Transverse Geometry Research Articles

A new slant on photonic crystal fibers

We present the novel use of microstructured optical fibers not as "light-pipes", but in a transverse geometry to manipulate the light propagating across the fiber. Fundamental and higher-order bandgaps were observed experimentally in this geometry using a number of techniques. The comparison of the measured spectra with photonic band structure and Finite-Difference Time-Domain simulations provide strong evidence that the spectral features are a result of the periodic nature of the fiber microstructure in the transverse direction.

Rankin—Cohen Brackets and the Hopf Algebra of Transverse Geometry

We settle in this paper a question left open in our paper “Modular Hecke algebras and their Hopf symmetry”, by showing how to extend the Rankin-Cohen brackets from modular forms to modular Hecke algebras. More generally, our procedure yields such brackets on any associative algebra endowed with an action of the Hopf algebra of transverse geometry in codimension one, such that the derivation corresponding to the Schwarzian derivative is inner. Moreover, we show in full generality that these Rankin-Cohen brackets give rise to associative deformations.

Modular Hecke Algebras and Their Hopf Symmetry

We introduce and begin to analyse a class of algebras, associated to congruence subgroups, that extend both the algebra of modular forms of all levels and the ring of classical Hecke operators. At the intuitive level, these are algebras of `polynomial coordinates' for the `transverse space' of lattices modulo the action of the Hecke correspondences. Their underlying symmetry is shown to be encoded by the same Hopf algebra that controls the transverse geometry of codimension 1 foliations. Its action is shown to span the `holomorphic tangent space' of the noncommutative space, and each of its three basic Hopf cyclic cocycles acquires a specific meaning. The Schwarzian 1-cocycle gives an inner derivation implemented by the level 1 Eisenstein series of weight 4. The Hopf cyclic 2-cocycle representing the transverse fundamental class provides a natural extension of the first Rankin-Cohen bracket to the modular Hecke algebras. Lastly, the Hopf cyclic version of the Godbillon-Vey cocycle gives rise to a 1-cocycle on PSL(2, Q) with values in Eisenstein series of weight 2, which when coupled with the `period' cocycle yields a representative of the Euler class.

On Harmonic Theory in Flows

AbstractRecently [8], a harmonic theory was developed for a compact contact manifold from the viewpoint of the transversal geometry of contact flow. A contact flow is a typical example of geodesible flow. As a natural generalization of the contact flow, the present paper develops a harmonic theory for various flows on compact manifolds. We introduce the notions of H-harmonic and H*-harmonic spaces associated to a Hörmander flow. We also introduce the notions of basic harmonic spaces associated to a weak basic flow. One of our main results is to show that in the special case of isometric flow these harmonic spaces are isomorphic to the cohomology spaces of certain complexes. Moreover, we find an obstruction for a geodesible flow to be isometric.

Electromagnetic Weibel instability in intense charged particle beams with large energy anisotropy

In plasmas with strongly anisotropic distribution functions, collective instabilities may develop if there is sufficient coupling between the transverse and longitudinal degrees of freedom. Our previous numerical and theoretical studies of intense charged particle beams with large temperature anisotropy [E. A. Startsev, R. C. Davidson, and H. Qin, Phys. Rev. ST Accel. Beams 6, 084401 (2003); Phys. Plasmas 9, 3138 (2002)] demonstrated that a fast, electrostatic, Harris-type instability develops, and saturates nonlinearly, for sufficiently large temperature anisotropy (T⊥b/T∥b≫1). The total distribution function after saturation, however, is still far from equipartitioned. In this paper the linearized Vlasov–Maxwell equations are used to investigate detailed properties of the transverse electromagnetic Weibel-type instability for a long charge bunch propagating through a cylindrical pipe of radius rw. The kinetic stability analysis is carried out for azimuthally symmetric perturbations about a two-temperature thermal equilibrium distribution in the smooth-focusing approximation. The most unstable modes are identified, and their eigenfrequencies, radial mode structure and instability thresholds are determined. The stability analysis shows that, although there is free energy available to drive the electromagnetic Weibel instability, the finite transverse geometry of the charged particle beam introduces a large threshold value for the temperature anisotropy [(T⊥b/T∥b)Weibel≫(T⊥b/T∥b)Harris] below which the instability is absent. Hence, unlike the case of an electrically neutral plasma, the Weibel instability is not expected to play as significant a role in the process of energy isotropization of intense unneutralized charged particle beams as the electrostatic Harris-type instability.

