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Lévi glass

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Summary

Lévy glass is a new material which is obtained by assembly of scattering elements. The result is a bulk material which has different degrees of elasticity (from rigid as window glass to flexible as certain plastics) and which can be shaped into any desired form. In particular, Lévy glasses have special optical properties not present in other materials. The name Lévy glass is based on the fact that under certain conditions light waves perform a Lévy walk inside these materials and the resulting optical transport is superdiffusive. From a structural point of view a Lévy glass is a random fractal. We will review the optical and structural properties of Lévy glasses, their design criteria, as well as their experimental realization. Also we will go into various fundamental aspects of Lévy-type light transport, discussing both superdiffusion as well as interference effects like weak and strong localization.

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References

  1. Einstein A., Investigations of the Theory of Brownian Movement (Dover) 1956.

  2. Sheng P., Introduction to Wave Scattering, Localization and Mesoscopic Phenomena, 2nd edition (Springer) 2006.

  3. Mandelbrot B., The Fractal Geometry of Nature (V.H. Freeman & Co.) 1977.

  4. Metzler R., Chechkin A. V., Gonchar V. Y. and Klafter J., Chaos, Solitons Fractals, 34 (2007) 129 http://dx.doi.org/10.1016/j.chaos.2007.01.055.

    Article  ADS  MathSciNet  Google Scholar 

  5. Mantegna R. N. and Stanley H. E., Phys. Rev. Lett., 73 (1994) 2946.

    Article  ADS  MathSciNet  Google Scholar 

  6. Reynolds A. M., Smith A. D., Menzel R., Greggers U., Reynolds D. R. and Riley J. R., Ecology, 88 (2007) 1955.

    Article  Google Scholar 

  7. Barthelemy P., Bertolotti J. and Wiersma D. S., Nature, 453 (2008) 495.

    Article  ADS  Google Scholar 

  8. Kallenberg O., Foundations of Modern Probability, 2nd edition (Springer) 1997.

  9. Bouchaud J. and Georges A., Phys. Rep., 195 (1990) 127.

    Article  ADS  MathSciNet  Google Scholar 

  10. Metzler R. and Klafter J., Phys. Rep., 339 (2000) 1.

    Article  ADS  Google Scholar 

  11. Metzler R. and Klafter J., J. Phys. A: Math. Gen., 37 (2004) R161.

    Article  ADS  Google Scholar 

  12. Klages R., Radons G. and Sokolov I. M., Anomalous Transport (Wiley-VHC) 2008.

  13. Lévy P., Théorie de l’addition des variables aléatoires (Gauthier-Villars) 1937.

  14. Nolan J. P., Stable Distributions — Models for Heavy Tailed Data (Birkhauser) 2011, Chapter 1 online at academic2.american.edu/~jpnolan.

  15. Gnedenko B. V. and Kolmogorov A. N., Limit Distributions for Sums of Independent Random Variables (Addison-Wesley) 1954.

  16. Mandelbrot B., J. Business, 36 (1963) 394.

    Article  Google Scholar 

  17. Limonov M. and De La Rue R., Optical Properties of Photonic Structures: Interplay of Order and Disorder, Series in Optics and Optoelectronics (CRC Press) 2012, ISBN 9781439871928, https://books.google.it/books?id=g1PNBQAAQBAJ.

  18. Shlesinger M., West B. and Klafter J., Phys. Rev. Lett., 58 (1987) 1100.

    Article  ADS  MathSciNet  Google Scholar 

  19. Zumofen G. and Klafter J., Phys. Rev. E, 47 (1993) 851.

    Article  ADS  Google Scholar 

  20. Corral A., Phys. Rev. Lett., 97 (2006) 178501.

    Article  ADS  Google Scholar 

  21. Mantegna R. N. and Stanley H. E., Nature, 376 (1995) 46.

    Article  ADS  Google Scholar 

  22. Ghashghaie S., Breymann W., Peinke J., Talkner P. and Dodge Y., Nature, 381 (1996) 767.

    Article  ADS  Google Scholar 

  23. Brockmann D., Hufnagel L. and Geisel T., Nature, 439 (2006) 462.

    Article  ADS  Google Scholar 

  24. Caspi A., Granek R. and Elbaum M., Phys. Rev. E, 66 (2002) 011916.

    Article  ADS  Google Scholar 

  25. Bartumeus F., Da Luz M. G. E., Viswanathan G. M. and Catalan J., Ecology, 86 (2005) 3078.

    Article  Google Scholar 

  26. Solomon T. H., Weeks E. R. and Swinney H. L., Phys. Rev. Lett., 71 (1993) 3975.

    Article  ADS  Google Scholar 

  27. Mercadier N., Guerin W., Chrevrollier M. and Kaiser R., Nat. Phys., 5 (2009) 602.

    Article  Google Scholar 

  28. Bardou F., Bouchaud J.-P., Aspect A. and Cohen-Tannoudji C., Lévy Statistics and Laser Cooling: How Rare Events Bring Atoms to Rest (Cambridge University Press) 2002.

