×

Making light of mathematics. (English) Zbl 1025.78001

From the text: Summary of AMS Gibbs Lecture, delivered at San Diego, CA, January 6, 2002. The author describes some mathematical phenomena and their optical counterparts – turning mathematics into light, as it were. The lecture was based almost entirely on images – photographic, computer-simulated and demonstrated live – and is impossible to reproduce in print. Here a summary with references is given, organised by the mathematical phenomena (stable singularities of gradient maps, divergent series, …).

MSC:

78-02 Research exposition (monographs, survey articles) pertaining to optics and electromagnetic theory
28A80 Fractals
78A45 Diffraction, scattering
78A97 Mathematically heuristic optics and electromagnetic theory (must also be assigned at least one other classification number in Section 78-XX)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)

Software:

DLMF
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] G. B. Airy, On the intensity of light in the neighbourhood of a caustic, Trans. Camb. Phil. Soc. 6 (1838), 379-403.
[2] V. I. Arnol\(^{\prime}\)d, Catastrophe theory, 2nd ed., Springer-Verlag, Berlin, 1986. Translated from the Russian by G. S. Wassermann; Based on a translation by R. K. Thomas.
[3] V. I. Arnol\(^{\prime}\)d, Critical points of smooth functions, and their normal forms, Uspehi Mat. Nauk 30 (1975), no. 5(185), 3 – 65 (Russian).
[4] V. I. Arnol\(^{\prime}\)d, Normal forms of functions near degenerate critical points, the Weyl groups \?_{\?},\?_{\?},\?_{\?} and Lagrangian singularities, Funkcional. Anal. i Priložen. 6 (1972), no. 4, 3 – 25 (Russian).
[5] V. I. Arnol\(^{\prime}\)d, Normal forms of functions in the neighborhood of degenerate critical points, Uspehi Mat. Nauk 29 (1974), no. 2(176), 11 – 49 (Russian). Collection of articles dedicated to the memory of Ivan Georgievič Petrovskiĭ (1901 – 1973), I.
[6] I. S. Averbukh and N. F. Perelman, Fractional quantum revivals: Universality in the long-term evolution of quantum wave packets beyond the correspondence principle dynamics, Phys. Lett. A. 139 (1989), 443-453.
[7] J. Barrow, The Book of Nothing, Jonathan Cape, London, 2000.
[8] M. V. Berry, Exuberant interference: Rainbows, tides, edges, (de)coherence, Phil. Trans. Roy. Soc. Lond. A 360 (2002), 1023-1037. · Zbl 1254.78003
[9] M. V. Berry, Focusing and twinkling: critical exponents from catastrophes in non-Gaussian random short waves, J. Phys. A 10 (1977), no. 12, 2061 – 2081. · Zbl 0382.58008
[10] M. V. Berry, Fractal modes of unstable lasers with polygonal and circular mirrors, Optics Commun. 200 (2001), 321-330.
[11] Michael Berry, Knotted zeros in the quantum states of hydrogen, Found. Phys. 31 (2001), no. 4, 659 – 667. Invited papers dedicated to Martin C. Gutzwiller, Part V. · doi:10.1023/A:1017521126923
[12] M. V. Berry, Mode degeneracies and the Petermann excess-noise factor for unstable lasers, J. Mod. Opt. (2003), in press. · Zbl 1026.78016
[13] M. V. Berry, Much ado about nothing: Optical dislocation lines (phase singularities, zeros, vortices...), M. S. Soskin, ed., Singular Optics, Proceedings of SPIE, 3487, 1998, pp. 1-5.
[14] M. V. Berry, Natural focusing, R. Gregory, J. Harris, P. Heard and D. Rose, eds., The Artful Eye, Oxford University Press, 1995, pp. 311-323.
[15] M. V. Berry, Pancharatnam, virtuoso of the Poincaré sphere: An appreciation, Current Science 67 (1994), 220-223.
[16] M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. Roy. Soc. London Ser. A 392 (1984), no. 1802, 45 – 57. · Zbl 1113.81306
[17] M. V. Berry, Quantum fractals in boxes, J. Phys. A 29 (1996), no. 20, 6617 – 6629. · Zbl 0910.60087 · doi:10.1088/0305-4470/29/20/016
[18] M. V. Berry, Singularities in waves and rays, R. Balian, M. Kléman and J.-P. Poirier, eds., Les Houches Lecture Series Session 35, North-Holland: Amsterdam, 1981, pp. 453-543.