Magnetoresistance of Ga1-xMnxAs Epitaxial Layers Doped by Be

Magnetoresistance (MR) measurements were performed on Ga1-xMnxAs (x=0.03) additionally doped with Be. At low temperatures and in low magnetic fields, a positive MR signal was observed for measurements in the transverse (B⊥I) geometry. However, at high temperatures, the MR becomes negative for all fields and all field orientations. The value of the negative MR has a maximum near the Curie temperature, and its magnitude depends strongly on the concentration of free holes. The observed MR behavior can be described by the magnetoimpurity scattering model in both the paramagnetic and the ferromagnetic temperature regions. Quantitative analysis of the Ga1-xMnxAs MR data yields the value of the p–d exchange energy as |N0β|≈1.6 eV.

Magnetic anisotropy of aligned magnetic stripe domains in FePd studied by soft x-ray resonant magnetic scattering, magnetic force microscopy and micromagnetic modeling

FePd thin-film samples with different perpendicular magnetic anisotropies have been studied with magnetic force microscopy (MFM), micromagnetic calculations and soft x-ray resonant magnetic scattering (SXRMS). The competition between perpendicular magnetic anisotropy (PMA) and shape anisotropy leads to the formation of highly ordered stripe domain patterns with a magnetization component perpendicular to the film plane. The magnetic stripes with a period of ∼100 nm, which are seen in the MFM images and can be modeled by micromagnetic calculations, give rise to magnetic peaks in the diffraction pattern. Closure domains occur in samples with a low to medium PMA, while a high PMA inhibits their formation. The in-plane magnetization component of the closure domains is not observable with MFM. In the presence of closure domains the interference between the scattering amplitudes in SXRMS from perpendicular and in-plane magnetized domains gives rise to a circular dichroism in the transverse geometry, where the scattering plane is along the stripes. We also recorded the magnetic speckle pattern from an 8 μm FePd wire using coherent x-rays and CCD detection. A high degree of coherence was obtained as evidenced from the observed intensity fluctuations. The speckle pattern can in principle provide information about the local disorder of the magnetic stripe domains.

Magnetic anisotropy of aligned magnetic stripe domains in FePd studied by soft x-ray resonant magnetic scattering, magnetic force microscopy and micromagnetic modeling*1

FePd thin-film samples with different perpendicular magnetic anisotropies have been studied with magnetic force microscopy (MFM), micromagnetic calculations and soft x-ray resonant magnetic scattering (SXRMS). The competition between perpendicular magnetic anisotropy (PMA) and shape anisotropy leads to the formation of highly ordered stripe domain patterns with a magnetization component perpendicular to the film plane. The magnetic stripes with a period of ∼100 nm, which are seen in the MFM images and can be modeled by micromagnetic calculations, give rise to magnetic peaks in the diffraction pattern . Closure domains occur in samples with a low to medium PMA, while a high PMA inhibits their formation. The in-plane magnetization component of the closure domains is not observable with MFM. In the presence of closure domains the interference between the scattering amplitudes in SXRMS from perpendicular and in-plane magnetized domains gives rise to a circular dichroism in the transverse geometry, where the scattering plane is along the stripes. We also recorded the magnetic speckle pattern from an 8 μm FePd wire using coherent x-rays and CCD detection. A high degree of coherence was obtained as evidenced from the observed intensity fluctuations. The speckle pattern can in principle provide information about the local disorder of the magnetic stripe domains.

Exact connections between effective magnetostriction and effective elastic moduli of fibrous composites and polycrystals

The effective magnetostriction of a two-phase fibrous composite and a two-dimensional polycrystal assembled from cylindrical magnetostrictive cubic crystals is studied. For the considered systems, we show that there exist exact microstructure-independent relations between the effective magnetostriction and the effective elastic moduli. The fibers could be aligned identically or randomly oriented in the transverse plane. There is no restriction on the cross-sectional shapes of the fibers, nor on the arrangement of transverse geometry of the composite aggregate. These connections imply that knowledge of the effective elastic moduli will readily provide the effective magnetostriction of the composite medium.

MAGNETIC BEHAVIOR OF MAGNESIUM DIBORIDE SUPERCONDUCTING THIN FILMS

We report the experimental results of the field H and temperature T dependent magnetization M measurements of MgB 2 superconducting thin films deposited by pulsed laser deposition. MgB 2 thin films were obtained from a precursor layer deposited in UHV at room temperaure from stoichiometric target, then an annealing treatment in Mg atmosphere allowed to crystallize the superconducting phase. From the hysteresis loops at each temperature of measurement the critical current density Jc(H) has been extracted according to Brandt's model for a transverse geometry. At low temperatures an anomalous behavior has been observed in the hysteresis loops: as the exernal magnetic field approaches the zero field region on the upper and lower branches, a double peak of the magnetization occurs. This magnetic flux jumping can be explained within an adiabatic critical state model, and can be ascribed to thermomagnetic instabilities as in bulk samples.