  29. Wiersma D. S., Bartolini P., Lagendijk A. and Righini R., Nature, 390 (1997) 671.

    Article  ADS  Google Scholar 

  30. Schuurmans F. J., Vanmaekelbergh D., van de Lagemaat J. and Lagendijk A., Science, 284 (1999) 141.

    Article  ADS  Google Scholar 

  31. Störzer M., Gross P., Aegerter C. M. and Maret G., Phys. Rev. Lett., 96 (2006) 063904.

    Article  ADS  Google Scholar 

  32. García P. D., Sapienza R., Blanco A. and López C., Adv. Mater., 19 (2007) 2597.

    Article  Google Scholar 

  33. Gottardo S., Sapienza R., García P. D., Blanco A., Wiersma D. S. and López C., Nat. Photon, 2 (2008) 429.

    Article  Google Scholar 

  34. Barthelemy P., Bertolotti J. and Wiersma D. S., Nature, 453 (2008) 495.

    Article  ADS  Google Scholar 

  35. ben Avraham D. and Havlin S., Diffusion and Reactions in Fractals and Disordered Systems (Cambridge University Press) 2000.

  36. Bertolotti J., Vynck K., Pattelli L., Barthelemy P., Lepri S. and Wiersma D. S., Adv. Funct. Mater., 20 (2010) 965.

    Article  Google Scholar 

  37. Mazzolo A., J. Math. Phys., 44 (2003) 6195.

    Article  ADS  MathSciNet  Google Scholar 

  38. Van Albada M. P. and Lagendijk A., Phys. Rev. Lett., 55 (1985) 2692.

    Article  ADS  Google Scholar 

  39. Wolf P.-E. and Maret G., Phys. Rev. Lett., 55 (1985) 2696.

    Article  ADS  Google Scholar 

  40. Buldyrev S., Havlin S., Kazakov A., da Luz M., Raposo E., Stanley H. and Viswanathan G., Phys. Rev. E, 64 (2001) 041108.

    Article  ADS  Google Scholar 

  41. Akkermans E. and Montambaux G., Mesoscopic Physics of Electrons and Photons (Cambridge University Press) 2007.

  42. Kutner R. and Maass P., J. Phys. A: Math. Gen., 31 (1998) 2603.

    Article  ADS  Google Scholar 

  43. Beenakker C. W. J., Groth C. W. and Akhmerov A. R., Phys. Rev. B, 79 (2009) 024204.

    Article  ADS  Google Scholar 

  44. Burioni R., Caniparoli L., Lepri S. and Vezzani A., Phys. Rev. E, 81 (2010) 011127.

    Article  ADS  Google Scholar 

  45. Barthelemy P., Bertolotti J., Vynck K., Lepri S. and Wiersma D. S., Phys. Rev. E, 82 (2010) 011101.

    Article  ADS  Google Scholar 

  46. Janicki A. and Weron A., Simulation and Chaotic Behavior of α-stable Stochastic Processes (Marcel Dekker, Inc.) 1994.

