Abstract

Monochromatic coherent light traversing a disordered photonic medium evolves into a random field whose statistics are dictated by the disorder level. Here we demonstrate experimentally that light statistics can be deterministically tuned in certain disordered lattices, even when the disorder level is held fixed, by controllably breaking the excitation symmetry of the lattice modes. We exploit a lattice endowed with disorder-immune chiral symmetry in which the eigenmodes come in skew-symmetric pairs. If a single lattice site is excited, a “photonic thermalization gap” emerges: the realm of sub-thermal light statistics is inaccessible regardless of the disorder level. However, by exciting two sites with a variable relative phase, as in a traditional two-path interferometer, the chiral symmetry is judiciously broken and interferometric control over the light statistics is exercised, spanning sub-thermal and super-thermal regimes. These results may help develop novel incoherent lighting sources from coherent lasers.

© 2016 Optical Society of America

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References

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2015 (6)

P. Lai, L. Wang, J. W. Tay, and L. V. Wang, “Photoacoustically guided wavefront shaping for enhanced optical focusing in scattering media,” Nat. Photonics 9, 126–132 (2015).
[Crossref]

B. Redding, A. Cerjan, X. Huang, M. L. Lee, A. D. Stone, M. A. Choma, and H. Cao, “Low spatial coherence electrically pumped semiconductor laser for speckle-free full-field imaging,” Proc. Natl. Acad. Sci. USA 112, 1304–1309 (2015).
[Crossref]

H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “A photonic thermalization gap in disordered lattices,” Nat. Phys. 11, 930–935 (2015).
[Crossref]

T. Meany, M. Gräfe, R. Heilmann, A. Perez-Leija, S. Gross, M. J. Steel, M. J. Withford, and A. Szameit, “Laser written circuits for quantum photonics,” Laser Photon. Rev. 9, 363–384 (2015).
[Crossref]

H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “Discrete Anderson speckle,” Optica 2, 201–209 (2015).
[Crossref]

Y. Gilead, M. Verbin, and Y. Silberberg, “Ensemble-averaged quantum correlations between path-entangled photons undergoing Anderson localization,” Phys. Rev. Lett. 115, 133602 (2015).
[Crossref]

2014 (4)

T. E. Matthews, M. Medina, J. R. Maher, H. Levinson, W. J. Brown, and A. Wax, “Deep tissue imaging using spectroscopic analysis of multiply scattered light,” Optica 1, 105–111 (2014).
[Crossref]

S. Mookherjea, J. R. Ong, X. Luo, and L. Guo-Qiang, “Electronic control of optical Anderson localization modes,” Nat. Nanotechnol. 9, 365–371 (2014).
[Crossref]

E. H. Zhou, H. Ruan, C. Yang, and B. Judkewitz, “Focusing on moving targets through scattering samples,” Optica 1, 227–232 (2014).
[Crossref]

Y. Bromberg and H. Cao, “Generating non-Rayleigh speckles with tailored intensity statistics,” Phys. Rev. Lett. 112, 213904 (2014).
[Crossref]

2013 (6)

K. A. Denault, M. Cantore, S. Nakamura, S. P. DenBaars, and R. Seshadri, “Efficient and stable laser-driven white lighting,” AIP Adv. 3, 072107 (2013).
[Crossref]

B. Judkewitz, Y. M. Wang, R. Horstmeyer, A. Mathy, and C. Yang, “Speckle-scale focusing in the diffusive regime with time reversal of variance-encoded light (TROVE),” Nat. Photonics 7, 300–305 (2013).
[Crossref]

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[Crossref]

M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics 7, 197–204 (2013).
[Crossref]

G. Di Giuseppe, L. Martin, A. Perez-Leija, R. Keil, F. Dreisow, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, “Einstein-Podolsky-Rosen spatial entanglement in ordered and Anderson photonic lattices,” Phys. Rev. Lett. 110, 150503 (2013).
[Crossref]

A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
[Crossref]

2012 (5)

A. F. Abouraddy, G. Di Giuseppe, D. N. Christodoulides, and B. E. A. Saleh, “Anderson localization and colocalization of spatially entangled photons,” Phys. Rev. A 86, 040302 (2012).
[Crossref]

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref]

O. Katz, E. Small, and Y. Silberberg, “Looking around corners and through thin turbid layers in real time with scattered incoherent light,” Nat. Photonics 6, 549–553 (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]

B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6, 355–359 (2012).
[Crossref]

2011 (2)

Y. Lahini, Y. Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y. Silberberg, “Hanbury Brown and Twiss correlations of Anderson localized waves,” Phys. Rev. A 84, 041806 (2011).
[Crossref]

L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, “Anderson localization in optical waveguide arrays with off-diagonal coupling disorder,” Opt. Express 19, 13636–13646 (2011).
[Crossref]

