Investigation on an ultra-compact 1\times 2 polymer electro-optic switch using cross-coupling 2N+1 vertical-turning serial-coupled microrings

Generic model and thorough investigation are proposed for a novel $1\times 2$ 1 × 2 polymer electro-optic (EO) switch based on one-group $2N+1$ 2 N + 1 vertical-turning serial-coupled microrings. For realizing boxlike flat spectrum as well as low crosstalk and insertion loss, resonance order and coupling gaps are optimized. The MRR switches with $N \ge 1$ N ≥ 1 reveal favorable boxlike spectrum as when compared with the simple device with only one microring ( $N = 0$ N = 0 ). For obtaining $<-30\,\text{ dB }$ < - 30 dB crosstalk under through-state, the dependency of switching voltage on $N$ N is determined as $7.19 \times \text{ exp }(-N/0.72) + 1.72\,(\text{ V })$ 7.19 × exp ( - N / 0.72 ) + 1.72 ( V ) . Under the operation voltages of 0 V (drop state) and the predicted switching voltages (through state), the device performances are analyzed, and $1 \le N \le 10$ 1 ≤ N ≤ 10 is required for dropping the insertion loss (drop state) below 10 dB. The crosstalk of the ten devices ( $N = 1-10$ N = 1 - 10 ) are $< -19.5\,\text{ dB }$ < - 19.5 dB under drop state and $< -28.7\,\text{ dB }$ < - 28.7 dB under through state, and the insertion losses of the devices ( $N = 1-10$ N = 1 - 10 ) are $< 9.715\,\text{ dB }$ < 9.715 dB under drop state and $< 1.573\,\text{ dB }$ < 1.573 dB under through state. The device also has ultra-compact footprint size of only 0.33–1.06 mm, which is only 1/10–1/3 of those of our previously reported polymer EO switches based on directional coupler or Mach–Zehnder interferometer structures. Therefore, the proposed device is capable of highly integration onto optical networks-on-chip.     

4D Einstein equations in a 5D general Kaluza–Klein space

Based on the new point of view on space–time–matter theory developed in our paper (Bejancu, Gen Rel Grav, 2013), we obtain the $4D$ 4 D Einstein equations in a general $5D$ 5 D Kaluza–Klein space with electromagnetic potentials. In particular, we recover the $4D$ 4 D Einstein equations obtained by Wesson and Ponce de Leon (J Math Phys 33:3883, 1992) in case the electromagnetic potentials vanish identically on $\bar{M}$ M ¯ . The Riemannian horizontal connection and the $4D$ 4 D tensor calculus on $\bar{M}$ M ¯ , are the main tools in the study.     

Poincaré-Dulac Normal Form Reduction for Unconditional Well-Posedness of the Periodic Cubic NLS

We implement an infinite iteration scheme of Poincaré-Dulac normal form reductions to establish an energy estimate on the one-dimensional cubic nonlinear Schrödinger equation (NLS) in ${C_tL^2(\mathbb{T})}$ C t L 2 ( T ) , without using any auxiliary function space. This allows us to construct weak solutions of NLS in ${C_tL^2(\mathbb{T})}$ C t L 2 ( T ) with initial data in ${L^2(\mathbb{T})}$ L 2 ( T ) as limits of classical solutions. As a consequence of our construction, we also prove unconditional well-posedness of NLS in ${H^s(\mathbb{T})}$ H s ( T ) for ${s \geq \frac{1}{6}}$ s ≥ 1 6 .     

An interacting dark energy model in a non-flat universe

In this paper, an interacting dark energy model in a non-flat universe is studied, with taking interaction form $C=\alpha H\rho _{de}$ C = α H ρ d e . And in this study a property for the mysterious dark energy is aforehand assumed, i.e. its equation of state $w_{\Lambda }=-1$ w Λ = - 1 . After several derivations, a power-law form of dark energy density is obtained $\rho _{\Lambda } \propto a^{-\alpha }$ ρ Λ ∝ a - α , here $a$ a is the cosmic scale factor, $\alpha $ α is a constant parameter introducing to describe the interaction strength and the evolution of dark energy. By comparing with the current cosmic observations, the combined constraints on the parameter $\alpha $ α is investigated in a non-flat universe. For the used data they include: the Union2 data of type Ia supernova, the Hubble data at different redshifts including several new published datapoints, the baryon acoustic oscillation data, the cosmic microwave background data, and the observational data from cluster X-ray gas mass fraction. The constraint results on model parameters are $\Omega _{K}=0.0024\,(\pm 0.0053)^{+0.0052+0.0105}_{-0.0052-0.0103}, \alpha =-0.030\,(\pm 0.042)^{+0.041+0.079}_{-0.042-0.085}$ Ω K = 0.0024 ( ± 0.0053 ) - 0.0052 - 0.0103 + 0.0052 + 0.0105 , α = - 0.030 ( ± 0.042 ) - 0.042 - 0.085 + 0.041 + 0.079 and $\Omega _{0m}=0.282\,(\pm 0.011)^{+0.011+0.023}_{-0.011-0.022}$ Ω 0 m = 0.282 ( ± 0.011 ) - 0.011 - 0.022 + 0.011 + 0.023 . According to the constraint results, it is shown that small constraint values of $\alpha $ α indicate that the strength of interaction is weak, and at $1\sigma $ 1 σ confidence level the non-interacting cosmological constant model can not be excluded.     

