Skip to main content Accessibility help
Hostname: page-component-66d7dfc8f5-vmgfl Total loading time: 3.097 Render date: 2023-02-08T18:37:30.827Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true


Published online by Cambridge University Press:  05 May 2022

Luca Dal Negro
Boston University
Get access


Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Waves in Complex Media , pp. 650 - 689
Publisher: Cambridge University Press
Print publication year: 2022

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Whitehead, A. N., Introduction to Mathematics. Henry Holt and Company, New York, 1911.Google Scholar
Falk, R. and Konold, C., “Making sense of randomness: implicit encoding as a basis for judgment,” Psychological Review, vol. 104, no. 2, pp. 301318, 1997.CrossRefGoogle Scholar
Dehaene, S., The Number Sense: How the Mind Creates Mathematics. Oxford University Press, New York, 2011.Google Scholar
Poincaré, H., “Sur le probléme des trois corps et les équations de la dynamique,” Acta Mathematica, vol. 13, pp. A3–A270, 1890.Google Scholar
Steinhardt, P. J., The Second Kind of Impossible: The Extraordinary Quest for a New Form of Matter. Simon and Schuster, New York, 2018.Google Scholar
Torquato, S., “Hyperuniform states of matter,” Physics Reports, vol. 745, pp. 195, 2018.CrossRefGoogle Scholar
Allouche, J. P. and Shallit, J. O., Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, New York, 2009.Google Scholar
Schroeder, M., Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity. Springer-Verlag, Berlin, 2009.Google Scholar
Dal Negro, L., Chen, Y., and Sgrignuoli, F., “Aperiodic photonics of elliptic curves,” Crystals, vol. 9, pp. 482509, 2019.CrossRefGoogle Scholar
Sgrignuoli, F., Gorsky, S., Britton, W. A., Zhang, R., Riboli, F., and Dal Negro, L., “Multi-fractality of light in photonic arrays based on algebraic number theory,” Communications Physics, vol. 3, pp. 19, 2020.CrossRefGoogle Scholar
Janot, C., Quasicrystals: A Primer. Oxford University Press, Oxford, 1994.Google Scholar
[12] de Lange, C. and Janssen, T., “Incommensurability and recursivity: lattice dynamics of modulated crystals,” Journal of Physics C: Solid State Physics, vol. 14, no. 34, pp. 52695292, December 1981. [Online]. Available:–3719%2F14%2F34%2F009CrossRefGoogle Scholar
[13] Hofstadter, D. R., “Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields,” Physical Review B, vol. 14, pp. 22392249, September 1976. [Online]. Available: Scholar
[14] Harper, P. G., “The general motion of conduction electrons in a uniform magnetic field, with application to the diamagnetism of metals,” Proceedings of the Physical Society. Section A, vol. 68, no. 10, pp. 879892, October 1955. [Online]. Available: https://doi .org/10.1088%2F0370–1298%2F68%2F10%2F305CrossRefGoogle Scholar
[15] Evangelou, S. N. and Pichard, J.-L., “Critical quantum chaos and the one-dimensional harper model,” Physical Review Letters, vol. 84, pp. 16431646, February 2000. [Online]. Available: ScholarPubMed
[16] Wang, R., Röntgen, M., Morfonios, C. V., Pinheiro, F. A., Schmelcher, P., and Dal Negro, L., “Edge modes of scattering chains with aperiodic order,” Optics Letters, vol. 43, no. 9, pp. 19861989, May 2018. [Online]. Available: ScholarPubMed
[17] Dareau, A., Levy, E., Aguilera, M. B., et al., “Revealing the topology of quasicrystals with a diffraction experiment,” Physical Review Letters, vol. 119, p. 215-304, November 2017. [Online]. Available: ScholarPubMed
[18] Baboux, F., Levy, E., Lemaître, A., et al., “Measuring topological invariants from generalized edge states in polaritonic quasicrystals,” Physics Review B, vol. 95, p. 161-114, April 2017. [Online]. Available: .95.161114CrossRefGoogle Scholar
Abe, S. and Hiramoto, H., “Fractal dynamics of electron wave packets in one-dimensional quasiperiodic systems,” Physics Review A, vol. 36, pp. 53495352, 1987.CrossRefGoogle ScholarPubMed
Ketzmerick, R., Kruse, K., Kraut, S., and Geisel, T., “What determines the spreading of a wave packet?Physical Review Letters, vol. 79, pp. 19591963, 1997.CrossRefGoogle Scholar
[21] Ketzmerick, R., Petschel, G., and Geisel, T., “Slow decay of temporal correlations in quantum systems with cantor spectra,” Physical Review Letters, vol. 69, pp. 695698, August 1992. [Online]. Available: ScholarPubMed
Dal Negro, L. and Inampudi, S., “Fractional transport of photons in deterministic aperiodic structures,” Scientific Reports, vol. 7, no. 1, p. 2259, 2017.CrossRefGoogle ScholarPubMed
Gardner, M., Penrose Tiles to Trapdoor Ciphers. W. H. Freeman, New York, 1989.Google Scholar
Sheng, P., Introduction to Wave Scattering, Localization and Mesoscopic Phenomena, 2nd ed. Springer, Berlin, 2006.Google Scholar
Akkermans, E. and Montambaux, G., Mesoscopic Physics of Electrons and Photons. Cambridge University Press, New York, 2007.CrossRefGoogle Scholar
Joannopoulos, J. D., Johnson, S. G., Winn, J. N., and Meade, R. D., Photonic Crystals: Molding the Flow of Light, 2nd ed. Princeton University Press, Princeton, 2008.Google Scholar
Sakoda, K., Optical Properties of Photonic Crystals, 2nd ed. Springer, Berlin, 2005.CrossRefGoogle Scholar
Brillouin, L., Wave Propagation in Periodic Structures. McGraw-Hill, New York, 1946.Google Scholar
Mishchenko, M. I., “125 years of radiative transfer: enduring triumphs and persisting misconceptions,” AIP Conference Proceedings, vol. 1531, no. 1, pp. 1118, 2013. [Online]. Available at: Scholar
Chandrasekhar, S., Radiative Transfer. Dover, New York, 1960.Google Scholar
Anderson, P. W., “Absence of diffusion in certain random lattices,” Physical Review, vol. 109, pp. 14921505, 1958.CrossRefGoogle Scholar
Anderson, P. W., “The question of classical localization: a theory of white paint?Phylosophical Magazine B, vol. 52, no. 3, pp. 505509, 1985.CrossRefGoogle Scholar
Kuga, Y. and Ishimaru, A., “Retroreflectance from a dense distribution of spherical particles,” Journal of the Optical Society of America A, vol. 1, p. 831, 1984.CrossRefGoogle Scholar
van Albada, M. P. and Lagendijk, A., “Observation of weak localization of light in a random medium,” Physical Review Letters, vol. 55, p. 2692, 1985.Google Scholar
Wolf, P. and Maret, G., “Weak localization and coherent backscattering of photons in disordered media,” Physical Review Letters, vol. 55, p. 2696, 1985.CrossRefGoogle ScholarPubMed
John, S., “Electromagnetic absorption in a disordered medium near a photon mobility edge,” Physical Review Letters, vol. 53, p. 2169, 1984.CrossRefGoogle Scholar
Shechtman, D., Blech, I., Gratias, D., and Cahn, J. W., “Metallic phase with long-range orientational order and no tranlsational symmetry,” Physical Review Letters, vol. 53, pp. 19511953, 1984.CrossRefGoogle Scholar
Levine, D. and Steinhardt, P. J., “Quasicrystals: a new class of ordered structures,” Physical Review Letters, vol. 26, pp. 24772480, 1984.CrossRefGoogle Scholar
Senechal, M., Quasicrystals and Geometry. Cambridge University Press, Cambridge, 1995.Google Scholar
Janssen, T., Chapuis, G., and de Boissieu, M., Aperiodic Crystals: From Modulated Phases to Quasicrystals. Oxford University Press, Oxford, 2007.CrossRefGoogle Scholar
Merlin, R., Bajema, K., Clarke, R., Juang, F. Y., and Bhattacharya, P. K., “Quasiperiodic GaAs-AIAs heterostructures,” Physical Review Letters, vol. 55, pp. 17681770, 1985.CrossRefGoogle ScholarPubMed
Kohmoto, B., Sutherland, H., and Iguchi, K., “Localization of optics: quasiperiodic media,” Physical Review Letters, vol. 58, p. 2436, 1987.CrossRefGoogle ScholarPubMed
Born, M. and Wolf, E., Principles of Optics, 7th ed. Cambridge University Press, Cambridge, 1999.CrossRefGoogle Scholar
[44] Dal Negro, L., Oton, C. J., Gaburro, Z., et al., “Light transport through the band-edge states of fibonacci quasicrystals,” Physical Review Letters, vol. 90, p. 055-501, February 2003. [Online]. Available: ScholarPubMed
Kohmoto, M., Kadanoff, L. P., and Tang, C., “Localization problem in one dimension: mapping and escape,” Physical Review Letters, vol. 50, pp. 18701872, 1983.CrossRefGoogle Scholar
[46] Kolá, M. and Ali, M. K., “Attractors of some volume-nonpreserving fibonacci trace maps,” Physical Review A, vol. 39, pp. 65386544, June 1989. [Online]. Available: ScholarPubMed
Gellerman, W., Kohmoto, M., Sutherland, B., and Taylor, P. C., “Localization of light waves in fibonacci dielectric multilayers,” Physical Review Letters, vol. 72, pp. 633636, 1994.CrossRefGoogle Scholar
Schreiber, M. and Grussbach, H., “Multifractal wave functions at the Anderson transition,” Physical Review Letters, vol. 67, no. 5, pp. 607610, 1991.CrossRefGoogle ScholarPubMed
[49] Faez, S., Strybulevych, A., Page, J. H., Lagendijk, A., and van Tiggelen, B. A., “Observation of multifractality in Anderson localization of ultrasound,” Physical Review Letters, vol. 103, p. 155-703, October 2009. [Online]. Available: ScholarPubMed
[50] Fujiwara, T., Kohmoto, M., and Tokihiro, T., “Multifractal wave functions on a fibonacci lattice,” Physical Review B, vol. 40, pp. 74137416, October 1989. [Online]. Available: ScholarPubMed
Kolar, M., Ali, M. K., and Nori, F., “Generalized Thue–Morse chains and their physical properties,” Physical Review B, vol. 43, pp. 10341047, 1991.CrossRefGoogle Scholar
[52] Cheng, Z., Savit, R., and Merlin, R., “Structure and electronic properties of Thue–Morse lattices,” Physical Review B, vol. 37, pp. 43754382, March 1988. [Online]. Available: ScholarPubMed
[53] Liu, N.-h., “Propagation of light waves in Thue–Morse dielectric multilayers,” Physical Review B, vol. 55, pp. 35433547, February 1997. [Online]. Available: Scholar
Dal Negro, L., Stolfi, M., Yi, Y., et al., “Photon band gap properties and omnidirectional reflectance in Si/SiO Thue–Morse quasicrystals,” Applied Physics Letters, vol. 84, no. 25, pp. 51865188, 2004.CrossRefGoogle Scholar
[55] Cheng, S.-F. and Jin, G.-J., “Trace map and eigenstates of a Thue–Morse chain in a general model,” Physical Review B, vol. 65, p. 134-206, March 2002. [Online]. Available: Scholar
[56] Jiang, X., Zhang, Y., Feng, S., Huang, K. C., Yi, Y., and Joannopoulos, J. D., “Photonic band gaps and localization in the Thue–Morse structures,” Applied Physics Letters, vol. 86, no. 20, p. 201110, 2005. [Online]. Available: Scholar
[57] Kohmoto, M., “Localization problem and mapping of one-dimensional wave equations in random and quasiperiodic media,” Physical Review B, vol. 34, pp. 50435047, October 1986. [Online]. Available: ScholarPubMed
[58] Tang, C. and Kohmoto, M., “Global scaling properties of the spectrum for a quasiperiodic Schrödinger equation,” Physical Review B, vol. 34, pp. 20412044, August 1986. [Online]. Available: ScholarPubMed
Kohmoto, M., Sutherland, B., and Tang, C., “Critical wave functions and a cantorset spectrum of a one-dimensional quasicrystal model,” Physical Review B, vol. 35, pp. 10201033, 1987.CrossRefGoogle Scholar
[60] Baake, M., Grimm, U., and Joseph, D., “Trace maps, invariants, and some of their applications,” International Journal of Modern Physics B, vol. 07, no. 06n07, pp. 15271550, 1993. [Online]. Available: Scholar
[61] Kohmoto, M. and Oono, Y., “Cantor spectrum for an almost periodic Schrodinger equation and a dynamical map,” Physics Letters A, vol. 102, no. 4, pp. 145148, 1984. [Online]. Available: Scholar
Esaki, K., Sato, M., and Kohmoto, M., “Wave propagation through Cantor-set media: chaos, scaling, and fractal structures,” Physical Review E, vol. 79, p. 056226, 2009.CrossRefGoogle ScholarPubMed
[63] Kolář, M. and Ali, M. K., “One-dimensional generalized fibonacci tilings,” Physical Review B, vol. 41, pp. 71087112, April 1990. [Online]. Available: Scholar
[64] Kolář, M. and Nori, F., “Trace maps of general substitutional sequences,” Physical Review B, vol. 42, pp. 10621065, July 1990. [Online]. Available: Scholar
Maciá, E. and Domínguez-Adame, F., “Physical nature of critical wave functions in Fibonacci systems,” Physical Review Letters, vol. 76, pp. 29572960, 1997.CrossRefGoogle Scholar
[66] Desideri, J.-P., Macon, L., and Sornette, D., “Observation of critical modes in quasiperiodic systems,” Physical Review Letters, vol. 63, pp. 390393, July 1989. [Online]. Available: ScholarPubMed
[67] Ferralis, N., Szmodis, A. W., and Diehl, R. D., “Diffraction from one- and two-dimensional quasicrystalline gratings,” American Journal of Physics, vol. 72, no. 9, pp. 12411246, 2004. [Online]. Available: Scholar
