Far-field radiation from a cleaved cylindrical dielectric waveguide

Angular spread in the far-field radiation pattern of a cleaved dielectric waveguide is determined from the modal structure at the surface of the waveguide using the Smythe vector integral formulation. Essential features: First, a mode exists in the fiber that has no wavelength cutoff--the so-called HE{sub 11} mode. This mode arises when non-azimuthal angular dependence of the incoming radiation is present. Second, the energy flow from this hybrid mode fills the fiber face and is not annularly shaped as opposed to the symmetric TE and TM modes. Third, the HE{sub 11} mode is not polarization dependent in contrast to the TE and TM modes. Fourth, for small differences in the refractive indices between the core and cladding regions, only the HE{sub 11} mode will be supported until the next modes appear around 3.33{lambda}. At this point, three new modes can propagate and the model structure of the radiation becomes more complicated. Fifth, the far-field radiation pattern will have negligibly small angular dependence in the phases of the vector fields when only the lowest mode is present; the amplitude has an overall angular dependent form factor. Furthermore, when other modes are present (above 3.33{lambda}), the phase of the vector fields will acquire an angular dependence

Closed-form solutions of the Schroedinger equation for a class of smoothed Coulomb potentials

An infinite family of closed-form solutions is exhibited for the Schroedinger equation for the potential $V(r) = -Z/\sqrt{|r|^{2} + a^{2}}$. Evidence is presented for an approximate dynamical symmetry for large values of the angular momentum $l$.Comment: 13 pages LaTeX, uses included Institute of Physics style files, 3 PostScript figures. In press at J. Phys. B: At. Mol. Opt. Phys. (1997

Bessel-Zernike Discrete Variable Representation Basis

The connection between the Bessel discrete variable basis expansion and a specific form of an orthogonal set of Jacobi polynomials is demonstrated. These so-called Zernike polynomials provide alternative series expansions of suitable functions over the unit interval. Expressing a Bessel function in a Zernike expansion provides a straightforward method of generating series identities. Furthermore, the Zernike polynomials may also be used to efficiently evaluate the Hankel transform for rapidly decaying functions or functions with finite support

Explicitly correlated Gaussian functions with shifted-center and projection techniques in pre-Born-Oppenheimer calculations

Numerical projection methods are elaborated for the calculation of eigenstates of the non-relativistic many-particle Coulomb Hamiltonian with selected rotational and parity quantum numbers employing shifted explicitly correlated Gaussian functions, which are, in general, not eigenfunctions of the total angular momentum and parity operators. The increased computational cost of numerically projecting the basis functions onto the irreducible representations of the three dimensional rotation-inversion group is the price to pay for the increased flexibility of the basis functions. This increased flexibility allowed us to achieve a substantial improvement for the variational upper bound to the Pauli-allowed ground-state energy of the H$_3^+=\{$p$^+,$p$^+,$p$^+,$e$^-,$e$^-\}$ molecular ion treated as an explicit five-particle system. We compare our pre-Born-Oppenheimer result for this molecular ion with rovibrational results including non-adiabatic corrections.Comment: 29 pages, 3 figures, 4 table