Abstract

Hyperuniform structures possess the ability to confine and drive light, although their fabrication is extremely challenging. Here we demonstrate that speckle patterns obtained by a superposition of randomly arranged sources of Bessel beams can be used to generate hyperunifrom scalar fields. By exploiting laser light tailored with a spatial filter, we experimentally produce (without requiring any computational power) a speckle pattern possessing maxima at locations corresponding to a hyperuniform distribution. By properly filtering out intensity fluctuation from the same speckle pattern, it is possible to retrieve an intensity profile satisfying the hyperuniformity requirements. Our findings are supported by extensive numerical simulations.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (2)

M. Jang, Y. Horie, A. Shibukawa, J. Brake, Y. Liu, S. M. Kamali, A. Arbabi, H. Ruan, A. Faraon, and C. Yang, “Wavefront shaping with disorder-engineered metasurfaces,” Nat. Photonics 12(2), 84–90 (2018).
[Crossref] [PubMed]

D. G. Pires, A. F. Sonsin, A. J. Jesus-Silva, and E. J. S. Fonseca, “Three-Dimensional Speckle Light Self-Healing-Based Imaging System,” Sci. Rep. 8(1), 563 (2018).
[Crossref] [PubMed]

2017 (6)

G. Ruocco, B. Abaie, W. Schirmacher, A. Mafi, and M. Leonetti, “Disorder-induced single-mode transmission,” Nat. Commun. 8(14571), 14571 (2017).
[Crossref] [PubMed]

R. Fickler, M. Ginoya, and R. W. Boyd, “Custom-tailored spatial mode sorting by controlled random scattering,” Phys. Rev. B 95(16), 161108 (2017).
[Crossref]

Z. Ma and S. Torquato, “Random Scalar Fields and Hyperuniformity,” J. Appl. Phys. 121(24), 244904 (2017).
[Crossref]

S. R. Sellers, W. Man, S. Sahba, and M. Florescu, “Local self-uniformity in photonic networks,” Nat. Commun. 8, 14439 (2017).
[Crossref] [PubMed]

N. Muller, J. Haberko, C. Marichy, and F. Scheffold, “Photonic hyperuniform networks obtained by silicon double inversion of polymer templates,” Optica 4(3), 361–366 (2017).
[Crossref]

J. Gateau, H. Rigneault, and M. Guillon, “Complementary Speckle Patterns: deterministic interchange of intrinsic vortices and maxima through Scattering Media,” Phys. Rev. Lett. 118(4), 043903 (2017).
[Crossref] [PubMed]

2016 (11)

S. Torquato, “Disordered hyperuniform heterogeneous materials,” J. Phys. Condens. Matter 28(41), 414012 (2016).
[Crossref] [PubMed]

R. Degl’Innocenti, Y. D. Shah, L. Masini, A. Ronzani, A. Pitanti, Y. Ren, D. S. Jessop, A. Tredicucci, H. E. Beere, and D. A. Ritchie, “Hyperuniform disordered terahertz quantum cascade laser,” Sci. Rep. 6(1), 19325 (2016).
[Crossref] [PubMed]

O. Leseur, R. Pierrat, and R. Carminati, “High-density hyperuniform materials can be transparent,” Optica 3(7), 763 (2016).
[Crossref]

L. S. Froufe-Pérez, M. Engel, P. F. Damasceno, N. Muller, J. Haberko, S. C. Glotzer, and F. Scheffold, “Role of short-range order and hyperuniformity in the formation of band gaps in disordered photonic materials,” Phys. Rev. Lett. 117(5), 053902 (2016).
[Crossref] [PubMed]

D. Di Battista, D. Ancora, M. Leonetti, and G. Zacharakis, “Tailoring non-diffractive beams from amorphous light speckles,” Appl. Phys. Lett. 109(12), 121110 (2016).
[Crossref]

S. Yu, X. Piao, J. Hong, and N. Park, “Metadisorder for designer light in random systems,” Sci. Adv. 2(10), e1501851 (2016).
[Crossref] [PubMed]