[68] Hattori, T., Tsurumachi, N., Kawato, S., and Nakatsuka, H., “Photonic dispersion relation in a one-dimensional quasicrystal,” Physical Review Letters, B, vol. 50, pp. 4220–4223, August 1994. [Online]. Available: Scholar
Noh, H., Yang, J., Boriskina, S. V., et al., “Lasing in Thue–Morse structures with optimized aperiodicity,” Applied Physics Letters, vol. 98, p. 201109, 2011.CrossRefGoogle Scholar
Dal Negro, L., Optics of Aperiodic Structures: Fundamentals and Device Applications. Pan Stanford Publishing, Singapore, 2014.Google Scholar
Dal Negro, L. and Boriskina, S. V., “Deterministic aperiodic nanostructures for photonics and plasmonics applications,” Laser Photonics Review, vol. 6, pp. 141, 2011.Google Scholar
Vardeny, Z. V., Nahata, A., and Agrawal, A., “Optics of photonic quasicrystals,” Nature Photonics, vol. 7, pp. 177187, 2013.CrossRefGoogle Scholar
Steurer, W. and Sutter-Widmer, D., “Photonic and phononic quasicrystals,” Journal of Physics D: Applied Physics, vol. 40, pp. R229–R247, 2007.CrossRefGoogle Scholar
Lifshitz, R., Arie, A., and Bahabad, A., “Photonic quasicrystals for nonlinear optical frequency conversion,” Physical Review Letters, vol. 95, p. 133901, 2005.CrossRefGoogle ScholarPubMed
Wiersma, D. S., “Disordered photonics,” Nature Photonics, vol. 7, pp. 188196, May 2013.CrossRefGoogle Scholar
Pauli, W., Theory of Relativity. Dover Publications, New York, 1958.Google Scholar
Messiah, A., Quantum Mechanics. Dover Publications, New York, 1999.Google Scholar
Saleh, B. E. A. and Teich, M. C., Fundamentals of Photonics, 2nd ed. John Wiley, Hoboken, 2007.Google Scholar
Yariv, A. and Pochi, Y., Photonics: Optical Electronics in Modern Communications, 6th ed. Oxford University Press, New York, 2007.Google Scholar
[80] Maxwell, J. C., “On physical lines of force,” Philosophical Magazine, vol. 21 (parts I and II) and 23 (parts III and IV), pp. 148, 1861–1862.Google Scholar
Faraday, M., Experimental Researches in Electricity. Dover Publications, New York, 1965.Google Scholar
Jackson, J. D., Classical Electrodynamics, 3rd ed. John Wiley, 1998.Google Scholar
Garg, A., Classical Electromagnetism in a Nutshell. Princeton University Press, Princeton, 2012.Google Scholar
Jackson, J. D., “From Lorenz to Coulomb and other explicit gauge transformations,” American Journal of Physics, vol. 70, p. 917, 2002.CrossRefGoogle Scholar
Cohen-Tannoudji, C., Dupont-Roc, J., and Grynberg, G., Photons and Atoms. Introduction to Quantum Electrodynamics. Wiley-VCH, Strauss GmbH, Morlenbach 2004.Google Scholar
Brill, O. L. and Goodman, B., “Causality in the Coulomb Gauge,” American Journal of Physics, vol. 35, pp. 832837, 1967.CrossRefGoogle Scholar
Scully, M. O. and Zubairy, M. S., Quantum Optics. Cambridge University Press, Cambridge, 2002.Google Scholar
Mandel, L. and Wolf, E., Optical Coherence and Quantum Optics. Cambridge University Press, New York, 1995.CrossRefGoogle Scholar
Courant, R. and Hilbert, D., Methods of Mathematical Physics, 6th ed., vols. 1–2. Interscience Publishers, New York, 1966.Google Scholar
Newton, R. G., Scattering Theory of Waves and Particles. Dover Publications, New York, 2002.Google Scholar
Zettili, N., Quantum Mechanics: Concepts and Applications, 2nd ed. John Wiley, UK, 2009.Google Scholar
Novotny, L. and Hecht, B., Principles of Nano-Optics, 2nd ed. Cambridge University Press, New York, 2012.CrossRefGoogle Scholar
Robinson, F. N. H., Macroscopic Electromagnetism, 3rd ed. Pergamon Press, Oxford, 1973.Google Scholar
Landau, E. M., Lifshitz, E. M., and Pitaevskii, L. P., Electrodynamics of Continuous Media. Elsevier, Amsterdam, 1984.Google Scholar
Shen, Y. R., The Principles of Nonlinear Optics. John Wiley, Menlo Park, 1984.Google Scholar
Feynman, R. P., Leighton, R. B., and Sands, M., The Feynman Lectures on Physics,vol.2. Addison-Wesley, Palo Alto, 1964.CrossRefGoogle Scholar
Maier, S. A., Plasmonics: Fundamentals and Applications. Springer, New York, 2007.CrossRefGoogle Scholar
Shahbazyan, T. V. and Stockman, M. I., Plasmonics: Theory and Applications. Springer, Dordrecht, 2013.CrossRefGoogle Scholar
Brongersma, M. L. and Kik, P. G., Surface Plasmon Nanophotonics. Springer, 2007.CrossRefGoogle Scholar
Tai, C. T., Dyadic Green’s Functions in Electromagnetic Theory, 2nd ed. IEEE Press, New York, 1993.Google Scholar
Chew, W. C., Waves and Fields in Inhomogeneous Media. IEEE Press, New York, 1995.Google Scholar
Bladel, J., “Some remarks on green’s dyadic for infinite space,” IEEE Transactions on Antennas and Propagation, vol. AP-9, pp. 563566, 1961.CrossRefGoogle Scholar
Harrington, R. F., Field Computation by Moment Methods. Macmillan, New York, 1968.Google Scholar
[104] Livesay, D. and Chen, K., “Electromangetic fields induced inside arbitrarily shaped biological bodies,” IEEE Transactions on Microwave Theory and Techniques, vol. MTT-22, no. 12, pp. 12731280, 1974.CrossRefGoogle Scholar
van Bladel, J., Electromagnetic Fields, 2nd ed. IEEE Press and John Wiley, Hoboken, 2007.CrossRefGoogle Scholar
van Bladel, J., Singular Electromagnetic Fields and Sources. IEEE Press, Piscataway, 1991.Google Scholar
Rubinacci, G. and Tamburrino, A., “A broadband volume integral formulation based on edge-elements for full-wave analysis of lossy interconnects,” IEEE Transactions on Antennas and Propagation, vol. 54, no. 10, pp. 29772989, 2006.CrossRefGoogle Scholar
Miano, G., Rubinacci, G., and Tamburrino, A., “Numerical modelling of the interaction of nanoparticles with electromagnetic waves,” Compel, vol. 26, no. 3, pp. 586599, 2007.CrossRefGoogle Scholar
Miano, G., Rubinacci, G., and Tamburrino, A., “Numerical modeling for the analysis of plasmon oscillations in metallic nanoparticles,” IEEE Transactions on Antennas and Propagation, vol. 58, no. 9, pp. 29202933, 2010.CrossRefGoogle Scholar
Dal Negro, L., Miano, G., Rubinacci, G., Tamburrino, A., and Ventre, S., “A fast computation method for the analysis of an array of metallic nanoparticles,” IEEE Transactions on Magnetics, vol. 45, no. 3, pp. 16181621, 2009.CrossRefGoogle Scholar
Lorentz, H. A., The Theory of Electrons and Its Applications to the Phenomena of Light and Radiant Heat, 2nd ed. Stechert and Co, New York, 1916.Google Scholar
Johnson, P. B. and Christy, R. W., “Optical constants of the noble metals,” Physical Review B, vol. 6, no. 12, pp. 43704379, 1972.CrossRefGoogle Scholar
Peskin, M. E. and Schroeder, D. V., An Introduction to Quantum Field Theory. Westview Press, Boulder, 1995.Google Scholar
Landau, L. D., “Über die bewegung der elektronen in kristallgitter,” Zeitschrift fur Physik Sowjetunion, vol. 3, pp. 644645, 1933.Google Scholar
Mattuck, R. D., A Guide to Feynman Diagrams in the Many-Body Problem, 2nd ed. Dover Publications, New York, 1976.Google Scholar
Grosso, G. and Pastori Parravicini, G., Solid State Physics, 2nd ed. Academic Press, Oxford, 2014.Google Scholar
Haug, H. and Koch, S. W., Quantum Theory of the Optical and Electronic Properties of Semiconductors, 4th ed. World Scientific Publishing, Singapore, 2004.CrossRefGoogle Scholar
Agranovich, V. M. and Ginzburg, V. L., Crystal Optics with Spatial Dispersion and Excitons. Springer Verlag, Berlin, 1984.CrossRefGoogle Scholar
Homola, J., Yee, S. S., and Gauglitz, G., “Surface plasmon resonance sensors: review,” Sensors and Actuators B, vol. 54, pp. 315, 1999.CrossRefGoogle Scholar
Raether, H., Surface Plasmons on Smooth and Rough Surfaces and on Gratings. Springer-Verlag, Berlin, 1988.CrossRefGoogle Scholar
Palik, E. D., Handbook of Optical Constants. Academic Press, Orlando, 1985.Google Scholar
McPeak, K. M., Jayanti, S. V., Kress, J. P., et al., “Plasmonic films can easily be better: rules and recipes,” ACS Photonics, vol. 2, no. 12, pp. 326333, 2015.CrossRefGoogle ScholarPubMed
Sihvola, A., Electromagnetic Mixing Formulas and Applications. Institution on Engineering and Technology, London, 2008.Google Scholar
Maxwell-Garnett, J. C., “Colours in metal glasses and in metallic films,” Philosophical Transactions of the Royal Society Series A, vol. 203, pp. 385420, 1904.Google Scholar
Bruggeman, D. A. G., “Berechnung verschiedener physikalischer konstanten von heterogenen substanzen. i. dielektrizitätskonstanten und leitfähigkeiten der mischkörper aus isotropen substanzen,” Annals of Physics, vol. 416, pp. 636664, 1935.CrossRefGoogle Scholar
[126] Mallet, P., Guérin, C. A., and Sentenac, A., “Maxwell-Garnett mixing rule in the presence of multiple scattering: derivation and accuracy,” Physical Review B, vol. 72, pp. 014 205–1–014 205–9, 2005.CrossRefGoogle Scholar
Draine, B. T., “The discrete-dipole approximation and its application to interstellar graphite grains,” Astrophysical Journal, vol. 333, pp. 848872, 1988.CrossRefGoogle Scholar
Tsang, L., Kong, J. A., and Shin, R. T., Theory of Microwave Remote Sensing. John Wiley and Sons, New York, 1985.Google Scholar
Tsang, L. and Kong, J. A., “Multiple scattering of electromagnetic waves by random distributions of discrete scatterers with coherent potential and quantum mechanical formalism,” Journal of Applied Physics, vol. 51, pp. 34653485, 1980.CrossRefGoogle Scholar
Frisch, V., Wave Propagation in Random Medium, in Probabilistic Methods in Applied Mathematics, vol. 1, Bharuch-Reid Ed. Academic Press, New York, 1968.Google Scholar
Guérin, C. A., Mallet, P., and Sentenac, A., “Effective-medium theory for finite-size aggregates,” Journal of the Optical Society of America A, vol. 23, pp. 349358, 2006.CrossRefGoogle ScholarPubMed
Tsang, L., Kong, J. A., and Ding, K., Scattering of Electromangetic Waves, vol. III. John Wiley, New York, 2000.Google Scholar
Wu, Y., Li, J., Zhang, Z., and Chan, C. T., “Effective medium theory for magnetodielectric composites: beyond the long-wavelength limit,” Physical Review B, vol. 74, no. 085111, pp. 19, 2006.CrossRefGoogle Scholar
Slovick, B. A., Yu, Z. G., and Krishnamurthy, S., “Generalized effective-medium theory for metamaterials,” Physical Review B, vol. 89, no. 155118, pp. 15, 2014.CrossRefGoogle Scholar
[135] Torquato, S. and Kim, J., Nonlocal Effective Electromagnetic Wave Characteristics of Composite Media: Beyond the Quasistatic Regime, Phys. Rev. X 11, 021002–2021.CrossRefGoogle Scholar
[136] Rechtsman, M. C. and Torquato, S., “Effective dielectric tensor for electromagnetic wave propagation in random media,” Journal of Applied Physics, vol. 103, no. 8, p. 084901, 2008. [Online]. Available: Scholar
Chen, Y., Lu, L., Karniadakis, G. E., and Dal Negro, L., “Physics-informed neural networks for inverse problems in nano-optics and metamaterials,” Optics Express, vol. 28, no. 8, pp. 1161811633, 2020.CrossRefGoogle ScholarPubMed
Solymar, L. and Shamonina, E., Waves in Metamaterials. Oxford University Press, 2009.Google Scholar
Shalaev, V. M. and Sarychev, A. K., Electrodynamics of Metamaterials. World Scientific, 2007.Google Scholar
Engheta, N. and Ziolkowski, R. W., Metamaterials. Physics and Engineering Explorations. John Wiley and IEEE Press, Canada, 2006.CrossRefGoogle Scholar
[141] Silveirinha, M. and Engheta, N., “Tunneling of electromagnetic energy through subwavelength channels and bends using ɛ-near-zero materials,” Physical Review Letters, vol. 97, no. 157403, 2006.CrossRefGoogle Scholar
Engheta, N., “Pursuing near-zero response,” Science, vol. 340, pp. 286287, 2013.CrossRefGoogle ScholarPubMed
Shelby, R. A., Smith, D. R., and Schultz, S., “Experimental verification of a negative index of refraction,” Science, vol. 292, pp. 7779, 2001.CrossRefGoogle ScholarPubMed
Pendry, J. B., “Negative refraction makes a perfect lens,” Physical Review Letters, vol. 85, pp. 39663969, 2000.CrossRefGoogle ScholarPubMed
Smith, D. R., Padilla, W. J., Vier, D. C., Nemat-Nasser, S. C., and Schultz, S., “Composite medium with simultaneously negative permeability and permittivity,” Physical Review Letters, vol. 84, pp. 41844187, 2000.CrossRefGoogle ScholarPubMed
Veselago, V. G., “The electrodynamics of substances with simultaneously negative values of ɛ and μ,” Soviet Physics Uspekhi, vol. 10, pp. 509514, 1967.CrossRefGoogle Scholar
Bose, J. C., “On the rotation of plane of polarization of electric waves by a twisted structure,” Proceedings of the Royal Society, vol. 63, pp. 146152, 1898.Google Scholar
Forestiere, C., Pasquale, A. J., Capretti, A., et al., “Genetically optimized plasmonic nanoarrays,” Nano Letters, vol. 12, no. 4, pp. 20372044, 2012.CrossRefGoogle ScholarPubMed
Dong, Y. and Liu, S., “Topology optimization of patch-typed left-handed metamaterial configurations for transmission performance within the radio frequency band based on the genetic algorithm,” Journal of Optics, vol. 14, no. 105101, pp. 19, 2012.CrossRefGoogle Scholar