[19] M. V. Berry, Stokes’ phenomenon; smoothing a Victorian discontinuity, Inst. Hautes Études Sci. Publ. Math. 68 (1988), 211 – 221 (1989). · Zbl 0701.58012
[20] M. V. Berry, Uniform asymptotic smoothing of Stokes’s discontinuities, Proc. Roy. Soc. London Ser. A 422 (1989), no. 1862, 7 – 21. · Zbl 0683.33004
[21] M. V. Berry, Why are special functions special?, Physics Today (April 2001), 11-12.
[22] M. V. Berry, R. Bhandari and S. Klein, Black plastic sandwiches demonstrating biaxial optical anisotropy, Eur. J. Phys. (1999), 1-14. · Zbl 1078.78500
[23] M. V. Berry and M. R. Dennis, Knotted and linked phase singularities in monochromatic waves, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 457 (2001), no. 2013, 2251 – 2263. · Zbl 1013.35016 · doi:10.1098/rspa.2001.0826
[24] M. V. Berry and M. R. Dennis, Knotting and unknotting of phase singularities: Helmholtz waves, paraxial waves and waves in 2+1 spacetime, J. Phys. A 34 (2001), no. 42, 8877 – 8888. · Zbl 0997.35097 · doi:10.1088/0305-4470/34/42/311
[25] M. V. Berry and M. R. Dennis, Phase singularities in isotropic random waves, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456 (2000), no. 2001, 2059 – 2079. · Zbl 0978.35072 · doi:10.1098/rspa.2000.0602
[26] M. V. Berry and J. Goldberg, Renormalisation of curlicues, Nonlinearity 1 (1988), no. 1, 1 – 26. · Zbl 0662.10029
[27] M. V. Berry and S. Klein, Integer, fractional and fractal Talbot effects, J. Modern Opt. 43 (1996), no. 10, 2139 – 2164. · Zbl 0941.78524 · doi:10.1080/095003496154761
[28] M. V. Berry and Z. V. Lewis, On the Weierstrass-Mandelbrot fractal function, Proc. Roy. Soc. London Ser. A 370 (1980), no. 1743, 459 – 484. · Zbl 0435.28008 · doi:10.1098/rspa.1980.0044
[29] M. V. Berry, I. Marzoli and W. P. Schleich, Quantum carpets, carpets of light, Physics World 14 (6) (2001), 39-44.
[30] M. V. Berry, J. F. Nye and F. J. Wright, The elliptic umbilic diffraction catastrophe, Phil. Trans. Roy. Soc. A291 (1979), 453-484.
[31] M. V. Berry and D. H. J. O’Dell, Diffraction by volume gratings with imaginary potentials, J. Phys. A 31 (1998), no. 8, 2093 – 2101. · Zbl 0905.35089 · doi:10.1088/0305-4470/31/8/019
[32] M. V. Berry, C. Storm and W. van Saarloos, Theory of unstable laser modes: Edge waves and fractality, Optics Commun. 197 (2001), 393-402.
[33] M. V. Berry and C. Upstill, Catastrophe optics: Morphologies of caustics and their diffraction patterns, Progress in Optics 18 (1980), 257-346.
[34] M. V. Berry and M. Wilkinson, Diabolical points in the spectra of triangles, Proc. Roy. Soc. London Ser. A 392 (1984), no. 1802, 15 – 43. · Zbl 1113.81307
[35] Max Born and Emil Wolf, Principles of optics: Electromagnetic theory of propagation, interference and diffraction of light, Pergamon Press, New York-London-Paris-Los Angeles, 1959. · Zbl 0086.41704
[36] D. Braun, F. Haake and W. Strunz, Universality of decoherence, Phys. Rev. Lett. 86 (2001), 2193-2197.
[37] J. Courtial and M. J. Padgett, Monitor-outside-a-monitor effect and self-similar fractal structure in the eigenmodes of unstable optical resonators, Phys. Rev. Lett. 85 (2000), 5320-5323.
[38] R. B. Dingle, Asymptotic expansions: their derivation and interpretation, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], London-New York, 1973. · Zbl 0279.41030
[39] DLMF, Digital Library of Mathematical Functions (2002), http://dlmf.nist.gov.
[40] J. Écalle, Cinq applications des fonctions résurgentes, 1984. Preprint 84T62 (Orsay).
