The singularities of light: intensity, phase, polarization

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1 submitted to Notices of the International Congress of Chinese Mathematicians January 2014 Extended summaries, including references, of lectures at Mathematical Sciences Center, Tsinghua University, Beijing, May M V Berry, H H Wills Physics Laboratory, University of Bristol, UK lecture 1 The singularities of light: intensity, phase, polarization Geometry dominates modern optics, in particular through understanding light in terms of its singularities. There are different levels of description in optics, each characterised by different singularities. Analogous considerations apply to other types of wave: quantum, acoustic, elastic, water. The coarsest level is geometrical optics, in which light fields are described by families of rays. Here the singularities are caustics: focal lines and surfaces, that is, the envelopes of ray families, on which the intensity diverges. These bright-light singularities are classified by the mathematics of catastrophe theory [1-4], providing a list of the geometric forms of caustics that are stable under generic perturbations. Many phenomena are described by caustics: rainbows [5], the bright lines of focused sunlight on the bottoms of swimming-pools, bright-edged shadows of floating insects [6], and twinkling starlight [7, 8] (whose statistics involve a competition between singularities [9]). In wave optics. these singularities are smoothed by diffraction, which decorates them with rich and ubiquitous interference patterns [10, 11], described by a new class of special functions ( diffraction catastrophes [12]), 1

2 represented by oscillatory integrals (chapter 36 of [13], [14, 15]. In white light, caustics display interesting colours [16, 17]. Wave optics, when represented by complex scalar wavefunctions, also introduces the additional concept of phase, which has its own singularities. Equivalent terms for phase singularities are optical vortices, nodal manifolds or wavefront dislocations [18, 19]. On phase singularities, the light intensity is zero, so these are the singularities of dark light. Geometrically, they are lines in space, or points in the plane, around which the phase changes by a multiple (generically 1) of 2π. Phase singularities are complementary to caustics, not only because the former are dark (zero intensity) and the latter are bright (infinite intensity), but also in the sense of Bohr: caustics are prominent features in the short-wave asymptotic regime, in which phase singularities are too close to be clearly resolved because these are fine-scale features, clearly discernable only in the opposite case of high magnification, where caustics are smoothed out and so are no longer distinct features. Phase singularities can form intricate patterns, for example as fine detail in diffraction catastrophes [10, 11, 20] and near spiral phase plates [21, 22]. They can organise the coloured interference patterns formed by white light [23-25]. They occur in all types of quantum [26-32] or classical (e.g. acoustic [33] and tide [34-38]) waves and have been extensively reviewed [39-41]. In three dimensions, phase singularity lines can be linked and knotted [42-47]. Incorporating the vector (electromagnetic) nature of light brings further singularities, corresponding to the new physical property thereby introduced, namely polarization. Polarization singularities are lines in three dimensions, of two types [48-51]: C singularities, on which the polarization is purely circular, and L singularities, on which the 2

3 polarization in purely linear. In direction space, polarization singularities play a central role in crystal optics [52-57] (notably conical refraction [58, 59]), and in the pattern of polarized light in the blue sky [60]. The C and L lines are different for the electric and magnetic fields [51], but coincide for paraxial fields [61]. Historically, all three levels of singularity can be considered to have originated in the same decade: the 1830s [62]. As well as representing physics at each level, these optical and wave geometries illustrate the idea of asymptotically emergent phenomena [63]. The levels form a hierarchy, with each deeper level of theory eliminating the earlier singularities and generating new ones. Consequences of this approach are predictions that the phase singularities of scalar light will have quantum cores [64, 65], and large momentum transfers to small particles [66]. References 1. Arnold, V. I.,1986, Catastrophe Theory (Springer, Berlin) 2. Berry, M. V. & Upstill, C.,1980, Catastrophe optics: morphologies of caustics and their diffraction patterns Progress in Optics 18, Berry, M. V.,1981, Singularities in Waves and rays in Les Houches Lecture Series Session 35 eds. Balian, R., Kléman, M. & Poirier, J.-P. (North-Holland: Amsterdam, pp Berry, M. V.,1976, Waves and Thom's theorem Advances in Physics 25,