Della Giavampaola, C. and Engheta, N., “Digital metamaterials,” Nature Materials, vol. 13, pp. 11151121, 2014.CrossRefGoogle Scholar
Cui, T. J., Qi, M. Q., Zhao, J., and Cheng, Q., “Coding metamaterials, digital metamaterials and programmable metamaterials,” Light Science and Applications, vol. 3, no. 218, pp. 19, 2014.CrossRefGoogle Scholar
Whitham, G. B., Linear and Nonlinear Waves. John Wiley and Sons, Hoboken, 1999.CrossRefGoogle Scholar
Ostrovsky, L. A. and Potapov, A. I., Modulated Waves. Theory and Applications. Johns Hopkins University Press, Baltimore, 1999.Google Scholar
Murphy, P. K., Machine Learning. A Probabilistic Perspective. MIT Press, Cambridge, 2012.Google Scholar
Goodfellow, I., Bengio, Y., and Courville, A., Deep Learning. MIT Press, Cambridge, 2016.Google Scholar
Funahashi, K., “On the approximate realization of continuous mappings by neural networks,” Neural Networks, vol. 2, pp. 183192, 1989.CrossRefGoogle Scholar
Hornik, K., Stinchcombe, M., and White, H., “Multilayer feedforward networks are universal approximators,” Neural Networks, vol. 2, pp. 359366, 1989.CrossRefGoogle Scholar
Haykin, S., Neural Networks and Learning Machines, 3rd ed. Pearson, New York, 2009.Google Scholar
[159] Sun, Y., Xia, Z., and Kamilov, U. S., “Efficient and accurate inversion of multiple scattering with deep learning,” Optics Express, vol. 26, no. 11, pp. 14 678–14 688, 2018.CrossRefGoogle ScholarPubMed
Sanghvi, Y., Kalepu, Y., and Khankhoje, U. K., “Embedding deep learning in inverse scattering problems,” IEEE Transactions on Computational Imaging, vol. 6, pp. 4656, 2019.CrossRefGoogle Scholar
Raissi, M., Perdikaris, P., and Karniadakis, G. E., “Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” Journal of Computational Physics, vol. 378, pp. 686707, 2019.CrossRefGoogle Scholar
Lu, L., Meng, X., Mao, Z., and Karniadakis, G. E., “Deepxde: a deep learning library for solving differential equations,” eprint arXiv:1907.04502, 2019.Google Scholar
Sommerfeld, A., Optics, vol.4,Lectures on Theoretical Physics. Academic Press, New York, 1954.Google Scholar
Wang, A. and Prata, A. J., “Lenslet analysis by rigorous vector diffraction theory,” Journal of the Optical Society of America A, vol. 12, no. 5, pp. 11611169, 1995.CrossRefGoogle Scholar
Marathay, A. S. and McCalmont, J. F., “Vector diffraction theory for electromagnetic waves,” Journal of the Optical Society of America A, vol. 18, no. 10, pp. 25852593, 2001.CrossRefGoogle ScholarPubMed
Braat, J. and Török, P., Imaging Optics. Cambridge University Press, Cambridge, 2019.CrossRefGoogle Scholar
Török, P., Munro, P. R. T., and Kriezis, E. E., “Rigorous near- to far-field transformation for vectorial diffraction calculations and its numerical implementation,” Journal of the Optical Society of America A, vol. 23, no. 3, pp. 713722, 2006.CrossRefGoogle ScholarPubMed
Hsu, W. and Barakat, R., “Stratton-chu vectorial diffraction of electromangetic fields by apertures with application to small-fresnel-number systems,” Journal of the Optical Society of America A, vol. 11, no. 2, pp. 623629, 1994.CrossRefGoogle Scholar
Kim, J., Wang, Y., and Zhang, X., “Calculation of vectorial diffraction in optical systems,” Journal of the Optical Society of America A, vol. 35, no. 4, pp. 526535, 2018.CrossRefGoogle ScholarPubMed
Bethe, H., “Theory of diffraction by small holes,” Physical Review, vol. 66, pp. 163182, 1944.CrossRefGoogle Scholar
Kirchhoff, G., “Zur theorie der lichtstrahlen,” Weidemann Ann., vol. 2, no. 18, pp. 663695, 1883.Google Scholar
Mukunda, N., “Consistency of Rayleigh’s diffraction formulas with Kirchhoff’s boundary conditions,” Journal of the Optical Society of America, vol. 52, no. 3, pp. 336337, 1962.CrossRefGoogle Scholar
Goodman, J. W., Introduction to Fourier Optics, 4th ed. W. H. Freeman, New York, 2017.Google Scholar
Stark, H., Applications of Optical Fourier Transforms. Academic Press, New York, 1982.Google Scholar
Balanis, C. A., Antenna Theory, 4th ed. John Wiley, Hoboken 2016.Google Scholar
Rechtsman, M. C., Zeuner, J. M., Plotnik, Y., et al., “Photonic floquet topological insulators,” Nature, vol. 496, pp. 196200, 2013.CrossRefGoogle ScholarPubMed
Stützer, S., Plotnik, Y., Lumer, Y., et al. “Photonic topological Anderson insulators,” Nature, vol. 560, no. 7719, pp. 461465, 2018.CrossRefGoogle ScholarPubMed
Kolner, B. H., “Space-time duality and the theory of temporal imaging,” IEEE Journal of Quantum Electronics, vol. 30, no. 8, pp. 19511963, 1994.CrossRefGoogle Scholar
Poon, T. and Kim, T., Engineering Optics with Matlab, 2nd ed. World Scientific Publishing, Singapore, 2018.Google Scholar
Ozaktas, H. M., Zalevsky, Z., and Alper Kutay, M., The Fractional Fourier Transform with Applications in Optics and Signal Processing. John Wiley, New York, 2001.Google Scholar
Mendlovic, H. and Ozaktas, H. M., “Fractional fourier transforms and their optical implementation: I,” Journal of the Optical Society of America A, vol. 10, pp. 18751881, 1993.CrossRefGoogle Scholar
Mendlovic, H. and Ozaktas, H. M., “Fractional fourier transforms and their optical implementation: Ii,” Journal of the Optical Society of America A, vol. 10, pp. 25222531, 1993.CrossRefGoogle Scholar
West, B. J., Bologna, M., and Grigolini, P., Physics of Fractal Operators. Springer, New York, 2003.CrossRefGoogle Scholar
Intonti, F., Caselli, N., Lawrence, N., Trevino, J., Wiersma, D. S., and Dal Negro, L., “Near-field distribution and propagation of scattering resonances in vogel spiral arrays of dielectric nanopillars,” New Journal of Physics, vol. 15, no. 8, p. 085023, 2013.CrossRefGoogle Scholar
Stratton, J. A. and Chu, L. J., “Diffraction theory of electromagnetic waves,” Physical Review, vol. 56, pp. 99107, 1939.CrossRefGoogle Scholar
Stratton, J. A., Electromagnetic Theory. McGraw-Hill, New York, 1941.Google Scholar
Holland, A. S. B., Introduction to the Theory of Entire Functions. Academic Press, New York and London, 1973.Google Scholar
Boas, R. P., Entire Functions. Academic Press, New York, 1954.Google Scholar
Lindberg, J., “Mathematical concepts of optical superresolution,” Journal of Optics, vol. 14, no. 083001, pp. 123, 2012.CrossRefGoogle Scholar
den Dekker, A. J. and van den Bos, A., “Resolution: a survey,” Journal of the Optical Society of America A, vol. 14, pp. 547557, 1997.CrossRefGoogle Scholar
Vijayakumar, A. and Bhattacharya, S., Design and Fabrication of Diffractive Optical Elements with MATLAB. SPIE Press, Bellingham, 2017.CrossRefGoogle Scholar
Chen, Y., Britton, W., and Dal Negro, L., “Phase-modulated axilenses for infrared multiband spectroscopy,” Optics Letters, vol. 45, no. 8, pp. 23712374, 2020.CrossRefGoogle ScholarPubMed
Britton, W. A., Chen, Y., Sgrignuoli, F., and Dal Negro, L.. “Phase-modulated axilenses as ultracompact spectroscopic tools,” ACS Photonics, vol. 7, no. 10, 27312738, 2020.CrossRefGoogle Scholar
Chen, Y., Britton, W., and Dal Negro, L., “Design of infrared microspectrometer based on phase-modulated axilenses,” Applied Optics, vol. 59, pp. 55325538, 2020.CrossRefGoogle Scholar
Harvey, J. E. and Forgham, L., “The spot of arago: new relevance for an old phenomenon,” American Journal of Physics, vol. 52, pp. 243247, 1984.CrossRefGoogle Scholar
Iizuka, K., Engineering Optics, 3rd ed. Springer, New York, 2008.Google Scholar
Berry, M. V. and Dennis, M. R., “Natural superoscillations in monochromatic waves in d dimensions,” Journal of Physics A: Mathematical and Theoretical, vol. 42, no. 2, p. 022003, 2009.CrossRefGoogle Scholar
Huang, F. M., Chen, Y., F. Javier Garcia de Abajo, and N. I. Zheludev, “Optical superresolution through super-oscillations,” Journal of Optics A: Pure and Applied Optics, vol. 9, pp. S285–S288, 2007.CrossRefGoogle Scholar
Rogers, E. T. F., Lindbergn, J., Roy, T., et al., “A super-oscillatory lens optical microscope for subwavelength imaging,” Nature Materials, vol. 11, pp. 432435, 2012.CrossRefGoogle ScholarPubMed
Huang, F. M. and Zheludev, N. I., “Super-resolution without evanescent waves,” Nano Letters, vol. 9, no. 3, pp. 12491254, 2009.CrossRefGoogle ScholarPubMed
Berry, M. V. and Popescu, S., “Evolution of quantum superoscillations and optical superresolution without evanescent waves,” Journal of Physics A: Mathematical and General, vol. 39, pp. 69656977, 2006.CrossRefGoogle Scholar
Kempf, A., “Black holes, bandwidths and Beethoven,” Journal of Mathematical Physics, vol. 41, pp. 23602374, 2000.CrossRefGoogle Scholar
Ferraira, P. J. S. G. and Kempf, A., “Superoscillations: faster than the Nyquist rate,” IEEE Transactions on Signal Processing, vol. 54, no. 10, pp. 37323740, 2006.CrossRefGoogle Scholar
Rogers, E. T. F. and Zheludev, N. I., “Optical super-oscillations: sub-wavelength light focusing and super-resolution imaging,” Journal of Optics, vol. 15, no. 094008, pp. 123, 2013.CrossRefGoogle Scholar
Wang, R., Pinheiro, F. A., and Dal Negro, L., “Spectral statistics and scattering resonances of complex primes arrays,” Physical Review B, vol. 97, no. 024202, pp. 111, 2018.Google Scholar
Vasara, A., Turunen, J., and Friberg, A. T., “Realization of general nondiffracting beams with computer-generated holograms,” Journal of the Optical Society of America A,vol.6, no. 11, pp. 17481754, 1989.CrossRefGoogle ScholarPubMed
Vijayakumar, A., Parthasarathi, P., Iyengar, S. S., et al., “Conical Fresnel zone lens for optical trapping,” International Conference on Optics and Photonics 2015, vol. 9654, 2015.Google Scholar
Chen, W. T., Zhu, A. Y., Sanjeev, V., Khorasaninejad, M., Shi, Z., Lee, E., and Capasso, F., “A broadband achromatic metalens for focusing and imaging in the visible,” Nature Nanotechnology, vol. 13, no. 3, p. 220, 2018.CrossRefGoogle ScholarPubMed
Zhang, S., Soibel, A., Keo, S. A., et al., “Solid-immersion metalenses for infrared focal plane arrays,” Applied Physics Letters, vol. 113, no. 11, p. 111104, 2018.Google Scholar
[210] Britton, W. A., Chen, Y., Sgrignuoli, F., and Dal Negro, L., “Phase-modulated axilenses as ultracompact spectroscopic tools,” ACS Photonics, vol. 7, no. 10, pp. 27312738, 2020. [Online]. Available: Scholar
Nye, J. F., Natural Focusing and Fine Structure of Light. Institute of Physics Publishing, 1999.Google Scholar
Salem, R., Foster, M. A., Turner, A. C., Geraghty, D. F., Lipson, M., and Gaeta, A. L., “Optical time lens based on four-wave mixing on a silicon chip,” Optics Letters, vol. 15, no. 33, pp. 10471049, 2008.CrossRefGoogle Scholar
Klein, A., Yaron, T., Preter, E., Duadi, H., and Fridman, M., “Temporal depth imaging,” Optica, vol. 4, no. 5, pp. 502506, 2017.CrossRefGoogle Scholar
Mendonça, J. T. and Shukla, P. K., “Time refraction and time reflection: two basic concepts,” Physica Scripta, vol. 65, no. 2, pp. 160163, 2002.CrossRefGoogle Scholar
Xiao, Y., Maywar, D. N., and Agrawal, G. P., “Reflection and transmission of electromagnetic waves at a temporal boundary,” Optics Letters, vol. 39, no. 3, pp. 574577, 2014.CrossRefGoogle Scholar
Shaltout, A. M., Lagoudakis, K. G., van de Groep, J., et al., “Spatiotemporal light control with frequency-gradient metasurfaces,” Science, vol. 365, pp. 374377, 2019.CrossRefGoogle ScholarPubMed
[217] Zhou, Y., Alam, M. Z., Karimi, M., et al., “What is the temporal analog of reflection and refraction of optical beams?” Nature Communications, vol. 11, no. 2180, 2020.Google Scholar
[218] Plansinis, B. W., Donaldson, W. R., and Agrawal, G. P., “What is the temporal analog of reflection and refraction of optical beams?” Physical Review Letters, vol. 115, no. 183901, 2015.CrossRefGoogle ScholarPubMed
Ye, J. and Cundiff, S. T. e., Femtosecond Optical Frequency Comb: Principle,Operation and Applications. Springer Science + Business Media, Boston, 2005.CrossRefGoogle Scholar
Picqué, N. and Hänsch, T. W., “Frequency comb spectroscopy,” Nature Photonics, vol. 13, pp. 146157, 2019.CrossRefGoogle Scholar
Lakshminarayanan, V., Ghatak, A. K., and Thyagarajan, K., Lagrangian Optics. Springer Science, New York, 2002.CrossRefGoogle Scholar
Berry, M. V. and Upstill, C., “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Progress in Optics, vol. 18, pp. 257346, 1980.CrossRefGoogle Scholar
Dupré, S., “Optics, pictures and evidence: Leonardo’s drawings of mirrors and machinery,” Early Science and Medicine, vol. 10, no. 2, p. 211236, 2005.CrossRefGoogle Scholar
Arnold, V. I., Catastrophe Theory, 2nd ed. Springer, Berlin, 1986.CrossRefGoogle Scholar
