Abstract

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

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References

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2019 (1)

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]

2017 (5)

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]

S. Rotter and S. Gigan, “Light fields in complex media: mesoscopic scattering meets wave control,” Rev. Mod. Phys. 89, 015005 (2017).
[Crossref]

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]

M. Dąbrowski, M. Parniak, and W. Wasilewski, “Einstein-Podolsky-Rosen paradox in a hybrid bipartite system,” Optica 4, 272–275 (2017).
[Crossref]

S. Smartsev, D. Eger, N. Davidson, and O. Firstenberg, “Continuous generation of delayed light,” J. Phys. B 50, 215003 (2017).
[Crossref]

2016 (1)

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]

2014 (1)

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]

2013 (3)

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]

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]

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]

2012 (2)

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]

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]

2011 (1)

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]

2010 (2)

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]

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nat. Photonics 4, 721–726 (2010).
[Crossref]

2009 (2)

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]

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]

2008 (5)

D. V. Vasilyev, I. V. Sokolov, and E. S. Polzik, “Quantum memory for images: a quantum hologram,” Phys. Rev. A 77, 020302 (2008).
[Crossref]

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]

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]

M. Shuker, O. Firstenberg, R. Pugatch, A. Ron, and N. Davidson, “Storing images in warm atomic vapor,” Phys. Rev. Lett. 100, 223601 (2008).
[Crossref]

A. Gatti, D. Magatti, and F. Ferri, “Three-dimensional coherence of light speckles: theory,” Phys. Rev. A 78, 063806 (2008).
[Crossref]

2007 (4)

R. Cerbino, “Correlations of light in the deep Fresnel region: an extended van Cittert and Zernike theorem,” Phys. Rev. A 75, 053815 (2007).
[Crossref]

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]

D. S. Grebenkov, “NMR survey of reflected Brownian motion,” Rev. Mod. Phys. 79, 1077–1137 (2007).
[Crossref]

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, “Topological stability of stored optical vortices,” Phys. Rev. Lett. 98, 203601 (2007).
[Crossref]

2005 (2)

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]

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]

2002 (1)

H. Kiesel, A. Renz, and F. Hasselbach, “Observation of Hanbury Brown-Twiss anticorrelations for free electrons,” Nature 418, 392–394 (2002).
[Crossref]

2000 (1)

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]

1999 (3)

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]

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]

J. Kikkawa and D. Awschalom, “Lateral drag of spin coherence in gallium arsenide,” Nature 397, 139–141 (1999).
[Crossref]

1996 (1)

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]

1995 (2)

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]

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]

1991 (1)

1987 (1)

J. Durnin, J.-J. Miceli, and J.-H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

1982 (1)

H. M. Pedersen, “Intensity correlation metrology: a comparative study,” Opt. Acta: Int. J. Opt. 29, 105–118 (1982).
[Crossref]

1981 (1)

Y. Ohtsuka, “Non-modified propagation of optical mutual intensity in the Fresnel diffraction region,” Opt. Commun. 39, 283–286 (1981).
[Crossref]

1956 (2)

R. H. Brown and R. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178, 1046–1048 (1956).
[Crossref]

R. H. Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[Crossref]

1938 (1)

F. Zernike, “The concept of degree of coherence and its application to optical problems,” Physica 5, 785–795 (1938).
[Crossref]

1934 (1)

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]

1921 (1)

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]

Aihara, K.

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]

André, A.

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]

Asakura, T.

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]

Aspect, A.

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]

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]

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]

Awschalom, D.

J. Kikkawa and D. Awschalom, “Lateral drag of spin coherence in gallium arsenide,” Nature 397, 139–141 (1999).
[Crossref]

Baldwin, K.

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]

Betz, T.

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]

Boas, D. A.

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]

Boiron, D.

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]

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]

Bromberg, Y.

Y. Bromberg, Y. Lahini, E. Small, and Y. Silberberg, “Hanbury Brown and Twiss interferometry with interacting photons,” Nat. Photonics 4, 721–726 (2010).
[Crossref]

Brown, R. H.

R. H. Brown and R. Twiss, “A test of a new type of stellar interferometer on Sirius,” Nature 178, 1046–1048 (1956).
[Crossref]

R. H. Brown and R. Q. Twiss, “Correlation between photons in two coherent beams of light,” Nature 177, 27–29 (1956).
[Crossref]

Bücker, R.

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]

Byer, R. L.

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]

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]

Campbell, C.

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]

Campbell, L.

