Gauss Functions Research Articles

Apodized photonic crystal waveguide gratings

Apodized photonic crystal (PC) waveguide gratings are proposed to suppress sidelobes which appear in reflection spectra of usual PC waveguide gratings. By using specific functions (Gauss and Gauss-cosine functions) for the longitudinal refractive index distribution, it is possible to suppress sidelobes in the reflection spectra of PC waveguide gratings efficiently. The apodization is realized by simply changing diameters of dielectric pillars adjacent to the PC waveguide core. It is shown that by using Gauss-cosine functions for the apodization, Bragg frequency of the waveguide grating becomes insensitive to the magnitude of perturbation leading to the possibility of designing waveguide gratings with arbitrarily Bragg frequency and bandwidth by modulating geometrical parameters only.

The ABC of hyper recursions

Each member of the family of Gauss hypergeometric functions f n = 2 F 1 ( a + ε 1 n , b + ε 2 n ; c + ε 3 n ; z ) , where a , b , c and z do not depend on n, and ε j = 0 , ± 1 (not all ε j equal to zero) satisfies a second order linear difference equation of the form A n f n - 1 + B n f n + C n f n + 1 = 0 . Because of symmetry relations and functional relations for the Gauss functions, the set of 26 cases (for different ε j values) can be reduced to a set of 5 basic forms of difference equations. In this paper the coefficients A n , B n and C n of these basic forms are given. In addition, domains in the complex z-plane are given where a pair of minimal and dominant solutions of the difference equation have to be identified. The determination of such a pair asks for a detailed study of the asymptotic properties of the Gauss functions f n for large values of n, and of other Gauss functions outside this group. This will be done in a later paper.

Differential Role of the α1C Subunit Tails in Regulation of the Cav1.2 Channel by Membrane Potential, β Subunits, and Ca2+ Ions

Voltage-gated Ca(v)1.2 channels are composed of the pore-forming alpha1C and auxiliary beta and alpha2delta subunits. Voltage-dependent conformational rearrangements of the alpha1C subunit C-tail have been implicated in Ca2+ signal transduction. In contrast, the alpha1C N-tail demonstrates limited voltage-gated mobility. We have asked whether these properties are critical for the channel function. Here we report that transient anchoring of the alpha1C subunit C-tail in the plasma membrane inhibits Ca2+-dependent and slow voltage-dependent inactivation. Both alpha2delta and beta subunits remain essential for the functional channel. In contrast, if alpha1C subunits with are expressed alpha2delta but in the absence of a beta subunit, plasma membrane anchoring of the alpha1C N terminus or its deletion inhibit both voltage- and Ca2+-dependent inactivation of the current. The following findings all corroborate the importance of the alpha1C N-tail/beta interaction: (i) co-expression of beta restores inactivation properties, (ii) release of the alpha1C N terminus inhibits the beta-deficient channel, and (iii) voltage-gated mobility of the alpha1C N-tail vis a vis the plasma membrane is increased in the beta-deficient (silent) channel. Together, these data argue that both the alpha1C N- and C-tails have important but different roles in the voltage- and Ca2+-dependent inactivation, as well as beta subunit modulation of the channel. The alpha1C N-tail may have a role in the channel trafficking and is a target of the beta subunit modulation. The beta subunit facilitates voltage gating by competing with the N-tail and constraining its voltage-dependent rearrangements. Thus, cross-talk between the alpha1C C and N termini, beta subunit, and the cytoplasmic pore region confers the multifactorial regulation of Ca(v)1.2 channels.

Size-controlled hydroxyapatite nanoparticles as self-organized organic–inorganic composite materials

