Partially coherent fields are abundant in many physical systems. While the propagation of partially coherent light undergoing diffraction is well understood, its evolution in the presence of coherent diffusion (i.e., diffusion of complex fields) remains largely unknown. Here we develop an analytic model describing the diffusion of partially coherent beams and study it experimentally. Our model is based on a diffusion analog of the famous Van Cittert–Zernike theorem. Experimentally, we use a four-wave mixing scheme with electromagnetically induced transparency to couple optical speckle patterns to diffusing atoms in a warm vapor. The spatial coherence properties of the speckle fields are monitored under diffusion and are compared to our model and to the familiar evolution of spatial coherence of light speckles under diffraction. We identify several important differences between the evolution dynamics of the spatial coherence under diffraction and diffusion. Our findings shed light on the propagation of partially coherent fields in media where multiple scattering or thermal motion lead to coherent diffusion.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Spatial correlations exist in many different physical systems, and the study of their origin and evolution is one of the primary roles of statistical physics. In optics, the propagation of spatial coherence of partially coherent light sources has attracted much attention ever since the early days of modern optics [1–3]. One of the central theorems in optics is the Van Cittert–Zernike (VCZ) theorem [4,5], which describes the spatial coherence sufficiently far from a spatially incoherent source. This theorem shows a Fourier relation between the intensity distribution on the surface of a spatially incoherent source and the spatial coherence sufficiently far from it. Physically, this implies that the boundaries of the source dictate the coherence properties of the illuminated light sufficiently far from the source. This property of the spatial coherence has been famously exploited in Michelson’s stellar interferometer to measure the size of distant radiation sources [6,7]. Hanbury Brown and Twiss (HBT) later showed that similar stellar information can be extracted by measuring intensity correlations [8–10]. While the classical theorems describe the spatial coherence at large distances from the source, more recent studies consider short propagation distances, in the region near the source (deep Fresnel region) [11,12], and show that therein the spatial coherence is propagation invariant [13–15].
The concepts and theorems derived in linear optics were later extended to interacting photons , as well as to atomic and condensed matter systems, demonstrating partial spatial coherence of electrons [17–19] and cold atoms [20–23]. The underlying assumption in all these systems is that they only exhibit ballistic transport, while any diffusive transport is negligible. Although this assumption is well justified in many cases, it does not always hold, and often diffusive transport must be taken into consideration [24–30]. Here we thoroughly investigate coherent diffusion of spatial correlations (spatial coherence) of partially coherent fields, theoretically and experimentally, and compare between diffraction and coherent diffusion of partial spatial coherence. We first present experimental results comparing the two and then present a detailed model explaining the results. Our model is general and is not limited to our specific system.
The comparison between these two distinct physical mechanisms, i.e., diffusion and diffraction, is based on the mathematical similarity between their governing equations:1) presents the familiar diffusion equation (Fick’s second law of diffusion); but as opposed to the traditional textbook examples of diffusion, such as diffusion of heat or concentration, here is complex valued, rather than real [25,26,31,32]. Hence, this equation describes coherent diffusion.
Clearly the two equations above are identical under the transformation , and accordingly diffraction can be considered as diffusion in imaginary time [27,28]. This mathematical similarity implies exciting physical analogies, where various well-known optical phenomena find their natural analogs in diffusion of complex vector fields [27,33–35]. For example, optical vortices are protected and will not unwind in both diffusion and diffraction .
In the past, coherent diffusion of complex fields was demonstrated and studied in various systems, including nuclear magnetic resonance , electromagnetically induced transparency (EIT) [27,29], and spintronics . Here, we exploit EIT based on a unique four-wave mixing scheme that was recently presented , to study diffusion of partial spatial coherence. Using this scheme, spatially patterned light beams are coherently imprinted onto the spin states of atoms in a hot vapor, such that the spins acquire the spatial amplitude and phase of the incoming light fields. After a temporal duration that is equal to the group delay in the medium, the spatially patterned spin state is coherently imprinted onto the outgoing light field, which is then detected by a camera. Since the atoms diffuse during this temporal duration, the retrieved signal indicates the atoms’ diffusion dynamics.
One of the main advantages of this scheme, as opposed to traditional EIT, is that it can be used to reach relatively long diffusion times with high signal-to-noise ratios. Therefore, it can be used to observe also high spatial frequencies for long diffusion times, which is a key component for this study.
