Abstract

We investigate the number of vortices embedded in a carrier beam needed to produce a speckle pattern and the necessary conditions in terms of their initial distribution and topological charges. A spatial light modulator is used to imprint arrays of vortices in a Gaussian beam, which is propagated in free space for a given distance and then focused in order to induce interaction among the vortices in the focal region. The resulting optical field is analyzed after propagation up to a transverse plane where the carrier beam would recover its initial size in the absence of vortices. The role of different control parameters for obtaining ordered and disordered patterns is discussed. Our experimental study is complemented with a thorough numerical analysis, from which the statistical properties of the disordered patterns are characterized, and the conditions for obtaining well-developed speckle are determined. We also discuss the creation and annihilation of vortex pairs, depending on the initial conditions.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  6. V. G. Shvedov, A. V. Rode, Y. V. Izdebskaya, A. S. Desyatnikov, W. Krolikowski, and Y. S. Kivshar, “Selective trapping of multiple particles by volume speckle field,” Opt. Express 18, 3137–3142 (2010).
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    [Crossref]
  10. R. A. Arizaga, N. L. Cap, H. J. Rabal, and M. Trivi, “Display of local activity using dynamical speckle patterns,” Opt. Eng. 41, 287–295 (2002).
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  11. R. Bates, “Astronomical speckle imaging,” Phys. Reports 90, 203–297 (1982).
    [Crossref]
  12. S.-H. Jiang and J. G. Walker, “Speckle-illuminated fluorescence confocal microscopy, using a digital micro-mirror device,” Meas. Sci. Technol. 20, 065501 (2009).
    [Crossref]
  13. M. Berry, “Disruption of wavefronts: statistics of dislocations in incoherent gaussian random waves,” J. Phys. A: Math. Gen. 11, 27–37 (1978).
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    [Crossref]
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    [Crossref]
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  23. M. Dennis, “Local phase structure of wave dislocation lines: twist and twirl,” J. Opt. A: Pure Appl. Opt. 6, S202–S208 (2004).
    [Crossref]
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    [Crossref]
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  26. N. Heckenberg, M. Vaupel, J. Malos, and C. Weiss, “Optical-vortex pair creation and annihilation and helical astigmatism of a nonplanar ring resonator,” Phys. Rev. A 54, 2369–2378 (1996).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  30. F. S. Roux, “Optical vortex density limitation,” Opt. Commun. 223, 31–37 (2003).
    [Crossref]
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    [Crossref]
  32. F. S. Roux, “Canonical vortex dipole dynamics,” JOSA B 21, 655–663 (2004).
    [Crossref]
  33. D. Rozas, C. Law, and G. Swartzlander, “Propagation dynamics of optical vortices,” JOSA B 14, 3054–3065 (1997).
    [Crossref]
  34. D. Rozas, Z. Sacks, and G. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
    [Crossref]
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2018 (2)

F. Xu, J. Wang, D. Zhu, and Q. Tu, “Speckle noise reduction technique for lidar echo signal based on self-adaptive pulse-matching independent component analysis,” Opt. Lasers Eng. 103, 92–99 (2018).
[Crossref]

L. De Angelis and L. Kuipers, “Screening and fluctuation of the topological charge in random wave fields,” Opt. Lett. 43, 2740–2743 (2018).
[Crossref]

2017 (1)

L. De Angelis, F. Alpeggiani, A. Di Falco, and L. Kuipers, “Persistence and lifelong fidelity of phase singularities in optical random waves,” Phys. Rev. Lett. 119, 203903 (2017).
[Crossref]

2014 (1)

2010 (1)

2009 (1)

S.-H. Jiang and J. G. Walker, “Speckle-illuminated fluorescence confocal microscopy, using a digital micro-mirror device,” Meas. Sci. Technol. 20, 065501 (2009).
[Crossref]

2008 (1)

K. O’Holleran, M. R. Dennis, F. Flossmann, and M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 053902 (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] [PubMed]

2006 (1)

2005 (1)

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902 (2005).
[Crossref] [PubMed]

2004 (4)

I. Yamaguchi, K. Kobayashi, and L. P. Yaroskavsky, “Measurement of surface roughness by speckle correlation,” Opt. Eng. 43, 2753–2762 (2004).
[Crossref]

M. Dennis, “Local phase structure of wave dislocation lines: twist and twirl,” J. Opt. A: Pure Appl. Opt. 6, S202–S208 (2004).
[Crossref]