  47. Kuga Y. and Ishimaru A., JOSA A, 1 (1984) 831.

    Article  ADS  Google Scholar 

  48. Van Albada M. P. and Lagendijk A., Phys. Rev. Lett., 55 (1985) 2692.

    Article  ADS  Google Scholar 

  49. Wolf P.-E. and Maret G., Phys. Rev. Lett., 55 (1985) 2696.

    Article  ADS  Google Scholar 

  50. John S., Phys. Rev. Lett., 53 (1984) 2169.

    Article  ADS  Google Scholar 

  51. Anderson P. W., Philos. Mag. B, 52 (1985) 505.

    Article  ADS  Google Scholar 

  52. Kaveh M., Rosenbluh M., Edrei I. and Freund I., Phys. Rev. Lett., 57 (1986) 2049.

    Article  ADS  Google Scholar 

  53. Wiersma D. S., van Albada M. P., van Tiggelen B. A. and Lagendijk A., Phys. Rev. Lett., 74 (1995) 4193.

    Article  ADS  Google Scholar 

  54. Labeyrie G., Müller C., Wiersma D., Miniatura C. and Kaiser R., J. Opt. B: Quantum Semiclass. Opt., 2 (2000) 672.

    Article  ADS  Google Scholar 

  55. Gurioli M., Bogani F., Cavigli L., Gibbs H., Khitrova G. and Wiersma D., Phys. Rev. Lett., 94 (2005) 183901.

    Article  ADS  Google Scholar 

  56. Freund I., Rosenbluh M., Berkovits R. and Kaveh M., Phys. Rev. Lett., 61 (1988) 1214.

    Article  ADS  Google Scholar 

  57. Wiersma D. S., van Albada M. P. and Lagendijk A., Phys. Rev. Lett., 75 (1995) 1739.

    Article  ADS  Google Scholar 

  58. Vlasov D., Zubkov L., Orekhova N. and Romanov V., JETP Lett., 48 (1988) 91.

    ADS  Google Scholar 

  59. Sapienza R., Mujumdar S., Cheung C., Yodh A. and Wiersma D., Phys. Rev. Lett., 92 (2004) 033903.

    Article  ADS  Google Scholar 

  60. Kuzmin L., Romanov V. and Zubkov L., Phys. Rev. E, 54 (1996) 6798.

    Article  ADS  Google Scholar 

  61. Koenderink A. F., Megens M., van Soest G., Vos W. L. and Lagendijk A., Phys. Lett. A, 268 (2000) 104.

    Article  ADS  Google Scholar 

  62. Huang J., Eradat N., Raikh M., Vardeny Z., Zakhidov A. and Baughman R., Phys. Rev. Lett., 86 (2001) 4815.

    Article  ADS  Google Scholar 

  63. Akkermans E., Wolf P. and Maynard R., Phys. Rev. Lett., 56 (1986) 1471.

    Article  ADS  Google Scholar 

  64. Van Der Mark M. B., van Albada M. P. and Lagendijk A., Phys. Rev. B, 37 (1988) 3575.

    Article  ADS  Google Scholar 

  65. Akkermans E., Wolf P., Maynard R. and Maret G., J. Phys. (Paris), 49 (1988) 77.

    Article  ADS  Google Scholar 

  66. Vollhardt D. and Wölfle P., Phys. Rev. B, 22 (1980) 4666.

    Article  ADS  Google Scholar 

  67. Chaves A. S., Phys. Lett. A, 239 (1998) 13.

    Article  ADS  MathSciNet  Google Scholar 

  68. Yanovsky V. V., Chechkin A. V., Schertzer D. and Tur A. V., Physica A, 282 (2000) 13.

    Article  ADS  Google Scholar 

  69. Oliveira E. and Machado J., Math. Prob. Engin., 2014 (2014) 238459.

    Article  Google Scholar 

  70. Chechkin A. V., Metzler R., Gonchar V. Y., Klafter J. and Tanatarov L. V., J. Phys. A: Math. Gen., 36 (2003) L537.

    Article  ADS  Google Scholar 

  71. Zumofen G. and Klafter J., Phys. Rev. E, 51 (1995) 2805.

    Article  ADS  Google Scholar 

  72. Drysdale P. M. and Robinson P. A., Phys. Rev. E, 58 (1998) 5382.

    Article  ADS  Google Scholar 

  73. Krepysheva N., Di Pietro L. and Nel M.-C., Phys. Rev. E, 73 (2006) 021104.

    Article  ADS  MathSciNet  Google Scholar 

  74. Podlubny I., Chechkin A., Skovranek T., Chen Y. and Vinagre Jara B. M., J. Comput. Phys., 228 (2009) 3137.

    Article  ADS  MathSciNet  Google Scholar 

  75. Gorenflo R., De Fabritiis G. and Mainardi F., Physica A, 269 (1999) 79.

    Article  ADS  MathSciNet  Google Scholar 

  76. Zoia A., Rosso A. and Kardar M., Phys. Rev. E, 76 (2007) 021116.

    Article  ADS  MathSciNet  Google Scholar 

  77. Bertolotti J., Vynck K. and Wiersma D. S., Phys. Rev. Lett., 105 (2010) 163902.

    Article  ADS  Google Scholar 

  78. Sokolov I. M. and Metzler R., Phys. Rev. E, 67 (2003) 010101(R).

    Article  ADS  Google Scholar 

  79. Meerschaert M., Benson D., Scheffler H. and Becker-Kern P., Phys. Rev. E, 66 (2002) 060102.

    Article  ADS  Google Scholar 

  80. Akkermans E., Wolf P.-E., Maynard R. and Maret G., J. Phys. (Paris), 49 (1988) 77.

    Article  ADS  Google Scholar 

  81. Wiersma D. S., van Albada M. P. and Lagendijk A., Rev. Sci. Instrum., 66 (1995) 5473.

    Article  ADS  Google Scholar 

  82. Burresi M., Radhalakshmi V., Savo R., Bertolotti J., Vynck K. and Wiersma D. S., Phys. Rev. Lett., 108 (2012) 110604.