4 5. Lee, R. & Fraser, A.,2001, The rainbow bridge: rainbows in art, myth and science (Pennsylvania State University and SPIE press, Bellingham, Washington) 6. Berry, M. V. & Hajnal, J. V.,1983, The shadows of floating objects and dissipating vortices Optica Acta 30, Berry, M. V.,1977, Focusing and twinkling: critical exponents from catastrophes in non-gaussian random short waves J. Phys. A 10, Walker, J. G., Berry, M. V. & Upstill, C.,1983, Measurement of twinkling exponents of light focused by randomly rippling water Optica Acta 30, Berry, M. V.,2000, Spectral twinkling in Proc International School of Physics Enrico Fermi eds. G. Casati, I. G., U. Smilansky (IOS Press, Amsterdam, Varenna), Vol. CLXIII, pp Berry, M. V., Nye, J. F. & Wright, F. J.,1979, The elliptic umbilic diffraction catastrophe Phil.Trans.Roy.Soc. A291, Nye, J. F.,2006, Dislocation lines in the hyperbolic umbilic diffraction catastrophe Proc. R. Soc. A 462, Trinkaus, H. & Drepper, F.,1977, On the Analysis of Diffraction Catastrophes J. Phys. A. 10, L-11-L DLMF,2010, NIST Handbook of Mathematical Functions (University Press, Cambridge) 4

5 14. Duistermaat, J. J.,1974, Oscillatory Integrals, Lagrange Immersions and Unfolding of Singularities Communs Pure App. Math. 27, Varchenko, A. N.,1976, Newton polyhedra and estimation of oscillating integrals Funkt.Anal.i Prilozhen (Moscow) 10, Berry, M. V. & Wilson, A. N.,1994, Black-and-white fringes and the colours of caustics Applied Optics 33, Berry, M. V. & Klein, S.,1996, Colored diffraction catastrophes Proc. Natl. Acad. Sci. USA 93, Nye, J. F. & Berry, M. V.,1974, Dislocations in wave trains Proc. Roy. Soc. Lond. A336, Nye, J. F.,1999, Natural focusing and fine structure of light: Caustics and wave dislocations (Institute of Physics Publishing, Bristol) 20. Nye, J. F.,2003, Evolution from a Fraunhofer to a Pearcey pattern J. Opt. A. 5, Berry, M. V.,2004, Optical vortices evolving from helicoidal integer and fractional phase steps J.Optics. A 6, Leach, J., Yao, E. & Padgett, M.,2004, Observation of the vortex structure of a non-integer vortex beam New Journal of Physics 6, Berry, M. V.,2002, Coloured phase singularities New Journal of Physics 4,

6 24. Berry, M. V.,2002, Exploring the colours of dark light New Journal of Physics 4, Leach, J. & Padgett, M. J.,2003, Observation of chromatic effects near a white-light vortex New. J. Phys. 5, Riess, J.,1970, Nodal structure of Schroedinger wave functions and its physical significance Ann. Phys. (NY) 57, Riess, J.,1970, Nodal structure, nodal flux fields, and flux quantization in stationary quantum states Phys. Rev.D. 2, Hirschfelder, J. O., Goebel, C. J. & Bruch, L. W.,1974, Quantized vortices around wavefunction nodes. II J. Chem. Phys. 61, Hirschfelder, J. O., Christoph, A. C. & Palke, W. E.,1974, Quantum mechanical streamlines. 1. Square potential barrier J. Chem. Phys. 61, Hirschfelder, J. O. & Christoph, A. C.,1974, Quantum mechanical streamlines. 1. Square potential barrier J. Chem. Phys. 61, Hirschfelder, J. O. & Tang, K. T.,1976, Quantum mechanical streamlines. III Idealized reactive atom-diatomic molecule collision J. Chem. Phys. 64, Hirschfelder, J. O. & Tang, K. T.,1976, Quantum mechanical streamlines. IV. Collision of two spheres with square potential wells or barriers J. Chem. Phys. 65,