This paper presents some results concerning the size-controlled hydroxyapatite nanoparticles obtained in aqueous media in a biopolymer matrix from soluble precursors salts. Taking the inspiration from nature, where composite materials made of a polymer matrix and inorganic fillers are often found, e.g. bone, shell of crustaceans, shell of eggs, etc., the feasibility on making composite materials containing chitosan and nanosized hydroxyapatite was investigated. A stepwise co-precipitation approach was used to obtain different types of composites by means of different ratio between components. The synthesis of hydroxyapatite was carried out in the chitosan matrix from calcium chloride and sodium dihydrogenphosphate in alkaline solutions at moderate pH of 10–11 for 24 h. Our research is focused on studying and understanding the structure of this class of composites, aiming at the development of novel materials, controlled at the nanolevel scale. The X-ray diffraction technique was employed in order to study the kinetic of hydroxyapatite formation in the chitosan matrix as well as to determine the HAp crystallite sizes in the composite samples. The hydroxyapatite synthesized using this route was found to be nano-sized (15–50 nm). Moreover, applying an original approach to analyze the (0 0 2) XRD diffraction peak profile of hydroxyapatite by using a sum of two Gauss functions, the bimodal distribution of nanosized hydroxyapatite within the chitosan matrix was revealed. Two types of size distribution domains such as cluster-like (between 200 and 400 nm), which are the habitat of ‘‘small’’ hydroxyapatite nanocrystallites and scattered-like, which are the habitat of ‘‘large’’ hydroxyapatite nanocrystallites was probed by TEM and CSLM. The structural features of composites suggest that self-assembly processes might be involved. The composites contain nanosized hydroxyapatite with structural features close to those of biological apatites that make them attractive for bone tissue engineering applications.

Numerical calculation of cutoff frequencies of optical fibers by a variational technique

A numerical method for calculating cutoff frequencies of modes of weakly guiding optical fibers is described. Starting from a properly constructed field representation, the method follows a variational approach to obtain an eigenvalue problem for the cutoff frequencies which can be easily solved by a standard numerical routine. The field representation uses an expansion in terms of the Laguerre–Gauss functions in the fiber core and limiting form for small arguments of modified Bessel functions in the fiber cladding. Fibers with power-law refractive index profiles and fibers with a profile showing a dip on the axis are analyzed. Results obtained for modes of arbitrary order (both azimuthal and radial) show that our method is very efficient and accurate.

Valeurs Exceptionnelles de Fonctions Hyperg�om�triques d?Appell (Exceptional Values of Appell Hypergeometric Functions)

In his article [18], J. Wolfart studied the following exceptionnal set \({\cal E}:=\{x\in\bar{\rm Q} : F(a;b;c;x)\in\bar{\rm Q}\}\) where F is the classical, or Gauss hypergeometric function. The first aim of the present article is to describe the exceptional set in the case of Appell hypergeometric functions, which are a generalization to two variables of the Gauss functions. The link will then be made between, on the one hand, the distribution of complex multiplication points (described by Appell function in the article [5] of P. Cohen and J. Wolfart) on a fixed modular variety, using a Andre-Oort conjecture, and on the other hand, the arithmeticity of the monodromy group related to this function. Lastly, we will see how the localization of certain complex multiplication points leads to the transcendance of the values of Appell hypergeometric functions, at algebraic points.

Analysis of EEG and MEG Signals with Matching Pursuit

Introduction: A wide spectrum of methods has been used for analysis of biomedical signals. A relatively new and promising approach to signal analysis is matching pursuit (MP). The main advantage of the MP algorithm is its ability to create a concise description of an analyzed signal with a relatively small number of components localized in the time-frequency space with maximal possible, adaptive resolution. We applied MP to real MEG and EEG signals in order to present its possible applications. Methods: MP creates signal approximations based on a linear combination of a small number of vectors (atoms) chosen from a bigger, redundant set (dictionary). The atoms are iteratively chosen in order to best match the signal structure. In the first iteration the best matching atom is subtracted from the signal. This creates a signal residuum. The next step is to find the best matching atom with the residuum. This procedure is repeated until the desired approximation energy level is reached. With a dictionary of Gabor time-frequency atoms (scaled, translated and modulated Gauss functions) MP is an adaptive time-frequency transformation. The Gabor atoms are described by five parameters. These parameters localize the atom in the time-frequency space. We applied MP to real MEG and EEG data. Neuromagnetic fields were recorded over the contralateral somatosensory cortex with a 31-channel biomagnetometer. Constant current 0.2 ms square wave pulses were delivered to the right wrist at a stimulation rate of 4Hz. The individual sensory and motor thresholds were determined and the stimulation strength was set to a value equal to the sum of sensory and motor thresholds. Simultaneously electric scalp potentials were recorded using 32 electrodes. We analyzed the so-called 600Hz component and filtered out artefact components. Conclusions: The description of the 600Hz component obtained using MP was consistent with the one found in a previous study, where a Gabor filter was used for the analysis of the same data. The advantage of MP is that the signals were well described with only 15 components. The parameterization of the signals makes further processing very simple. The possible applications of MP are filtering, i.e., artefact rejection (stimulus artefacts or 50Hz powerline noise) or separation of the signal components that are of interest (e.g., 600Hz components) and automatic signal recognition through detection of characteristic signal components or component groups.