Poston, T. and Stewart, I., Catastrophe Theory and Its Applications. Pitman, London, 1978.Google Scholar
Gilmore, R., Catastrophe Theory for Scientists and Engineers. John Wiley, New York, 1981.Google Scholar
Thom, R., Structural Stability and Morphogenesis. An Outline of a General Theory of Models. W. A. Benjamin Inc., Reading, 1975.Google Scholar
Siviloglou, G. A. and Christodoulides, D. N., “Accelerating finite energy airy beams,” Optics Letters, vol. 32, no. 8, p. 979981, 2007.CrossRefGoogle ScholarPubMed
Berry, M. V. and Balazs, N., “Nonspreading wave packets,” American Journal of Physics, vol. 47, pp. 264267, 1979.CrossRefGoogle Scholar
[230] Siviloglou, G. A., Broky, J., Dogariu, A., and Christodoulides, D. N., “Observation of accelerating airy beams,” Physical Review Letters, vol. 99, no. 213901, 2007.CrossRefGoogle ScholarPubMed
Baumgartl, J., Mazilu, M., and Dholakia, K., “Optically mediated particle clearing using airy wavepackets,” Nature Photonics, vol. 2, pp. 675678, 2008.CrossRefGoogle Scholar
Nye, J. F., “The motion and structure of dislocations in wavefronts,” Proceedings of the Royal Society of London. Series A, vol. 378, pp. 219239, 1981.Google Scholar
Nye, J. F. and Berry, M., “Dislocations in wave trains,” Proceedings of the Royal Society of London. Series A, vol. 336, pp. 165190, 1974.Google Scholar
Dennis, M. R., O’Holleran, K., and Padgett, M. J., “Singular optics: optical vortices and polarization singularities,” Progress in Optics, vol. 53, pp. 293363, 2009.CrossRefGoogle Scholar
Berry, M., Nye, J. F., and Wright, F., “The elliptic umbilic diffraction catastrophe,” Philosophical Transactions of the Royal Society, vol. 291, pp. 453484, 1979.Google Scholar
[236] Kharif, C. and Pelinovsky, E., “Physical mechanisms of the rogue wave phenomenon,” European Journal of Mechanics – B/Fluids, vol. 22, pp. 603–63 437, 2003.CrossRefGoogle Scholar
Osborne, A. R., Nonlinear Ocean Waves and the Inverse Scattering Transform. Academic Press, New York, 2010.Google Scholar
Solli, D. R., Ropers, C., Koonath, P., and Jalali, B., “Optical rogue waves,” Nature, vol. 450, pp. 10541057, 2007.CrossRefGoogle ScholarPubMed
Dudley, J. M., Dias, F., Erkintalo, M., and Genty, G., “Instabilities, breathers and rogue waves in optics,” Nature Photonics, vol. 8, pp. 755764, 2014.CrossRefGoogle Scholar
[240] Metzger, J. J., Fleischmann, R., and Geisel, T., “Statistics of extreme waves in random media,” Physical Review Letters, vol. 112, p. 203903, May 2014. [Online]. Available: Scholar
[241] Mathis, A., Froehly, L., Toenger, S., Dias, F., Genty, G., and Dudley, J. M., “Caustics and rogue waves in an optical sea,” Scientific Reports, vol. 5, no. 12822, 2015.CrossRefGoogle Scholar
[242] Safari, A., Fickler, R., Padgett, M. J., and Boyd, R. W., “Generation of caustics and rogue waves from nonlinear instability,” Physical Review Letters, vol. 119, no. 203901, 2017.CrossRefGoogle ScholarPubMed
Coles, S., An Introduction to Statistical Modeling of Extreme Values. Springer-Verlag, London, 2001.CrossRefGoogle Scholar
[244] Sgrignuoli, F., Chen, Y., Gorsky, S., Britton, W. A., and Dal Negro, L., “Optical rogue waves in multifractal photonic arrays,” Physical Review B, vol. 103, no. 19, 2021.CrossRefGoogle Scholar
Coullet, P., “Optical vortices,” Optics Communications, vol. 73, pp. 403408, 1989.CrossRefGoogle Scholar
Yao, A. M. and Padgett, M. J., “Orbital angular momentum: origins, behavior and applications,” Advances in Optics and Photonics, vol. 3, pp. 161204, 2011.CrossRefGoogle Scholar
Torres, J. P. and Torner, L., Eds., Twisted Photons. Applications of Light with Orbital Angular Momentum. Wiley-VCH, Weinheim, 2011.CrossRefGoogle Scholar
Allen, L., Beijersbergen, M. W., Spreeuw, R. J. C., and Woerdman, J. P., “Orbital angular-momentum of light and the transformation of Laguerre–Gaussian laser modes,” Physical Review A, vol. 45, pp. 81858189, 1992.CrossRefGoogle ScholarPubMed
Chavez-Cerda, S., Padgett, M., Allison, I., et al., “Holographic generation and orbital angular momentum of high-order mathieu beams,” Journal of Optics B: Quantum Semiclassical Optics, vol. 4, pp. S52S57, 2002.CrossRefGoogle Scholar
Liew, S. F., Noh, H., Trevino, J., Dal Negro, L., and Cao, H., “Localized photonic band edge modes and orbital angular momenta of light in a golden-angle spiral,” Optics Express, vol. 19, pp. 2363123642, 2011.CrossRefGoogle Scholar
Lawrence, N., Trevino, J., and Dal Negro, L., “Control of optical orbital angular momentum by vogel spiral arrays of metallic nanoparticles,” Optics Letters, vol. 37, pp. 50765078, 2012.CrossRefGoogle ScholarPubMed
Molina-Terriza, G., Torres, J. P., and Torner, L., “Twisted photons,” Nature Physics,vol.3, pp. 3015–310, 2007.CrossRefGoogle Scholar
Grier, D., “A revolution in optical manipulation,” Nature Physics, vol. 424, pp. 810816, 2003.CrossRefGoogle ScholarPubMed
Mair, A., Vaziri, A., Weihs, G., and Zeilinger, A., “Entanglement of the orbital angular momentum states of photons,” Nature, vol. 412, pp. 313316, 2001.CrossRefGoogle ScholarPubMed
[255] Baev, A., Prasad, P. N., Ren, H., Samoc, M., and Wegener, M., “Metaphotonics: an emerging field with opportunities and challenges,” Physics Reports, vol. 594, pp. 160, 2015. [Online]. Available: Scholar
Capasso, F., “The future and promise of flat optics: a personal perspective,” Nanophotonics, vol. 7, no. 6, pp. 953957, 2018.CrossRefGoogle Scholar
Yu, N. and Capasso, F., “Flat optics with designer metasurfaces,” Nature Materials, vol. 13, pp. 139150, 2014.CrossRefGoogle ScholarPubMed
Yu, N., Genevet, P., Kats, M. A., et al., “Light propagation with phase discontinuities: generalized laws of reflection and refraction,” Science, vol. 334, pp. 333337, 2011.CrossRefGoogle ScholarPubMed
Ni, X., Emani, N. K., Kildishev, A. V., Boltasseva, A., and Shalaev, V. M., “Broadband light bending with plasmonic nanoantennas,” Science, vol. 335, p. 427, 2012.CrossRefGoogle ScholarPubMed
Lin, D., Fan, P., Hasman, E., and Brongersma, M. L., “Dielectric gradient metasurface optical elements,” Science, vol. 345, pp. 298301, 2014.CrossRefGoogle ScholarPubMed
Wu, C., Arju, N., Kelp, G., et al., “Spectrally selective chiral silicon metasurfaces based on infrared fano resonances,” Nature Communications, vol. 5, no. 3892, pp. 19, 2014.CrossRefGoogle ScholarPubMed
Khorasaninejad, M. and Capasso, F., “Metalenses: versatile multifunctional photonic components,” Science, vol. 358, no. 1146, pp. 18, 2017.CrossRefGoogle ScholarPubMed
Genevet, P. and Capasso, F., “Holographic optical metasurfaces: a review of current progress,” Reports on Progress in Physics, vol. 78, no. 024401, pp. 119, 2015.CrossRefGoogle ScholarPubMed
Ding, F., Pors, A., and Bozhevolnyi, S. I., “Gradient metasurfaces: a review of fundamentals and applications,” Reports on Progress in Physics, vol. 81, no. 2, p. 026401, 2017.Google Scholar
Zhao, Y., Liu, X., and Alù, A., “Recent advances on optical metasurfaces,” Journal of Optics, vol. 16, no. 123001, pp. 114, 2014.CrossRefGoogle Scholar
Mahan, G. D., Many-Particle Physics. Plenum Press, New York, 1990.CrossRefGoogle Scholar
Devaney, A. J., Mathematical Foundation of Imaging, Tomography and Wavefield Inversion. Cambridge University Press, Cambridge, 2012.Google Scholar
Liu, H., Liu, D., Mansour, H., Boufounos, P. T., Waller, L., and Kamilov, U. S., “Seagle: sparsity-driven image reconstruction under multiple scattering,” IEEE Transactions on Computational Imaging, vol. 4, no. 1, pp. 7386, 2018.CrossRefGoogle Scholar
Colton, D. and Kress, R., “Looking back on inverse scattering theory,” SIAM Review, vol. 60, no. 4, pp. 779807, 2018.CrossRefGoogle Scholar
Jin, J.-M., Theory and Computation of Electromangetic Fields, 2nd ed. John Wiley & Sons, Hoboken, 2010.CrossRefGoogle Scholar
Kaku, M., Quantum Field Theory. A Modern Introduction. Oxford University Press, New York, 1993.Google Scholar
Sementilli, P. J., Hunt, B. R., and Nadar, M. S., “Analysis of the limit to superresolution in incoherent imaging,” Journal of the Optical Society of America A, vol. 10, pp. 22652276, 1993.CrossRefGoogle Scholar
Newton, R. G., “Optical theorem and beyond,” American Journal of Physics, vol. 44, no. 7, p. 639, 1976.CrossRefGoogle Scholar
Feenberg, E., “The scattering of slow electrons by neutral atoms,” Physical Review, vol. 40, p. 40, 1932.CrossRefGoogle Scholar
Levine, H. and Schwinger, J., “On the theory of diffraction by an aperture in an infinite plane screen I,” Physical Review, vol. 74, p. 958, 1948.CrossRefGoogle Scholar
van de Hulst, H. C., “On the attenuation of plane waves by obstacles of arbitrary size and form,” Physica, vol. 15, p. 740, 1949.CrossRefGoogle Scholar
Lock, J. A., Hodges, J. T., and Gouesbet, G., “Failure of the optical theorem for Gaussianbeam scattering by a spherical particle,” Journal of the Optical Society of America A, vol. 12, pp. 27082715, 1995.CrossRefGoogle Scholar
Berg, M. J., Sorensen, C. M., and Chakrabarti, A., “Extinction and the optical theorem. Part I. Single particles,” Journal of the Optical Society of America A, vol. 25, no. 7, pp. 15041513, 2008.CrossRefGoogle ScholarPubMed
Berg, M. J., Sorensen, C. M., and Chakrabarti, A., “Extinction and the optical theorem. Part II. Multiple particles,” Journal of the Optical Society of America A, vol. 25, no. 7, pp. 15141520, 2008.CrossRefGoogle ScholarPubMed
Mishchenko, M. I. and Travis, L. D., Multiple Scattering of Light by Particles. Cambridge Univesrity Press, New York, 2006.Google Scholar
Bohren, C. F. and Huffman, D. R., Absorption and Scattering of Light by Small Particles. Wiley-VCH, Weinheim, 2004.Google Scholar
Reali, G. C., “Reflection from dielectric materials,” American Journal of Physics, vol. 50, no. 12, pp. 11331136, 1982.CrossRefGoogle Scholar
Ballenegger, V. C., “The Ewald–Oseen extinction theorem and extinction lengths,” American Journal of Physics, vol. 67, no. 7, pp. 599605, 1999.CrossRefGoogle Scholar
Goodman, J. W., Statistical Optics, 2nd ed. John Wiley and Sons., Greenwood Village, 2007.Google Scholar
Wolf, E., Introduction to the Theory of Coherence and Polarization of Light. Cambridge University Press, 2007.Google Scholar
Wolf, E., “A macroscopic theory of interference and diffraction of light from finite sources ii. fields with a spectral range of arbitrary width.” Proceedings of the Royal Society of London, vol. 230, p. 246265, 1955.Google Scholar
Wolf, E., “Unified theory of coherence and polarization of random electromagnetic beams,” Physical Letters A, vol. 312, pp. 263267, 2003.CrossRefGoogle Scholar
Wolf, E., “Correlation-induced changes in the degree of polarization, the degree of coherence and the spectrum of random electromagnetic beams on propagation,” Optics Letters, vol. 28, pp. 10781080, 2003.CrossRefGoogle ScholarPubMed
Tervo, J., Setälä, T., and Friberg, A. T., “Theory of partially coherent electromagnetic fields in the space–frequency domain,” Journal of the Optical Society of America A, vol. 21, pp. 22052215, 2005.CrossRefGoogle Scholar
Gbur, G. and Visser, T. D., “The structure of partially coherent fields, in progress in optics, Emil Wolf ed.” Progess in Optics, vol. 55, pp. 285341, 2010.CrossRefGoogle Scholar
Labeyrie, A., Lipson, S. G., and Nisenson, P., An Introduction to Optical Stellar Interferometry. Cambridge University Press, Cambridge, 2006.CrossRefGoogle Scholar
Dogariu, A. and Wolf, E., “Spectral changes produced by static scattering on a system of particles,” Optics Letters, vol. 23, pp. 13401342, 1998.CrossRefGoogle ScholarPubMed
Gbur, G. and Wolf, E., “Determination of density correlation functions from scattering of polychromatic light,” Optics Communications, vol. 168, pp. 3945, 1999.CrossRefGoogle Scholar
Schell, A. C., “Multiple plate antenna,” Ph.D. Thesis, Massachusetts Institute of Technology, 1961.Google Scholar
Nugent, K. A., “A generalization of Schell’s theorem,” Optics Communications, vol. 79, pp. 267269, 1990.CrossRefGoogle Scholar
Ewald, P. P., “Introduction to the dynamical theory of X-ray diffraction,” Acta Crystallographica, vol. A25, pp. 103108, 1969.CrossRefGoogle Scholar
Goodstein, D. L., States of Matter. Dover Publications, Mineola, 1985.Google Scholar
Torquato, S. and Stillinger, F. H., “Local density fluctuations, hyperuniformity, and order metrics,” Physical Review E, vol. 68, p. 041113, 2003.Google ScholarPubMed
Torquato, S., Zhang, G., and Stillinger, F. H., “Ensemble theory for stealthy hyperuniform disordered ground states,” Physical Review X, vol. 5, p. 021010, 2015.CrossRefGoogle Scholar