2010 (2)

2008 (1)

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref]

2007 (1)

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[Crossref]

2003 (2)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[Crossref]

S. N. Evangelou and D. E. Katsanos, “Spectral statistics in chiral-orthogonal disordered systems,” J. Phys. A 36, 3237–3254 (2003).
[Crossref]

1993 (1)

R. Gade, “Anderson localization for sublattice models,” Nucl. Phys. B 398, 499–515 (1993).
[Crossref]

1991 (1)

R. Gade and F. Wegner, “The n = 0 replica limit of U(n) and U(n)/SO(n) models,” Nucl. Phys. B 360, 213–218 (1991).
[Crossref]

1981 (1)

C. M. Soukoulis and E. N. Economou, “Off-diagonal disorder in one-dimensional systems,” Phys. Rev. B 24, 5698–5702 (1981).
[Crossref]

1958 (1)

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[Crossref]

Abouraddy, A. F.

H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “A photonic thermalization gap in disordered lattices,” Nat. Phys. 11, 930–935 (2015).
[Crossref]

H. E. Kondakci, A. F. Abouraddy, and B. E. A. Saleh, “Discrete Anderson speckle,” Optica 2, 201–209 (2015).
[Crossref]

G. Di Giuseppe, L. Martin, A. Perez-Leija, R. Keil, F. Dreisow, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, “Einstein-Podolsky-Rosen spatial entanglement in ordered and Anderson photonic lattices,” Phys. Rev. Lett. 110, 150503 (2013).
[Crossref]

A. F. Abouraddy, G. Di Giuseppe, D. N. Christodoulides, and B. E. A. Saleh, “Anderson localization and colocalization of spatially entangled photons,” Phys. Rev. A 86, 040302 (2012).
[Crossref]

L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, “Anderson localization in optical waveguide arrays with off-diagonal coupling disorder,” Opt. Express 19, 13636–13646 (2011).
[Crossref]

Anderson, P. W.

P. W. Anderson, “Absence of diffusion in certain random lattices,” Phys. Rev. 109, 1492–1505 (1958).
[Crossref]

Avidan, A.

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref]

Bartal, G.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[Crossref]

Bertolotti, J.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref]

Blum, C.

J. Bertolotti, E. G. van Putten, C. Blum, A. Lagendijk, W. L. Vos, and A. P. Mosk, “Non-invasive imaging through opaque scattering layers,” Nature 491, 232–234 (2012).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge University, 1999).

Bromberg, Y.

Y. Bromberg and H. Cao, “Generating non-Rayleigh speckles with tailored intensity statistics,” Phys. Rev. Lett. 112, 213904 (2014).
[Crossref]

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[Crossref]

Y. Lahini, Y. Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y. Silberberg, “Hanbury Brown and Twiss correlations of Anderson localized waves,” Phys. Rev. A 84, 041806 (2011).
[Crossref]

Brown, W. J.

Cantore, M.

K. A. Denault, M. Cantore, S. Nakamura, S. P. DenBaars, and R. Seshadri, “Efficient and stable laser-driven white lighting,” AIP Adv. 3, 072107 (2013).
[Crossref]

Cao, H.

B. Redding, A. Cerjan, X. Huang, M. L. Lee, A. D. Stone, M. A. Choma, and H. Cao, “Low spatial coherence electrically pumped semiconductor laser for speckle-free full-field imaging,” Proc. Natl. Acad. Sci. USA 112, 1304–1309 (2015).
[Crossref]

Y. Bromberg and H. Cao, “Generating non-Rayleigh speckles with tailored intensity statistics,” Phys. Rev. Lett. 112, 213904 (2014).
[Crossref]

B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6, 355–359 (2012).
[Crossref]

Cerjan, A.

B. Redding, A. Cerjan, X. Huang, M. L. Lee, A. D. Stone, M. A. Choma, and H. Cao, “Low spatial coherence electrically pumped semiconductor laser for speckle-free full-field imaging,” Proc. Natl. Acad. Sci. USA 112, 1304–1309 (2015).
[Crossref]

Choma, M. A.

B. Redding, A. Cerjan, X. Huang, M. L. Lee, A. D. Stone, M. A. Choma, and H. Cao, “Low spatial coherence electrically pumped semiconductor laser for speckle-free full-field imaging,” Proc. Natl. Acad. Sci. USA 112, 1304–1309 (2015).
[Crossref]

B. Redding, M. A. Choma, and H. Cao, “Speckle-free laser imaging using random laser illumination,” Nat. Photonics 6, 355–359 (2012).
[Crossref]

Christodoulides, D. N.