Delocalization and Diffusion Profile for Random Band Matrices

We consider Hermitian and symmetric random band matrices H = (h _xy ) in ${d\,\geqslant\,1}$ d ? 1 dimensions. The matrix entries h _xy , indexed by ${x,y \in (\mathbb{Z}/L\mathbb{Z})^d}$ x , y ∈ ( Z / L Z ) d , are independent, centred random variables with variances ${s_{xy} = \mathbb{E} |h_{xy}|^2}$ s x y = E | h x y | 2 . We assume that s _xy is negligible if |x ? y| exceeds the band width W. In one dimension we prove that the eigenvectors of H are delocalized if ${W\gg L^{4/5}}$ W ? L 4 / 5 . We also show that the magnitude of the matrix entries ${|{G_{xy}}|^2}$ | G x y | 2 of the resolvent ${G=G(z)=(H-z)^{-1}}$ G = G ( z ) = ( H - z ) - 1 is self-averaging and we compute ${\mathbb{E} |{G_{xy}}|^2}$ E | G x y | 2 . We show that, as ${L\to\infty}$ L → ∞ and ${W\gg L^{4/5}}$ W ? L 4 / 5 , the behaviour of ${\mathbb{E} |G_{xy}|^2}$ E | G x y | 2 is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions.     

Long-time Asymptotic for the Derivative Nonlinear Schrödinger Equation with Step-like Initial Value

We study long-time asymptotics of the solution to the Cauchy problem for the Gerdjikov-Ivanov type derivative nonlinear Schrödinger equation i q t + q xx ? i q 2 q ? x + 1 2 | q | 4 q = 0 $$iq_{t}+q_{xx}-iq^{2}\bar{q}_{x}+\frac{1}{2}|q|^{4}{q}=0 $$ with step-like initial data q ( x , 0 ) = 0 $q(x,0)=0$ for x ≤ 0 $x \leqslant 0$ and q ( x , 0 ) = A e ? 2 iBx $q(x,0)=A\mathrm {e}^{-2iBx}$ for x > 0 $x>0$ , where A > 0 $A>0$ and B ∈ ? $B\in \mathbb R$ are constants. We show that there are three regions in the half-plane { ( x , t ) | ? ∞ < x < ∞ , t > 0 } $\{(x,t) | -\infty <x<\infty , t>0\}$ , on which the asymptotics has qualitatively different forms: a slowly decaying self-similar wave of Zakharov-Manakov type for x > ? 4 tB $x>-4tB$ , a plane wave region: x > ? 4 t B + 2 A 2 B + A 2 4 $x<-4t\left (B+\sqrt {2A^{2}\left (B+\frac {A^{2}}{4}\right )}\right )$ , an elliptic region: ? 4 t B + 2 A 2 B + A 2 4 > x > ? 4 tB $-4t\left (B+\sqrt {2A^{2}\left (B+\frac {A^{2}}{4}\right )}\right )<x<-4tB$ . Our main tools include asymptotic analysis, matrix Riemann-Hilbert problem and Deift-Zhou steepest descent method.     

Consequences of f(R) theories of gravity on gravitational leptogenesis

$f(R)$ f ( R ) -theories of gravity are reviewed in the framework of the matter-antimatter asymmetry in the Universe. The asymmetry is generated by the gravitational coupling of heavy (Majorana) neutrinos with the Ricci scalar curvature. In order that the mechanism works, a time varying non-zero Ricci curvature is necessary. The latter is provided by $f(R)$ f ( R ) cosmology, whose Lagrangian density is of the form $\mathcal{L}(R)\sim f(R)$ L ( R ) ～ f ( R ) , with $f(R)\sim R+\alpha R^n$ f ( R ) ～ R + α R n .     