Sorensen, C. M., “Light scattering by fractal aggregates: a review,” Aerosol Science and Technology, vol. 35, pp. 648687, 2001.CrossRefGoogle Scholar
Mishchenko, M. I., Travis, L. D., and Lacis, A. A., Scattering, Absorption and Emission of Light by Small Particles. Cambridge Univesrity Press, Edinburgh, 2002.Google Scholar
Conley, G. M., Burresi, M., Pratesi, F., Vynck, K., and Wiersma, D. S., “Light transport and localization in two-dimensional correlated disorder,” Physical Review Letters, vol. 112, p. 143901, 2014.CrossRefGoogle ScholarPubMed
Sivia, D. S., Elementry Scattering Theory. Oxford University Press, New York, 2011.CrossRefGoogle Scholar
Hansen, J. and McDonald, I. R., Theory of Simple Liquids with Applications to Soft Matter, 4th ed. Academic Press, San Diego, 2013.Google Scholar
Khare, K., Fourier Optics and Computational Imaging. John Wiley and Ane Books Pvt. Ltd., 2016.Google Scholar
Cowley, J. M., Difraction Physics. Elsevier, 1995.Google Scholar
Inui, T., Tanabe, Y., and Onodera, Y., Group Theory and Its Applications in Physics. Sprringer-Verlag, Berlin and Heidelberg, 1990.CrossRefGoogle Scholar
Kritikos, H. N., “Radiation symmetries of antenna arrays,” Journal of the Franklin Institute, vol. 295, no. 4, pp. 283292, 1973.CrossRefGoogle Scholar
[309] Lee, S. Y. K., Amsden, J. J., et al., “Spatial and spectral detection of protein mono-layers with deterministic aperiodic arrays of metal nanoparticles,” PNAS, vol. 107, pp. 12 086–12 090, 2010.CrossRefGoogle Scholar
[310] Boriskina, S. V., Lee, S. Y. K., Amseden, J. J., Omenetto, F., and Dal Negro, L., “Formation of colorimetric fingerprints on nano-patterned deterministic aperiodic surfaces,” Optics Express, vol. 18, no. 14, pp. 14 568–14 576, 2010.CrossRefGoogle ScholarPubMed
Trevino, J., Forestiere, C., Di Martino, G., Yerci, S., Priolo, F., and Dal Negro, L., “Plasmonic-photonic arrays with aperiodic spiral order for ultra-thin film solar cells,” Optics Express, vol. 20, pp. A418–A430, 2012.CrossRefGoogle ScholarPubMed
Pierro, V., Galdi, V., Castaldi, G., Pinto, I. M., and Felsen, L. B., “Radiation properties of planar antenna arrays based on certain categories of aperiodic tilings,” IEEE Transactions on Antennas and Propagation, vol. 53, no. 2, pp. 635644, 2005.CrossRefGoogle Scholar
Lang, S., Algebraic Number Theory. Addison-Wesley, Reading, 1970.Google Scholar
Cohen, H., A Course in Computational Algebraic Number Theory. Springer-Verlag, Berlin, 1993.CrossRefGoogle Scholar
Dekker, T. J., “Prime numbers in quadratic fields,” CWI Quarterly, vol. 7, pp. 367394, 1994.Google Scholar
Baake, M. and Grimm, U., Aperiodic Order,vol.1A Mathematical Invitation. Cambridge University Press, New York, 2013.CrossRefGoogle Scholar
Dal Negro, L., Henderson, D. T., Sgrignuoli, F., “Wave transport and localization in prime number landscapes”, Frontiers in Physics, 9, 490 (2021).CrossRefGoogle Scholar
Berthier, S., Iridescences: The Physical Colors of Insects. Springer, New York, 2007.Google Scholar
Kinoshita, S., Structural Colors in the Realm of Nature. World Scientific, 2008.CrossRefGoogle Scholar
[320] Gorsky, S., Zhang, R., Gok, A., et al., “Directonal light emission enhancement from led-phosphor converters using dielectric vogel spiral arrays,” Applied Physics Letters Photonics, vol. 3, pp. 126 103–126 114, 2018.Google Scholar
Gorsky, S., Britton, W. A., Chen, Y., et al., “Engineered hyperuniformity for directional light extraction,” Applied Physics Letters Photonics, no. 4, p. 110801, 2019.Google Scholar
Levine, D. and Steinhardt, P. J., “Quasicrystals. I. Definition and structure,” Physical Review B, vol. 34, pp. 596616, 1986.CrossRefGoogle ScholarPubMed
Maciá-Barber, E., Quasicrystals: Fundamentals and Applications. CRC Press, Boca Raton, 2021.Google Scholar
Queffélec, M., Substitution Dynamical Systems – Spectral Analysis. Springer-Verlag, Berlin, 2010.CrossRefGoogle Scholar
Schroeder, M., Fractals, Chaos, Power Laws. W. H. Freeman, New York, 1991.Google Scholar
Luck, J. M., “Cantor spectra and scaling of gap widths in deterministic aperiodic systems,” Physical Review B, vol. 39, pp. 58345849, 1989.CrossRefGoogle ScholarPubMed
Maciá-Barber, E., Aperiodic Structures in Condensed Matter: Fundamentals and Applications. CRC Press Taylor and Francis, Boca Raton, 2009.Google Scholar
Maciá, E., “The role of aperiodic order in science and technology,” Reports on Progress in Physics, vol. 69, pp. 397441, 2006.CrossRefGoogle Scholar
Dulea, M., Johansson, M., and Riklund, R., “Localization of electrons and electromagnetic waves in a deterministic aperiodic system,” Physical Review B, vol. 45, pp. 105114, 1992.CrossRefGoogle Scholar
Dulea, M., Johansson, M., and Riklund, R., “Unusual scaling of the spectrum in a deterministic aperiodic tight-binding model,” Physical Review B, vol. 47, pp. 85478551, 1993.CrossRefGoogle Scholar
Dal Negro, L., Feng, N. N., and Gopinath, A., “Electromagnetic coupling and plasmon localization in deterministic aperiodic arrays,” Journal of Optics A: Pure and Applied Optics, vol. 10, p. 064013, 2008.Google Scholar
Kroon, L., Lennholm, E., and Riklund, R., “Localization-delocalization in aperiodic systems,” Physical Review B, vol. 66, p. 094204, 2002.CrossRefGoogle Scholar
Kroon, L. and Riklund, R., “Absence of localization in a model with correlation measure as a random lattice,” Physical Review B, vol. 69, p. 094204, 2004.CrossRefGoogle Scholar
García de Abajo, F. J., Gómez-Medina, R., and Sáenz, J. J., “Full transmission through perfect-conductor subwavelength hole arrays,” Physical Review E, vol. 72, no. 016608, pp. 14, 2005.CrossRefGoogle ScholarPubMed
García de Abajo, F. J., Sáenz, J. J., Campillo, I., and Dolado, J. S., “Site and lattice resonances in metallic hole arrays,” Optics Express, vol. 14, no. 1, pp. 718, 2006.CrossRefGoogle ScholarPubMed
[336] Zou, S. and Schatz, G. C., “Narrow plasmonic/photonic extinction and scattering line shapes for one and two dimensional silver nanoparticles arrays,” Journal of Chemical Physics, vol. 121, pp. 12 606–12 612, 2004.CrossRefGoogle Scholar
[337] Zou, S., Janel, N., and Schatz, G. C., “Silver nanoparticle array structures that produce remarkable narrow plasmon lineshapes,” Journal of Chemical Physics, vol. 120, pp. 10 871–10 875, 2004.CrossRefGoogle Scholar
Ebbesen, T. W., Lezec, H. J., Ghaemi, H. F., Thio, T., and Wolff, P. A., “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature, vol. 391, pp. 667669, 1998.CrossRefGoogle Scholar
García de Abajo, F. J., “Colloquium: light scattering by particles and hole arrays,” Reviews of Modern Physics, vol. 79, pp. 12671290, 2007.CrossRefGoogle Scholar
Authier, A., Dynamical Theory of X-Ray Diffraction. Oxford University Press, Oxford, 2001.Google Scholar
Wood, R. W., “Anomalous diffraction gratings,” Physical Review, vol. 48, pp. 928936, 1935.CrossRefGoogle Scholar
Fano, U., “The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld’s waves),” Journal of the Optical Society of America, vol. 31, pp. 213222, 1941.CrossRefGoogle Scholar
Fano, U., “Effects of configuration interaction on intensities and phase shifts,” Physical Review, vol. 124, pp. 18661878, 1961.CrossRefGoogle Scholar
Matsui, T., Agrawal, A., Nahata, A., and Vardeny, Z. V., “Transmission resonances through aperiodic arrays of subwavelength apertures,” Nature, vol. 446, pp. 517521, 2007.CrossRefGoogle ScholarPubMed
[345] Przybilla, F., Genet, C., and Ebbesen, T. W., “Enhanced transmission through Penrose subwavelength hole arrays,” Applied Physics Letters, vol. 89, no. 121115, 2006.CrossRefGoogle Scholar
Bellissard, J. V., Bovier, A., and Ghez, J., “Gap labelling theorems for one-dimensional discrete Schrödinger operators,” Reviews in Mathematical Physics, vol. 4, pp. 137, 1992.CrossRefGoogle Scholar
[347] Dal Negro, L. and Feng, N. N., “Spectral gaps and mode localization in Fibonacci chains of metal nanoparticles,” Optics Express, vol. 15, pp. 14 396–14 403, 2007.CrossRefGoogle ScholarPubMed
[348] Huang, F. M., Zheludev, N. I., Chen, Y., and García de Abajo, F. J., “Focusing of light by a nano-hole array,” Applied Physics Letters, vol. 90, no. 091119, 2007.CrossRefGoogle Scholar
Sgrignuoli, F., Wang, R., Pinheiro, F., and Dal Negro, L., “Localization of scattering resonances in aperiodic Vogel spirals,” Physical Review B, vol. 99, p. 104202, 2019.CrossRefGoogle Scholar
[350] Chen, Y., Sgrignuoli, F., and Dal Negro, L., “Optical superoscillations of prime number arrays,” in preparation.Google Scholar
Illian, J., Penttinen, A., Stoyan, H., and Stoyan, D., Statistical Analysis and Modelling of Spatial Point Patterns. John Wiley, Chichester, 2008.Google Scholar
Torquato, S., Scardicchio, A., and Zachary, C. E., “Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory,” Journal of Statistical Mechanics, 2008.CrossRefGoogle Scholar
Gabrielli, A., Jancovici, B., Joyce, M., Lebowitz, J. L., Pietronero, L., and Sylos Labini, F., “Generation of primordial cosmological perturbations from statistical mechanical models,” Physical Review D, vol. 67, p. 043506, 2003.CrossRefGoogle Scholar
Torquato, S., Random Heterogeneous Materials: Microstructure and Macroscopic Properties. Springer-Verlag, New York, 2012.Google Scholar
Huang, K., Statistical Mechanics, 2nd ed. John Wiley and Sons, California, 1987.Google Scholar
Isihara, A., Condensed Matter Physics. Oxford University Press, New York and Oxford, 1991.Google Scholar
[357] Ornstein, L. S. and Zernike, F., “Accidental deviations of density and opalescence at the critical point of a single substance,” Royal Netherlands Academy of Arts and Sciences (KNAW). Proceedings, vol. 17, p. 793806, 1914.Google Scholar
Percus, J. K. and Yevick, G. J., “Analysis of classical statistical mechanics by means of collective coordinates,” Physical Review, vol. 110, p. 113, 1958.CrossRefGoogle Scholar
Tsang, L. and Kong, K., Ding, J. A., Scattering of Electromangetic Waves: Numerical Simulations, vol. 3. John Wiley, New York, 2001.Google Scholar
Wertheim, M. S., “Exact solution of the Percus–Yevick integral equation for hard spheres,” Physical Review Letters, vol. 20, p. 321323, 1963.CrossRefGoogle Scholar
ben Avraham, D. and Havlin, S., Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press, Cambridge, 2000.CrossRefGoogle Scholar
Goodman, J. W., Speckle Phenomena in Optics: Theory and Applications. Ben Roberts and Company, US, 2015.Google Scholar
Papoulis, A., Probability, Random Variables, and Stochastic Processes, 3rd ed. McGraw-Hill, New York, 1991.Google Scholar
Jakeman, E. and Ridley, K. D., Modeling Fluctuations in Scattered Waves. Taylorand Francis, New York, 2006.CrossRefGoogle Scholar
Kallenberg, O., Foundations of Modern Probability, 2nd ed. Springer-Verlag, New York, 2002.CrossRefGoogle Scholar
Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions. Dover, New York, 1965.Google Scholar
van Rossum, M. C. W. and Nieuwenhuizen, T. M., “Multiple scattering of classical waves: microscopy, mesoscopy and diffusion,” Reviews of Modern Physics, vol. 71, pp. 313371, 1999.CrossRefGoogle Scholar
[368] Scheffold, F. and Maret, G., “Universal conductance fluctuations of light,” Physical Review Letters, vol. 81, pp. 58005803, December 1998. [Online]. Available: Scholar
[369] Shapiro, B., “Large intensity fluctuations for wave propagation in random media,” Physical Review Letters, vol. 57, pp. 21682171, October 1986. [Online]. Available: ScholarPubMed
[370] Feng, S., Kane, C., Lee, P. A., and Stone, A. D., “Correlations and fluctuations of coherent wave transmission through disordered media,” Physical Review Letters, vol. 61, pp. 834837, August 1988. [Online]. Available: PhysRevLett.61.834CrossRefGoogle ScholarPubMed
[371] Feng, S. and Lee, P. A., “Mesoscopic conductors and correlations in laser speckle patterns,” Science, vol. 251, no. 4994, pp. 633639, 1991. [Online]. Available: ScholarPubMed
[372] Bertolotti, J., van Putten, E. G., Blum, C., Lagendijk, A., Vos, W. L., and Mosk, A. P., “Non-invasive imaging through opaque scattering layers,” Nature, vol. 491, no. 7423, pp. 232234, November 2012. [Online]. Available: ScholarPubMed
[373] Vellekoop, I. M., Lagendijk, A., and Mosk, A. P., “Exploiting disorder for perfect focusing,” Nature Photonics, vol. 4, no. 5, pp. 320322, May 2010. [Online]. Available: Scholar
[374] Mosk, A. P., Lagendijk, A., Lerosey, G., and Fink, M., “Controlling waves in space and time for imaging and focusing in complex media,” Nature Photonics, vol. 6, no. 5, pp. 283292, May 2012. [Online]. Available: Scholar