D. C. Adler, T. H. Ko, and J. G. Fujimoto, “Speckle reduction in optical coherence tomography images by use of a spatially adaptive wavelet filter,” Opt. Lett. 29, 2878–2880 (2004).
[Crossref]

F. S. Roux, “Canonical vortex dipole dynamics,” JOSA B 21, 655–663 (2004).
[Crossref]

2003 (1)

F. S. Roux, “Optical vortex density limitation,” Opt. Commun. 223, 31–37 (2003).
[Crossref]

2002 (3)

A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, “Generation of lattice structures of optical vortices,” JOSA B 19, 550–556 (2002).
[Crossref]

K. Z. Abd-Elmoniem, A.-B. Youssef, and Y. M. Kadah, “Real-time speckle reduction and coherence enhancement in ultrasound imaging via nonlinear anisotropic diffusion,” IEEE Transactions on Biomed. Eng. 49, 997–1014 (2002).
[Crossref]

R. A. Arizaga, N. L. Cap, H. J. Rabal, and M. Trivi, “Display of local activity using dynamical speckle patterns,” Opt. Eng. 41, 287–295 (2002).
[Crossref]

2000 (1)

M. Berry and M. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. Lond. A 456, 2059–2079 (2000).
[Crossref]

1998 (1)

G.-H. Kim, J.-H. Jeon, Y.-C. Noh, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147, 131–137 (1998).
[Crossref]

1997 (2)

D. Rozas, C. Law, and G. Swartzlander, “Propagation dynamics of optical vortices,” JOSA B 14, 3054–3065 (1997).
[Crossref]

D. Rozas, Z. Sacks, and G. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
[Crossref]

1996 (1)

N. Heckenberg, M. Vaupel, J. Malos, and C. Weiss, “Optical-vortex pair creation and annihilation and helical astigmatism of a nonplanar ring resonator,” Phys. Rev. A 54, 2369–2378 (1996).
[Crossref] [PubMed]

1995 (1)

K. Staliunas, A. Berzanskis, and V. Jarutis, “Vortex statistics in optical speckle fields,” Opt. Commun. 120, 23–28 (1995).
[Crossref]

1994 (2)

I. Freund, “Optical vortices in gaussian random wave fields: statistical probability densities,” JOSA A 11, 1644–1652 (1994).
[Crossref]

N. Shvartsman and I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[Crossref] [PubMed]

1993 (2)

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[Crossref]

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[Crossref]

1992 (1)

K. Staliunas, “Dynamics of optical vortices in a laser beam,” Opt. Commun. 90, 123–127 (1992).
[Crossref]

1982 (1)

R. Bates, “Astronomical speckle imaging,” Phys. Reports 90, 203–297 (1982).
[Crossref]

1978 (1)

M. Berry, “Disruption of wavefronts: statistics of dislocations in incoherent gaussian random waves,” J. Phys. A: Math. Gen. 11, 27–37 (1978).
[Crossref]

1974 (1)

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336, 165–190 (1974).
[Crossref]

Abd-Elmoniem, K. Z.

K. Z. Abd-Elmoniem, A.-B. Youssef, and Y. M. Kadah, “Real-time speckle reduction and coherence enhancement in ultrasound imaging via nonlinear anisotropic diffusion,” IEEE Transactions on Biomed. Eng. 49, 997–1014 (2002).
[Crossref]

Adler, D. C.

Alpeggiani, F.

L. De Angelis, F. Alpeggiani, A. Di Falco, and L. Kuipers, “Persistence and lifelong fidelity of phase singularities in optical random waves,” Phys. Rev. Lett. 119, 203903 (2017).
[Crossref]

Arizaga, R. A.

R. A. Arizaga, N. L. Cap, H. J. Rabal, and M. Trivi, “Display of local activity using dynamical speckle patterns,” Opt. Eng. 41, 287–295 (2002).
[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] [PubMed]

Bates, R.

R. Bates, “Astronomical speckle imaging,” Phys. Reports 90, 203–297 (1982).
[Crossref]

Berry, M.

M. Berry and M. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. Lond. A 456, 2059–2079 (2000).
[Crossref]

M. Berry, “Disruption of wavefronts: statistics of dislocations in incoherent gaussian random waves,” J. Phys. A: Math. Gen. 11, 27–37 (1978).
[Crossref]

Berry, M. V.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336, 165–190 (1974).
[Crossref]

Berzanskis, A.

