Tight focal spots using azimuthally polarised light from a Fresnel cone

When focusing a light beam at high numerical aperture, the resulting electric field profile in the focal plane depends on the transverse polarisation profile, as interference between different parts of the beam needs to be taken into account. It is well known that radial polarised light produces a longitudinal polarisation component and can be focused below the conventional diffraction limit for homogeneously polarised light, and azimuthally polarised light that carries one unit of angular momentum can achieve even tighter focal spots. This is of interest for example for enhancing resolution in scanning microscopy. There are numerous ways to generate such polarisation structures, however, setups can be expensive and usually rely on birefringent components, hence prohibiting broadband operation. We have recently demonstrated a passive, low-cost technique using a simple glass cone (Fresnel cone) to generate beams with structured polarisation. We show here that the polarisation structure generated by Fresnel cones focuses better than radial polarised light at all numerical apertures. Furthermore, we investigate in detail the application of polarised light structures for two-photon microscopy. Specifically we demonstrate a method that allows us to generate the desired polarisation structure at the back aperture of the microscope by pre-compensating any detrimental phase shifts using a combination of waveplates

Is the angular momentum of an electron conserved in a uniform magnetic field?

We show that an electron moving in a uniform magnetic field possesses a time-varying ``diamagnetic'' angular momentum. Surprisingly this means that the kinetic angular momentum of the electron may vary with time, despite the rotational symmetry of the system. This apparent violation of angular momentum conservation is resolved by including the angular momentum of the surrounding fields

Analytical Quartic Centrifugal Distortion Constants By Fourth-order Rayleigh SchrÖdinger Perturbation Theory

Recent advances in microwave spectroscopy, allowing for the measurement and fitting of thousands of spectral lines for a given chemical system, have prompted a need for high accuracy predictions of spectroscopic constants. The quartic Centrifugal Distortion (CD) constants are derived at fourth-order in Rayleigh-Schr\"{o}dinger Vibrational Perturbation Theory (VPT4). Analytical expressions are presented. The constants are implemented in the CFOUR software package in both an explicit sum-over-states form and the analytical (i.e., algebraic) form. The expression for VPT4 quartic CD can be broken into ten distinct contributions, involving products of force constants, Coriolis constants, and coefficients in the expansion of the inverse moment of inertia tensor. It is considerably more complicated than the VPT2 vibration-rotation interaction constants and the VPT4 sextic CD constants. The quartic CD constants first appear at VPT2. The VPT4 level of approximation introduces corrections that are linear in the vibrational quantum numbers. Approximately linear relationships have been identified in analyses of microwave spectra, which allow for direct comparison with the computed CD constants. The VPT4 quartic CD constants require a partial quartic force field, containing all force constants except those for which all indices are different (i.e., $\phi$$_{ijkl}$). As this truncation of quartic force field is usually computed for VPT2 vibrational frequencies, it will be possible to obtain the CD constants alongside routine VPT2 frequencies with negligible added cost

Canonical formulation of the embedded theory of gravity equivalent to Einstein's General Relativity

We study the approach in which independent variables describing gravity are functions of the space-time embedding into a flat space of higher dimension. We formulate a canonical formalism for such a theory in a form, which requires imposing additional constraints, which are a part of Einstein's equations. As a result, we obtain a theory with an eight-parameter gauge symmetry. This theory becomes equivalent to Einstein's general relativity either after partial gauge fixing or after rewriting the metric in the form that is invariant under the additional gauge transformations. We write the action for such a theory.Comment: LaTeX, 17 page

A functional non-central limit theorem for jump-diffusions with periodic coefficients driven by stable Levy-noise

We prove a functional non-central limit theorem for jump-diffusions with periodic coefficients driven by strictly stable Levy-processes with stability index bigger than one. The limit process turns out to be a strictly stable Levy process with an averaged jump-measure. Unlike in the situation where the diffusion is driven by Brownian motion, there is no drift related enhancement of diffusivity.Comment: Accepted to Journal of Theoretical Probabilit