Magnetotransport properties in \text{ Ni }_{81}\text{ Fe }_{19}\text{/Zr } multi-layers

In this work, we present a study of the magneto transport properties in magnetic multilayered structure $\text{ Ni }_{81}\text{ Fe }_{19}\text{/Zr }$ Ni 81 Fe 19 /Zr . The magnetic $(\text{ Ni }_{81}\text{ Fe }_{19})$ ( Ni 81 Fe 19 ) and non magnetic (Zr) layer thickness $(\mathbf{t}_\mathbf{NiFe}, \mathbf{t}_\mathbf{zr})$ ( t NiFe , t zr ) effects on the magneto resistance (MR) are discussed theoretically in the framework of the Johnson–Camley semi classical approach based on the Boltzmann transport equation. A comparison between calculated and measured MR is obtained. The observed MR ratio oscillates for Zr layer thickness with an average period of 7Å. A generally weak $\text{ MR }(\text{ t }_{\mathrm{NiFe}})$ MR ( t NiFe ) ratio for fixed $\mathbf{t}_\mathbf{zr}$ t zr is obtained and it shows a maxima peak of the MR with a value of 1.8 % located at $\mathbf{t}_\mathbf{NiFe}= 80$ t NiFe = 80 Å.     

Noncommutative Geometry of the Moyal Plane: Translation Isometries, Connes’ Distance on Coherent States, Pythagoras Equality

We study the metric aspect of the Moyal plane from Connes’ noncommutative geometry point of view. First, we compute Connes’ spectral distance associated with the natural isometric action of ${\mathbb{R}^2}$ R 2 on the algebra of the Moyal plane ${\mathcal{A}}$ A . We show that the distance between any state of ${\mathcal{A}}$ A and any of its translated states is precisely the amplitude of the translation. As a consequence, we obtain the spectral distance between coherent states of the quantum harmonic oscillator as the Euclidean distance on the plane. We investigate the classical limit, showing that the set of coherent states equipped with Connes’ spectral distance tends towards the Euclidean plane as the parameter of deformation goes to zero. The extension of these results to the action of the symplectic group is also discussed, with particular emphasis on the orbits of coherent states under rotations. Second, we compute the spectral distance in the double Moyal plane, intended as the product of (the minimal unitization of) ${\mathcal{A}}$ A by ${\mathbb{C}^2}$ C 2 . We show that on the set of states obtained by translation of an arbitrary state of ${\mathcal{A}}$ A , this distance is given by the Pythagoras theorem. On the way, we prove some Pythagoras inequalities for the product of arbitrary unital and non-degenerate spectral triples. Applied to the Doplicher- Fredenhagen-Roberts model of quantum spacetime [DFR], these two theorems show that Connes’ spectral distance and the DFR quantum length coincide on the set of states of optimal localization.     

Global Existence for the Critical Dissipative Surface Quasi-Geostrophic Equation

In this article, we study the critical dissipative surface quasi-geostrophic equation (SQG) in ${\mathbb{R}^2}$ R 2 . Motivated by the study of the homogeneous statistical solutions of this equation, we show that for any large initial data θ ₀ liying in the space ${\Lambda^{s} (\dot{H}^{s}_{uloc}(\mathbb{R}^2)) \cap L^\infty(\mathbb{R}^2)}$ Λ s ( H ˙ u l o c s ( R 2 ) ) ∩ L ∞ ( R 2 ) the critical (SQG) has a global weak solution in time for 1/2 <  s <  1. Our proof is based on an energy inequality verified by the equation ${(SQG)_{R,\epsilon}}$ ( S Q G ) R , ? which is nothing but the (SQG) equation with truncated and regularized initial data. By classical compactness arguments, we show that we are able to pass to the limit ( ${R \rightarrow \infty}$ R → ∞ , ${\epsilon \rightarrow 0}$ ? → 0 ) in ${(SQG)_{R,\epsilon}}$ ( S Q G ) R , ? and that the limit solution has the desired regularity.     