2. EXPERIMENTAL RESULTS
The experimental arrangement used to characterize the diffusion of partially coherent fields is illustrated in Fig. 1(a). We use , which diffuses in 10 Torr of buffer gas, rendering a diffusion coefficient of . The vapor cell is illuminated by two spatially overlapping “control” beams, which are separated by a slight angle, and by a third weak “probe” beam, which is oriented along one of the control beams. Consequently, a fourth beam, denoted as “signal,” is generated in a four-wave mixing process along the orientation of the second control beam. We set the optical frequencies of the probe and control beams such that they couple, respectively, the lower and upper hyperfine states to the excited states and of the transition. The incoming probe beam is shaped using a spatial light modulator. The outgoing signal is imaged onto a CCD camera, and we use digital Fourier filtering to improve the signal-to-noise ratio. Further details regarding the experimental arrangement are given in Supplement 1, and full characterization and analysis of the generation process are described in .
We use the Fourier transformation for the transverse coordinates , , and focus on the weak EIT regime with confined spatial frequencies . In this regime, the motional broadening of the generation spectrum due to diffusion translates linearity to the generation amplitude, and one finds 
The propagator in Fourier space implies diffusion in real space. It follows from Eq. (2) that a structured probe beam in our system can continuously generate a signal which underwent diffusion for an effective temporal duration . The easiest way to control in experiment is by changing the two-photon detuning . Figure 1(b) presents a representative retrieved signal for an input Gaussian speckle field, under large detuning (i.e., short diffusion time, ), and Fig. 1(d) shows the retrieved signal for the same speckle pattern under small detuning (i.e., long diffusion time, ). Figures 1(c) and 1(e) show the autocorrelation of the retrieved intensity patterns. Based on such measurements and for various values of , we can study the effect of diffusion on the coherence of speckle fields.
Figure 2(a) shows a 1D cross section of a 2D Gaussian speckle pattern as a function of diffusion time, while normalizing the total intensity distribution at every moment, so as to account for the dissipation of the field under diffusion. Figure 2(b) presents the autocorrelation of the speckle pattern as a function of , and Fig. 2(c) shows the width of the autocorrelation versus . As evident, the speckles grow in size with diffusion time , and the area of the autocorrelation function grows linearly with time, i.e., .
It is well known that speckles can be propagation invariant under diffraction, if they are made by a random superposition of Bessel beams [14,37,38], namely if the field can be presented in the form , where is the radial coordinate and is the azimuthal coordinate, is a complex constants, is a Bessel function of order , and is the radial wave vector. In Fourier space, the last expression transforms to a field with random phases and an intensity distribution in the shape of annular ring. As explained in Supplement 1, Bessel beams are also invariant under diffusion, and therefore a random superposition of Bessel beams with the same radial wave vector would result in a diffusion-invariant speckle field . Consequently, the spatial coherence of such a speckle field would be diffusion invariant as well. Indeed, as demonstrated in Fig. 2(c) (purple squares), the size of these speckles is preserved and does not increase significantly with diffusion time (see Supplement 1 for details). Experimentally, the non-diffracting speckles are generated by encoding axicon phases on to the spatial light modulator (see Fig. 1), with additional phases that are constant in the radial direction and vary randomly with azimuth angle.
Generally, there are great differences between the diffusion and diffraction of speckles. To show this explicitly, we also measure the free-space optical propagation of Gaussian speckles (see Supplement 1 for details). We measure and analyze the propagation of Gaussian speckles in steps of 0.5 mm, over a total distance of 100 mm. The results of these measurements are shown in Figs. 2(d)–2(f). As evident, there are two regimes of propagation distances. Near the source, at the deep Fresnel region, the size of a typical speckle is constant. Far from the source, at the VCZ region, the speckles start pulling apart from one another, and the size of a typical speckle grain gradually grows. In diffusion, on the other hand, Gaussian speckles continuously expand with diffusion time, and the relative intensity of small speckle grains drops while the large speckle grains “take over.” Consequently, the area of the coherence region continuously grows with diffusion time.