[41] Jean Écalle, Les fonctions résurgentes. Tome I, Publications Mathématiques d’Orsay 81 [Mathematical Publications of Orsay 81], vol. 5, Université de Paris-Sud, Département de Mathématique, Orsay, 1981 (French). Les algèbres de fonctions résurgentes. [The algebras of resurgent functions]; With an English foreword. Jean Écalle, Les fonctions résurgentes. Tome II, Publications Mathématiques d’Orsay 81 [Mathematical Publications of Orsay 81], vol. 6, Université de Paris-Sud, Département de Mathématique, Orsay, 1981 (French). Les fonctions résurgentes appliquées à l’itération. [Resurgent functions applied to iteration]. Jean Écalle, Les fonctions résurgentes. Tome I, Publications Mathématiques d’Orsay 81 [Mathematical Publications of Orsay 81], vol. 5, Université de Paris-Sud, Département de Mathématique, Orsay, 1981 (French). Les algèbres de fonctions résurgentes. [The algebras of resurgent functions]; With an English foreword. Jean Écalle, Les fonctions résurgentes. Tome II, Publications Mathématiques d’Orsay 81 [Mathematical Publications of Orsay 81], vol. 6, Université de Paris-Sud, Département de Mathématique, Orsay, 1981 (French). Les fonctions résurgentes appliquées à l’itération. [Resurgent functions applied to iteration]. Jean Écalle, Les fonctions résurgentes. Tome III, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 85, Université de Paris-Sud, Département de Mathématiques, Orsay, 1985 (French). L’équation du pont et la classification analytique des objects locaux. [The bridge equation and analytic classification of local objects].
[42] J. W. Gibbs, On double refraction in perfectly transparent media which exhibit the phenomenon of circular polarization, Amer. J. Sci. (ser. 3) 23 (1882), 460-476.
[43] G. H. Hardy and J. E. Littlewood, Some problems of Diophantine approximation, Acta. Math. 37 (1914), 192-239. · JFM 45.0305.03
[44] G. Herzberg and H. C. Longuet-Higgins, Intersection of potential-energy surfaces in polyatomic molecules, Disc. Far. Soc. 35 (1963), 77-82.
[45] P. Horwitz, Asymptotic theory of unstable resonator modes, J. Opt. Soc. Amer. 63 (1973), 1528-1543.
[46] Robert Kaplan, The nothing that is, Oxford University Press, New York, 2000. A natural history of zero. · Zbl 1010.01001
[47] G. P. Karman, G. S. McDonald, G. H. C. New and J. P. Woerdman, Fractal modes in unstable resonators, Nature 402 (1999), 138.
[48] G. P. Karman, G. S. McDonald, J. P. Woerdman and G. H. C. New, Excess-noise dependence on intracavity aperture shape, Appl. Opt. 38 (1999), 6874-6878.
[49] G. P. Karman and J. P. Woerdman, Fractal structure of eigenmodes of unstable-cavity lasers, Opt. Lett. 23 (1998), 1909-1911.
[50] Nahum Kipnis, History of the principle of interference of light, Science Networks. Historical Studies, vol. 5, Birkhäuser Verlag, Basel, 1991. · Zbl 0757.01007
[51] D. Lee, The J. W. Gibbs Fan Club Homepage (2001), http://www.stanford.edu/ dalee/gibbs.html
[52] R. Lee and A. Fraser, The Rainbow Bridge: Rainbows in Art, Myth and Science, Pennsylvania State University and SPIE press, Bellingham, WA, 2001.
[53] M. S. Longuet-Higgins, Reflection and refraction at a random moving surface. I. Pattern and paths of specular points, J. Opt. Soc. Amer. 50 (1960), 838 – 844. · doi:10.1364/JOSA.50.000838
[54] M. S. Longuet-Higgins, Reflection and refraction at a random moving surface. II. Number of specular points in a Gaussian surface, J. Opt. Soc. Amer. 50 (1960), 845 – 850. · doi:10.1364/JOSA.50.000845
[55] M. S. Longuet-Higgins, Reflection and refraction at a random moving surface. III. Frequency of twinkling in a Gaussian surface, J. Opt. Soc. Amer. 50 (1960), 851 – 856. · doi:10.1364/JOSA.50.000851
[56] G. S. McDonald, G. P. Karman, G. H. C. New and J. P. Woerdman, Kaleidoscope laser, J. Opt. Soc. Amer. B. 17 (2000), 524-529.
[57] G. S. McDonald, G. H. C. New and J. P. Woerdman, Excess-noise in low Fresnel number unstable resonators, Opt. Commun. 164 (1999), 285-295.
[58] G. H. C. New, The origin of excess noise, J. Modern Optics 42 (1995), 799-810.
[59] G. H. C. New, M. A. Yates, J. P. Woerdman and G. S. McDonald, Diffractive origin of fractal resonator modes, Optics Letters (2001), in press.