J. Bingi and V. M. Murukeshan, “Speckle lithography for fabricating Gaussian, quasi-random 2D structures and black silicon structures,” Sci. Rep. 5(1), 18452 (2016).
[Crossref] [PubMed]

D. B. Phillips, R. He, Q. Chen, G. M. Gibson, and M. J. Padgett, “Non-diffractive computational ghost imaging,” Opt. Express 24(13), 14172–14182 (2016).
[Crossref] [PubMed]

P. Ni, P. Zhang, X. Qi, J. Yang, Z. Chen, and W. Man, “Light localization and nonlinear beam transmission in specular amorphous photonic lattices,” Opt. Express 24(3), 2420–2426 (2016).
[Crossref] [PubMed]

S. Torquato, “Hyperuniformity and its generalizations,” Phys. Rev. E 94(2), 022122 (2016).
[Crossref] [PubMed]

S. Atkinson, G. Zhang, A. B. Hopkins, and S. Torquato, “Critical slowing down and hyperuniformity on approach to jamming,” Phys Rev E 94(1), 012902 (2016).
[Crossref] [PubMed]

2015 (7)

S. Torquato, G. Zhang, and F. H. Stillinger, “Ensemble theory for stealthy hyperuniform disordered ground states,” Phys. Rev. X 5(2), 021020 (2015).
[Crossref]

C. Liu, R. E. Van Der Wel, N. Rotenberg, L. Kuipers, T. F. Krauss, A. Di Falco, and A. Fratalocchi, “Triggering extreme events at the nanoscale in photonic seas,” Nat. Phys. 11(4), 358–363 (2015).
[Crossref]

T. Amoah and M. Florescu, “High-Q optical cavities in hyperuniform disordered materials,” Phys. Rev. B 91(2), 020201 (2015).
[Crossref]

P. Hsieh, C. Chung, J. F. McMillan, M. Tsai, M. Lu, N. C. Panoiu, and C. W. Wong, “Photon transport enhanced by transverse Anderson localization in disordered superlattices,” Nat. Phys. 11(3), 268–274 (2015).
[Crossref]

R. Fischer, I. Vidal, D. Gilboa, R. R. B. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, J. Hickman, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
[Crossref] [PubMed]

J. H. Weijs, R. Jeanneret, R. Dreyfus, and D. Bartolo, “Emergent Hyperuniformity in Periodically Driven Emulsions,” Phys. Rev. Lett. 115(10), 108301 (2015).
[Crossref] [PubMed]

C. De Rosa, F. Auriemma, C. Diletto, R. Di Girolamo, A. Malafronte, P. Morvillo, G. Zito, G. Rusciano, G. Pesce, and A. Sasso, “Toward hyperuniform disordered plasmonic nanostructures for reproducible surface-enhanced Raman spectroscopy,” Phys. Chem. Chem. Phys. 17(12), 8061–8069 (2015).
[Crossref] [PubMed]

2014 (8)

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8(10), 784–790 (2014).
[Crossref]

N. Muller, J. Haberko, C. Marichy, and F. Scheffold, “Silicon hyperuniform disordered photonic materials with a pronounced gap in the shortwave infrared,” Advanced Optical Materials 2(2), 115–119 (2014).
[Crossref]

Y. Bromberg and H. Cao, “Generating non-Rayleigh speckles with tailored intensity statistics,” Phys. Rev. Lett. 112(21), 213904 (2014).
[Crossref]

G. Volpe, G. Volpe, and S. Gigan, “Brownian motion in a speckle light field: tunable anomalous diffusion and selective optical manipulation,” Sci. Rep. 4(3936), 3936 (2014).
[PubMed]

G. Volpe, L. Kurz, A. Callegari, G. Volpe, and S. Gigan, “Speckle optical tweezers: micromanipulation with random light fields,” Opt. Express 22(15), 18159–18167 (2014).
[Crossref] [PubMed]