To validate and complement our experimental results, we ran numerical simulations comparing between diffraction and diffusion of speckle fields. The calculations were performed by propagating an initial speckle field using the known propagators, shown in the first row of Table 1. Figure 3 compares between diffusion and diffraction and qualitatively agrees with the experimental results of Fig. 2. As evident, the linear relation between the width squared and diffusion time holds even long after the temporal regime measured experimentally (Fig. 2). Furthermore, Figs. 3(c) and 3(f) compare the width of the autocorrelation of a single speckle pattern, to the equivalent width of coherence calculated by averaging many uncorrelated speckle patterns. As evident, the two methods are equivalent, as expected.
3. DISCUSSION AND THEORETICAL ANALYSIS
We now establish the theoretical framework needed to explain the experimental and numerical results. Due to the strong correspondence between speckle theory and the theory of partial spatial coherence [40–42], we explain the results of diffusion of speckles as diffusion of spatial coherence. We therefore begin with the familiar formalism of diffraction of partially coherent beams and then extend this formalism to diffusion of such beams.
Consider an extended quasi-monochromatic pseudothermal source, which generates a complex field amplitude at axial distance from the source and at a 2D transverse coordinate . The field spatial correlations are given by the mutual intensity40–42].
In the following we consider a planar quasi-homogeneous source, namely a source whose area is very large compared to the coherence area on the source, and any variation in intensity on the source occurs on area scales that are of order . If the two scales are well separated, , the mutual intensity at the plain of the source can then be factorized:Supplement 1. Here and are the Fourier transforms of and , respectively, and and are the Fourier coefficients. The evolution of the mutual intensity with propagation distance can therefore be calculated using the propagator . In the near vicinity of the source , the propagator is negligible, resulting in . Therefore, in this region, the mutual intensity is constant and does not vary with propagation distance [13–15]. Notice this is true, even for propagation distances that are long compared to the Rayleigh distance of a single speckle grain, , where is a typical size of a single speckle, . Indeed, the experimental data of Figs. 2(d)–2(f) show that in this region, the spatial coherence of the speckles is constant, and does not vary with propagation distance.
Recall that , where and are the Fourier transforms of and , respectively. For and , the function varies very slowly as compared to , and therefore is approximately constant. In this case the absolute value of the mutual intensity depends only on , and therefore the coherence region grows as . Indeed, Fig. 2(f) shows a slope of , which agrees with the expected slope of .
We now turn to derive equivalent expressions to describe diffusion of partially coherent fields. As before, we begin with Eq. (6), but now we express the fields using the diffusion coherence propagator , yielding in Fourier spaceSupplement 1. This equation is the diffusion analog of Eq. (7), as they both describe the evolution of the spatial coherence in Fourier space. In diffraction, the local variations described by and the global variations described by are “mixed” by the propagator , and consequently the mutual intensity in Eq. (7) cannot be factorized. In contrast, in diffusion the propagator can be factorized. Therefore, the local and global coherence features undergo diffusion without “mixing.” The spatial coherence propagators under diffusion and diffraction are presented in the second row of Table 1, and are compared to the field propagators in the first row.
For , Eq. (9) implies that the width along the relative coordinate of the mutual intensity at any point in time depends mainly on the width of the coherence region on the surface of the source , even far from the source. This is very different from the behavior of the mutual intensity under diffraction, where near the source (, even if ) the spatial coherence is almost constant; and sufficiently far from the source (), when the boundary of the source begins to play a role, the spatial coherence is dominated by the shape and size of the source. This leads to a significant difference between diffusion and diffraction in a Michelson or Hanbury Brown and Twiss type of interferometer. In diffraction, a HBT interferometer can be used to measure the size of a distant spatially incoherent object, but cannot be used to retrieve information regarding the original size of coherent regions on the source. However, in diffusion the picture is reversed: measuring the spatial coherence indicates the size of coherence regions at a distant source (albeit with accuracy that decays with diffusion time) and will supply little information regarding the size of the source itself.
To see this more formally, consider a Gaussian speckle field with Gaussian envelope, , . In this case, Eq. (9) yields
Notice that Fig. 2(c) indeed shows a slope of , which agrees with an independent measurement of the diffusion coefficient, , as described in Supplement 1. Also, comparing Figs. 1(c) and 3 indicates that the small discrepancies between theory and experiment are similar to those between theory and simulation, and are dominated by the limited number of speckles analyzed.
The number of speckles at the plane of the source is estimated by . While is conserved under diffraction, it is not conserved under diffusion. In diffusion decreases with diffusion time, .