[60] J. F. Nye, Natural focusing and fine structure of light, Institute of Physics Publishing, Bristol, 1999. Caustics and wave dislocations. · Zbl 0984.78002
[61] J. F. Nye and M. V. Berry, Dislocations in wave trains, Proc. Roy. Soc. London Ser. A 336 (1974), 165 – 190. · Zbl 0289.76046 · doi:10.1098/rspa.1974.0012
[62] Roland Omnès, Consistent interpretations of quantum mechanics, Rev. Modern Phys. 64 (1992), no. 2, 339 – 382. · doi:10.1103/RevModPhys.64.339
[63] K. Patorski, The self-imaging phenomenon and its applications, Progress in Opt. 27 (1989), 1-108.
[64] T. Pearcey, The structure of an electromagnetic field in the neighbourhood of a cusp of a caustic, Phil. Mag. 37 (1946), 311-317.
[65] Tim Poston and Ian Stewart, Catastrophe theory and its applications, Pitman, London-San Francisco, Calif.-Melbourne: distributed by Fearon-Pitman Publishers, Inc., Belmont, Calif., 1978. With an appendix by D. R. Olsen, S. R. Carter and A. Rockwood; Surveys and Reference Works in Mathematics, No. 2. Tim Poston and Ian Stewart, Catastrophe theory and its applications, Dover Publications, Inc., Mineola, NY, 1996. With an appendix by D. R. Olsen, S. R. Carter and A. Rockwood; Reprint of the 1978 original.
[66] K. Sabbagh, Dr. Riemann’s Zeros, Atlantic Books, London, 2002.
[67] Alfred Shapere and Frank Wilczek , Geometric phases in physics, Advanced Series in Mathematical Physics, vol. 5, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989. · Zbl 0914.00014
[68] A. E. Siegman, Lasers, University Science Books, Mill Valley, CA, 1986.
[69] M. S. Soskin, V. N. Gorshkov, M. V. Vasnetsov, J. T. Malos and N. R. Heckenberg, Topological charge and angular momentum of light beams carrying optical vortices, Phys. Rev. A 56 (1997), 4064-4075.
[70] M. S. Soskin and M. V. Vasnetsov, Singular optics, Progress in Optics 42 (2001), 219-276.
[71] M. S. E. Soskin, Singular Optics, Proceedings of SPIE, 3487, 1998.
[72] W. H. Southwell, Unstable-resonator-mode derivation using virtual-source theory, J. Opt. Soc. Amer. 3 (1986), 1885-1891.
[73] G. G. Stokes, On the discontinuity of arbitrary constants which appear in divergent developments, Trans. Camb. Phil. Soc. 10 (1864), 106-128.
[74] G. G. Stokes, On the numerical calculation of a class of definite integrals and infinite series, Trans. Camb. Phil. Soc. 9 (1847), 379-407.
[75] H. F. Talbot, Facts relating to optical science. No. IV, Phil. Mag. 9 (1836), 401-407.
[76] K. Uhlenbeck, Generic properties of eigenfunctions, Amer. J. Math. 98 (1976), no. 4, 1059 – 1078. · Zbl 0355.58017 · doi:10.2307/2374041
[77] A. N. Varchenko, Newton polyhedra and estimation of oscillating integrals, Funkt. Anal. i Prilozhen (Moscow) 10 (1976), 13-38.
[78] M. Vasnetsov and K. Staliunas, eds., Optical Vortices, Nova Science Publishers, Commack, NY, 1999.
[79] F. J. Wright and M. V. Berry, Wavefront dislocations in the sound-field of a pulsed circular piston radiator, J. Acoust. Soc. Amer. 75 (1984), 733-748.
[80] J. A. Yeazell and C. R. J. Stroud, Observation of fractional revivals in the evolution of a Rydberg atomic wave packet, Phys. Rev. A 43 (1991), 5153-5156.
[81] T. Young, The Bakerian Lecture. Experiments and calculations relative to physical optics, Phil. Trans. Roy. Soc. Lond. 94 (1804), 1-16.
[82] T. Young, The Bakerian Lecture: On the theory of light and colours, Phil. Trans. Roy. Soc. 92 (1802), 12-48.
[83] W. H. Zurek, Decoherence, chaos, quantum-classical correspondence, and the algorithmic arrow of time, Physica Scripta 76 (1998), 186-198. · Zbl 1219.81058
[84] W. H. Zurek and J. P. Paz, Decoherence, chaos and the 2nd law, Phys. Rev. Lett. 72 (1994), 2508-2511.
[85] W. H. Zurek and J. P. Paz, Quantum chaos–a decoherent definition, Physica D 83 (1995), 300-308. · Zbl 1194.81111
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.