[375] Yilmaz, H., van Putten, E. G., Bertolotti, J., Lagendijk, A., Vos, W. L., and Mosk, A. P., “Speckle correlation resolution enhancement of wide-field fluorescence imaging,” Optica, vol. 2, no. 5, pp. 424429, May 2015. [Online]. Available: Scholar
[376] Sahoo, S. K., Tang, D., and Dang, C., “Single-shot multispectral imaging with a monochromatic camera,” Optica, vol. 4, no. 10, pp. 12091213, 2017. [Online]. Available: Scholar
Klages, R., Radons, G., and Igor, M. Sokolov, (eds.), Anomalous Transport: Foundations and Applications. Wiley-VCH Verlag GmbH and Co. KGaA, Weinheim, 2008.CrossRefGoogle Scholar
de Gennes, P., “On a relation between percolation theory and the elasticity of gels,” Journal de Physique Lettres, vol. 37, pp. 12, 1976.CrossRefGoogle Scholar
Gouyet, J., Physics and Fractal Structures. Masson and Springer, Paris, 1996.Google Scholar
Lévy, P., Calcul des probabilités. Gauthier-Villars, Paris, 1925.Google Scholar
Mandelbrot, B., “Stable paretian random functions and the multiplicative variation of income,” Econometrica, vol. 29, no. 4, p. 517543, 1961.CrossRefGoogle Scholar
Bouchaud, J. and Potters, J., Theory of Financial Risk and Derivative Pricing, 2nd ed. Cambridge University Press, Cambridge 2003.CrossRefGoogle Scholar
Gnedenko, B. V. and Kolmogorov, A. N., Limit Distributions for Sums of Independent Random Variables. Addison-Wesley, Cambridge, 1954.Google Scholar
Montroll, E. W. and Weiss, G. H., “Random walks on lattices,” Journal of Mathematical Physics, vol. 6, pp. 167181, 1965.CrossRefGoogle Scholar
Scher, H. and Montroll, E. W., “Anomalous transit-time dispersion in amorphous solids,” Physical Review B, vol. 12, pp. 24552477, 1975.CrossRefGoogle Scholar
Klafter, J. and Sokolov, I. M., First Steps in Random Walks. Oxford University Press, Oxford, 2011.CrossRefGoogle Scholar
Barthelemy, P., Bertolotti, J., and Wiersma, D. S., “A lévy flight for light,” Nature, vol. 453, pp. 495498, 2008.CrossRefGoogle ScholarPubMed
Bertolotti, J., Light Transport beyond Diffusion. Ph.D. Thesis, University of Florence, 2007.Google Scholar
[389] Sgrignuoli, F. and Dal Negro, L., “Subdiffusive light transport in three-dimensional subrandom arrays,” Physical Review B, vol. 101, no. 214204, 2020.CrossRefGoogle Scholar
Chen, Y., Fiorentino, A., and Dal Negro, L., “A fractional diffusion random laser,” Scientific Reports, vol. 9, no. 1, pp. 114, 2019.Google ScholarPubMed
[391] Metzler, R. and Klafter, J., “The random walk’s guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, pp. 177, 2000. [Online]. Available: Scholar
[392] Gorenflo, R. and Mainardi, F., “Fractional calculus: integral and differential equations of fractional order,” arXiv preprint arXiv:0805.3823, 2008.Google Scholar
[393] Kwaśnicki, M., “Ten equivalent definitions of the fractional Laplace operator,” Fractional Calculus and Applied Analysis, vol. 20, no. 1, pp. 751, September 2017. [Online]. Available:; Scholar
[394] Lischke, A., Pang, G., Gulian, M., et al., “What is the fractional Laplacian?” November 2018. [Online]. Available:; Scholar
[395] Mainardi, F., Mura, A., Pagnini, G., and Gorenflo, R., “Sub-diffusion equations of fractional order and their fundamental solutions,” in Mathematical Methods in Engineering, K. Taḑ, J. A. Tenreiro Machado, and D. Baleanu, Eds. Springer, 2007, pp. 2355.CrossRefGoogle Scholar
[396] Mainardi, F. and Pagnini, G., “The Wright functions as solutions of the time-fractional diffusion equation,” Applied Mathematics and Computation, vol. 141, no. 1, pp. 5162, August 2003. [Online]. Available: Scholar
[397] Mainardi, F., Luchko, Y., and Pagnini, G., “The fundamental solution of the space-time fractional diffusion equation,” arXiv preprint cond-mat/0702419, 2007.Google Scholar
[398] Blumen, A., Zumofen, G., and Klafter, J., “Transport aspects in anomalous diffusion: Lévy walks,” Physical Review A, vol. 40, pp. 39643973, October 1989. [Online]. Available: ScholarPubMed
Chaves, A., “A fractional diffusion equation to describe Lévy flights,” Physics Letters A, vol. 239, no. 1, pp. 13 – 16, 1998. [Online]. Available: Scholar
[400] Mainardi, F., Mura, A., Pagnini, G., and Gorenflo, R., “Time-fractional diffusion of distributed order,” Journal of Vibration and Control, vol. 14, no. 9–10, pp. 12671290, 2008. [Online]. Available: Scholar
[401] Chechkin, A. V., Gorenflo, R., and Sokolov, I. M., “Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations,” Physical Review E, vol. 66, p. 046129, October 2002. [Online]. Available: ScholarPubMed
[402] Florescu, M., Torquato, S., and Steinhardt, P. J., “Designer disordered materials with large, complete photonic band gaps,” PNAS, vol. 106, no. 49, pp. 20 658–20 663, 2009.CrossRefGoogle ScholarPubMed
Batten, R. D., Stillinger, F. H., and Torquato, S., “Classical disordered ground states: super-ideal gasses and stealth and equi-luminous materials,” Journal of Applied Physics, vol. 104, p. 033504, 2008.CrossRefGoogle Scholar
Zachary, C. E. and Torquato, S., “Hyperuniformity in point patterns and two-phase random heterogeneous media,” Journal of Statistical Mechanics, p. P12015, 2009.CrossRefGoogle Scholar
[405] Jiao, Y., Lau, T., Hatzikirou, H., Meyer-Hermann, M., Corbo, J. C., and Torquato, S., “Avian photoreceptor patterns represent a disordered hyperuniform solution to a multiscale packing problem,” Physical Review E, vol. 89, p. 022721, February 2014. [Online]. Available: ScholarPubMed
Torquato, S., Zhang, G., and de Courcy-Ireland, M., “Uncovering multiscale order in the prime numbers via scattering,” Journal of Statistical Mechanics, no. 093401, pp. 115, 2018.Google Scholar
[407] Torquato, S., Zhang, G., and Courcy-Ireland, M. D., “Hidden multiscale order in the primes,” Journal of Physics A: Mathematical and Theoretical, vol. 52, no. 13, p. 135002, March 2019. [Online]. Available:–8121%2Fab0588CrossRefGoogle Scholar
Ma, Z. and Torquato, S., “Random scalar fields and hyperuniformity,” Journal of Applied Physics, vol. 121, no. 244904, pp. 115, 2017.CrossRefGoogle Scholar
Hardy, G. H. and Wright, E. M., An Introduction to the Theory of Numbers. Oxford University Press, New York, 2008.Google Scholar
Leseur, O., Pierrat, R., and Carminati, R., “High-density hyperuniform materials can be transparent,” Optica, vol. 3, no. 7, pp. 763767, 2016.CrossRefGoogle Scholar
Kendall, D. G., “On the number of lattice points inside a random oval,” Quarterly Journal of Mathematics, vol. 19, pp. 126, 1948.CrossRefGoogle Scholar
Kendall, D. G. and Rankin, R. A., “On the number of points of a given lattice in a random hypersphere,” Quarterly Journal of Mathematics, vol. 4, pp. 178189, 1953.CrossRefGoogle Scholar
Hardy, G. H., “On the expression of a number as the sum of two squares,” Quarterly Journal of Mathematics, vol. 46, pp. 263283, 1915.Google Scholar
Beck, J., “Irregularities of distribution i,” Acta Mathematica, vol. 159, pp. 149, 1987.CrossRefGoogle Scholar
Gabrielli, A., Joyce, M., and Torquato, S., “Tilings of space and superhomogeneous point processes,” Physical Review E, vol. 77, p. 031125, 2008.CrossRefGoogle ScholarPubMed
[416] Kim, J. and Torquato, S., “Methodology to construct large realizations of perfectly hyperuniform disordered packings,” Physical Review E, vol. 99, p. 052141, May 2019. [Online]. Available: ScholarPubMed
Froufe-Pérez, L. S., Engel, M., Saénz, J. J., and Scheffold, F., “Band gap formation and anderson localization in disordered photonic materials with structural correlations,” PNAS, vol. 114, no. 36, pp. 95709574, 2017.CrossRefGoogle ScholarPubMed
[418] Aubry, G. J., Froufe-Pérez, L. S., Kuhl, U., Legrand, O., Scheffold, F., and Mortessagne, F., “Experimental evidence for transparency, band gaps and anderson localization in two-dimensional hyperuniform disordered photonic materials,” arXiv preprint arXiv:2003.00913, 2020.Google Scholar
[419] Sgrignuoli, F., Torquato, S. and Dal Negro, L., “Localization in three-dimensional stealthy hyperuniform disordered systems,” arXiv:2109.03894, 2021.Google Scholar
Oğuz, E. C., Socolar, J. E. S., Steinhardt, P. J., and Torquato, S., “Hyperuniformity of quasicrystals,” Physical Review B, vol. 95, no. 054119, pp. 110, 2017.CrossRefGoogle Scholar
Oğuz, E. C., Socolar, J. E. S., Steinhardt, P. J., and Torquato, S., “Hyperuniformity and anti-heperuniformity in one-dimensional substitution tilings,” Acta Crystallographica, vol. A75, pp. 313, 2019.Google Scholar
Korobov, N. M., Exponential Sums and Their Applications. Springer, Dordrecht, 1992.CrossRefGoogle Scholar
Lemieux, C., Monte Carlo and Quasi-Monte Carlo Sampling. Springer, New York, 2009.Google Scholar
Weyl, H., “Über die gleichverteilung von zahlen mod. eins.” Mathematische Annalen, vol. 77, pp. 313352, 1916.CrossRefGoogle Scholar
Kuipers, L. and Niederreiter, H., Uniform Distribution of Sequences. John Wiley, New York, 1974.Google Scholar
Miller, S. J. and Takloo-Bighash, R., An Invitation to Modern Number Theory. Princeton University Press, Princeton, 2006.CrossRefGoogle Scholar
Niederreiter, H., “Low-discrepancy and low-dispersion sequences,” Journal of Number Theory, vol. 30, pp. 5170, 1988.CrossRefGoogle Scholar
Mckay, M., Beckman, R., and Conover, W., “A comparison of three methods for selecting values of input variables in the analysis of output from a computer code,” Technometrics, vol. 21, pp. 239245, 1979.Google Scholar
Bohigas, O., Haq, R. U., and Pandey, A., “Higher-order correlations in spectra of complex systems,” Physical Review Letters, vol. 54, no. 15, p. 1645, 1985.CrossRefGoogle ScholarPubMed
Mehta, M. L., Random Matrices. Elesvier, Amsterdam, 2004.Google Scholar
Gardner, M., “Mathematical games: the remarkable lore of the prime number,” Scientific American, vol. 210, pp. 120128, 1964.CrossRefGoogle Scholar
Hardy, G. H. and Littlewood, J. E., “Some problems of partitions numerorum III: on the expression of a number as a sum of primes,” Acta Mathematica, vol. 44, no. 1, pp. 170, 1923.CrossRefGoogle Scholar
Radin, C., Miles of Tiles. AMS, Providence, 1999.CrossRefGoogle Scholar
Radin, C., “The pinwheel tilings of the plane,” Annals of Mathematics, vol. 139, pp. 661702, 1994.CrossRefGoogle Scholar
Sgrignuoli, F. and Dal Negro, L., “Hyperuniformity and wave localization in pinwheel scattering,” Physics Review B, vol. 103, no. 22, 224202, 2021.CrossRefGoogle Scholar
Schwarz, W. and Spilker, J., Arithmetical Functions. An Introduction to Elementary and Anlytic Properties of Arithmetic Functions and to Some of Their Almost-Periodic Properties. Cambridge University Press, Cambridge, 1994.Google Scholar
Sander, J., Steuding, J., and Steuding, R., From Arithmetic to Zeta-Functions. Springer International Publishing AG, Switzerland, 2016.CrossRefGoogle Scholar
Ramanujan, S., “On certain trigonometrical sums and their applications in the theory of numbers,” Transactions of the Cambridge Philosophical Society, vol. 22, no. 13, pp. 259276, 1918.Google Scholar
Apostol, T. M., Introduction to Analytic Number Theory. Springer-Verlag, New York, 1976.Google Scholar
Wendt, H., Abry, P., and Jaffard, S., “Bootstrap for empirical multifractal analysis,” IEEE Signal Processing Magazine, vol. 1053, no. July, pp. 3848, 2007.CrossRefGoogle Scholar
Wendt, H. and Abry, P., “Multifractality tests using bootstrapped wavelet leaders,” IEEE Transactions on Signal Processing, vol. 55, no. 10, pp. 48114820, 2007.CrossRefGoogle Scholar
Mallat, S., A Wavelet Tour of Signal Processing, 3rd ed. Elsevier, 2009.Google Scholar
Falconer, K., Fractal Geometry: Mathematical Foundations and Applications, 3rded. John Wiley and Sons Ltd, Chichester, 2014.Google Scholar
Arneodo, A., Bacry, E., and Muzy, J. F., “The thermodynamics of fractals revisited with wavelets,” Physica A: Statistical Mechanics and Its Applications, vol. 213, pp. 232275, 1995.CrossRefGoogle Scholar
Arneodo, A., Grasseau, G., and Holschneider, M., “Wavelet transform of multifractals,” Physical Review Letters, vol. 61, pp. 22812284, 1988.CrossRefGoogle ScholarPubMed
Jaffard, S., “Wavelet techniques in multifractal analysis,” Proceedings of Symposia Pure Mathematics, Americal Mathematical Society, vol. 72, no. 2, pp. 91152, 2004.CrossRefGoogle Scholar
[447] Jaffard, S., Lashermes, B., and Abry, P., “Wavelet leaders in multifractal analysis,” in Wavelet Analysis and Applications, T. Quian, M. I. Vai, X. Yuesheng, Eds. Birkhäuser, pp. 219264, 2006.Google Scholar
