Abstract

Interactions of structured light and structured media belong to the topical trends of modern optics. Dynamic vector speckle fields induced by the feedback interaction with the dynamic “optical-damage” effect provide, for the first time, real possibilities to produce and investigate the topological regularities of structured photorefractive media and coherent structured light interactions [Proc. SPIE 6905, 690505 (2008)]. The experiments provide evidence that dynamic topological speckle fields form ergodic systems, which develop through loop and chain trajectories in accordance with paired nucleation and annihilation of optical singularities. The loop/chain trajectories are realized for even/odd numbers of involved singular C points. Each loop reaction starts from a nucleated pair of C points and occurs in the same speckle up to their annihilation. A chain reaction starts from one C point of a nucleated C-point pair. It leaves the starting speckle and moves through the speckle field to other C-point pairs and annihilates with one of these. The second C point moves through the varying speckle field to the next pair of C points and so on. The conserved topological charge is equal to +(1/2) or (1/2) for L-type and S-type chain reactions, respectively. Loop trajectories with zero topological charges are short-lived in contrast with the chain reactions that are theoretically not limited in space and time. The chain reactions consist of long links with a standard topological structure. The results obtained are important for fundamentals and applications. This paper is the first study of the optical dynamic topology.

© 2014 Optical Society of America

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References

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  1. V. I. Vasil’ev and M. S. Soskin, “Dynamic singular optics,” Proc. SPIE 6905, 690505 (2008).
    [CrossRef]
  2. J. F. Nye and M. V. Berry, “Dislocations in wave trains,” Proc. Royal Soc. London A 336, 165–190 (1974).
    [CrossRef]
  3. J. F. Nye, Natural Focusing and Finer Structure of Light (Institute of Physics, 1999).
  4. M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
    [CrossRef]
  5. S. Odoulov and B. I. Sturman, “Photorefraction with the photovoltaic charge transport,” in Progress in Photorefractive Nonlinear Optics (Taylor & Francis, 2002), pp. 113–132.
  6. V. Vasil’ev and M. Soskin, “Topological and morphological transformations of developing singular paraxial vector light fields,” Opt. Commun. 281, 5527–5540 (2008).
    [CrossRef]
  7. M. Soskin and V. Vasil’ev, “Topological regularities of scalar/vector singular dynamic speckle fields,” in Coherence and Quantum Optics X, E. Bigelow and C. Stroud, eds., (Optical Society of America, 2013), pp. 112–120.
  8. J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Robert & Company, 2007).
  9. C. Dainty, ed., Laser Speckle and Related Phenomena (Springer, 1984).
  10. M. Born and E. Wolf, Principle of Optics, 7th ed. (Pergamon, 1999).
  11. M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A: Math. Gen. 10, 1809–1821 (1977).
    [CrossRef]
  12. M. Soskin and V. Vasil’ev, “Topological ergodic dynamics of optical singularities in laser-induced speckle fields following ‘optical damage’ of photorefractive LiNbO3:Fe crystal,” Proc. SPIE 8637, 86370U (2012).
    [CrossRef]
  13. V. I. Vasil’ev and M. S. Soskin, “Tangled nonlinear driven chain reactions of all optical singularities,” Proc. SPIE 8637, 820740W (2013).
  14. I. Freund, “Optical diabolos: configurations, nucleations, transformations, and reactions,” Opt. Commun. 272, 293–309 (2007).
    [CrossRef]
  15. M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572–2574 (2008).
    [CrossRef]
  16. M. S. Soskin and V. I. Vasil’ev, “Space–time design of the tangled C-points and optical vortex chain and loop reactions in paraxial dynamic elliptic speckle fields,” J. Opt. 15, 044022 (2013).
    [CrossRef]
  17. P. Walters, An Introduction Ergodic Theory (Springer, 1982).
  18. F. Flossmann, O. O’Holleran, M. Dennis, and M. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 203902 (2008).
    [CrossRef]
  19. M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
    [CrossRef]
  20. M. Soskin and V. Vasil’ev, “Dynamic singular vector speckle fields and their Hurst exponent time analysis,” in Contemporary Optoelectronics: From (Meta)Materials to Device Applications, O. V. Shulika and I. A. Suchoivanov, eds. (Springer Science + Business Media B. V., 2014), p. 156.
  21. V. Vasil’ev, V. Ponevchinsky, and M. Soskin, “Chain topological reactions in developing random light fields,” Proc. SPIE 7227, 72270A (2009).
    [CrossRef]
  22. J. Lamarsh and A. Baratta, Introduction to Nuclear Engineering (Prentice Hall, 2001).
  23. R. H. Petrucci and D. J. Goss, “Chemical kinetics,” in General Chemistry Principles & Modern Applications, 9th ed. (Pearson Prentice Hall, 2007), p. 77.