THE MODIFIED TETRAHEDRON EQUATION AND ITS SOLUTIONS

A large class of 3-dimensional integrable lattice spin models is constructed. The starting point is an invertible canonical mapping operator [Formula: see text] in the space of a triple Weyl algebra. [Formula: see text] is derived postulating a current branching principle together with a Baxter Z-invariance. The tetrahedron equation for [Formula: see text] follows without further calculation. If the Weyl parameter is taken to be a root of unity, [Formula: see text] decomposes into a matrix conjugation operator R1,2,3 and a c-number functional mapping [Formula: see text]. The operator R1,2,3 satisfies a modified tetrahedron equation (MTE) in which the "rapidities" are solutions of a classical integrable Hirota-type equations. R1,2,3 can be represented in terms of the Bazhanov-Baxter Fermat curve cyclic functions, or alternatively in terms of Gauss functions. The paper summarizes several recent publications on the subject.

Implementing a dynamical ammonia emission parameterization in the large‐scale air pollution model ACDEP

Currently, very simple parameterizations for the seasonal variation of the ammonia (NH3) emissions are used in most air pollution models. This limits the capability to reproduce observed seasonal variations in the NH3 concentrations in rural regions with agricultural activity. A dynamical parameterization of the temporal variation of NH3 emission for application into air pollution models has therefore been proposed. The obtained improvements are demonstrated by implementing a dynamical NH3 emission parameterization into the large‐scale air pollution model ACDEP. Meteorology, information about agricultural practice, and a simple crop growth model are the basis for the parameterization. Fifteen additive functions are used to describe the different parts of the NH3 emissions from agriculture, and a 16th function is used to describe NH3 emission from catalysts used in road traffic. The new parameterization is tested by comparison of model results with NH3 measurements from Danish monitoring stations situated in agricultural areas during the years 1999–2001. Improvements in correlation coefficient of calculated diurnal mean concentrations from 0.42 to 0.70 have been achieved, and a monthly correlation coefficient up to 0.98 has been obtained for the Tange station in 2000. This validation shows that the new parameterization gives a substantial improvement in the model calculations of the NH3 concentrations in areas with high agricultural activities. Finally, the validation showed that optimum results are obtained by including a temperature correction on the gauss functions representing the emission sources.

Truncation rules for modelling discontinuities with Galerkin method in electromagnetic theory

It is well known that apparently similar discretization schemes of Maxwell's equations in Fourier series may provide very different convergence performances because of truncation. We argue that this work performed in grating theory can be applied to other electromagnetic theories relying on expansions over series different from Fourier series. This generalization is supported by an intuitive argument and by a simple numerical example with Hermite–Gauss functions.

Characterization of monoclinic crystals in tablets by pattern-fitting procedure using X-ray powder diffraction data

The purpose of this study is to characterize the monoclinic crystals in tablets by using X-ray powder diffraction data and to evaluate the deformation feature of crystals during compression. The monoclinic crystals of acetaminophen and benzoic acid were used as the samples. The observed X-ray diffraction intensities were fitted to the analytic expression, and the fitting parameters, such as the lattice parameters, the peak-width parameters, the preferred orientation parameter and peak asymmetric parameter were optimized by a non-linear least-squares procedure. The Gauss and March distribution functions were used to correct the preferred orientation of crystallites in the tablet. The March function performed better in correcting the modification of diffraction intensity by preferred orientation of crystallites, suggesting that the crystallites in the tablets had fiber texture with axial orientation. Although a broadening of diffraction peaks was observed in acetaminophen tablets with an increase of compression pressure, little broadening was observed in the benzoic tablets. These results suggest that “acetaminophen is a material consolidating by fragmentation of crystalline particles and benzoic acid is a material consolidating by plastic deformation then occurred rearrangement of molecules during compression”. A pattern-fitting procedure is the superior method for characterizing the crystalline drugs of monoclinic crystals in the tablets, as well as orthorhombic isoniazid and mannitol crystals reported in the previous paper.