K. Staliunas, A. Berzanskis, and V. Jarutis, “Vortex statistics in optical speckle fields,” Opt. Commun. 120, 23–28 (1995).
[Crossref]

Callegari, A.

Cap, N. L.

R. A. Arizaga, N. L. Cap, H. J. Rabal, and M. Trivi, “Display of local activity using dynamical speckle patterns,” Opt. Eng. 41, 287–295 (2002).
[Crossref]

Chang, J.-S.

G.-H. Kim, J.-H. Jeon, Y.-C. Noh, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147, 131–137 (1998).
[Crossref]

Chervenkov, S.

A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, “Generation of lattice structures of optical vortices,” JOSA B 19, 550–556 (2002).
[Crossref]

De Angelis, L.

L. De Angelis and L. Kuipers, “Screening and fluctuation of the topological charge in random wave fields,” Opt. Lett. 43, 2740–2743 (2018).
[Crossref]

L. De Angelis, F. Alpeggiani, A. Di Falco, and L. Kuipers, “Persistence and lifelong fidelity of phase singularities in optical random waves,” Phys. Rev. Lett. 119, 203903 (2017).
[Crossref]

Dennis, M.

M. Dennis, “Local phase structure of wave dislocation lines: twist and twirl,” J. Opt. A: Pure Appl. Opt. 6, S202–S208 (2004).
[Crossref]

M. Berry and M. Dennis, “Phase singularities in isotropic random waves,” Proc. R. Soc. Lond. A 456, 2059–2079 (2000).
[Crossref]

Dennis, M. R.

K. O’Holleran, M. R. Dennis, F. Flossmann, and M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 053902 (2008).
[Crossref]

Desyatnikov, A. S.

Di Falco, A.

L. De Angelis, F. Alpeggiani, A. Di Falco, and L. Kuipers, “Persistence and lifelong fidelity of phase singularities in optical random waves,” Phys. Rev. Lett. 119, 203903 (2017).
[Crossref]

Dreischuh, A.

A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, “Generation of lattice structures of optical vortices,” JOSA B 19, 550–556 (2002).
[Crossref]

Erf, R.

R. Erf, Speckle metrology (Elsevier, 2012).

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] [PubMed]

Flossmann, F.

K. O’Holleran, M. R. Dennis, F. Flossmann, and M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 053902 (2008).
[Crossref]

Freilikher, V.

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[Crossref]

Freund, I.

I. Freund, “Optical vortices in gaussian random wave fields: statistical probability densities,” JOSA A 11, 1644–1652 (1994).
[Crossref]

N. Shvartsman and I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[Crossref] [PubMed]

I. Freund, N. Shvartsman, and V. Freilikher, “Optical dislocation networks in highly random media,” Opt. Commun. 101, 247–264 (1993).
[Crossref]

Fujimoto, J. G.

Gigan, S.

Goodman, J. W.

J. W. Goodman, Speckle phenomena in optics: theory and applications (Roberts and Company Publishers, 2007).

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw Hill Book, 1996).

Hanson, S. G.

W. Wang, T. Yokozeki, R. Ishijima, M. Takeda, and S. G. Hanson, “Optical vortex metrology based on the core structures of phase singularities in laguerre-gauss transform of a speckle pattern,” Opt. Express 14, 10195–10206 (2006).
[Crossref] [PubMed]

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902 (2005).
[Crossref] [PubMed]

Heckenberg, N.

N. Heckenberg, M. Vaupel, J. Malos, and C. Weiss, “Optical-vortex pair creation and annihilation and helical astigmatism of a nonplanar ring resonator,” Phys. Rev. A 54, 2369–2378 (1996).
[Crossref] [PubMed]

Indebetouw, G.

G. Indebetouw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[Crossref]

Ishijima, R.

Izdebskaya, Y. V.

Jarutis, V.

K. Staliunas, A. Berzanskis, and V. Jarutis, “Vortex statistics in optical speckle fields,” Opt. Commun. 120, 23–28 (1995).
[Crossref]

Jeon, J.-H.

G.-H. Kim, J.-H. Jeon, Y.-C. Noh, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147, 131–137 (1998).
[Crossref]

Jiang, S.-H.

S.-H. Jiang and J. G. Walker, “Speckle-illuminated fluorescence confocal microscopy, using a digital micro-mirror device,” Meas. Sci. Technol. 20, 065501 (2009).
[Crossref]

Kadah, Y. M.

