s experiment [17] are employed, no correlation will be observed when these two detectors are in the symmetrical positions. It is noteworthy that when the light is strong and PIN diodes are employed to detect the intensities, the background can be removed by employing DC-blocks after these two detectors and before multiplying these two signals [29].

With our experimental and theoretical results, the spatial interference experiments in a HOM interferometer with entangled-photon pairs or thermal light can be understood better. For instance, Kim *et al.* claimed by changing the angle of the mirror in their experiments, the observed temporal HOM dip can be changed to a peak. Hence they concluded that “both the fermionic and bosonic properties of the twin photons can be found [14].” This statement is not accurate. The true two-photon coincidence count comes from the same entangled-photon pair in the entangled-photon pair experiments. Changing the angle of the mirror in their experiments is equivalent to changing the relative position of these two detectors. When the two detectors are in the symmetrical positions, temporal HOM dip is always observed due to the destructive interference. When the relative distance between these two detector increases, the observed *g*^{(2)}(*x*_{1} − *x*_{2}) are modulated by a cosine function as shown in Fig. 2(d) in our experiments. If we fix the relative position of these two detectors when *g*^{(2)}(*x*_{1} − *x*_{2}) gets its maximums, just as Kim *et al.* did in their experiments, temporal HOM dip can be changed into a peak. However, this is not fermionic properties of entangled-photon pairs. It is bosonic properties of entangled-photon pairs.

As a second example, let us employ our results to analyze the experiment with thermal light in a HOM interferometer by Brida *et al.*[17]. Based on the conclusion that there is no correlation when two thermal light beams are incident to a HOM interferometer, the authors give an interesting scenario that an illusionist can employ a third beam to distinguish whether there is a beam splitter or not when two thermal light beams are mixed. Their method is necessary only when the following conditions: (I) these two thermal sources have equal sizes and are symmetrical about the beam splitter, (II) the distances between the thermal sources and the detection planes are equal, (III) the polarizations of these two beams are parallel, are satisfied. In this special case, there is no correlation for all the relative positions of these two detectors when there is a beam splitter as shown in Fig. 3(a). Hence a third beam is needed to verify whether there is a beam splitter or not as suggested by Brida *et al.*[17]. However, in most cases, the illusionist does not need the third beam to distinguish whether there is a beam splitter or not. For as verified in our experiments, when there is a beam splitter and the two detectors are not in the symmetrical positions, the correlation is usually not zero. When there is no beam splitter, the correlation is always zero, no matter whether the two detectors are in the symmetrical positions or not. Hence the illusionist can distinguish these two situations by measuring the correlation functions when these two detectors are not in the symmetrical positions.

## 5. Conclusions

In conclusion, we have observed the spatial second-order interference pattern with two independent pseudothermal light beams in a HOM interferometer. Two-photon interference theory based on Feynman’s path integral theory is employed to interpret our experiments. It is suggested that the background in the interference pattern can be eliminated by employing sum frequency generation or two-photon absorption. It is also predicted that *g*^{(2)}(*x*_{1} − *x*_{2}) is a constant when the two sources have equal size and are symmetrical about the 50:50 non-polarized BS and photons emitted by these two thermal sources have the same polarization.

## Acknowledgments

The authors acknowledge the helpful comments from the anonymous reviewers. This project is supported by the National Basic Research Program of China (973 Program) under Grant No. 2009CB623306, International Science & Technology Cooperation Program of China under Grant No. 2010DFR50480, and the Fundamental Research Funds for the Central Universities.

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