2013

V. I. Vasil’ev and M. S. Soskin, “Tangled nonlinear driven chain reactions of all optical singularities,” Proc. SPIE 8637, 820740W (2013).

M. S. Soskin and V. I. Vasil’ev, “Space–time design of the tangled C-points and optical vortex chain and loop reactions in paraxial dynamic elliptic speckle fields,” J. Opt. 15, 044022 (2013).
[CrossRef]

2012

M. Soskin and V. Vasil’ev, “Topological ergodic dynamics of optical singularities in laser-induced speckle fields following ‘optical damage’ of photorefractive LiNbO3:Fe crystal,” Proc. SPIE 8637, 86370U (2012).
[CrossRef]

2010

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[CrossRef]

2009

V. Vasil’ev, V. Ponevchinsky, and M. Soskin, “Chain topological reactions in developing random light fields,” Proc. SPIE 7227, 72270A (2009).
[CrossRef]

2008

F. Flossmann, O. O’Holleran, M. Dennis, and M. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef]

M. R. Dennis, “Polarization singularity anisotropy: determining monstardom,” Opt. Lett. 33, 2572–2574 (2008).
[CrossRef]

V. I. Vasil’ev and M. S. Soskin, “Dynamic singular optics,” Proc. SPIE 6905, 690505 (2008).
[CrossRef]

V. Vasil’ev and M. Soskin, “Topological and morphological transformations of developing singular paraxial vector light fields,” Opt. Commun. 281, 5527–5540 (2008).
[CrossRef]

2007

I. Freund, “Optical diabolos: configurations, nucleations, transformations, and reactions,” Opt. Commun. 272, 293–309 (2007).
[CrossRef]

2001

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

1977

M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A: Math. Gen. 10, 1809–1821 (1977).
[CrossRef]

1974

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

Baratta, A.

J. Lamarsh and A. Baratta, Introduction to Nuclear Engineering (Prentice Hall, 2001).

Berry, M. V.

M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A: Math. Gen. 10, 1809–1821 (1977).
[CrossRef]

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

Born, M.

M. Born and E. Wolf, Principle of Optics, 7th ed. (Pergamon, 1999).

Dennis, M.

F. Flossmann, O. O’Holleran, M. Dennis, and M. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef]

Dennis, M. R.

Flossmann, F.

F. Flossmann, O. O’Holleran, M. Dennis, and M. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef]

Freund, I.

I. Freund, “Optical diabolos: configurations, nucleations, transformations, and reactions,” Opt. Commun. 272, 293–309 (2007).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Robert & Company, 2007).

Goss, D. J.

R. H. Petrucci and D. J. Goss, “Chemical kinetics,” in General Chemistry Principles & Modern Applications, 9th ed. (Pearson Prentice Hall, 2007), p. 77.

Hannay, J. H.

M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A: Math. Gen. 10, 1809–1821 (1977).
[CrossRef]

Hasan, M. Z.

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[CrossRef]

Kane, C. L.

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[CrossRef]

Lamarsh, J.

J. Lamarsh and A. Baratta, Introduction to Nuclear Engineering (Prentice Hall, 2001).

Nye, J. F.

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

J. F. Nye, Natural Focusing and Finer Structure of Light (Institute of Physics, 1999).

O’Holleran, O.

F. Flossmann, O. O’Holleran, M. Dennis, and M. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef]

Odoulov, S.

S. Odoulov and B. I. Sturman, “Photorefraction with the photovoltaic charge transport,” in Progress in Photorefractive Nonlinear Optics (Taylor & Francis, 2002), pp. 113–132.

Padgett, M.

F. Flossmann, O. O’Holleran, M. Dennis, and M. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef]

Petrucci, R. H.

R. H. Petrucci and D. J. Goss, “Chemical kinetics,” in General Chemistry Principles & Modern Applications, 9th ed. (Pearson Prentice Hall, 2007), p. 77.

Ponevchinsky, V.

V. Vasil’ev, V. Ponevchinsky, and M. Soskin, “Chain topological reactions in developing random light fields,” Proc. SPIE 7227, 72270A (2009).
[CrossRef]

Soskin, M.

M. Soskin and V. Vasil’ev, “Topological ergodic dynamics of optical singularities in laser-induced speckle fields following ‘optical damage’ of photorefractive LiNbO3:Fe crystal,” Proc. SPIE 8637, 86370U (2012).
[CrossRef]

V. Vasil’ev, V. Ponevchinsky, and M. Soskin, “Chain topological reactions in developing random light fields,” Proc. SPIE 7227, 72270A (2009).
[CrossRef]

V. Vasil’ev and M. Soskin, “Topological and morphological transformations of developing singular paraxial vector light fields,” Opt. Commun. 281, 5527–5540 (2008).
[CrossRef]

M. Soskin and V. Vasil’ev, “Topological regularities of scalar/vector singular dynamic speckle fields,” in Coherence and Quantum Optics X, E. Bigelow and C. Stroud, eds., (Optical Society of America, 2013), pp. 112–120.