K. Z. Abd-Elmoniem, A.-B. Youssef, and Y. M. Kadah, “Real-time speckle reduction and coherence enhancement in ultrasound imaging via nonlinear anisotropic diffusion,” IEEE Transactions on Biomed. Eng. 49, 997–1014 (2002).
[Crossref]

Kim, G.-H.

G.-H. Kim, J.-H. Jeon, Y.-C. Noh, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147, 131–137 (1998).
[Crossref]

Kivshar, Y. S.

Ko, K.-H.

G.-H. Kim, J.-H. Jeon, Y.-C. Noh, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147, 131–137 (1998).
[Crossref]

Ko, T. H.

Kobayashi, K.

I. Yamaguchi, K. Kobayashi, and L. P. Yaroskavsky, “Measurement of surface roughness by speckle correlation,” Opt. Eng. 43, 2753–2762 (2004).
[Crossref]

Krolikowski, W.

Kuipers, L.

L. De Angelis and L. Kuipers, “Screening and fluctuation of the topological charge in random wave fields,” Opt. Lett. 43, 2740–2743 (2018).
[Crossref]

L. De Angelis, F. Alpeggiani, A. Di Falco, and L. Kuipers, “Persistence and lifelong fidelity of phase singularities in optical random waves,” Phys. Rev. Lett. 119, 203903 (2017).
[Crossref]

Kurz, L.

Law, C.

D. Rozas, C. Law, and G. Swartzlander, “Propagation dynamics of optical vortices,” JOSA B 14, 3054–3065 (1997).
[Crossref]

Lee, J.-H.

G.-H. Kim, J.-H. Jeon, Y.-C. Noh, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147, 131–137 (1998).
[Crossref]

Malos, J.

N. Heckenberg, M. Vaupel, J. Malos, and C. Weiss, “Optical-vortex pair creation and annihilation and helical astigmatism of a nonplanar ring resonator,” Phys. Rev. A 54, 2369–2378 (1996).
[Crossref] [PubMed]

Miyamoto, Y.

W. Wang, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Experimental investigation of local properties and statistics of optical vortices in random wave fields,” Phys. Rev. Lett. 94, 103902 (2005).
[Crossref] [PubMed]

Moon, H.-J.

G.-H. Kim, J.-H. Jeon, Y.-C. Noh, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147, 131–137 (1998).
[Crossref]

Neshev, D.

A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, “Generation of lattice structures of optical vortices,” JOSA B 19, 550–556 (2002).
[Crossref]

Noh, Y.-C.

G.-H. Kim, J.-H. Jeon, Y.-C. Noh, K.-H. Ko, H.-J. Moon, J.-H. Lee, and J.-S. Chang, “An array of phase singularities in a self-defocusing medium,” Opt. Commun. 147, 131–137 (1998).
[Crossref]

Nye, J. F.

J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. R. Soc. Lond. A 336, 165–190 (1974).
[Crossref]

O’Holleran, K.

K. O’Holleran, M. R. Dennis, F. Flossmann, and M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 053902 (2008).
[Crossref]

Padgett, M. J.

K. O’Holleran, M. R. Dennis, F. Flossmann, and M. J. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 053902 (2008).
[Crossref]

Paulus, G. G.

A. Dreischuh, S. Chervenkov, D. Neshev, G. G. Paulus, and H. Walther, “Generation of lattice structures of optical vortices,” JOSA B 19, 550–556 (2002).
[Crossref]

Rabal, H. J.

R. A. Arizaga, N. L. Cap, H. J. Rabal, and M. Trivi, “Display of local activity using dynamical speckle patterns,” Opt. Eng. 41, 287–295 (2002).
[Crossref]

Rode, A. V.

Roux, F. S.

F. S. Roux, “Canonical vortex dipole dynamics,” JOSA B 21, 655–663 (2004).
[Crossref]

F. S. Roux, “Optical vortex density limitation,” Opt. Commun. 223, 31–37 (2003).
[Crossref]

Rozas, D.

D. Rozas, C. Law, and G. Swartzlander, “Propagation dynamics of optical vortices,” JOSA B 14, 3054–3065 (1997).
[Crossref]

D. Rozas, Z. Sacks, and G. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
[Crossref]

Sacks, Z.

D. Rozas, Z. Sacks, and G. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
[Crossref]

Schwartz, T.

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] [PubMed]

Segev, M.