Conductance fluctuations in a-Si : H: effects of alloying and device structure

We present measurements of conductance noise in undoped a-Si : H and a-SiGe : H thin films in both a transverse and coplanar electrode geometry. For a-Si : H with coplanar electrodes, the noise spectrum is not a pure f −α power law but consists of two linear regions with different slope parameters α. The spectral shape and its temperature dependence are similar for all samples, regardless of the growth technique. Adding Ge results in qualitatively similar spectra; however, α at high frequencies and the temperature dependence are altered. For both a-Si : H and a-SiGe : H with transverse electrodes, the noise spectra are pure f −α power laws, and α decreases with the Ge content.

Detecting spin-polarized currents in ballistic nanostructures.

We demonstrate a mesoscopic spin polarizer/analyzer system that allows the spin polarization of current from a quantum point contact in a large in-plane magnetic field to be measured. A transverse electron focusing geometry is used to couple current from an emitter point contact into a collector point contact. At large in-plane fields, with the point contacts biased to transmit only a single spin (g<e(2)/h), the voltage across the collector depends on the spin polarization of the current incident on it. Spin polarizations of >70% are found for both emitter and collector at 300 mK and 7 T in-plane field.

An isomorphic Fourier transform approach to light propagation in AWGs, FBGs, and photonic crystals

We have identified a comprehensive isomorphism between arrayed-waveguide grating (AWG), fibre Bragg grating (FBG) and photonic crystal (PC) devices, which allows the design characteristics of the former transverse geometry to be mapped on to the latter longitudinal structures. In this paper, we present a unified Fourier transform (FT) approach to the study of these important passive optical grating-based devices.

Exchange coupling in Fe/NiO/Co film studied by soft x-ray resonant magnetic reflectivity

A 10 nm Fe/5 nm NiO/10 nm Co polycrystalline trilayer prepared by magnetron sputtering has been investigated by resonant magnetic reflectivity of linearly polarized soft x rays (transversal magneto-optical Kerr effect geometry) at the L 3 edges of Fe, Ni, and Co. The exchange coupling in the trilayer, introduced during synthesis, has been altered by cooling through the Neel temperature T N of NiO at three different cooling fields H F C . We show that the chemical structure at the interfaces, the grain size and the magnetic properties of the trilayer are governed by factors such as successive heat treatment, magnetic field applied during synthesis and cooling through T N of NiO. Clear evidence for oxidation of Fe and Co states has been obtained, suggestive for the formation of mixed oxides at the Fe/NiO and NiO/Co interfaces. Oxidized Fe and Co are found to carry a net magnetic moment, which couples to the moments of the metallic states. We find that the coupling between oxidized and metallic Co spins is ferromagnetic, whereas the coupling between oxidized and metallic Fe spins is antiferromagnetic. We observe the presence of two types of excess Ni spins. A rigid excess Ni magnetic moment is induced at the interface during synthesis as follows from the strong asymmetry of the x-ray Kerr loops of the oxidized Co spins at H F C = 0 Oe and metallic Co spins at H F C =240 Oe. Heat treatment leads to appearence of a soft excess Ni magnetic moment, which is ferromagnetically coupled to the metallic Fe and Co spins and exhibits a conventional hysteresis loop. Competing interactions lead to frustrated magnetization at the NiO/Fe interface and cause a tilt of the Fe spins away from the direction of applied magnetic field. This manifests itself in the presence of characteristic humps in the Fe Kerr loops.

String tensions and three dimensional confining gauge theories

In the context of gauge/gravity duality, we try to understand better the proposed duality between the fractional D2-brane supergravity solutions of (Nucl. Phys. B 606 (2001) 18, hep-th/0101096) and a confining 2+1 dimensional gauge theory. Based on the similarities between this fractional D2-brane solution and D3-brane supergravity solutions with more firmly established gauge theory duals, we conjecture that a confining q-string in the 2+1 dimensional gauge theory is dual to a wrapped D4-brane. In particular, the D4-brane looks like a string in the gauge theory directions but wraps a S**3 in S**4 in the transverse geometry. For one of the supergravity solutions, we find a near quadratic scaling law for the tension: $T \sim q (N-q)$. Based on the tension, we conjecture that the gauge theory dual is SU(N) far in the infrared. We also conjecture that a quadratic or near quadratic scaling is a generic feature of confining 2+1 dimensional SU(N) gauge theories.