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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).
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O. Firstenberg, M. Shuker, A. Ron, and N. Davidson, “Colloquium: coherent diffusion of polaritons in atomic media,” Rev. Mod. Phys. 85, 941–960 (2013).
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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).
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Supplementary Material (1)

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Figures (3)

Fig. 1.
Fig. 1. Experiment and representative results. (a) Experimental arrangement for diffusion of speckle fields. BS, beam splitter; SLM, spatial light modulator. (b), (d) Detected intensity distribution at the center cross section of the vapor cell for (b) large and (d) small detuning, manifesting short and long diffusion time, respectively. (c), (e) Corresponding autocorrelation of the detected speckle pattern.
Fig. 2.
Fig. 2. Experimental comparison between (a)–(c) diffusion and (d)–(f) diffraction of speckle intensity distributions. (a) Linear cross section of the detected speckle pattern, as a function of diffusion time. (b) Corresponding autocorrelation function of the detected speckle pattern, as a function of diffusion time. Note that, for clarity, this color map presents numerical interpolation of the data circumventing for uneven time steps. (c) Corresponding 1/e width squared of the autocorrelation in (b) (with no interpolation) as a function of diffusion time (red circles), as well as the width squared of the autocorrelation for non-diffusive speckles (purple squares). Dashed lines present linear fit to the data. (d) Linear cross section of the detected speckle pattern, as a function of propagation distance. (e) Corresponding autocorrelation function of the detected speckle pattern, as a function of propagation distance. (f) Corresponding 1/e width squared of the autocorrelation function in (e) as a function of propagation distance (red solid line), together with a linear fit in the linear regime zzVCZ (dashed blue line). Dashed vertical line denotes the propagation distance z=zVCZ. The data in (b), (c), (e), and (f) represent the radial mean of the autocorrelation. The errors of these measurements are smaller than the thickness of the data points in (c) and of the solid red line in (f).
Fig. 3.
Fig. 3. Numerical calculations comparing between (a)–(c) diffusion and (d)–(f) diffraction of speckle intensity distributions, and of their spatial coherence. (a) Calculated intensity distribution of a speckle pattern as a function of normalized diffusion time τ/τR, where τR is the diffusion analog of the Rayleigh distance in diffraction τR=w02/4D. (b) Corresponding autocorrelation function of the simulated speckle pattern, as a function of normalized diffusion time. (c) Corresponding 1/e width squared of the autocorrelation in (b) as a function of normalized diffusion time (solid red line). (d) Calculated intensity distribution of a speckle pattern as a function of normalized propagation distance z/zR, where zR=πw02/λ is the Rayleigh distance. (e) Corresponding autocorrelation function of the simulated speckle pattern, as a function of normalized propagation distance. (f) Corresponding 1/e width squared of the autocorrelation function in (e) as a function of normalized propagation distance (red solid line). The inset in (b) compares between experiment (circles), simulation (solid red line), and analytic expression (dashed black line) for short diffusion times. The calculated intensity correlations of many such speckle realization (ensemble average) are presented in (c) and (f) as well (dashed blue line). The cross sections and widths were obtained by considering the radial mean of the autocorrelation.

Tables (1)

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Table 1. Diffusion Propagators Derived in the Paper, Compared to Well-Known Diffraction Propagators

Equations (11)

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coherent diffusion:ϕ(r,t)t=Dr2ϕ(r,t),paraxial diffraction:ϕ(r,t)z=iλ4πr2ϕ(r,t),
E˜S(q)=E˜S(q=0)E˜in(q=0)·E˜in(q)eDτq2,
τ=τ1+(Δ2pτ)2,
G(1)(r1,r2;z)=E*(r1;z)E(r2;z),
G(2)(r1,r2;z)=I(r1;z)I(r2;z)+|G(1)(r1,r2;z)|2.
G0(1)(r1,r2)=I0(r¯)μ0(Δr),
G˜(1)(q¯,Δq;z)=G˜0(1)(q¯,Δq)eiλzq¯·Δq/2π,
G(1)(r¯,Δr;z)=(2πλz)2G˜0(1)(2πλzr¯,2πλzΔr)e(2πi/λz)r¯Δr,
G˜(1)(q¯,Δq;τ)=G˜0(1)(q¯,Δq)eDτ(q¯2+Δq2)=[μ˜0(q¯)eDτq¯2]·[I˜0(Δq)eDτΔq2],
G(1)(r¯,Δr;τ)=exp(Δr2lc2+4Dτ)·exp(r¯2L2+4Dτ).
E(r,τ)exp(r2w0+4Dτ),