M. Segev, Y. Silberberg, and D. N. Christodoulides, “Anderson localization of light,” Nat. Photonics 7, 197–204 (2013).
[Crossref]

G. Di Giuseppe, L. Martin, A. Perez-Leija, R. Keil, F. Dreisow, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, “Einstein-Podolsky-Rosen spatial entanglement in ordered and Anderson photonic lattices,” Phys. Rev. Lett. 110, 150503 (2013).
[Crossref]

A. F. Abouraddy, G. Di Giuseppe, D. N. Christodoulides, and B. E. A. Saleh, “Anderson localization and colocalization of spatially entangled photons,” Phys. Rev. A 86, 040302 (2012).
[Crossref]

Y. Lahini, Y. Bromberg, Y. Shechtman, A. Szameit, D. N. Christodoulides, R. Morandotti, and Y. Silberberg, “Hanbury Brown and Twiss correlations of Anderson localized waves,” Phys. Rev. A 84, 041806 (2011).
[Crossref]

L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, “Anderson localization in optical waveguide arrays with off-diagonal coupling disorder,” Opt. Express 19, 13636–13646 (2011).
[Crossref]

Y. Lahini, A. Avidan, F. Pozzi, M. Sorel, R. Morandotti, D. N. Christodoulides, and Y. Silberberg, “Anderson localization and nonlinearity in one-dimensional disordered photonic lattices,” Phys. Rev. Lett. 100, 013906 (2008).
[Crossref]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817–823 (2003).
[Crossref]

Crespi, A.

A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
[Crossref]

Davidson, N.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[Crossref]

De Nicola, F.

A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
[Crossref]

Denault, K. A.

K. A. Denault, M. Cantore, S. Nakamura, S. P. DenBaars, and R. Seshadri, “Efficient and stable laser-driven white lighting,” AIP Adv. 3, 072107 (2013).
[Crossref]

DenBaars, S. P.

K. A. Denault, M. Cantore, S. Nakamura, S. P. DenBaars, and R. Seshadri, “Efficient and stable laser-driven white lighting,” AIP Adv. 3, 072107 (2013).
[Crossref]

Di Giuseppe, G.

G. Di Giuseppe, L. Martin, A. Perez-Leija, R. Keil, F. Dreisow, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, “Einstein-Podolsky-Rosen spatial entanglement in ordered and Anderson photonic lattices,” Phys. Rev. Lett. 110, 150503 (2013).
[Crossref]

A. F. Abouraddy, G. Di Giuseppe, D. N. Christodoulides, and B. E. A. Saleh, “Anderson localization and colocalization of spatially entangled photons,” Phys. Rev. A 86, 040302 (2012).
[Crossref]

L. Martin, G. Di Giuseppe, A. Perez-Leija, R. Keil, F. Dreisow, M. Heinrich, S. Nolte, A. Szameit, A. F. Abouraddy, D. N. Christodoulides, and B. E. A. Saleh, “Anderson localization in optical waveguide arrays with off-diagonal coupling disorder,” Opt. Express 19, 13636–13646 (2011).
[Crossref]

Dreisow, F.

Economou, E. N.

C. M. Soukoulis and E. N. Economou, “Off-diagonal disorder in one-dimensional systems,” Phys. Rev. B 24, 5698–5702 (1981).
[Crossref]

Evangelou, S. N.

S. N. Evangelou and D. E. Katsanos, “Spectral statistics in chiral-orthogonal disordered systems,” J. Phys. A 36, 3237–3254 (2003).
[Crossref]

Fazio, R.

A. Crespi, R. Osellame, R. Ramponi, V. Giovannetti, R. Fazio, L. Sansoni, F. De Nicola, F. Sciarrino, and P. Mataloni, “Anderson localization of entangled photons in an integrated quantum walk,” Nat. Photonics 7, 322–328 (2013).
[Crossref]

Fink, M.

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]

Fishman, S.

T. Schwartz, G. Bartal, S. Fishman, and M. Segev, “Transport and Anderson localization in disordered two-dimensional photonic lattices,” Nature 446, 52–55 (2007).
[Crossref]

Friesem, A. A.

M. Nixon, O. Katz, E. Small, Y. Bromberg, A. A. Friesem, Y. Silberberg, and N. Davidson, “Real-time wavefront shaping through scattering media by all-optical feedback,” Nat. Photonics 7, 919–924 (2013).
[Crossref]

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AIP Adv. (1)

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Nat. Photonics (9)

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[Crossref]

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Figures (5)