Interaction of light waves of arbitrary polarisation on a photorefractive grating in a cubic gyrotropic crystal in an external alternating electric field

The symmetric two-wave interaction is considered, for small modulation indices of an interference light pattern, on a transmission photorefractive grating with stationary amplitude produced in a crystal of symmetry group 23 in an external alternating electric field. For an arbitrary off-duty ratio of the external field or arbitrary elliptic polarisation of incident light waves, the space-charge field is shown to contain both shifted and unshifted components, while the polarisation state and intensity of a weak light wave change upon the external field switching. The space-charge-field amplitude and the weak-wave intensity gain are analysed as functions of the interaction wavelength, the off-duty ratio of the external field, and the photorefractive-grating spacing for the longitudinal and transverse interaction geometry at a wavelength of 633 nm in a Bi12TiO20 crystal.

Interface Magnetoresistance of Fe/Ag Multilayers

Magnetoresistance and magnetic behaviour of a vacuum evaporated multilayer consisting of 1.4 nm Fe and 2 nm Ag layers were studied. The magnetisation of the multilayer shows normal ferromagnetic behaviour while the resistance decreases both in parallel and transversal geometry up to 12 T magnetic field and can be well described by squared Langevin functions. The giant magnetoresistance (GMR) is attributed to a small amount of Fe clusters and/or single impurities in the Ag matrix. This explanation is supported by similar results on a 8 nm Ag/25 nm Fe/8 nm Ag trilayer. The concentration distribution due to interface mixing was studied by Rutherford backscattering spectrometry (RBS) measurements.

ZnSe-Based Room Temperature Low-Threshold Electron-Beam Pumped Semiconductor Laser

We report on room temperature operation of blue-green ZnSe-based electron-beam pumped semiconductor lasers grown by molecular beam epitaxy. The lowest ever reported values of threshold current density in transverse excitation geometry (3-3.5 A/cm 2 ), which is independent of the electron beam energy in the 15-35 kV range, have been obtained. Possible ways to improve the electron-beam pumped laser characteristics are discussed.

Multiple-order adaptive dispersion compensation using polynomially-chirped grating devices

Devices such as the planar arrayed-waveguide grating or the distributed Bragg reflector (AWG and DBR, respectively) are assuming increasing importance in the areas of fibre point-to-point communication and networking. In the particular context of dense wavelength-division multiplexing (DWDM), these devices play a well-established role as wavelength-selective elements. More recently, chirped variants have found use as dispersion compensators, offering wideband reduction of both basic and higher-order departures from constant group delay. However, up to the present time, the existence of a systematic approach to higher-order dispersion compensation has not been recognised. Additionally, we have identified a comprehensive isomorphism between AWG and DBR devices that allows the design characteristics of the former transverse geometry to be mapped on to the latter longitudinal structure. Here, we present a systematic study of these important passive optical grating-based devices which considers, firstly, analytic solutions for 2nd-order dispersion compensation (DC), and then 3rd- and 4th-order dispersion compensators with 100 GHz bandwidth. We then review optimisation strategies for DC, 3 dB bandwidth, and passband dispersion ripple, as determined by device chirp, apodisation, coupling strength and polyphase transfer function superposition. We conclude with a preliminary discussion of grating parametric sensitivity as evidenced by Monte Carlo analysis.

Influence of microscopic defects in type-II superconducting thin films on the magnetic flux penetration

The magnetic flux penetration into thin type-II superconducting films with circular defects is investigated. The artificial circular defects (diameter $=40 \ensuremath{\mu}\mathrm{m})$ in an ${\mathrm{YBa}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ thin film (thickness $\ensuremath{\approx}300 \mathrm{nm})$ were prepared by pulse-laser irradiation. The flux penetration into the zero-field-cooled superconducting film was visualized by means of the magneto-optic method. A stepwise increase of the external magnetic field allowed a detailed investigation of the influence of local defects on the flux penetration. For a magnetic field parallel to a long sample (longitudinal geometry) with a long cylindrical defect a single parabolic discontinuity line appears. Also in the case of a thin superconducting film exposed to a transverse magnetic field (transverse geometry), a single parabolic discontinuity line has been supposed in the vicinity of a local defect. On the contrary, our investigations show that the flux and current distribution around a single defect in a superconducting thin film can be determined not by a single, but by two discontinuity parabolas. In thin superconducting films in transverse geometry screening currents in the Meissner region $(j&lt;{j}_{c})$ are present in contrast to extended infinitely long samples in the longitudinal geometry. We explain our experimental results by the influence of these Meissner screening currents on the temporal formation of the shape of an approaching flux front.