Simulation and optimization of reflection optical module design for single LED

Illuminance uniformity and illuminating efficiency are always the key problems of light emitting diode (LED) lighting system design. Based on the new design of the reflection optical module, illuminance uniformity and illuminating efficiency are investigated simultaneously in this paper. At the first stage, a reflector with various profile designs is installed to improve the uniformity and efficiency of irradiances collected through the receiver. Using the macro commands, an effective process is presented through the optimal design of the reflector geometries, and the merit function is used as the optical quality objective to find the optimal design value of $K2$ K 2 and $R2$ R 2 parameters of the reflector. At the second stage, based on the optimal design result, the displacement between LED source and reflector is further adjusted automatically by the macro commands too. The results from the ray tracing simulation indicate that the optimal design can be achieved through the compromise of Illuminance uniformity and illuminating efficiency. Finally, we obtain the merit function value of $1.81$ 1.81 with $K2=-1.09, R2=1.11\,\hbox {mm}$ K 2 = - 1.09 , R 2 = 1.11 mm , and the source position $Z1= 1.68\,\hbox {mm}$ Z 1 = 1.68 mm . The results shown in this paper could be beneficial for machine vision systems which are heavily demanded in the light source applications for the uniform and efficient illuminance.     

Almost Commuting Unitary Matrices Related to Time Reversal

The behavior of fermionic systems depends on the geometry of the system and the symmetry class of the Hamiltonian and observables. Almost commuting matrices arise from band-projected position observables in such systems. One expects the mathematical behavior of almost commuting Hermitian matrices to depend on two factors. One factor will be the approximate polynomial relations satisfied by the matrices. The other factor is what algebra the matrices are in, either ${{\bf M}_n(\mathbb{A})}$ M n ( A ) for ${\mathbb{A} = \mathbb{R}}$ A = R , ${\mathbb{A} = \mathbb{C}}$ A = C or ${\mathbb{A} = \mathbb{H}}$ A = H , the algebra of quaternions. There are potential obstructions keeping k-tuples of almost commuting operators from being close to a commuting k-tuple.We consider two-dimensional geometries and so this obstruction lives in ${KO_{-2}(\mathbb{A})}$ K O - 2 ( A ) . This obstruction corresponds to either the Chern number or spin Chern number in physics. We show that if this obstruction is the trivial element in K-theory then the approximation by commuting matrices is possible.     

Asymptotic analysis of perturbed dust cosmologies to second order

Nonlinear perturbations of Friedmann–Lemaitre cosmologies with dust and a cosmological constant $\Lambda >0$ Λ > 0 have recently attracted considerable attention. In this paper our first goal is to compare the evolution of the first and second order perturbations by determining their asymptotic behaviour at late times in ever-expanding models. We show that in the presence of spatial curvature K or a cosmological constant, the density perturbation approaches a finite limit both to first and second order, but the rate of approach depends on the model, being power law in the scale factor if $\Lambda >0$ Λ > 0 but logarithmic if $\Lambda =0$ Λ = 0 and $K<0$ K < 0 . Scalar perturbations in general contain a growing and a decaying mode. We find, somewhat surprisingly, that if $\Lambda >0$ Λ > 0 the decaying mode does not die away, i.e. it contributes on an equal footing as the growing mode to the asymptotic expression for the density perturbation. On the other hand, the future asymptotic regime of the Einstein–de Sitter universe ( $K=\Lambda =0$ K = Λ = 0 ) is completely different, as exemplified by the density perturbation which diverges; moreover, the second order perturbation diverges faster than the first order perturbation, which suggests that the Einstein–de Sitter universe is unstable to perturbations, and that the perturbation series do not converge towards the future. We conclude that the presence of spatial curvature or a cosmological constant stabilizes the perturbations. Our second goal is to derive an explicit expression for the second order density perturbation that can be used to study the effects of including a cosmological constant and spatial curvature.     

Cold, ultracold and Nariai black holes with quintessence

In this paper, we study the properties of the charged black hole surrounded by the quintessence. The solution space for the horizons for various values of the mass $M$ M , charge $Q$ Q , and the quintessence parameter $\alpha $ α are studied in detail. Special focus in given to the degenerate horizons: we obtain cold, ultracold and Nariai black holes which has similar topologies as for the Reissner–Nordstrom-de Sitter black holes. We also study the lukewarm black hole with the quintessence in this paper.     

Low-threshold, high SMSR tunable external cavity quantum cascade laser around 4.7\,\upmu \hbox {m}

Room-temperature pulsed and continuous-wave (cw) operation of a tunable external cavity (EC) quantum cascade laser (QCL) at an emitting wavelength of $4.7\,\upmu \hbox {m}$ 4.7 μ m was presented. The effect of different external cavity lengths and grating angles of the EC–QCL system were analyzed numerically. A wide tuning range greater than $131\,\hbox {cm}^{-1}$ 131 cm - 1 was obtained in pulsed mode at room temperature. Without the anti-reflection coating procedure, single-mode cw operation with a side-mode suppression ratio (SMSR) above 20 dB and a wide tuning range greater than $116\, \hbox {cm}^{-1}$ 116 cm - 1 were achieved. Near the center region, SMSR about 30 dB was also realized through designing the external cavity length. Strain-compensation combined with two-phonon resonance in an active region design and the high-reflection coating promised low threshold current density. A record low threshold current density of $0.901\,\hbox {kA/cm}^{2}$ 0.901 kA/cm 2 for an EC–QCL operated in cw mode was realized.     