4. CONCLUDING REMARKS
We analyzed diffusion of partially coherent complex fields and compared between diffusion and diffraction of the spatial coherence. For this purpose, we presented a new theoretical model, which we derived in analogy and comparison to the familiar theorems in diffraction. Our theoretical analysis is general and applies to any diffusive physical system. We showed, both in theory and experiment, that the complex field and the spatial coherence of partially coherent beams both undergo diffusion in a similar manner, and that in the case of a Gaussian input their widths grow with the square root of diffusion time. This is in contrast to the well-known linear dependence under diffraction. While diffraction of partially coherent beams behaves differently in the deep Fresnel region and the VCZ region, diffusion of the partial coherence behaves the same in all temporal regimes.
As we showed, diffusion of partial coherence leads to a diffusion analog of the classical diffraction Michelson or HBT interferometers. In this diffusion analog, the boundary of the source has little effect on the spatial coherence, and measuring the spatial coherence far from the source can be considered as a measurement of the original region of coherence at the source. This behavior is, of course, very different than the classical Michelson of HBT interferometers, where the boundary of the source dominates the spatial coherence far from the source, while any information regarding the coherence region on the surface of the source is lost.
Finally, we showed that while the number of speckle grains in a field is conserved under diffraction, in diffusion it decreases with diffusion time. The exception of diffusion-free speckle fields, where the width of the spatial coherence and the number of speckles remains constant, was explored as well.
The work presented here extended concepts and theorems from statistical optics to the field of coherent diffusion. While we focused here on polariton diffusion, our analysis is general and provides a first step in applying the VCZ theory and HBT interferometry to various diffusive systems, such as astronomical stellar atmospheres  and imaging through turbulent or complex scattering media [44–46]. Furthermore, our model can be highly relevant for multi-pixel quantum memories [47–49], where realistic systems are expected to suffer from partial spatial coherence. In such systems it is therefore extremely important to understand the diffusion dynamics of spatial coherence, to appropriately estimate the temporal duration over which information can be stored. Recently, dissipation has been exploited for computational resources in advanced photonic systems [50–52]. It has been shown that dissipation in highly non-linear systems can rapidly anneal a system to a global ground state and serve as a physical simulator for hard computational problems [53–55]. Such systems are often governed by diffusive transport, and they are often only partially spatially coherent. The model we established and verified in this work could serve as an important step in fully understanding the physical mechanisms behind such applications and will help develop next-generation computational resources.
Israel Science Foundation; United States–Israel Binational Science Foundation; Pazi Foundation.
See Supplement 1 for supporting content.
1. J. W. Goodman, Statistical Optics (Wiley, 2015).
2. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
3. M. Lahiri, “Coherence and statistical optics,” in Photonics: Scientific Foundations, Technology and Applications, Vol. 1, D. L. Andrews, ed. (Wiley, 2015).
4. P. H. van Cittert, “Die wahrscheinliche schwingungsverteilung in einer von einer lichtquelle direkt oder mittels einer linse beleuchteten ebene,” Physica 1, 201–210 (1934). [CrossRef]
5. F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938). [CrossRef]
6. A. A. Michelson and F. G. Pease, “Measurement of the diameter of Alpha-Orionis by the interferometer,” Proc. Natl. Acad. Sci. U. S. A. 7, 143–146 (1921). [CrossRef]