F. Riboli, N. Caselli, S. Vignolini, F. Intonti, K. Vynck, P. Barthelemy, A. Gerardino, L. Balet, L. H. Li, A. Fiore, M. Gurioli, and D. S. Wiersma, “Engineering of light confinement in strongly scattering disordered media,” Nat. Mater. 13(7), 720–725 (2014).
[Crossref] [PubMed]

G. M. Conley, M. Burresi, F. Pratesi, K. Vynck, and D. S. Wiersma, “Light transport and localization in two-dimensional correlated disorder,” Phys. Rev. Lett. 112(14), 143901 (2014).
[Crossref] [PubMed]

I. Lesanovsky and J. P. Garrahan, “Out-of-equilibrium structures in strongly interacting Rydberg gases with dissipation,” Phys. Rev. A 90(1), 011603 (2014).
[Crossref]

2013 (4)

M. Florescu, P. J. Steinhardt, and S. Torquato, “Optical cavities and waveguides in hyperuniform disordered photonic solids,” Phys. Rev. B 87(16), 165116 (2013).
[Crossref]

M. Boguslawski, S. Brake, J. Armijo, F. Diebel, P. Rose, and C. Denz, “Analysis of transverse Anderson localization in refractive index structures with customized random potential,” Opt. Express 21(26), 31713–31724 (2013).
[Crossref] [PubMed]

D. S. Wiersma, “Disordered photonics,” Nat. Photonics 7(3), 188–196 (2013).
[Crossref]

W. Man, M. Florescu, E. P. Williamson, Y. He, S. R. Hashemizad, B. Y. C. Leung, D. R. Liner, S. Torquato, P. M. Chaikin, and P. J. Steinhardt, “Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids,” Proc. Natl. Acad. Sci. U.S.A. 110(40), 15886–15891 (2013).
[Crossref] [PubMed]

2012 (2)

F. Jendrzejewski, A. Bernard, K. Mueller, P. Cheinet, V. Josse, M. Piraud, L. Pezzé, L. Sanchez-Palencia, A. Aspect, and P. Bouyer, “Three-dimensional localization of ultracold atoms in an optical disordered potential,” Nat. Phys. 8(5), 398–403 (2012).
[Crossref]

E. Mudry, K. Belkebir, J. Girard, J. Savatier, E. Le Moal, C. Nicoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics 6(5), 312–315 (2012).
[Crossref]

2011 (2)

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106(19), 193905 (2011).
[Crossref] [PubMed]

M. Leonetti, C. Conti, and C. Lopez, “The mode-locking transition of random lasers,” Nat. Photonics 5(10), 615–617 (2011).
[Crossref]

2010 (2)

P. D. García, R. Sapienza, and C. López, “Photonic glasses: a step beyond white paint,” Adv. Mater. 22(1), 12–19 (2010).
[Crossref] [PubMed]

L. Sanchez-Palencia and M. Lewenstein, “Disordered quantum gases under control,” Nat. Phys. 6(2), 87–95 (2010).
[Crossref]

2009 (2)

M. Florescu, S. Torquato, and P. J. Steinhardt, “Designer disordered materials with large, complete photonic band gaps,” Proc. Natl. Acad. Sci. U.S.A. 106(49), 20658–20663 (2009).
[Crossref] [PubMed]

E. C. Zachary and S. Torquato, “Hyperuniformity in point patterns and two-phase random heterogeneous media,” J. Stat. Mech. 12(12), 12015 (2009).
[Crossref]

2007 (1)

C. Vanneste, P. Sebbah, and H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. 98(14), 143902 (2007).
[Crossref] [PubMed]

2004 (1)

A. P. Joglekar, H. H. Liu, E. Meyhöfer, G. Mourou, and A. J. Hunt, “Optics at critical intensity: Applications to nanomorphing,” Proc. Natl. Acad. Sci. U.S.A. 101(16), 5856–5861 (2004).
[Crossref] [PubMed]

2003 (1)

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[Crossref] [PubMed]

1997 (1)

1987 (1)

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
[Crossref] [PubMed]

Abaie, B.