7 33. Wright, F. J. & Berry, M. V.,1984, Wavefront dislocations in the sound-field of a pulsed circular piston radiator J. Acoust. Soc. Amer. 75, Berry, M. V.,2001, Geometry of phase and polarization singularities, illustrated by edge diffraction and the tides in Singular Optics 2000 eds. Soskin, M. (SPIE, Alushta, Crimea), Vol. 4403, pp Whewell, W.,1833, Essay towards a first approximation to a map of cotidal lines Phil. Trans. Roy. Soc. Lond. 123, Whewell, W.,1836, On the results of an extensive series of tide observations made on the coasts of Europe and America in June 1835 Phil. Trans. Roy. Soc. Lond. 126, Defant, A.,1961, Physical Oceanography (Pergamon, Oxford) 38. Berry, M. V.,2002, Exuberant interference: rainbows, tides, edges, (de)coherence Phil. Trans. Roy. Soc. Lond. A 360, Berry, M. V.,1998, Much ado about nothing: optical dislocation lines (phase singularities, zeros, vortices...) SPIE 3487, Dennis, M. R., O'Holleran, K. & Padgett, M. J.,2009, Singular optics: Optical Vortices and Polarization Singularities Progress in Optics 53, Soskin, M. S. & Vasnetsov, M. V.,2001, Singular Optics Progress in Optics 42,

8 42. Berry, M. V. & Dennis, M. R.,2001, Knotted and linked phase singularities in monochromatic waves Proc.Roy.Soc.Lond. 457, Berry, M. V. & Dennis, M. R.,2001, Knotting and unknotting of phase singularities in Helmholtz waves, paraxial waves and waves in 2+1 spacetime J. Phys. A 34, Berry, M. V.,2001, Knotted zeros in the quantum states of hydrogen Found. Phys. 31, Dennis, M. R.,2003, Braided nodal lines in wave superpositions New J. Phys. 5, O'Holleran, K., Padgett, M. J. & Dennis, M. R.,2006, Topology of optical vortex lines formed by the interference of three, four and five plane waves Optics Express 14, Leach, J., Dennis, M. R., Courtial, J. & Padgett, M. J.,2004, Laser beams: Knotted threads of darkness Nature 432, Nye, J. F. & Hajnal, J. V.,1987, The wave structure of monochromatic electromagnetic radiation Proc. Roy. Soc. Lond. A409, Hajnal, J. V.,1987, Singularities of the transverse fields of electromagnetic waves. I. Theory Proc. Roy. Soc. Lond. A414, Hajnal, J. V.,1987, Singularities in the transverse fields of electromagnetic waves. II. Observations on the electric field. Proc. Roy. Soc. Lond. A414,

9 51. Hajnal, J. V.,1990, Observation of singularities in the electric and magnetic fields of freely propagating microwaves. Proc. Roy. Soc. Lond. A430, Berry, M. V. & Dennis, M. R.,2003, The optical singularities of birefringent dichroic chiral crystals Proc. Roy. Soc. A. 459, Pancharatnam, S.,1955, The propagation of light in absorbing biaxial crystals - I. Theoretical Proc. Ind. Acad. Sci. XLII, Pancharatnam, S.,1955, The propagation of light in absorbing biaxial crystals - II. Experimental Proc. Ind. Acad. Sci. XLII, Berry, M. V.,2005, The optical singularities of bianisotropic crystals Proc. Roy. Soc. A 461, Berry, M. V., Bhandari, R. & Klein, S.,1999, Black plastic sandwiches demonstrating biaxial optical anisotropy Eur. J. Phys. 20, Berry, M. V. & Dennis, M. R.,2004, Black polarization sandwiches are nontrivial square roots of zero J.Optics A 6, S24-S Berry, M. V.,2004, Conical diffraction asymptotics: fine structure of Poggendorff rings and axial spike J.Optics A 6, Berry, M. V. & Jeffrey, M. R.,2007, Conical diffraction: Hamilton's diabolical point at the heart of crystal optics Progress in Optics 50,