7. P. Hariharan, Optical Interferometry (Academic, 2003).
8. R. H. Brown and R. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178, 1046–1048 (1956). [CrossRef]
9. R. H. Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956). [CrossRef]
10. R. Hanbury Brown, The Intensity Interferometer (Taylor and Francis, 1974), p. 199.
11. R. Cerbino, “Correlations of light in the deep Fresnel region: an extended van Cittert and Zernike theorem,” Phys. Rev. A 75, 053815 (2007). [CrossRef]
12. A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. A 78, 063806 (2008). [CrossRef]
13. Y. Ohtsuka, “Non-modified propagation of optical mutual intensity in the Fresnel diffraction region,” Opt. Commun. 39, 283–286 (1981). [CrossRef]
14. J. Turunen, A. Vasara, and A. T. Friberg, “Propagation invariance and self-imaging in variable-coherence optics,” J. Opt. Soc. Am. A 8, 282–289 (1991). [CrossRef]
15. M. Giglio, M. Carpineti, and A. Vailati, “Space intensity correlations in the near field of the scattered light: a direct measurement of the density correlation function g (r),” Phys. Rev. Lett. 85, 1416–1419 (2000). [CrossRef]
16. Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nat. Photonics 4, 721–726 (2010). [CrossRef]
17. W. D. Oliver, J. Kim, R. C. Liu, and Y. Yamamoto, “Hanbury Brown and Twiss-type experiment with electrons,” Science 284, 299–301 (1999). [CrossRef]
18. M. Henny, S. Oberholzer, C. Strunk, T. Heinzel, K. Ensslin, M. Holland, and C. Schönenberger, “The fermionic Hanbury Brown and Twiss experiment,” Science 284, 296–298 (1999). [CrossRef]
19. H. Kiesel, A. Renz, and F. Hasselbach, “Observation of Hanbury Brown-Twiss anticorrelations for free electrons,” Nature 418, 392–394 (2002). [CrossRef]
20. M. Schellekens, R. Hoppeler, A. Perrin, J. V. Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, “Hanbury Brown Twiss effect for ultracold quantum gases,” Science 310, 648–651 (2005). [CrossRef]
21. A. Perrin, R. Bücker, S. Manz, T. Betz, C. Koller, T. Plisson, T. Schumm, and J. Schmiedmayer, “Hanbury Brown and Twiss correlations across the Bose-Einstein condensation threshold,” Nat. Phys. 8, 195–198 (2012). [CrossRef]
22. T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, “Comparison of the Hanbury Brown-Twiss effect for bosons and fermions,” Nature 445, 402–405 (2007). [CrossRef]
23. R. Dall, S. Hodgman, A. Manning, M. Johnsson, K. Baldwin, and A. Truscott, “Observation of atomic speckle and Hanbury Brown-Twiss correlations in guided matter waves,” Nat. Commun. 2, 291 (2011). [CrossRef]
24. C. Jurczak, B. Desruelle, K. Sengstock, J.-Y. Courtois, C.-I. Westbrook, and A. Aspect, “Atomic transport in an optical lattice: an investigation through polarization-selective intensity correlations,” Phys. Rev. Lett. 77, 1727–1730 (1996). [CrossRef]
25. D. S. Grebenkov, “NMR survey of reflected Brownian motion,” Rev. Mod. Phys. 79, 1077–1137 (2007). [CrossRef]
26. M. Dąbrowski, M. Parniak, and W. Wasilewski, “Einstein-Podolsky-Rosen paradox in a hybrid bipartite system,” Optica 4, 272–275 (2017). [CrossRef]
27. O. Firstenberg, P. London, M. Shuker, A. Ron, and N. Davidson, “Elimination, reversal and directional bias of optical diffraction,” Nat. Phys. 5, 665–668 (2009). [CrossRef]
28. O. Firstenberg, M. Shuker, A. Ron, and N. Davidson, “Colloquium: coherent diffusion of polaritons in atomic media,” Rev. Mod. Phys. 85, 941–960 (2013). [CrossRef]
29. G. Heinze, C. Hubrich, and T. Halfmann, “Stopped light and image storage by electromagnetically induced transparency up to the regime of one minute,” Phys. Rev. Lett. 111, 033601 (2013). [CrossRef]
30. J. Kikkawa and D. Awschalom, “Lateral drag of spin coherence in gallium arsenide,” Nature 397, 139–141 (1999). [CrossRef]
31. Y. Xiao, M. Klein, M. Hohensee, L. Jiang, D. F. Phillips, M. D. Lukin, and R. L. Walsworth, “Slow light beam splitter,” Phys. Rev. Lett. 101, 043601 (2008). [CrossRef]
32. O. Firstenberg, M. Shuker, R. Pugatch, D.-R. Fredkin, N. Davidson, and A. Ron, “Theory of thermal motion in electromagnetically induced transparency: effects of diffusion, doppler broadening, and Dicke and Ramsey narrowing,” Phys. Rev. A 77, 043830 (2008). [CrossRef]
33. R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, “Topological stability of stored optical vortices,” Phys. Rev. Lett. 98, 203601 (2007). [CrossRef]
34. M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, “Storing images in warm atomic vapor,” Phys. Rev. Lett. 100, 223601 (2008). [CrossRef]
35. O. Firstenberg, P. London, D. Yankelev, R. Pugatch, M. Shuker, and N. Davidson, “Self-similar modes of coherent diffusion,” Phys. Rev. Lett. 105, 183602 (2010). [CrossRef]
36. S. Smartsev, D. Eger, N. Davidson, and O. Firstenberg, “Continuous generation of delayed light,” J. Phys. B 50, 215003 (2017). [CrossRef]
37. J. Durnin, J.-J. Miceli, and J.-H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987). [CrossRef]
38. K. Uno, J. Uozumi, and T. Asakura, “Speckle clustering in diffraction patterns of random objects under ring-slit illumination,” Opt. Commun. 114, 203–210 (1995). [CrossRef]
39. S. Smartsev, R. Chriki, D. Eger, O. Firstenberg, and N. Davidson are preparing a manuscript to be called “Diffusion free beams.”
