2D FDTD simulation of low loss small angle bend and Y branch configurations in a photonic crystal waveguide layout with a Mach–Zehnder device design configuration

A technique to produce low loss small angle bends in photonic crystal waveguides is presented. The technique consists of bridging parallel input and output waveguide segments with an inclined waveguide region of the same basic design that has a lateral dielectric shift. Results are presented for waveguides produced by enlarging the silicon gap along the central line, separating air holes in a square array photonic crystal for the TE polarization and an operating wavelength of λ_o = 1.55 μm. This low loss waveguide bending technique is applied to the design of Y branch and Mach–Zehnder photonic crystal structures. Simulation of the performance of the dielectric structures is achieved using 2-D FDTD, similar results are anticipated when applied to 3-D waveguide configurations and for other photonic crystal layouts.     

On the instability of the Riemann Hypothesis for varieties over finite fields

There exist perturbations of a rational function which remove zeroes and poles from a prescribed region as well as perturbations which add zeroes and poles to a prescribed region. We employ this to show the instability of the Riemann Hypothesis for zeta-functions of smooth projective varieties over finite fields.     

Bloch constants for planar harmonic mappings

We give a lower estimate for the Bloch constant for planar harmonic mappings which are quasiregular and for those which are open. The latter includes the classical Bloch theorem for holomorphic functions as a special case. Also, for bounded planar harmonic mappings, we obtain results similar to a theorem of Landau on bounded holomorphic functions.

     

Trapping of Terminal Platinapolyynes by Copper(I) Catalyzed Click Cycloadditions; Probes of Labile Intermediates in Syntheses of Complexes with Extended sp Carbon Chains,and Crystallographic Studies

The silylated hexatriynyl complex trans-(C₆F₅)(p-tol₃P)₂Pt(C≡C)₃SiEt₃ ( PtC₆TES ) is converted in situ to PtC₆H (wet n-Bu₄N⁺ F⁻, THF) and cross coupled with the diyne H(C≡C)₂SiEt₃ ( HC₄TES ; CuCl/TMEDA, O₂) to give PtC₁₀TES (71 %). This sequence is repeated twice to afford PtC₁₄TES (65 %) and then PtC₁₈TES (27 %). An analogous series of reactions starting with PtC₈TES gives PtC₁₂TES (60 %), then PtC₁₆TES (43 %), and then PtC₂₀TES (17 %). Similar cross couplings with H(C≡C)₂Si(i-Pr)₃ ( HC₄TIPS ) give PtC₁₂TIPS (68 %), PtC₁₄TIPS (68 %), and PtC₁₆TIPS (34 %). The trialkylsilyl species (up to PtC₁₈TES ) are converted to 3+2 “click” cycloadducts or 1,4-disubstituted 1,2,3-triazoles trans-(C₆F₅)(p-tol₃P)₂Pt(C≡C)_n-1C=CHN(CH₂C₆H₅)N=N (29–92 % after workups). The most general procedure involves generating the terminal polyynes PtC _x H (wet n-Bu₄N⁺ F⁻, THF) in the presence of benzyl azide in DMF and aqueous CuSO₄/ascorbic acid. All of the preceding complexes are crystallographically characterized and the structural and spectroscopic properties analyzed as a function of chain length. Two pseudopolymorphs of PtC₂₀TES are obtained, both of which feature molecules with parallel sp carbon chains in a pairwise head/tail packing motif with extensive sp/sp van der Waals contacts.     

Laboratory scale-up of two-stage laser chemistry separation of13C from CF2HCl

The laboratory scale-up of a two-stage laser enrichment process for carbon isotopes, involving infrared multiphoton dissociation of freon-22, is described. Unmodified commercial equipment and materials were used. An initial study of the effect of fluence, laser frequency, freon-22 pressure and pressure of argon, nitrogen and trifluoromethyl chloride was made in short irradiation cells (constant fluence) in order to define optimum process parameters. The process was then scaled to higher throughput in longer cells (1–5 m) in which compensation for beam-energy depletion by absorption was made by reduction in the beam area by focussing. From the scale-up experiments, measurements of yield and enrichment of the tetrafluoroethylene product gave demonstrated production rates. These, coupled with measurements of the absorption, allowed extrapolation to production rates assuming total utilization of the available output energy.Using a 100 W TEA CO₂ laser (10 J, 10 Hz) we have demonstrated production rates of 0.20 g h^–1 carbon-12 at 99.99% carbon-12, 11 mg h^–1 carbon-13 at 72% carbon-13 and 2 kg per annum carbon-13 at 50%. Energy absorption measurements imply a capability to produce 3 kg per annum carbon-13 at over 95% carbon-13 in a two-stage process.The apparatus was used to produce gram quantities of carbon-13 depleted freon-22 (99.99% carbon-12). A comparison of the infrared multiphoton dissociation of this material with that of natural freon-22 (1.11% carbon-13) showed that under the conditions required to give selective dissociation of¹³CF₂HCl that¹²CF₂HCl was excited as a result of a dominantly radiative interaction and that collisional transfer from¹³CF₂HCl molecules played a minor role.Issued as NRCC 23639Summer Student 1983, Acadia University, N.S., Canada     

Hopf bifurcations in time-delay systems with band-limited feedback

We investigate the steady-state solution and its bifurcations in time-delay systems with band-limited feedback. This is a first step in a rigorous study concerning the effects of AC-coupled components in nonlinear devices with time-delayed feedback. We show that the steady state is globally stable for small feedback gain and that local stability is lost, generically, through a Hopf bifurcation for larger feedback gain. We provide simple criteria that determine whether the Hopf bifurcation is supercritical or subcritical based on the knowledge of the first three terms in the Taylor-expansion of the nonlinearity. Furthermore, the presence of double-Hopf bifurcations of the steady state is shown, which indicates possible quasiperiodic and chaotic dynamics in these systems. As a result of this investigation, we find that AC-coupling introduces fundamental differences to systems of Ikeda-type [K. Ikeda, K. Matsumoto, High-dimensional chaotic behavior in systems with time-delayed feedback, Physica D 29 (1987) 223–235] already at the level of steady-state bifurcations, e.g. bifurcations exist in which limit cycles are created with periods other than the fundamental “period-2” mode found in Ikeda-type systems.     

A numerical model for the 3-D non-linear vibrations of an N-string

We present a non-linear numerical model describing the 3-D vibrations of a planar network of N sections of string which are connected together at one common mobile extremity. We call such a network N-string. For small-amplitude vibrations perpendicular to the N-string equilibrium plane, the numerical results coincide with the already known analytical solutions of the linear model. This non-linear model makes it possible to describe small- or large-amplitude 3-D vibrations of any kind of N-string subjected to an initial plucking. The equations of motion are also presented in a dimensionless form and a vast dimensionless physical parameter space is identified. The numerical model can be extended to more complex networks of strings.