    Article  ADS  Google Scholar 

  83. Berkovits R. and Kaveh M., Phys. Rev. B, 36 (1987) 9322.

    Article  ADS  Google Scholar 

  84. Edrei I. and Stephen M. J., Phys. Rev. B, 42 (1990) 110.

    Article  ADS  Google Scholar 

  85. Schuurmans F. J., Megens M., Vanmaekelbergh D. and Lagendijk A., Phys. Rev. Lett., 83 (1999) 2183.

    Article  ADS  Google Scholar 

  86. Ben-Avraham D. and Havlin S., Diffusion and Reactions in Fractals and Disordered Systems (Cambrige University Press) 2000.

  87. Cates M. E., J. Phys. (Paris), 46 (1985) 19.

    Article  MathSciNet  Google Scholar 

  88. Klages R., Radons G. and Sokolov I. M., Anomalous Transport: Foundations and Applications (Wiley-VCH) 2008, ISBN 3527407227.

  89. Condamin S., Benichou O., Tejedor V., Voituriez R. and Klafter J., Nature, 450 (2007) 77.

    Article  ADS  Google Scholar 

  90. Savo R., Burresi M., Svensson T., Vynck K. and Wiersma D. S., Phys. Rev. A, 90 (2014) 023839.

    Article  ADS  Google Scholar 

  91. Shah J., IEEE J. Quantum Electron., 24 (1988) 276.

    Article  ADS  Google Scholar 

  92. Svensson T., Savo R., Alerstam E., Vynck K., Burresi M. and Wiersma D. S., Opt. Lett., 38 (2013) 437.

    Article  ADS  Google Scholar 

  93. Wiersma D. S., Muzzi A., Colocci M. and Righini R., Phys. Rev. Lett., 83 (1999) 4321.

    Article  ADS  Google Scholar 

  94. Bertolotti J., Vynck K. and Wiersma D. S., Phys. Rev. Lett., 105 (2010) 163902.

    Article  ADS  Google Scholar 

  95. Burresi M., Radhalakshmi V., Savo R., Bertolotti J., Vynck K. and Wiersma D. S., Phys. Rev. Lett., 108 (2012) 110604.

    Article  ADS  Google Scholar 

  96. Akkermans É. and Montambaux G., Mesoscopic physics of electrons and photons (Cambridge University Press) 2007 ISBN 9780521855129.

  97. Svensson T., Vynck K., Grisi M., Savo R., Burresi M. and Wiersma D. S., Phys. Rev. E, 87 (2013) 022120.

    Article  ADS  Google Scholar 

  98. Elaloufi R., Carminati R. and Greffet J.-J., J. Opt. Soc. Am. A, 21 (2004) 1430.

    Article  ADS  Google Scholar 

  99. Buonsante P., Burioni R. and Vezzani A., Phys. Rev. E, 84 (2011) 021105.

    Article  ADS  Google Scholar 

  100. Ishimaru A., Wave Propagation and Scattering in Random Media, IEEE Press Series on Electromagnetic Wave Theory (IEEE Press) 1999, ISBN 9780780347175.

  101. van Rossum M. V. and Nieuwenhuizen T. M., Rev. Mod. Phys., 71 (1999) 313.

    Article  ADS  Google Scholar 

  102. Drysdale P. and Robinson P., Phys. Rev. E, 58 (1998) 5382.

    Article  ADS  Google Scholar 

  103. Groth C. W., Akhmerov A. R. and Beenakker C. W. J., Phys. Rev. E, 85 (2012) 021138.

    Article  ADS  Google Scholar 

  104. Burioni R., Ubaldi E. and Vezzani A., Phys. Rev. E, 89 (2014) 022135.

    Article  ADS  Google Scholar 

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Wiersma, D.S. Lévi glass. Riv. Nuovo Cim. 38, 271–310 (2015). https://doi.org/10.1393/ncr/i2015-10113-1

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