40. J. W. Goodman, “Role of coherence concepts in the study of speckle,” in Applications of Optical Coherence (International Society for Optics and Photonics, 1979), Vol. 194, pp. 86–95.
41. H. M. Pedersen, “Intensity correlation metrology: a comparative study,” Opt. Acta: Int. J. Opt. 29, 105–118 (1982). [CrossRef]
42. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Roberts and Company Publishers, 2007).
43. T. Ryabchikova, “Diffusion and its manifestation in stellar atmospheres,” in Determination of Atmospheric Parameters of B-, A-, F-and G-Type Stars (Springer, 2014), pp. 141–148.
44. D. A. Boas, L. Campbell, and A. G. Yodh, “Scattering and imaging with diffusing temporal field correlations,” Phys. Rev. Lett. 75, 1855–1858 (1995). [CrossRef]
45. A. P. Mosk, A. Lagendijk, G. Lerosey, and M. Fink, “Controlling waves in space and time for imaging and focusing in complex media,” Nat. Photonics 6, 283–292 (2012). [CrossRef]
46. S. Rotter and S. Gigan, “Light fields in complex media: mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017). [CrossRef]
47. M. Eisaman, A. André, F. Massou, M. Fleischhauer, A. Zibrov, and M. Lukin, “Electromagnetically induced transparency with tunable single-photon pulses,” Nature 438, 837–841 (2005). [CrossRef]
48. R. Zhao, Y. Dudin, S. Jenkins, C. Campbell, D. Matsukevich, T. Kennedy, and A. Kuzmich, “Long-lived quantum memory,” Nat. Phys. 5, 100–104 (2009). [CrossRef]
49. D. V. Vasilyev, I. V. Sokolov, and E. S. Polzik, “Quantum memory for images: a quantum hologram,” Phys. Rev. A 77, 020302 (2008). [CrossRef]
50. M. Nixon, E. Ronen, A. A. Friesem, and N. Davidson, “Observing geometric frustration with thousands of coupled lasers,” Phys. Rev. Lett. 110, 184102 (2013). [CrossRef]
51. A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nat. Photonics 8, 937–942 (2014). [CrossRef]
52. S. Mukherjee, D. Mogilevtsev, G. Y. Slepyan, T. H. Doherty, R. R. Thomson, and N. Korolkova, “Dissipatively coupled waveguide networks for coherent diffusive photonics,” Nat. Commun. 8, 1909 (2017). [CrossRef]
53. P. L. McMahon, A. Marandi, Y. Haribara, R. Hamerly, C. Langrock, S. Tamate, T. Inagaki, H. Takesue, S. Utsunomiya, K. Aihara, and R. L. Byer, “A fully programmable 100-spin coherent Ising machine with all-to-all connections,” Science 354, 614–617 (2016). [CrossRef]
54. V. Pal, C. Tradonsky, R. Chriki, A. A. Friesem, and N. Davidson, “Observing dissipative topological defects with coupled lasers,” Phys. Rev. Lett. 119, 013902 (2017). [CrossRef]
55. R. Hamerly, T. Inagaki, P. L. McMahon, D. Venturelli, A. Marandi, T. Onodera, E. Ng, C. Langrock, K. Inaba, T. Honjo, and K. Enbutsu, “Experimental investigation of performance differences between coherent Ising machines and a quantum annealer,” Sci. Adv. 5, eaau0823 (2019). [CrossRef]