Active control on high-order coherence and statistic characterization on random phase fluctuation of two classical point sources

Young’s double-slit or two-beam interference is of fundamental importance to understand various interference effects, in which the stationary phase difference between two beams plays the key role in the first-order coherence. Different from the case of first-order coherence, in the high-order optical coherence the statistic behavior of the optical phase will play the key role. In this article, by employing a fundamental interfering configuration with two classical point sources, we showed that the high- order optical coherence between two classical point sources can be actively designed by controlling the statistic behavior of the relative phase difference between two point sources. Synchronous position Nth-order subwavelength interference with an effective wavelength of λ/M was demonstrated, in which λ is the wavelength of point sources and M is an integer not larger than N. Interestingly, we found that the synchronous position Nth-order interference fringe fingerprints the statistic trace of random phase fluctuation of two classical point sources, therefore, it provides an effective way to characterize the statistic properties of phase fluctuation for incoherent light sources.

Optical high-order coherence effect was first reported by Hanbury Brown and Twiss (HBT) in 1956, where interference among randomly distributed first-order incoherent point emitters forms the bunching effect of a thermal light source in the far-field plane 1,2 . Since then, much attention has been paid to the field of optical high-order coherence, leading to discovery of many intriguing interference effects [3][4][5][6] and applications such as ghost imaging 7-10 , subwavelength interference and optical lithography 6,[11][12][13][14][15][16] , and super-resolving measurements 4,17-20 . In the field of optical coherence, one of the most important fundamental issues is two-beam interference, which is the basis to understand various interference effects observed in various interferometers, and it can be historically traced back to Young's double-slit interference, where the stationary phase difference between two beams plays the key role in the first-order coherence 21 . High-order optical coherence with various light sources [22][23][24][25][26][27][28] was extensively studied, and much attention was paid to the quantum or classical properties of light sources, offering the basic understanding of many novel high-order coherence effects [4][5][6] . In contrast to the case of first-order coherence, although high-order coherence of two completely independent light sources are well studied and understood 23,24 , effects of optical phase on the high-order optical coherence are largely ignored. Interestingly, recent progresses on high-order coherence of classical light show that modulation on the wavefront of classical light sources will generate novel interference effects such as subwavelength interference 16,17,[29][30][31][32] and non-Rayleigh speckles with tailored intensity statistics 33 , indicating the importance of optical phase in high-order optical coherence.
In this article, we studied the effects of optical phase on high-order optical coherence in a fundamental interfering configuration with two classical point sources. We showed that, different from the case in the first-order optical coherence where a stationary phase difference between two beams is important, in high-order optical coherence the statistic behavior of the optical phase difference between light beams will play an important role. Therefore, it is possible for one to actively design the high-order optical coherence for particular applications such as subwavelength interference and high spatial resolution optical lithography through control on the statistic behavior of optical phase of light sources. Furthermore, we found that the high-order optical interference pattern reveals the statistic properties of the relative phase fluctuation of interfering light sources, which is of fundamental importance in optical science.

Results
Theoretical Analysis. Figure 1 shows the fundamental scheme to study the first and high-order coherence effects between two spatially separated classical point sources S A and S B , in which is single-photon detector in the far-field observation plane, d is the distance between S A and S B , and z is the distance between the observation plane and the source plane, respectively. When the relative phase difference ϕ between the two point sources is stationary, they are first-order coherent with respect to each other, and the scheme is essentially a Young's double-slit interference scheme, which will show an intensity interference fringe I(x) ∝ 1 + cos(kdx/z) with k being the wave number and x being the observation coordinate in the far-field plane. However, due to the complimentary effect between the first-order coherence and the high-order coherence 23,24,34 , no high-order interference fringe can be observed in this case. In the other extreme case, when the two point sources are completely independent and the relative phase difference ϕ varies randomly in the range [0, 2π), it is evident that the two point sources are first-order incoherent, but one can observe high-order interference fringes, for example, two-photon interference fringe with respect to the spatial separation between two observation points (x 1 − x 2 ). One may already notice that the relative phase difference ϕ plays an important role in not only the first-order coherence but also the high-order coherence. In the following, we will show that the high-order coherence properties can be actively designed by controlling the statistic behavior of the relative phase difference ϕ.
