Showing papers by "Michael V Berry published in 2008"

Optical lattices with PT symmetry are not transparent

Abstract: In diffraction of a plane wave by a non-Hermitian PT symmetric optical lattice, the sum of the Bragg beam intensities need not be conserved, even though the gain and loss are equally distributed: the evolution is not unitary. Instead, different sums are conserved, in which the intensities are weighted with real numbers (positive or negative); several such sum rules are derived. Two-beam diffraction from a refractive index of the form constant −a cos x + ib sin x is studied in detail; the sum rule depends on the balance between the (real) Hermitian parameter a and the (real) anti-Hermitian parameter b.

Exact and geometrical optics energy trajectories in twisted beams

Abstract: Energy trajectories, that is, integral curves of the Poynting (current) vector, are calculated for scalar Bessel and Laguerre–Gauss beams carrying orbital angular momentum. The trajectories for the exact waves are helices, winding on cylinders for Bessel beams and hyperboloidal surfaces for Laguerre–Gauss beams. In the geometrical optics approximations, the trajectories for both types of beam are overlapping families of straight skew rays lying on hyperboloidal surfaces; the envelopes of the hyperboloids are the caustics: a cylinder for Bessel beams and two hyperboloids for Laguerre–Gauss beams.

Boundary-condition-varying circle billiards and gratings: the Dirichlet singularity

Abstract: Waves in a two-dimensional domain with Robin (mixed) boundary conditions that vary smoothly along the boundary exhibit unexpected phenomena. If the variation includes a 'D point' where the boundary condition is Dirichlet (vanishing wavefunction), a variety of arguments indicate that the system is singular. For a circle billiard, the boundary condition fails to determine a discrete set of levels, so the spectrum is continuous. For a diffraction grating defined by periodically varying boundary conditions on the edge of a half-plane, the phase of a diffracted beam amplitude remains undetermined. In both cases, the wavefunction on the boundary has a singularity at a D point, described by the polylogarithm function.

Three quantum obsessions

Abstract: Is there a connection between the Riemann zeros and the quantum physics of classical chaos? Can the relation between spin and statistics be understood within elementary quantum mechanics? How are the phase singularities in classical optics smoothed by quantum effects?

Tuck's incompressibility function: statistics for zeta zeros and eigenvalues

Abstract: For any function D(x) that is real for real x, positivity of Tuck's function Q(x) ≡ D'2(x)/(D'2(x) − D(x)D''(x)) is a condition for the absence of complex zeros close to the real axis. Study of the probability distribution PN(Q), for D(x) with N zeros corresponding to eigenvalues of the Gaussian unitary ensemble (GUE), supports Tuck's observation that large values of Q are very rare for the Riemann zeros. PN(Q) has singularities at Q = 0, Q = 1 and Q = N. The moments (averages of Qm) are much smaller for the GUE than for uncorrelated random (Poisson-distributed) zeros. For the Poisson case, the large-N limit of PN(Q) can be expressed as an integral with infinitely many poles, whose accumulation, requiring regularization with the Lerch transcendent, generates the singularity at Q = 1, while the large-Q decay is determined by the pole closest to the origin. Determining the large-N limit of PN(Q) for the GUE seems difficult.

Tuck's incompressibility function: statistics for zeta zeros and eigenvalues

Abstract: For any function that is real for real x, positivity of Tuck's function Q(x)=D'^2(x)/(D'^2(x)-D"(x) D(x)) is a condition for the absence of the complex zeros close to the real axis. Study of the probability distribution P(Q), for D(x) with N zeros corresponding to eigenvalues of the Gaussian unitary ensemble (GUE), supports Tuck's observation that large values of Q are very rare for the Riemann zeros. P(Q) has singularities at Q=0, Q=1 and Q=N. The moments (averages of Q^m) are much smaller for the GUE than for uncorrelated random (Poisson-distributed) zeros. For the Poisson case, the large-N limit of P(Q) can be expressed as an integral with infinitely many poles, whose accumulation, requiring regularization with the Lerch transcendent, generates the singularity at Q=1, while the large-Q decay is determined by the pole closest to the origin. Determining the large-N limit of P(Q) for the GUE seems difficult.

The Physical Tourist Physics in Bristol

Abstract: We trace the history of physics in Bristol, before and after the foundation of the University, describing the important locations and events and contributions by notable individuals. As well as the Nobel prizewinners Paul Dirac, Cecil Powell, and Nevill Mott, these include Arthur Tyndall, Charles Frank, Yakir Aharonov, and David Bohm.