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] [PubMed]

Shvartsman, N.

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S.-H. Jiang and J. G. Walker, “Speckle-illuminated fluorescence confocal microscopy, using a digital micro-mirror device,” Meas. Sci. Technol. 20, 065501 (2009).
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F. S. Roux, “Optical vortex density limitation,” Opt. Commun. 223, 31–37 (2003).
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[Crossref]

R. A. Arizaga, N. L. Cap, H. J. Rabal, and M. Trivi, “Display of local activity using dynamical speckle patterns,” Opt. Eng. 41, 287–295 (2002).
[Crossref]

Opt. Express (3)

Opt. Lasers Eng. (1)

F. Xu, J. Wang, D. Zhu, and Q. Tu, “Speckle noise reduction technique for lidar echo signal based on self-adaptive pulse-matching independent component analysis,” Opt. Lasers Eng. 103, 92–99 (2018).
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[Crossref] [PubMed]

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N. Shvartsman and I. Freund, “Vortices in random wave fields: nearest neighbor anticorrelations,” Phys. Rev. Lett. 72, 1008–1011 (1994).
[Crossref] [PubMed]

D. Rozas, Z. Sacks, and G. Swartzlander, “Experimental observation of fluidlike motion of optical vortices,” Phys. Rev. Lett. 79, 3399–3402 (1997).
[Crossref]

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R. Erf, Speckle metrology (Elsevier, 2012).

D. Voelz, Computational Fourier Optics, A MATLAB Tutorial (SPIE Press, 2011).
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J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw Hill Book, 1996).

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Figures (5)

Fig. 1
Fig. 1 (a) Experimental setup: Spatial Light Modulator (SLM), convergent lens(L), beam profiler (BP). The insets illustrate the phase mask for an ordered square lattice (left), where the red dots indicate the vortices, and the image of the propagated field obtained with the BP. (b)Different cases of arrays with N = 100 vortices. From left to right: Ordered Lattice (OL) with the Same Helicity (SH); Disordered Lattice (DL) with SH; OL with Random Helicity (RH); DL with RH; OL with Anticorrelated Helicity (AH); DL with AH. From top to bottom: schematics of the arrays, where vortices of opposite helicities are represented by red and blue dots; phase masks; simulations of the field at the observation plane z = 2f; experimental results. The phase levels in the masks vary from 0 (black) to 2π (white). The pitch of the OL is P = 13 pixels of the SLM (260μm) and the scales in the simulations are in mm, being the same for the experiments.
Fig. 2
Fig. 2 Intensity patterns generated by disordered arrays of vortices with (a) random helicities and (b) anticorrelated helicities, for different number of vortices, as indicated on the top of each figure. Simulations are compared with experimental results for each particular realization.
Fig. 3
Fig. 3 Simulations and experimental results for the intensity patterns generated by disordered arrays of N = 169 vortices with (a) random helicities and (b) anticorrelated helicities, for different values of the pitch parameter P expressed in SLM pixels, indicated on the top of each figure.
Fig. 4
Fig. 4 (a)–(b) Examples of the numerically calculated phase (left) and contour plots (right) of the real (red) and imaginary (blue) parts of the complex field amplitude for the initial mask with N = 100, and the propagated field, for RH (a) and AH (b). Vortices of opposite charges are indicated with red and blue markers. The phase levels vary from 0 (blue) to 2π (yellow). (c) Ensemble average of the final number of vortices 〈Nf〉 as a function of the initial number of vortices N, for RH (blue curve) and AH (green curve), for an analyzed area of radius rwin = 1.69 mm. The black line represents vortex number conservation: 〈Nf〉 = N.
Fig. 5
Fig. 5 Left: Numerical statistical analysis. Left: Examples of the analyzed fields (N = 900) in the region of interest. Center: Probability Density Functions of the intensity for N =25 (green), 100 (blue), 225 (purple), 400 (yellow) and 900 (red), and their comparison with a negative exponential function of a well-developed speckle (dashed black curves). Insets: semilog plots of the PDF’s. Right: Normalized autocorrelation of the intensity.

Equations (2)

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U ( x , y ) = exp ( ( x 2 + y 2 ) w 0 2 ) n = 1 N [ ( x x n ) + β n i ( y y n ) ] ,
C I ( Δ x , Δ y ) = ( I ( x , y ) I ¯ ) ( I ( x , y ) I ¯ ) ( Δ x , Δ y ) ,

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