M. Soskin and V. Vasil’ev, “Dynamic singular vector speckle fields and their Hurst exponent time analysis,” in Contemporary Optoelectronics: From (Meta)Materials to Device Applications, O. V. Shulika and I. A. Suchoivanov, eds. (Springer Science + Business Media B. V., 2014), p. 156.

Soskin, M. S.

V. I. Vasil’ev and M. S. Soskin, “Tangled nonlinear driven chain reactions of all optical singularities,” Proc. SPIE 8637, 820740W (2013).

M. S. Soskin and V. I. Vasil’ev, “Space–time design of the tangled C-points and optical vortex chain and loop reactions in paraxial dynamic elliptic speckle fields,” J. Opt. 15, 044022 (2013).
[CrossRef]

V. I. Vasil’ev and M. S. Soskin, “Dynamic singular optics,” Proc. SPIE 6905, 690505 (2008).
[CrossRef]

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

Sturman, B. I.

S. Odoulov and B. I. Sturman, “Photorefraction with the photovoltaic charge transport,” in Progress in Photorefractive Nonlinear Optics (Taylor & Francis, 2002), pp. 113–132.

Vasil’ev, V.

M. Soskin and V. Vasil’ev, “Topological ergodic dynamics of optical singularities in laser-induced speckle fields following ‘optical damage’ of photorefractive LiNbO3:Fe crystal,” Proc. SPIE 8637, 86370U (2012).
[CrossRef]

V. Vasil’ev, V. Ponevchinsky, and M. Soskin, “Chain topological reactions in developing random light fields,” Proc. SPIE 7227, 72270A (2009).
[CrossRef]

V. Vasil’ev and M. Soskin, “Topological and morphological transformations of developing singular paraxial vector light fields,” Opt. Commun. 281, 5527–5540 (2008).
[CrossRef]

M. Soskin and V. Vasil’ev, “Topological regularities of scalar/vector singular dynamic speckle fields,” in Coherence and Quantum Optics X, E. Bigelow and C. Stroud, eds., (Optical Society of America, 2013), pp. 112–120.

M. Soskin and V. Vasil’ev, “Dynamic singular vector speckle fields and their Hurst exponent time analysis,” in Contemporary Optoelectronics: From (Meta)Materials to Device Applications, O. V. Shulika and I. A. Suchoivanov, eds. (Springer Science + Business Media B. V., 2014), p. 156.

Vasil’ev, V. I.

M. S. Soskin and V. I. Vasil’ev, “Space–time design of the tangled C-points and optical vortex chain and loop reactions in paraxial dynamic elliptic speckle fields,” J. Opt. 15, 044022 (2013).
[CrossRef]

V. I. Vasil’ev and M. S. Soskin, “Tangled nonlinear driven chain reactions of all optical singularities,” Proc. SPIE 8637, 820740W (2013).

V. I. Vasil’ev and M. S. Soskin, “Dynamic singular optics,” Proc. SPIE 6905, 690505 (2008).
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

Walters, P.

P. Walters, An Introduction Ergodic Theory (Springer, 1982).

Wolf, E.

M. Born and E. Wolf, Principle of Optics, 7th ed. (Pergamon, 1999).

J. Opt.

M. S. Soskin and V. I. Vasil’ev, “Space–time design of the tangled C-points and optical vortex chain and loop reactions in paraxial dynamic elliptic speckle fields,” J. Opt. 15, 044022 (2013).
[CrossRef]

J. Phys. A: Math. Gen.

M. V. Berry and J. H. Hannay, “Umbilic points on Gaussian random surfaces,” J. Phys. A: Math. Gen. 10, 1809–1821 (1977).
[CrossRef]

Opt. Commun.

I. Freund, “Optical diabolos: configurations, nucleations, transformations, and reactions,” Opt. Commun. 272, 293–309 (2007).
[CrossRef]

V. Vasil’ev and M. Soskin, “Topological and morphological transformations of developing singular paraxial vector light fields,” Opt. Commun. 281, 5527–5540 (2008).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

F. Flossmann, O. O’Holleran, M. Dennis, and M. Padgett, “Fractality of light’s darkness,” Phys. Rev. Lett. 100, 203902 (2008).
[CrossRef]

Proc. Royal Soc. London A

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

Proc. SPIE

M. Soskin and V. Vasil’ev, “Topological ergodic dynamics of optical singularities in laser-induced speckle fields following ‘optical damage’ of photorefractive LiNbO3:Fe crystal,” Proc. SPIE 8637, 86370U (2012).
[CrossRef]

V. I. Vasil’ev and M. S. Soskin, “Tangled nonlinear driven chain reactions of all optical singularities,” Proc. SPIE 8637, 820740W (2013).