10 60. Berry, M. V., Dennis, M. R. & Lee, R. L. J.,2004, Polarization singularities in the clear sky New J. of Phys. 6, Berry, M. V.,2004, The electric and magnetic polarization singularities of paraxial waves J.Optics A 6, Berry, M. V.,2000, Making waves in physics: three wave singularities from the miraculous 1830s Nature 403, Berry, M. V.,1994, Asymptotics, singularities and the reduction of theories in Proc. 9th Int. Cong. Logic, Method., and Phil. of Sci. eds. Prawitz, D., Skyrms, B. & Westerståhl, D. (Elsevier Science B.V., pp Berry, M. V. & Dennis, M. R.,2004, Quantum cores of optical phase singularities J.Optics A 6, S178-S Barnett, S. M.,2008, On the quantum core of an optical vortex J. Mod. Opt. 55, Barnett, S. M. & Berry, M. V.,2013, Superweak momentum transfer near optical vortices J. Opt. 15, (6pp) 10

11 Lecture 2 Superoscillation and weak measurement Contrary to naive intuition, band-limited functions can oscillate arbitrarily faster than their fastest Fourier component, over arbitrarily long intervals [1]. There is no contradiction wih the uncertainty principle, because where such superoscillations occur the functions are exponentially weak [2-5]. In random waves with wavenumber k, such as typical monochromatic optical fields, superoscillations are unexpectedly common [6]: for substantial fractions of the domain (one-third in two dimensions [7, 8]), the local wavenumber (modulus of phase gradient [9]) exceeds k. Superoscillations are related to supergain and superdirectivity in antenna theory [10, 11]. They have implications for signal processing [1], and raise the possibility of sub-wavelength resolution microscopy without evanescent waves [12-16]. In quantum mechanics, where the concept originated, superoscillations correspond to weak measurements [17-19], in which pointer shifts [20, 21] can correspond to weak values of observables (e.g photon momenta [9, 22, 23]) far outside the range represented in the quantum state (i.e. outside the spectrum of the operator representing the observable). For typical quantum weak measurements, superweak values are unexpectedly common [24, 25]. For relativistic waves, superoscillations correspond to superluminal group velocities [26]. An application is to a recent (now discredited) experiment [27, 28] suggesting superluminal speeds for neutrinos; although in principle a weak measurement of neutrino speed could lead to a superluminal result without violating causality (analogous to an earlier 11

12 demonstration for light [29]), analysis showed [30] that the effect is too small to explain the speed that was claimed (see also [31-33]. References 1. Berry, M. V.,1994, Faster than Fourier in Quantum Coherence and Reality; in celebration of the 60th Birthday of Yakir Aharonov eds. Anandan, J. S. & Safko, J. L. (World Scientific, Singapore), pp Katzav, E. & Schwartz, M.,2013, Yield-Optimised Superoscillations IEEE Trans. Signal Processing 61, Berry, M. V.,1994, Evanescent and real waves in quantum billiards, and Gaussian Beams J. Phys. A 27, L391-L Kempf, A. & Ferreira, P. J. S. G.,2004, Unusual properties of superoscillating particles J. Phys. A 37, Berry, M. V.,2008, Waves near zeros in Coherence and Quantum Optics eds. Bigelow, N. P., Eberly, J. H. & Stroud, C. R. J. (Optical Society of America, Washington DC), pp Berry, M. V.,2013, A note on superoscillations associated with Bessel beams J.Opt. 15, (5pp) 7. Dennis, M. R., Hamilton, A. C. & Courtial, J.,2008, Superoscillation in speckle patterns Optics Letters 33, Berry, M. V. & Dennis, M. R.,2009, Natural superoscillations in monochromatic waves in D dimensions J. Phys. A 42, Berry, M. V.,2009, Optical currents J. Opt. A. 11,