Jaffard, S., “Multifractal formalism for functions. part 2: Self-similar functions,” SIAM Journal on Mathematical Analysis, vol. 28, no. 4, pp. 971998, 1997.CrossRefGoogle Scholar
[449] Tenenbaum, G., Introduction to Analytic and Probabilistic Number Theory, vol 46, Cambridge Studies in Advanced Mathematics. Cambridge University Press, 1995.Google Scholar
Mazur, B. and Stein, W., Prime Numbers and the Riemann Hypothesis. Cambridge University Press, New York, 2016.CrossRefGoogle Scholar
Schmidt, E., “Über die anzahl der primzahlen unter gegebener grenze,” Mathematische Annalen, vol. 57, pp. 195204, 1903.CrossRefGoogle Scholar
Hardy, G. H. and Littlewood, J. E., “Contributions to the theory of the riemann zeta-function and the theory of the distribution of primes,” Acta Mathematica, vol. 41, p. 119196, 1916.CrossRefGoogle Scholar
Schoenfeld, L., “Sharper bounds for the Chebyshev functions θ(x) and ψ(x). ii,” Mathematics of Computation, vol. 30, no. 134, pp. 337360, 1976.Google Scholar
Schumayer, D. and Hutchinson, D. A. W., “Colloquium: physics of the Riemann hypothesis,” Reviews of Modern Physics, vol. 83, no. 2, pp. 307330, 2011.CrossRefGoogle Scholar
Conrey, J. B., “The Riemann hypothesis,” Notices of the American Mathematical Society, vol. 50, no. 3, pp. 341353, 2003.Google Scholar
[456] Riemann, B., “Ueber die anzahl der primzahlen unter einer gegebenen grösse,” Monatsberichte der Berliner Akademie, pp. 1–9, 1859.Google Scholar
Tenenbaum, G. and France, M. M., The Prime Numbers and Their Distribution, vol.6. American Mathematical Society, 2000.CrossRefGoogle Scholar
Steuding, J., An Introdution to the Theory of L-Functions. A course given at the Autonoma University, Madrid, 2005.Google Scholar
Edwards, H. M., Riemann’s Zeta Function. Academic Press, New York and London, 1974.Google Scholar
Ivic, A., The Riemann Zeta Function: The Theory of the Riemann Zeta Function with Applications. John Wiley and Sons, New York, 1985.Google Scholar
Wu, H. and Sprung, D. W. L., “Riemann zeros and a fractal potential,” Physical Review E, vol. 48, no. 4, pp. 25952598, 1993.CrossRefGoogle Scholar
Schumayer, D., van Zyl, B. P., and Hutchinson, D. A. W., “Quantum mechanical potentials related to the prime numbers and Riemann zeros,” Physical Review E, vol. 78, no. 5, p. 056215, 2008.CrossRefGoogle Scholar
Bohr, H., “Über eine quasi-periodische eigenshaft dirichletscher reihen mit anwendung auf dirichletschen l-functione,” Mathematische Annalen, vol. 85, pp. 115122, 1922.CrossRefGoogle Scholar
Crandall, R. and Pomerance, C., Prime Numbers. A Computational Perspective, 2nd ed. Springer, New York, 2005.Google Scholar
Riesel, H. and Gohl, G., “Some calculations related to Riemann’s prime number formula,” Mathematics of Computations, vol. 24, no. 112, pp. 969983, 1970.Google Scholar
Bombieri, E., Problems of the Millennium: The Riemann Hypothesis. Clay Mathematics Institute, 2008.Google Scholar
Borwein, C. S. R. B. W. A., P., “Localization of waves,” in The Riemann Hypothesis. A Resource for the Afficionado and Virtuoso Alike, Borwein, C.S.R.B.W.A.,P.,Ed. Springer, New York, 2008, pp. 37.CrossRefGoogle Scholar
Odlyzko, A. M., “On the distribution of spacings between zeros of the zeta function,” Mathematics of Computation, vol. 48, no. 177, pp. 273308, 1987.CrossRefGoogle Scholar
[469] Goetschy, A. and Skipetrov, S. E., “Euclidean random matrices and their applications in physics,” arXiv:1303.2880, 2013.Google Scholar
Goetschy, A. and Skipetrov, S. E., “Non-Hermitian Euclidean random matrix theory,” Physical Review E, vol. 84, pp. 011 150–1–011 150–10, 2011.CrossRefGoogle ScholarPubMed
Mitchell, G. E., Richter, A., and Weidenmüller, H. A., “Random matrices and chaos in nuclear physics: nuclear reactions,” Reviews of Modern Physics, vol. 82, no. 4, pp. 28452901, 2010.CrossRefGoogle Scholar
[472] Bourgade, P. and Keating, J. P., “Quantum chaos, random matrix theory, and the Riemann ζ-function,” Séminaire Poincaré XIV, pp. 115–153, 2010.Google Scholar
Beenakker, C. W. J., “Random-matrix theory of quantum transport,” Reviews of Modern Physics, vol. 69, no. 3, pp. 731808, 1997.CrossRefGoogle Scholar
Guhr, T., Müller-Groeling, A., and Weidenmüller, H. A., “Random-matrix theories in quantum physics: common concepts,” Physics Reports, vol. 299, pp. 189425, 1998.CrossRefGoogle Scholar
Mirlin, A. D., “Statistics of energy levels and eigenfunctions in disordered systems,” Physics Reports, vol. 326, pp. 259382, 2000.CrossRefGoogle Scholar
[476] Skipetrov, S. E. and Sokolov, M. E., “Absence of Anderson localization of light in a random ensemble of point scatterers,” Physical Review Letters, vol. 112, pp. 023 905–1– 023 905–5, 2013.Google Scholar
[477] Skipetrov, S. E. and Goetschy, A., “Eigenvalue distributions of large Euclidean random matrices for waves in random media,” Journal of Physics A: Mathematical and Theoretical, vol. 44, pp. 065 102–065 127, 2011.CrossRefGoogle Scholar
[478] Goetschy, A. and Skipetrov, S. E., “Euclidean matrix theory of random lasing in a cloud of cold atoms,” European Physics Letters, vol. 96, pp. 34 005–p1–34 005–p6, 2011.CrossRefGoogle Scholar
[479] Timberlake, T. K. and Tucker, J. M., “Is there quantum chaos in the prime numbers?” arXiv:quant-ph/0708.2567, 2007.Google Scholar
Berry, M. V. and Robnik, M., “Semiclassical level spacings when regular and chaotic orbits coexist,” Journal of Physics A, vol. 17, no. 12, p. 2413, 1984.CrossRefGoogle Scholar
Steuding, J., Value-Distribution of L-Functions. Springer-Verlag, Berlin and Heidelberg, 2007.Google Scholar
Ireland, K. and Rosen, M., A Classical Introduction to Modern Number Theory, 2nd ed. Springer-Verlag, New York, 1990.CrossRefGoogle Scholar
Everest, G. and Ward, T., An Introduction to Number Theory. Springer-Verlag, London 2005.Google Scholar
Littlewood, J. E., “Distribution des nombres premiers,” Comptes rendus de l’Académie des Sciences, vol. 158, pp. 18691872, 1914.Google Scholar
Granville, A. and Martin, G., “Prime number races,” American Mathematical Monthly, vol. 113, no. 1, pp. 133, 2006.CrossRefGoogle Scholar
Rubinstein, M. and Sarnak, P., “Chebyshev’s bias,” Experimental Mathematics, vol.3, no. 3, pp. 173197, 1994.CrossRefGoogle Scholar
Lemke Oliver, R. J. and Soundararajan, K., “Unexpected biases in the distribution of consecutive primes,” PNAS, vol. 113, no. 31, pp. E4446–E4454, 2016.CrossRefGoogle ScholarPubMed
[488] Tao, T., “The dichotomy between structure and randomness, arithmetic progressions, and the primes,” arXiv:math/0512114v2, 2005.Google Scholar
Szemerédi, E., “On sets of integers containing no k elements in arithmetic progression,” Acta Arithmetica, vol. 27, pp. 299345, 1975.CrossRefGoogle Scholar
Tao, T., “The Gaussian primes contain arbitrarily shaped constellations,” Journal d’Analyse Mathématique, vol. 99, pp. 109176, 2006.CrossRefGoogle Scholar
Zhang, Y., “Bounded gaps between primes,” Annals of Mathematics, vol. 179, no. 3, pp. 11211174, 2014.CrossRefGoogle Scholar
[492] Zhang, G., Martelli, F., and Torquato, S., “The structure factor of primes,” Journal of Physics A: Mathematical and Theoretical, vol. 51, no. 11, p. 115001, February 2018. [Online]. Available:–8121%2Faaa52aCrossRefGoogle Scholar
Gallagher, P. X., “On the distribution of primes in short intervals,” Mathematika, vol. 23, no. 1, p. 49, 1976.CrossRefGoogle Scholar
Torquato, S., Scardicchio, A., and Zachary, C. E., “Point processes in arbitrary dimension from fermionic gases, random matrix theory, and number theory,” Journal of Statistical Mechanics, vol. 2008, no. 11, p. P11019, 2008.CrossRefGoogle Scholar
[495] Wolf, M., “Multifractality of prime numbers,” Physica A: Statistical Mechanics and Its Applications, vol. 160, no. 1, pp. 2442, 1989. [Online]. Available: www.sciencedirect .com/science/article/pii/0378437189904615CrossRefGoogle Scholar
[496] Wolf, M., “1/f noise in the distribution of prime numbers,” Physica A: Statistical Mechanics and Its Applications, vol. 241, no. 3, pp. 493499, 1997. [Online]. Available: Scholar
[497] Bak, P., Tang, C., and Wiesenfeld, K., “Self-organized criticality: an explanation of the 1/f noise,” Physical Review Letters, vol. 59, pp. 381384, July 1987. [Online]. Available: ScholarPubMed
[498] Ares, S. and Castro, M., “Hidden structure in the randomness of the prime number sequence?” Physica A: Statistical Mechanics and Its Applications, vol. 360, no. 2, pp. 285296, 2006. [Online]. Available: Scholar
Silverman, J. H., Friendly, A Introduction to Number Theory, 4th ed. Pearson Education Inc., 2013.Google Scholar
Lange, L. J., “An elegant continued fraction for π,” American Mathematical Monthly, vol. 106, no. 5, pp. 456458, 1999.Google Scholar
Cusick, T. W. and Flahive, M. E., The Markov and Lagrange Spectra. Mathematical Surveys and Monographs no. 30, American Mathematical Society, Providence, 1994.Google Scholar
Moreira, C. G., “Geometric properties of the Markov and Lagrange spectra,” Annals of Mathematics, vol. 188, no. 1, pp. 145170, 2018.CrossRefGoogle Scholar
Bohr, H., Almost Periodic Functions. Julius Springer, Berlin, 1933.Google Scholar
Bohr, H., “Zur theorie der fastperiodischen funktionen i,” Acta Mathematica, vol. 45, p. 29127, 1925.CrossRefGoogle Scholar
Besicovitch, A. S., Almost Periodic Functions. Dover Publications, New York, 1954.Google Scholar
Cooke, R. L., “Almost-periodic functions,” American Mathematical Monthly, vol. 88, no. 7, pp. 515526, 1981.CrossRefGoogle Scholar
Corduneanu, C., Almost Periodic Oscillations and Waves. Springer, Berlin and Heidelberg, 2008.Google Scholar
Berry, M. V. and Tabor, M., “Level clustering in the regular spectrum,” Proceedings of the Royal Society of London. Series A, vol. A356, pp. 375394, 1977.Google Scholar
Rudnick, Z., Sarnak, P., and Zaharescu, A., “The distribution of spacings between the fractional parts of n2α,” Inventiones Mathematicae, vol. 145, pp. 3757, 2001.CrossRefGoogle Scholar
Rudnick, Z. and Sarnak, P., “The pair correlation function of fractional parts of polynomials,” Communications in Mathematical Physics, vol. 194, pp. 6170, 1998.CrossRefGoogle Scholar
Beltrami, E., What Is Random? Copernicus, Springer-Verlag, New York, 1999.CrossRefGoogle Scholar
Bartlett, M. S., “Chance or chaos?Journal of the Royal Statistical Society, vol. 153, no. 3, pp. 321347, 1990.CrossRefGoogle Scholar
Rudnick, Z. and Zaharescu, A., “The distribution of spacings between fractional parts of lacunary sequences,” Forum Mathematicum, vol. 14, no. 5, pp. 691712, 2002.CrossRefGoogle Scholar
Rudnick, Z. and Zaharescu, A., “A metric result on the pair correlation of fractional parts of sequences,” Acta Arithmetica, vol. 89, no. 3, pp. 283293, 1999.CrossRefGoogle Scholar
Cohen, H., A Classical Invitation to Algebraic Numbers and Class Fields. Springer-Verlag, New York, 1978.CrossRefGoogle Scholar
Goldman, J. R., The Queen of Mathematics. A Historically Motivated Guide to Number Theory. A. K. Peters Ltd., Natick, 2004.Google Scholar
Stewart, I. and Tall, D., Algebraic Number Theory and Fermat’s Last Theorem, 3rded. A. K. Peters Ltd., Canada, 2002.Google Scholar
Gethner, E., Wagon, S., and Wick, B., “A stroll through the Gaussian primes,” American Mathematical Monthly, vol. 105, pp. 327333, 1998.CrossRefGoogle Scholar
Tsuchimura, N., “Computational results for Gaussian moat problem,” IEICE Transactions, vol. 88-A, pp. 12671273, 2005.Google Scholar
Vardi, I., “Prime percolation,” Experimental Mathematics, vol. 7, no. 3, pp. 275289, 1998.CrossRefGoogle Scholar
West, P. P. and Sittinger, B. D., “A further stroll into the Eisenstein primes,” American Mathematical Monthly, vol. 124, no. 7, pp. 609620, 2017.CrossRefGoogle Scholar
Bressoud, D. and Wagon, S., A Course in Computational Number Theory. John Wiley and Sons, Hoboken, 2000.Google Scholar
Renze, J., Wagon, S., and Wick, B., “The Gaussian zoo,” Experimental Mathematics, vol. 10, no. 2, pp. 161173, 2001.CrossRefGoogle Scholar
Weil, A., “On some exponential sums,” PNAS, vol. 34, no. 5, pp. 204207, 1948.CrossRefGoogle ScholarPubMed
Niederreiter, H. and Rivat, J., “On the correlation of pseudorandom numbers generated by inversive methods,” Monatshefte für Mathematik, vol. 153, pp. 251264, 2008.CrossRefGoogle Scholar
Trappe, W. and Washington, L. C., Introduction to Cryptography with Coding Theory, 2nd ed. Pearson Prentice Hall, Upper Saddle River, 2006.Google Scholar
Meijer, A. R., Algebra for Cryptologists. Springer International Publishing, Switzerland, 2016.CrossRefGoogle Scholar
Calinger, R. S., Leonhard Euler: Mathematical Genius in the Enlightenment. Princeton and Oxford University Press, Princeton, 2016.Google Scholar