G. Ruocco, B. Abaie, W. Schirmacher, A. Mafi, and M. Leonetti, “Disorder-induced single-mode transmission,” Nat. Commun. 8(14571), 14571 (2017).
[Crossref] [PubMed]

Akahane, Y.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[Crossref] [PubMed]

Akbulut, D.

E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106(19), 193905 (2011).
[Crossref] [PubMed]

Allain, M.

E. Mudry, K. Belkebir, J. Girard, J. Savatier, E. Le Moal, C. Nicoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics 6(5), 312–315 (2012).
[Crossref]

Amoah, T.

T. Amoah and M. Florescu, “High-Q optical cavities in hyperuniform disordered materials,” Phys. Rev. B 91(2), 020201 (2015).
[Crossref]

Ancora, D.

D. Di Battista, D. Ancora, M. Leonetti, and G. Zacharakis, “Tailoring non-diffractive beams from amorphous light speckles,” Appl. Phys. Lett. 109(12), 121110 (2016).
[Crossref]

Arbabi, A.

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E. G. van Putten, D. Akbulut, J. Bertolotti, W. L. Vos, A. Lagendijk, and A. P. Mosk, “Scattering lens resolves sub-100 nm structures with visible light,” Phys. Rev. Lett. 106(19), 193905 (2011).
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J. H. Weijs, R. Jeanneret, R. Dreyfus, and D. Bartolo, “Emergent Hyperuniformity in Periodically Driven Emulsions,” Phys. Rev. Lett. 115(10), 108301 (2015).
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D. Di Battista, D. Ancora, M. Leonetti, and G. Zacharakis, “Tailoring non-diffractive beams from amorphous light speckles,” Appl. Phys. Lett. 109(12), 121110 (2016).
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Nat. Mater. (1)

F. Riboli, N. Caselli, S. Vignolini, F. Intonti, K. Vynck, P. Barthelemy, A. Gerardino, L. Balet, L. H. Li, A. Fiore, M. Gurioli, and D. S. Wiersma, “Engineering of light confinement in strongly scattering disordered media,” Nat. Mater. 13(7), 720–725 (2014).
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M. Jang, Y. Horie, A. Shibukawa, J. Brake, Y. Liu, S. M. Kamali, A. Arbabi, H. Ruan, A. Faraon, and C. Yang, “Wavefront shaping with disorder-engineered metasurfaces,” Nat. Photonics 12(2), 84–90 (2018).
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E. Mudry, K. Belkebir, J. Girard, J. Savatier, E. Le Moal, C. Nicoletti, M. Allain, and A. Sentenac, “Structured illumination microscopy using unknown speckle patterns,” Nat. Photonics 6(5), 312–315 (2012).
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Opt. Express (4)

Opt. Lett. (1)

Optica (2)

Phys Rev E (1)

S. Atkinson, G. Zhang, A. B. Hopkins, and S. Torquato, “Critical slowing down and hyperuniformity on approach to jamming,” Phys Rev E 94(1), 012902 (2016).
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Phys. Chem. Chem. Phys. (1)

C. De Rosa, F. Auriemma, C. Diletto, R. Di Girolamo, A. Malafronte, P. Morvillo, G. Zito, G. Rusciano, G. Pesce, and A. Sasso, “Toward hyperuniform disordered plasmonic nanostructures for reproducible surface-enhanced Raman spectroscopy,” Phys. Chem. Chem. Phys. 17(12), 8061–8069 (2015).
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T. Amoah and M. Florescu, “High-Q optical cavities in hyperuniform disordered materials,” Phys. Rev. B 91(2), 020201 (2015).
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M. Florescu, P. J. Steinhardt, and S. Torquato, “Optical cavities and waveguides in hyperuniform disordered photonic solids,” Phys. Rev. B 87(16), 165116 (2013).
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R. Fickler, M. Ginoya, and R. W. Boyd, “Custom-tailored spatial mode sorting by controlled random scattering,” Phys. Rev. B 95(16), 161108 (2017).
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Phys. Rev. E (1)