13 10. Schelkunoff, S. A.,1943, A Mathematical Theory of Linear Arrays Bell. Syst. Tech. J. 22, Berry, M. V.,2012, in Yakir Aharonov 80th birthday celebration, eds. Tollaksen, J. (Springer, Chapman University). 12. Bucklew, J. A. & Saleh, B. E. A.,1985, Theorem for highresolution high-contrast image synthesis J. Opt. Soc. Amer. A 2, Berry, M. V. & Popescu, S.,2006, Evolution of quantum superoscillations, and optical superresolution without evanescent waves J.Phys.A. 39, Berry, M. V.,2013, Exact nonparaxial transmission of subwavelength detail using superoscillations J. Phys. A 46, (15pp) 15. Toraldo di Francia, G.,1952, SuperGain Antennas and Optical Resolving Power Nuovo Cimento Suppl. 9, Makris, K. G., El-Ganainy, R., Christodoulides, D. N. & Musslimani, Z. H.,2008, Beam Dynamics in PT Symmetric Optical Lattices Phys. Rev. Lett. 100, Aharonov, Y., Albert, D. Z. & Vaidman, L.,1988, How the result of a measurement of a component of the spin of a spin 1/2 particle can turn out to be 100 Phys. Rev. Lett. 60, Aharonov, Y. & Rohrlich, D.,2005, Quantum Paradoxes: Quantum Theory for the perplexed (Wiley-VCH, Weinheim) 13

14 19. Aharonov, Y., Popescu, S. & Tollaksen, J.,2010, A time-symmetric formulation of quantum mechanics Physics Today 63, issue 11, Berry, M. V. & Shukla, P.,2012, Pointer supershifts and superoscillations in weak measurements J.Phys.A. 45, (14pp) 21. Jozsa, R.,2007, Complex weak values in quantum measurement Phys, Rev. A 76, Barnett, S. M. & Berry, M. V.,2013, Superweak momentum transfer near optical vortices J. Opt. 15, (6pp) 23. Berry, M. V.,2013, Five momenta Eur. J. Phys. 44, Berry, M. V. & Shukla, P.,2010, Typical weak and superweak values J. Phys. A 43, (9pp) 25. Berry, M. V., Dennis, M. R., McRoberts, B. & Shukla, P.,2011, Weak value distributions for spin 1/2 J. Phys. A 44, Berry, M. V.,2012, Superluminal speeds for relativistic random waves J. Phys. A 45, (14pp) 27. Adam, T. & al., e.,2011, Measurement of the neutrino velocity with the OPERA detector in the CNGS beam Adamson, P. & al., e.,2008, Measurement of Neutrino Oscillations with the MINOS Detectors in the NuMI Beam Phys. Rev. Lett 101, (5pp) 14

15 29. Brunner, N., Scarani, V., Wegmüller, M., Legré, M. & Gisin, N.,2004, Direct measurement of superluminal group velocity and of signal velocity in an optical fiber Phys, Rev. Lett 93, Berry, M. V., Brunner, N., Popescu, S. & Shukla, P.,2011, Can apparent superluminal neutrino speeds be explained as a quantum weak measurement J. Phys. A 44, (5pp) 31. Morris, T. R.,2011, Superluminal group velocity through maximal neutrino oscillations.arxiv: v2 [hep-ph] 32. Tanimura, S.,2011, Apparent Superluminal Muon-neutrino Velocity as a Manifestation of Weak Value.arXiv: v1 [hep-ph] 33. Mecozzi, A. & Bellini, M.,2011, Superluminal group velocityof neutrinos.arxiv: v1 [hep-ph] 15

Coherence and Quantum Optics IX (Optical Society of America, Washington DC, 2008), Eds: N P Bigelow, J H Ebery & C R Stroud, pp37-41.

Coherence and Quantum Optics IX (Optical Society of America, Washington DC, 2008), Eds: N P Bigelow, J H Ebery & C R Stroud, pp37-41 1. Introduction Waves near zeros M V Berry H H Wills Physics Laboratory,