Cobeli, C. and Zaharescu, A., “On the distribution of primitive roots mod p,” Acta Arithmetica, vol. 83, no. 2, pp. 143153, 1998.CrossRefGoogle Scholar
Rudnick, Z. and Zaharescu, A., “The distribution of spacings between small powers of a primitive root,” Israel Journal of Mathematics, vol. 120, pp. 271287, 2000.CrossRefGoogle Scholar
Hoffstein, J., Pipher, J., and Silverman, J. H., An Introduction to Mathematical Cryptography. Springer, New York, 2008.Google Scholar
Lee, S. Y., Walsh, G. F., and Dal Negro, L., “Microfluidics integration of aperiodic plasmonic arrays for spatial-spectral optical detection,” Optics Express, vol. 21, no. 4, pp. 49454957, 2013.CrossRefGoogle ScholarPubMed
Silverman, J. H., The Arithmetic of Elliptic Curves. Springer Science & Business Media Berlin and New York, 2009.CrossRefGoogle Scholar
Washington, L. C., Elliptic Curves Number Theory and Cryptography. Chapman and Hall/CRC, Boca Raton, 2008.Google Scholar
[535] Clay Mathematics, Millennium Problems. Available online: Scholar
Taylor, R., “Automorphy for some -adic lifts of automorphic mod galois representations. ii.” Publications Mathematiques de l’Institut des Hautes Etudes Scientifiques, vol. 108, pp. 183239, 2008.CrossRefGoogle Scholar
Blum, M. and Micali, S., “How to generate cryptographically strong sequences of pseudorandom bits,” SIAM Journal on Computing, vol. 13, no. 4, pp. 850864, 1984.CrossRefGoogle Scholar
Blum, L., Blum, M., and Shub, M., “A simple unpredictable pseudo-random number generator,” SIAM Journal on Computing, vol. 15, no. 2, pp. 364383, 1986.CrossRefGoogle Scholar
Marsaglia, G., “Random numbers fall mainly in the planes,” PNAS, vol. 61, no. 1, pp. 2528, 1968.CrossRefGoogle ScholarPubMed
[540] Fenimore, E. E. and Cannon, T. M., “Coded aperture imaging with uniformly redundant arrays,” Applied Optics, vol. 17, no. 3, pp. 337347, February 1978. [Online]. Available: ScholarPubMed
[541] Gottesman, S. R. and Fenimore, E. E., “New family of binary arrays for coded aperture imaging,” Applied Optics, vol. 28, no. 20, pp. 43444352, October 1989. [Online]. Available: ScholarPubMed
Cieślak, M. J., Gamage, K. A. A., and Glover, R.,”Coded-apertureimagingsystems: past, present and future development past, present and future development – a review.” Radiation Measurements, vol. 92, pp. 5971, 2016.CrossRefGoogle Scholar
Chaitin, G. J., Algorithmic Information Theory. Cambridge University Press, 1987.CrossRefGoogle Scholar
[544] Chaitin, G. J., Thinking about Gödel and Turing. Essays on Complexity, 1970–2007. World Scientific, Singapore, 2007.CrossRefGoogle Scholar
Chaitin, G. J., The Limits of Mathematics. Springer-Verlag, Singapore, 1998.Google Scholar
Chaitin, G. J., “Information-theoretic limitations of formal systems,” Journal of the ACM, vol. 21, no. 3, pp. 403434, 1974.CrossRefGoogle Scholar
[547] Nagel, E. and N. J. R., Gödel’s Proof. Revised Edition. New York University Press, New York, 2001.Google Scholar
Smullyan, R. M., A Beginner’s Guide to Mathematical Logic. Dover Publications, Mineola, 2014.Google Scholar
Shannon, C. E., “A mathematical theory of communication,” Bell System Technical Journal, vol. 27, no. 3, pp. 379423, 1948.CrossRefGoogle Scholar
Pincus, S. M., “Approximate entropy as a measure of system complexity,” PNAS, vol. 88, pp. 22972301, 1991.CrossRefGoogle ScholarPubMed
Costa, M., Goldberger, A. L., and Peng, C. K., “Multiscale entropy analysis of complex physiologic time series,” Physical Review Letters, vol. 89, no. 6, p. 068102, 2002.CrossRefGoogle ScholarPubMed
[552] Costa, M., Goldberger, A. L., and Peng, C. K., “Multiscale entropy analysis of biological signals,” Physical Review E, vol. 71, no. 021906, 2005.CrossRefGoogle ScholarPubMed
Keeping, E. S., Introduction to Statistical Inference. Dover Publications Inc., New York, 1995.Google Scholar
Ramsey, F. P., “On a problem of formal logic,” Proceedings of the London Mathematical Society, vol. s2–30, pp. 264286, 1930.CrossRefGoogle Scholar
Micciancio, D. and Goldwasser, S., Complexity of Lattice Problems: A Cryptographic Perspective. Kluwer Academic Publishers, Dordrecht, 2002.CrossRefGoogle Scholar
[556] Zamir, R., Lattice Coding for Signals and Networks. Cambridge Univerity Press, Cambridge, 20014.Google Scholar
Pickover, C. A., The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling, New York, 2009.Google Scholar
Dunbabin, K. M., Mosaics of the Greek and Roman World. Cambridge Univeristy Press, Cambridge, 1999.Google Scholar
Broug, E., Islamic Geometric Patterns. Thames and Hudson, London, 2008.Google Scholar
[560] Kepler, J., Harmonice Mundi, Book II. Lincii, 1619.Google Scholar
[561] Fyodorov, E. S., “Simmetrija na ploskosti [symmetry in the plane],” Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo Obshchestva [Proceedings of the Imperial St. Petersburg Mineralogical Society], vol. 28, pp. 245291, 1891.Google Scholar
Grünbaum, B. and Shephard, G. C., Tilings and Patterns, 2nd ed. Dover, New York, 2016.Google Scholar
Coxeter, H. S. M., Regular Polytopes. Dover Publications Inc., New York, 1973.Google Scholar
Minkowski, H., Diophantische Approximationen. Druck und Verlag Von B. G. Teubner, Leipzig, 1907.CrossRefGoogle Scholar
Fedorov, E. S., Das Kristallreich: Tabellen zur Kristallochemischen Analyse. Academy of Sciences, St. Petersburg, 1920.Google Scholar
Schoenflies, A., Kristallsystem und Kristallstruktur. Teubner, 1891.Google Scholar
Barlow, W., “Über die geometrischen eigenschaften homogener starrer strukturen und ihre anwendung auf krystalle, [on the geometrical properties of homogeneous rigid structures and their application to crystals],” Zeitschrift für Krystallographie und Mineralogie, vol. 23, pp. 163, 1894.Google Scholar
Hiller, H., “The crystallographic restriction in higher dimensions,” Acta Crystallographica, vol. A41, pp. 541544, 1985.CrossRefGoogle Scholar
de Bruijn, N. G., “Algebraic theory of Penrose’s non-periodic tilings of the plane,” Indagationes Mathematicae, Proceedings of the Koninklijke Nederlandse Akademie van Wetenshappen Series, vol. A84, no. 1, pp. 3866, 1981.Google Scholar
Moody, R. V. and Patera, J., “Quasicrystals and icosians,” Journal of Physics A: Mathematical and General, vol. 26, pp. 28292853, 1993.CrossRefGoogle Scholar
van Smaalen, S., Incommensurate Crystallography. Oxford University Press, Oxford, 2007.CrossRefGoogle Scholar
Berger, R., “The undecidability of the domino problem,” Memoirs American Mathematical Society, vol. 66, pp. 172, 1966.Google Scholar
[573] Jeandel, E. and Rao, M., “An aperiodic set of 11 Wang tiles,” eprint arXiv:1506.06492, pp. 1–40, 2015.Google Scholar
Penrose, R., “Pentaplexity,” Bulletin of the Institute for Mathematics and Applications, vol. 10, pp. 266271, 1974.Google Scholar
de Wolff, P. M. and van Aalst, W., “The four-dimensional space group of γ − na2co3,” Acta Crystallographica A, vol. 28, p. 111, 1972.Google Scholar
de Wolff, P. M., “The pseudo-symmetry of modulated crystal structures,” Acta Crystallographica A, vol. 30, pp. 777785, 1974.CrossRefGoogle Scholar
Bieberbach, L., “Über die bewegungsgruppen der n-dimensional en euklidischen räume mit einem endlichen fundamental bereich,” Matematische Annallen, vol. 72, pp. 400412, 1912.CrossRefGoogle Scholar
Asher, E. and Janner, A., “Algebraic aspects of crystallography I: space groups as extensions,” Helvetica Physica Acta, vol. 38, pp. 551572, 1965.Google Scholar
Asher, E. and Janner, A., “Algebraic aspects of crystallography II: non-primitive translations in space groups,” Communications in Mathematical Physics, vol. 11, pp. 138167, 1968.Google Scholar
Fast, G. and Janssen, T., “Determination of n-dimensional space groups by means of an electronic computer,” Journal of Computational Physics, vol. 7, pp. 111, 1971.CrossRefGoogle Scholar
Icru report of the executive committee,” Acta Crystallographica A, vol. 48, p. 922, 1992.Google Scholar
Thompson, D. A. W., On Growth and Form. Dover, New York, 1992.CrossRefGoogle Scholar
Prusinkiewicz, P. and Lindenmayer, A., The Algorithmic Beauty of Plants. Springer-Verlag, New York, 1990.CrossRefGoogle Scholar
Ball, P., Nature’s Patterns. Oxford University Press, New York, 2009.Google Scholar
Mitchison, G. J., “Phyllotaxis and the Fibonacci series,” Science, vol. 196, pp. 270275, 1977.CrossRefGoogle ScholarPubMed
Adam, J. A., A Mathematical Nature Walk. Princeton University Press, Princeton, 2009.CrossRefGoogle Scholar
Jean, R. V., Phyllotaxis. Cambridge University Press, New York, 1995.Google ScholarPubMed
[588] Bonnet, C., Recherches sur l’usage des feuilles dans les plantes. E. Luzac, fils., Göttingen and Leyden, 1754.Google Scholar
Adler, I., Barabe, D., and Jean, R. V., “A history of the study of phillotaxis,” Annals of Bothany, vol. 80, pp. 231244, 1997.CrossRefGoogle Scholar
Turing, A. M., “The chemical basis of morphogenesis,” Philosophical Transactions of the Royal Society London, vol. 237B, pp. 3752, 1952.Google Scholar
Adler, I., “A model of contact pressure in phyllotaxis,” Journal of Theoretical Biology, vol. 45, pp. 179, 1974.CrossRefGoogle Scholar
√ [592] Naylor, M., “Golden, 2, and π flowers: a spiral story,” Mathematics Magazine, vol. 75, pp. 163172, 2002.Google Scholar
[593] Dal Negro, L., Lawrence, N., and Trevino, J., “Analytical light scattering and orbital angular momentum spectra of arbitrary Vogel spirals,” Optics Express, vol. 20, pp. 18 209–18 223, 2012.CrossRefGoogle ScholarPubMed
Simon, D. S., Lawrence, N., Trevino, J., Dal Negro, L., and Sergienko, A. V., “High Capacity quantum Fibonacci coding for key distribution,” Physical Review A, vol. 87, p. 032312, 2013.CrossRefGoogle Scholar
Stanley, H. E., “Multifractal phenomena in physics and chemistry (review),” Nature, vol. 335, pp. 405409, 1988.CrossRefGoogle Scholar
Mandelbrot, B. B., “An Introduction to multifractal distribution functions,” in Fluctuations and Pattern Formation, Stanley, H. E. and Ostrowsky, N., Ed. Kluwer, Dordrecht and Boston, 1988, 345360.Google Scholar
Frisch, U. and Parisi, G., “Fully developed turbulence and intermittency.” New York Academy of Sciences, Annals, vol. 357, 359367, 1980.CrossRefGoogle Scholar
Trevino, J., Liew, S. F., Noh, H., Cao, H., and Dal Negro, L., “Geometrical structure, multifractal spectra and localized optical modes of aperiodic Vogel spirals,” Optics Express, vol. 20, pp. 30153033, 2012.CrossRefGoogle ScholarPubMed
Dal Negro, L., Wang, R., and Pinheiro, F. A., “Structural and spectral properties of deterministic aperiodic optical structures,” Crystals, vol. 6, pp. 161195, 2016.CrossRefGoogle Scholar
Halsey, T., Jensen, M. H., Kadanoff, L. P., Procaccia, I., and Shraiman, B. I., “Fractal measures and their singularities: the characterization of strange sets,” Physical Review A., vol. 33, pp. 11411151, 1986.CrossRefGoogle ScholarPubMed
Chhabra, A. and Jensen, R. V., “Direct determination of the f (α) singularity spectrum,” Physical Review Letters, vol. 62, pp. 13271330, 1989.CrossRefGoogle Scholar
Pollard, M. E. and Parker, G. J., “Low-contrast bandgaps of a planar parabolic spiral lattice,” Optics Letters, vol. 34, pp. 28052807, 2009.CrossRefGoogle ScholarPubMed
Hof, A., “On diffraction by aperiodic structures,” Communications in Mathematical Physics, vol. 169, pp. 2543, 1995.CrossRefGoogle Scholar
Baake, M. and Grimm, U., “Mathematical diffraction of aperiodic structures,” Chemical Society Reviews, vol. 41, pp. 68216843, 2012.CrossRefGoogle ScholarPubMed
Baake, M. and Grimm, U., “Kinematic diffraction from a mathematical viewpoint,” Zeitschrift für Kristallographie, vol. 226, pp. 711725, 2011.CrossRefGoogle Scholar
Janner, A. and Janssen, T., “Symmetry of periodically distorted crystals,” Physical Review B, vol. 15, pp. 643658, 1977.CrossRefGoogle Scholar
Bombieri, E. and Taylor, J. E., “Quasicrystals, tilings, and algebraic number theory: some preliminary connections,” Contemporary Mathematics, vol. 64, pp. 241264, 1987.CrossRefGoogle Scholar
Moody, R. V., “Model sets: a survey”. In From Quasicrystals to More Complex Systems. Centre de Physique des Houches, vol 13, F. Axel, F. Dénoyer, J. P. Gazeau, Ed. Springer, Berlin, Heidelberg, 2000.Google Scholar
Baake, M. and Moody, R. V., “Weighted Dirac combs with pure point diffraction,” Journal für die reine und angewandte Mathematik (Crelle), vol. 573, pp. 6194, 2004.Google Scholar
Baake, M. and Grimm, U., “Diffraction of limit periodic point sets,” Philosophical Magazine, vol. 91, pp. 26612670, 2011.CrossRef