Let's first consider the simplest special case when the relative phase difference ϕ varies randomly between two critical values {θ 1 , θ 2 } with corresponding probability P 1 and P 2 , respectively. According to the complimentary rule, the high-order interference fringes can be observed when the intensity interference fringes between S A and S B are completely washed out, i.e., ϕ θ θ where  means an ensemble average. This is satisfied when one sets θ 2 − θ 1 = π and P 1 = P 2 = 0.5. Under this condition, the normalized second-order intensity correlation function can be calculated as where the subscript "2" in g 2 (2) indicates the number of the critical phase θ i . One sees that the second-order interference pattern is completely different from that discussed previously. Besides the well-known cosine fringe cos (kd(x 1 − x 2 )/z) with respect to (x 1 − x 2 ), an additional set of cosine fringe cos(kd(x 1 − x 2 )/z + 2θ 1 ) with respect to (x 1 + x 2 ) appears. The second-order interference pattern is the sum of these two kinds of interference fringes, which makes the total interference pattern quite richer as shown in Fig. 2. One notes that the second-order interference fringe is dependent on both the positions and the scanning mode of two photodetectors. For example, no second-order interference fringe can be observed when one fixes detector D 2 and scans detector D 1 along line Ia in Fig. 2(a), while a periodical cosine fringe with a visibility of 100%, which is usually thought as a property of two-photon interference with quantum single-photon sources 23,24 , is obtained when one scans the detector D 1 along line Ib (see Fig. 2(b)). More interestingly, when one sets the two detectors at exactly the same position x 1 = x 2 = x and scans them together along line II, synchronous position subwavelength two-photon interference fringe can be observed with an effective wavelength being half of the original wavelength of the point sources (see , and it evolves to Two elementary fringes in the form of are included in Eq. (4), which are phase-shifted by π with respect to each other. One sees that, it is the sum of these two elementary fringes that results in subwavelength interference with an effective wavelength of λ/2. Furthermore, the visibility of the interference fringes grows with N. Interestingly, the high-order interference fringe (N ≥ 2) reveals the statistic fluctuation of the relative phase difference ϕ between S A and S B . From Eq. (4), one can see that the peak amplitude and the peak position of the interference fringe are determined by the probability P i and the critical phase value θ i , respectively. Therefore, one can extract the statistic information of the phase variation ϕ between two point sources such as the critical relative phase value θ i and its probability P i simply by measuring the synchronous position Nth-order correlation function. Although optical noise may also be amplified during the correlation treatment, one should note that the signal is amplified much more as compared to optical noise, therefore, the signal-to-noise ratio and the contrast-to-noise ratio will be improved in high-order correlation measurements, which was well verified both experimentally and theoretically 35,36 .