V. Vasil’ev, V. Ponevchinsky, and M. Soskin, “Chain topological reactions in developing random light fields,” Proc. SPIE 7227, 72270A (2009).
[CrossRef]

V. I. Vasil’ev and M. S. Soskin, “Dynamic singular optics,” Proc. SPIE 6905, 690505 (2008).
[CrossRef]

Prog. Opt.

M. S. Soskin and M. V. Vasnetsov, “Singular optics,” Prog. Opt. 42, 219–276 (2001).
[CrossRef]

Rev. Mod. Phys.

M. Z. Hasan and C. L. Kane, “Colloquium: topological insulators,” Rev. Mod. Phys. 82, 3045–3067 (2010).
[CrossRef]

Other

M. Soskin and V. Vasil’ev, “Dynamic singular vector speckle fields and their Hurst exponent time analysis,” in Contemporary Optoelectronics: From (Meta)Materials to Device Applications, O. V. Shulika and I. A. Suchoivanov, eds. (Springer Science + Business Media B. V., 2014), p. 156.

P. Walters, An Introduction Ergodic Theory (Springer, 1982).

S. Odoulov and B. I. Sturman, “Photorefraction with the photovoltaic charge transport,” in Progress in Photorefractive Nonlinear Optics (Taylor & Francis, 2002), pp. 113–132.

J. F. Nye, Natural Focusing and Finer Structure of Light (Institute of Physics, 1999).

M. Soskin and V. Vasil’ev, “Topological regularities of scalar/vector singular dynamic speckle fields,” in Coherence and Quantum Optics X, E. Bigelow and C. Stroud, eds., (Optical Society of America, 2013), pp. 112–120.

J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications (Robert & Company, 2007).

C. Dainty, ed., Laser Speckle and Related Phenomena (Springer, 1984).

M. Born and E. Wolf, Principle of Optics, 7th ed. (Pergamon, 1999).

J. Lamarsh and A. Baratta, Introduction to Nuclear Engineering (Prentice Hall, 2001).

R. H. Petrucci and D. J. Goss, “Chemical kinetics,” in General Chemistry Principles & Modern Applications, 9th ed. (Pearson Prentice Hall, 2007), p. 77.

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Figures (8)

Fig. 1.
Fig. 1.

Setups for the creation of generic developing scalar (a) linearly polarized and (b) vector elliptically polarized] speckle fields due to the optical-damage effect in PRC. (c) Their real-time high-precision Stokes polarimetry.

Fig. 2.
Fig. 2.

Development of speckled scattering of the incident He–Ne smooth laser beam due to the interaction of the induced light scattering by refractive index inhomogeneities in a photorefractive lithium niobate crystal after (a) 510 s, (b) 900 s, and (c) 1400 s of laser beam irradiation. Arrows show orientations of the crystal c axis.

Fig. 3.
Fig. 3.

Two kinds of optical diabolos around C points with various distributions of ellipse axes a and b. Elliptic (a), (b) and hyperbolic (c), (d) diabolos are located on the tops (slopes) of the speckles. The nomenclature of C points is labeled in accordance with Fig. 4.

Fig. 4.
Fig. 4.

Nomenclature of C points three-level topological characteristics. Dashed line marks monstar and lemon, which transform mutually easily during singular speckle fields development [3].

Fig. 5.
Fig. 5.

Space-time evolution of the areas with right and left handedness of polarization ellipses and the delimiting L lines in the course of the development of dynamic singular speckle fields at (a) 510 s, (b) 900 s, and (c) 1400 s. Speckle heights and contours are shown by the gradation of gray color and the lines of equal areas of polarization ellipses. Topological characteristics of C points are shown in the same manner as in Fig. 4. All morphological and diabolic forms of C points are presented.

Fig. 6.
Fig. 6.

Loop trajectories of nucleated C point pairs in LH (a)–(c) and RH (d)–(i) polarized areas of a developing speckle field: (a) 300 s, (b) 315 s, (c) 330 s, (d) 795 s, (e) 810 s, (f) 840 s, (g) 870 s, (h) 930 s, and (i) 945 s. The full topological characteristics of all nucleated and moving C points are given in Fig. 4 nomenclature.

Fig. 7.
Fig. 7.

(a) L-type topological chain reaction in a developing vector speckle field as an example of a new kind of chain reactions with conservation of total topological charge. (b) S-type chain reaction.

Fig. 8.
Fig. 8.

Topological space-time development of two paired space-time limited L and S chain reactions started simultaneously and developing independently at (a) 330 s, (b) 345 s, (c) 510 s, (d) 720 s, (e) 810 s, and (f) 915 s.

Tables (1)

Tables Icon

Table 1. Ergodicity of Dynamic and Static Random Vector Speckle Fields

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