One-step hydrothermal synthesis of nitrogen and tungsten codoped TiO2 nanorods with high visible light photocatalytic activity

N,W codoped TiO 2 $\mathrm{TiO}_{2}$ nanorods were synthesized via a one-step hydrothermal method using ammonium metatungstate as the nitrogen and tungstate sources. The prepared samples were characterized by X-ray diffraction (XRD), scanning electron microscopy (SEM), transmission electron microscopy (TEM), UV–visible diffuse reflectance spectroscopy (DRS), and X-ray photoelectron spectroscopy (XPS). The results indicated that the N,W codoped TiO 2 $\mathrm{TiO}_{2}$ nanorods exhibited a higher photocatalytic activity under visible light irradiation compared with P25 and undoped TiO 2 $\mathrm{TiO}_{2}$ , because the codoping of N and W ions not only extended the visible light absorption but also promoted the separation of the photogenerated electrons and holes.     

Multiple Scattering in Random Mechanical Systems and Diffusion Approximation

This paper is concerned with stochastic processes that model multiple (or iterated) scattering in classical mechanical systems of billiard type, defined below. From a given (deterministic) system of billiard type, a random process with transition probabilities operator P is introduced by assuming that some of the dynamical variables are random with prescribed probability distributions. Of particular interest are systems with weak scattering, which are associated to parametric families of operators P _h , depending on a geometric or mechanical parameter h, that approaches the identity as h goes to 0. It is shown that (P _h ? I)/h converges for small h to a second order elliptic differential operator ${\mathcal{L}}$ L on compactly supported functions and that the Markov chain process associated to P _h converges to a diffusion with infinitesimal generator ${\mathcal{L}}$ L . Both P _h and ${\mathcal{L}}$ L are self-adjoint (densely) defined on the space ${L^2(\mathbb{H},\eta)}$ L 2 ( H , η ) of square-integrable functions over the (lower) half-space ${\mathbb{H}}$ H in ${\mathbb{R}^m}$ R m , where η is a stationary measure. This measure’s density is either (post-collision) Maxwell-Boltzmann distribution or Knudsen cosine law, and the random processes with infinitesimal generator ${\mathcal{L}}$ L respectively correspond to what we call MB diffusion and (generalized) Legendre diffusion. Concrete examples of simple mechanical systems are given and illustrated by numerically simulating the random processes.     

Mössbauer spectroscopy in the investigation of new mineral–related materials

New materials based on the composition of the mineral schafarzikite, FeSb $_{2}\textit {O}_{4}$ , have been synthesised. $^{57}$ Fe- and $^{121}$ Sb- Mössbauer spectroscopy shows that iron is present as Fe $^{2+}$ and that antimony is present as Sb $^{3+}$ . The presence of Pb $^{2+}$ on the antimony sites in materials of composition FeSb $_{1.5}$ Pb $_{0.5}\textit {O}_{4}$ induces partial oxidation of Fe $^{2+}_{}$ to Fe $^{3+}$ . The quasi-one-dimensional magnetic structure of schafarzikite is retained in FeSb $_{1.5}$ Pb $_{0.5}\textit {O}_{4}$ and gives rise to weakly coupled non-magnetic Fe $^{2+}$ ions coexisting with Fe $^{3+}$ ions in a magnetically ordered state. A similar model can be applied to account for the spectra recorded from the compound Co $_{0.5}$ Fe $_{0.5}$ Sb $_{1.5}$ Pb $_{0.5}\textit {O}_{4}$ .     

Warm-intermediate inflationary universe model with viscous pressure in high dissipative regime

Warm inflation model with bulk viscous pressure in the context of “intermediate inflation” where the cosmological scale factor expands as $a(t)=a_0\exp (At^f)$ , is studied. The characteristics of this model in slow-roll approximation and in high dissipative regime are presented in two cases: 1—Dissipative parameter $\Gamma $ as a function of scalar field $\phi $ and bulk viscous coefficient $\zeta $ as a function of energy density $\rho $ . 2— $\Gamma $ and $\zeta $ are constant parameters. Scalar, tensor perturbations and spectral indices for this scenario are obtained. The cosmological parameters appearing in the present model are constrained by recent observational data (WMAP7).