When the relative phase difference ϕ varies randomly among three critical values {θ 1 , θ 2 , θ 3 }, each with a probability P 1 , P 2 and P 3 , respectively. Under the condition that the first-order interference fringe between S A and S B is washed out, and the normalized synchronous position Nth-order interference fringes can be described as are included in Eq. (6). Again, one can see that, the fringe peak amplitude is determined by the probability P i and the peak position is determined by the critical phase value θ i . Eq. (5) can be satisfied with various sets of {P i , θ i } (see Methods for the general way to find out the solution {P i , θ i }), therefore, various interference fringes can be actively designed according to the requirement of practical applications. For example, by setting the relative phase difference ϕ to vary randomly among {0, − 2π/3, − 4π/3} with equal probability 1/3, one can get a set of synchronous position Nth-order interference fringe (N ≥ 3) with an effective wavelength of λ/3, where λ is the original wavelength of the point source, as shown in Fig. 3. On the other hand, one can get the statistic variation information (P i , θ i ) of the relative phase difference ϕ simply by measuring the synchronous position high-order interference fringes, as we will demonstrate experimentally in the following section. Note that, to clearly show the relationship between the peak amplitude and position of the interference fringe and the statistic parameters (P i , θ i ) of the relative phase difference ϕ, the vertical axis of Fig. 3 is scaled by a factor 2 N . Experimental verification. Figure 4 shows the schematic diagram to experimentally demonstrate active control on high-order coherence of two classical point sources. A 780-nm single-mode continuous-wave laser beam was focused to be a small point to mimic a point source. The beam was then input into a Michelson interferometer, where BS was a 50:50 non-polarized beam splitter, M was a reflection mirror and SLM represented a reflection-type phase-only spatial light modulator (HEO 1080P from HOLOEYE Photonics AG, Germany). By slightly tilting the end mirror M to deviate from the perpendicular position (with respect to the incident light on the arm) by a small angle α, one could generate two effective point sources S A and S B as shown in the inset of Fig. 4. The relative phase difference ϕ between S A and S B was controlled through the phase-only spatial light modulator SLM. By adjusting the tilting angle α, the distance d between S A and S B was set to be d = 1.43 mm, and the distance between the source plane (determined by S A and S B ) and the detection plane (a CCD camera) was set to be z = 136 cm, respectively. The intensity distribution on the detection plane for each relative phase difference ϕ was recorded by the CCD camera with an acquisition time of 1.0 ms. The high-order interference fringes were then calculated through an ensemble average over 1000 frames of the intensity distribution recorded by the CCD camera. Figure 5 shows the measured two-photon interference fringes with different position and scanning mode of detectors when the relative phase difference ϕ varied randomly between {0, π} with equal probability 0.5. Figure 5(a) corresponds to the case when one detector was fixed while the other detector was scanned along line Ia in Fig. 2(a), no interference fringe was observed in this scanning mode. Figure 5(b) is the case when one fixed one detector (but at different position from that in Fig. 5(a)) and scanned the other detector along line Ib in Fig. 2(a). One sees that two-photon interference fringes with a visibility of 98%, which is close to the theoretically predicted 100% (see Eq. (2) and Fig. 2(b)), could be observed in this configuration even with two classical point sources. Furthermore, synchronous position subwavelength two-photon interference fringes with an effective wavelength of λ/2 = 390 nm were observed when one scanned the two detectors simultaneously along line II in Fig. 2(a), in good accordance with the theoretical prediction by Eq. (2).
More interestingly, synchronous position subwavelength interference fringes with an effective wavelength of λ/3 = 260 nm were generated when the relative phase difference ϕ varied randomly among {0, 2π/3, 4π/3} with equal probability P 1 = P 2 = P 3 = 1/3, as shown in Fig. 6(a). As predicted theoretically by Eq. (6), the fringe pattern can be actively controlled simply by adjusting the critical value θ i and its probability P i as long as Eq. (5) is satisfied. Figure 6(b,c) Fig. 3, the vertical axes in Fig. 6 are again scaled by a factor 2 N . On the contrary, one can also extract the statistic parameters of the random fluctuation of ϕ from the measured synchronous position high-order interference fringes by fitting the experimental data based on Eq. (6). During the fitting process, we set the statistic parameters {P i , θ i } as free fitting parameters under the restriction ∑ = P 1 i , while the experimental values were taken for k, d and z. The extracted parameters were found to be ∈ .
.  Fig. 6(a) Fig. 6(b) Fig. 6(c), respectively. As compared to the corresponding theoreti- cal parameters, good agreement with an experimental deviation less than 10% (most of them were within 5%) is achieved, which verified the effectiveness of the proposed method. The slight mismatching is mainly due to the deviation of the relative phase difference ϕ set by SLM from the theoretically required ones because of the imperfect phase linearity and stability of SLM and the vibration instability of the environment. , the intensity interference fringes between S A and S B will be smeared out under the condition One sees that it is composed of M elementary fringes, and the ith elementary fringe is characterized by a probability amplitude P i and a phase shift θ i . Therefore, one can extract the statistic information of the random variation of the relative phase difference ϕ from the Nth-order interference fringe pattern. Moreover, by equally distributing θ i within [0, 2π) and each with a equal probability of 1/M, one can achieve subwavelength interference fringes with an effective wavelength of λ/M.