Fig. 1.
Fig. 1. Light-statistics interferometry in random networks. (a) Schematic of traditional two-path interferometry. Coherent fields with a relative phase θ interfere in a deterministic system (depicted here as a simple beam splitter). (b) The output intensity I varies sinusoidally with θ . The field remains coherent and g ( 2 ) = 1 . (c) Light-statistics interferometry in a disordered system. Just as in (a), two coherent fields with relative phase θ enter the system. (d) While the ensemble-averaged intensity I at the output is independent of θ , g ( 2 ) varies sinusoidally with it, resulting in a photon-statistics interferogram.
Fig. 2.
Fig. 2. Lattice model and photonic thermalization gap. (a) Coupled potential wells with random couplings and fixed site energies represent an off-diagonal disordered lattice. Assuming nearest-neighbor-only coupling, the Hamiltonian H is a tri-diagonal matrix, which can be rearranged in block off-diagonal form in the interaction picture, a signature of chiral symmetry. (b) A waveguide array with off-diagonal disorder is coherently excited at two sites. The color along the waveguides represents the calculated intensity in a logarithmic scale. (c) Calculated g ( 2 ) at x = 0 as a function of disorder level Δ C . The solid black line corresponds to single-waveguide excitation ( x = 0 ) in the steady state ( z ), while dashed lines represent g ( 2 ) at z C ¯ = 10 when two neighboring waveguides ( x = 0 and 1) are excited, E 1 ( 0 ) = e i θ E 0 ( 0 ) , for θ = 0 , π / 4 , and π / 2 . The ensemble size is 10 5 . (d) The mean mode-excitation amplitudes | c n | are asymmetric (left) around n = 0 for θ = 0 and π , and are symmetric (right) when θ = π / 2 and π / 2 .
Fig. 3.
Fig. 3. Experimental setup. A single-mode coherent beam from a He–Ne laser is split equally into two paths, a phase shift θ is introduced, and the two beams are then imaged into two neighboring waveguides within an array. The different disorder realizations are produced by translating the waveguide array along x and ensuring that the input beams are realigned for each configuration. After magnification, the waveguide-array output is imaged to a CCD camera, and a single waveguide at x = 0 is separately imaged to a multimode fiber.
Fig. 4.
Fig. 4. Dependence of the output intensity across the lattice on the input relative phase. (a) Color plots depicting the intensity distributions for nine different disordered lattice realizations captured by a CCD camera. Each plot represents a single realization when two input lattice sites are illuminated with the relative phase θ = 0 . (b) An ensemble average obtained from 30 realizations. (c)–(e) Color plots depicting the output intensity distributions I ( x , θ ) while varying the input relative phase θ for three different disorder realizations. Each row is generated by integration along the y direction of the CCD images [such as those in (a)]. The three color plots are normalized to the same peak value. The arrows at the top identify the input waveguides. (f) The ensemble average (30 realizations) of the intensity distribution I ( x , θ ) . The dashed white lines are guides to the eye highlighting the variation in the spatial offset of the mean intensity distribution with θ .
Fig. 5.
Fig. 5. Light-statistics interferometry. (a) The normalized intensity correlation g ( 2 ) is deterministically tuned by varying the input relative phase θ . Simulations (solid line) are in agreement with the data (squares). The gray shading is the calculated probability distribution of the expected g ( 2 ) values assuming a small ensemble size of 30 realizations (the size of the experimental ensemble), while the solid line is the average value of g ( 2 ) for an ensemble size of 10 5 realizations. The red-dashed line corresponds to the edge of the photonic thermalization gap and separates the sub- and super-thermal regimes. The dotted line at g ( 2 ) = 2.35 is the value produced at the output when only one input lattice site is illuminated (and no tuning is available). (b) The mean intensity as a function of θ . The small-amplitude oscillation in the simulation (solid line) is due to the finite array length. The gray shading is the probability distribution of the mean intensity calculated for a small ensemble of 30 realizations, while the solid line was calculated for an ensemble of 10 5 realizations.

Equations (7)

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I 0 ( z ) = I 0,0 ( z ) + I 0,1 ( z ) 2 sin θ E 0,0 ( z ) E 0,1 ( z ) ,
g 0 ( 2 ) ( θ ) = α β cos 2 θ .
E x ( z ) = { i E x , 0 ( z ) + e i θ E x , 1 ( z ) , x odd , E x , 0 ( z ) + i e i θ E x , 1 ( z ) , x even ,
E x , 0 ( z ) = { n = 1 N ϕ n ( x ) ϕ n ( 0 ) sin ( b n z ) , x odd , n = 1 N ϕ n ( x ) ϕ n ( 0 ) cos ( b n z ) , x even ,
E x , 1 ( z ) = { n = 1 N ϕ n ( x ) ϕ n ( 1 ) cos ( b n z ) , x  odd , n = 1 N ϕ n ( x ) ϕ n ( 1 ) sin ( b n z ) , x even ,
I x ( z ; θ ) = I x , 0 ( z ) + I x , 1 ( z ) 2 p sin θ I x , 0 ( z ) I x , 1 ( z ) ,
g ( 2 ) = 1 + 2 k , k = 1,2 , .

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