S. Torquato, “Hyperuniformity and its generalizations,” Phys. Rev. E 94(2), 022122 (2016).
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C. Vanneste, P. Sebbah, and H. Cao, “Lasing with resonant feedback in weakly scattering random systems,” Phys. Rev. Lett. 98(14), 143902 (2007).
[Crossref] [PubMed]

J. H. Weijs, R. Jeanneret, R. Dreyfus, and D. Bartolo, “Emergent Hyperuniformity in Periodically Driven Emulsions,” Phys. Rev. Lett. 115(10), 108301 (2015).
[Crossref] [PubMed]

G. M. Conley, M. Burresi, F. Pratesi, K. Vynck, and D. S. Wiersma, “Light transport and localization in two-dimensional correlated disorder,” Phys. Rev. Lett. 112(14), 143901 (2014).
[Crossref] [PubMed]

R. Fischer, I. Vidal, D. Gilboa, R. R. B. Correia, A. C. Ribeiro-Teixeira, S. D. Prado, J. Hickman, and Y. Silberberg, “Light with tunable non-Markovian phase imprint,” Phys. Rev. Lett. 115(7), 073901 (2015).
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Phys. Rev. X (1)

S. Torquato, G. Zhang, and F. H. Stillinger, “Ensemble theory for stealthy hyperuniform disordered ground states,” Phys. Rev. X 5(2), 021020 (2015).
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W. Man, M. Florescu, E. P. Williamson, Y. He, S. R. Hashemizad, B. Y. C. Leung, D. R. Liner, S. Torquato, P. M. Chaikin, and P. J. Steinhardt, “Isotropic band gaps and freeform waveguides observed in hyperuniform disordered photonic solids,” Proc. Natl. Acad. Sci. U.S.A. 110(40), 15886–15891 (2013).
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G. Volpe, G. Volpe, and S. Gigan, “Brownian motion in a speckle light field: tunable anomalous diffusion and selective optical manipulation,” Sci. Rep. 4(3936), 3936 (2014).
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D. G. Pires, A. F. Sonsin, A. J. Jesus-Silva, and E. J. S. Fonseca, “Three-Dimensional Speckle Light Self-Healing-Based Imaging System,” Sci. Rep. 8(1), 563 (2018).
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J. Bingi and V. M. Murukeshan, “Speckle lithography for fabricating Gaussian, quasi-random 2D structures and black silicon structures,” Sci. Rep. 5(1), 18452 (2016).
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S. Torquato, “Hyperuniform states of matter,” Phys. Rep. in press (2018).

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Figures (7)