It is worth noting that the mechanism to generate subwavelength interference here is completely different from that based on the concept of photonic de Broglie wavelength 6,15 , where entangled M-photon source is required and M-photon correlation measurement will be performed. In contrast, two classical point sources with specially designed random phase fluctuation are used and N-photon correlation measurement (N ≥ M) are performed in our case. The subwavelength interference demonstrated here is also different from those with chaotic thermal light sources, in which the subwavelength interference was realized in a configuration with respect to the position separations among different detectors, i.e., the detectors are located symmetrically and scanning in opposite directions 35,36,[37][38][39][40][41][42] . In our case, the synchronous position subwavelength interference is realized when the relative phase difference ϕ between two incoherent point sources changes randomly only among several discrete critical values 2mπ/M =  m M ( 1, 2, , ) with equal probability and all detectors are at the same spatial position and scanning in the same direction. One may also note that there will be no interference fringes at all with thermal light sources in the synchronous position scanning scheme 39 . In addition, we realized the second-order interference fringes with respect to position separation of detectors (x 1 − x 2 ) with a 100% visibility (see Figs 2(b) and 5(b)), while the visibility of the second-order interference fringes is always less than 50% with thermal light 35,36,[37][38][39][40][41][42] .
Oppel et al. 17 reported the (N − 1)th-fold subwavelength interference with thermal light through Nth-order correlation measurement by putting the (N − 1) detectors at the so-called magic angles, and Cao et al. 32  were realized in the configuration when detectors or light sources were at different positions, that is, the demonstrated subwavelength interference is with respect to the position difference among different detectors 17 or in the first-order intensity distribution 32 , respectively. While our subwavelength interference fringes are the synchronous position one, i.e., all detectors are exactly at the same spatial position and scanning in the same direction synchronously, which is of practical importance for applications such as optical lithography 43 .
In summary, we showed that it is possible to actively control the high-order coherence of two classical point sources simply by designing the statistic fluctuation of the relative phase difference between these two point sources, indicating that the optical phase difference also plays an important role in the high-order optical coherence. Therefore, by designing the statistic fluctuation of the relative phase difference between two classical point sources, we demonstrated synchronous position subwavelength interference with an effective wavelength much shorter than the original one of the point sources, which may have potential applications in optical lithography and holography with high spatial resolution. Furthermore, the use of classical light with much high intensity in our configuration may overcome the limitation of weak intensity with entangled quantum light source in quantum optical lithography. In addition, we found that the synchronous position high-order interference fringe pattern is actually a fingerprint of the statistic fluctuation of the relative phase difference between two point sources, therefore, one can characterize the statistic distribution of the relative optical phase between two first-order incoherent light sources simply by measuring their synchronous position high-order interference fringe pattern, which may be useful for optical metrology of random processes. can be met when a closed vectorial triangle is constructed by three vectors (P i , θ i ), as shown in Fig. 7. Note that Eq. (9) is also satisfied by rotating each vector (P i , θ i ) with the same arbitrary angle θ 0 , i.e., which is exactly the same as Eq. (5) when one sets θ 0 = kdx/z. Therefore, one can conclude that once the three vectors (P i , θ i ) construct a closed vectorial triangle, Eq. (5) is satisfied and the intensity interference fringes between S A and S B will be washed out.
To generalize, for arbitrary M vectors (P i , θ i ), Eq. (7) is satisfied when these M vectors form a closed structure. In this geometric viewpoint, for the extreme case of two fully independent point sources S A and S B with the relative phase difference ϕ distributed randomly and linearly within [0, 2π), the corresponding closed vectorial structure would actually approach to a circle.