Fig. 1
Fig. 1 Gauss modes distributed according to the χ=0.4 stealthy hyperuniform map. We compare the χ ˜ ( r ) and ψ ˜ ( k ) from the patterns composed of modes with different width σ: in panel (a) the σ is 20 pixels, in (b) is 60 pixels, in (c) is 90 pixels and in (d) is 320 pixels. The distance between the first two peaks in the χ ˜ ( r ) from (c) provides the average distance between the modes position, d ˜ =368 pixels. In (a), (b) and (c) σ d ˜ , for larger values of σ, χ ˜ ( r ) tends to a Bessel function (yellow dashed curves); he hyperuniform information from the map is maintained ( ψ ˜ ( k )0 when k0). Differently, in (d) we have  σ d ˜ , the speckle pattern results completely disordered.
Fig. 2
Fig. 2 ASPs are given by the superposition of Bessel beams. In panel (a) we show the pattern from random superposition of Bessel beams, in (b) the phasors are distributed according to the given hyperuniform map ( χ=0.4). In panel (c) we report the ASP from the experiment. χ ˜ ( r ) and ψ ˜ calculated from patterns (a) and (b) [blue curves] and patterns (c) [dashed black curves] are shown in (d) and (e); all the cases exhibit same pair statistic. The χ ˜ ( r ) is Bessel shaped, the width of the first peak gives the typical speckle grain size, while the distance between the peak provides the average distance d ˜ within the grains. In panel (e) the ψ ˜ ( k )0 in the annular region defined by k ˜ = 2π d ˜ and around k x , k y =0 where is strongly peaked. Outside the annular region ψ ˜ ( k )=0
Fig. 3
Fig. 3 Study of the distribution of the speckle pattern grains. We extrapolate the position of the speckle grains from a conventional speckle pattern (Gauss modes) in (a) and a synthetic ASP (superposition of Bessel modes) in (b). Yellow crosses in patterns (a) and (b) define intensity peaks (speckle grains) positions. Peaks position are used as maps to generate point distributions in (c) and (d). Their S(k) are shown in (e) for standard speckles and (f) for ASPs. we notice that S( k )0 when k0; the speckle grains (intensity peaks) distribution of ASPs respect the condition for hyperuniformity.
Fig. 4
Fig. 4 In (a) the typical ASP, in (b) the same pattern is corrected with a Gaussian filter. In (c) and (d) we show respectively 5X magnification from the patterns (a) and (b), after correction the pattern loses the intensity fluctuations, e.g. the bright and dark regions in the white circles. The small white crosses in (c) and (d) indicate the speckle grains positions, it is visible that the Gauss-filter does not modify the grains positions but correct for their intensity only. The χ ˜ ( k ) from (b) is shown with the inset in (e) and the blue curve is its intensity profile. We have that χ ˜ ( k )0 when k0.
Fig. 5
Fig. 5 On the left panel the comparison of the local scalar field fluctuation versus the windows radius from different speckle patterns: dotted blues curve from synthetic standard speckle pattern, dashed red curve from the synthetic ASP and the intensity corrected ASP [the one in Fig. 4(b)]. On the right panel the same comparison is presented for the experimental data.
Fig. 6
Fig. 6 In section a) the scheme of the experimental setup is depicted. A coherent laser beam impinges onto a scattering diffuser (D). The transmitted light at 1mm at the back of D is projected with a 4f system composed of lenses L1 and L2 and is magnified 10X on the CCD camera plane. In section b) we apply a spatial frequency selection of the forward scattered light introducing an annular aperture (ring filter RF) in the Fourier plane of the 4f system. In this last case the ASP is generated.
Fig. 7
Fig. 7 Standard speckle patterns analysis. We show the speckle pattern from the numerical simulation in panel (a) and experimental speckle patter (b). In (c) and (d) we report their correspondent auto-correlation function χ ˜ ( r ) and in (e) and (f) their spectral density ψ ˜ ( k ). Comparing the two cases we prove that our simulations respect the statistic from the real experiment.

Equations (14)

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σ N 2 ( R )~ R d1
lim | k |0 S( k ) = 0.
S( k )=0for0| k |K
χ( r )= | FT 1 { | FT{ C μ C } | 2 } |
C simulation = ( i=1 N f i ) 2 .
lim | k |0 ψ ˜ ( k ) = 0.
J α,i = ( z 2 ) α k=0 ( z i   2 4 ) k k!Γ( α+k+1 )  e i φ i
C simul. Bessel = ( i=1 N J α,i ) 2
h g ( n 1 , n 2 )= e ( n 1 2 + n 2 2 )/2 σ 2
h( n 1 , n 2 )= h g ( n 1 , n 2 ) n 1 n 2   h g
A * =FT{ A }hB= FT 1 { A * }
C( l,m )= A l,m / B l,m l[ 1,  n 1 ], m[ 1,  n 2 ]
f i = e ( x x i ) 2 + ( y y i ) 2 2 σ 2 e i φ i
C simulation = ( i=1 N f i ) 2 .

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