The general expression for the transition amplitude of two-photon ionization ofatomic hydrogen

Two-photon ionization of atomic hydrogen with an excess photon is revisited. Thenon-relativistic dipole approximation and Coulomb Green function (CGF)formalism are applied. Using the CGF Sturmian expansion straightforwardly, onegets the radial transition amplitude in the form of an infinite sum over Gausshypergeometric functions which are polynomials. It is convergent if allintermediate states are in the discrete spectrum. In the case of two-photonionization with an excess photon, when photoionization is also possible,intermediate states are in the continuum. We performed the explicit summationover intermediate states and got a simple general expression for the radialtransition amplitude in the form of a finite sum over Appell hypergeometricfunctions, which are not polynomials. An Appell function may be expressed as aninfinite sum over Gauss functions. In the case of ionization by an excess photon,Gauss functions are transformed to give a convergent radial transitionamplitude for the whole region. The generalized cross sections for two-photonabove-threshold ionization of atomic hydrogen in the ground state and excitedstates calculated by us agree very well with results of previous calculations.Generalized cross sections for two-photon ionization of positronium in theground state are obtained by scaling those for atomic hydrogen.

Adaptable method for modelling optical fields in VCSELs

A quasi-analytic formulation based on Laguerre–Gauss functions (LG-Fs) that appears to be suitable to analyse vertical-cavity surface-emitting lasers (VCSELs) is described. The paper focuses on demonstrating the attractive feature of LG-Fs in solving for optical field propagation in both diffracting type and guided-wave type configurations, which is particularly advantageous for modelling present-day VCSELs. The paper also outlines the LG-F expansion formalism to solve for both the optical field and the carrier density distribution to achieve a self-consistent VCSEL model.

Application of the rotation operator in the design of light-scattering experiments

The standard analysis of light-scattering experiments, which presumes single scattering, cannot be applied in highly turbid systems because of the presence of multiply scattered light. Cross-correlation techniques, however, enable to determine the intensity and the correlation function of singly scattered light even in the presence of multiple scattering. We present here a novel theoretical assessment of the influence of experimental conditions (intensity distribution of the laser beam, aperture of the fiber optics, scattering geometry) on the scattering intensity, and on the amplitude of the cross-correlation function for the three-dimensional (3D) coding cross-correlation experiment. The theory makes use of the rotation operator when transforming vectors and the coefficient matrices of the Gauss functions which represent the intensity distribution of the laser and the aperture function of the fiber optics. Rather simple analytical results are obtained for the scattering volume, the polarization of the scattered light, the overlap volume and the amplitude of the correlation function in the cross-correlation experiment. The effects of misalignment are assessed in a general way. The optimal setup is characterized by a small incidence angle of the laser beams (≈ 1°), equality of the intensities of the incident beams, and by equality of the widths of the laser beams of the apertures of the detection systems. Under those conditions the amplitude of the intensity cross-correlation function can approach the optimum of 1 4 for 90° scattering and shows a plateau in a wide region of the goniometer angles.

Luminance and chromatic cues in a spatial integration task

These experiments explore the way in which cues provided by luminance and chromatic contrast interact in the spatial integration of elements. The stimuli were composed of bidimensional and isotropic Gauss functions. The elements were placed so that when experimentally manipulating the separations between the lines, subjects could generate an oriented percept from the elements sharing luminance or chromaticity. Results showed that, in most cases, grouping elements that share chromatic content is possible, in spite of variations in luminance content. Grouping elements as a function of luminance is more difficult when chromaticity alternates from one element to another. Lastly, if competing groupings are generated, the stimulus is structured as a function of chromatic content and not of luminance content.

Investigation of the beam dependence of a fiber-optic voltage sensor by use of a LiNbO_3 crystal

To investigate the beam dependence of a fiber-optic voltage sensor by using a LiNbO3 crystal, I undertook a series of calculations of various beam conditions expressed as Gauss functions by using the plane-wave method, treating each linear electro-optic term as a perturbation of the birefringence of a uniaxial crystal caused by an off-axis beam. The approximation proved to be accurate to within 0.2%. I discovered that the performance of the sensor is closely dependent on the beam angle from the z axis and the beam angle on the x–y plane but that its fabrication requires an impractical degree of control of the collimated beam. I also found the optimal uncollimated beam condition, which provides immunity to temperature changes and variations in the direction of the propagating beam, compensating for the temperature dependence of the quartz 1/4 wave plate. The experimental data support the simulated results.

Experimental determination of the instrumental transparency function of texture goniometers

The instrumental transparency functions of five commercially available texture goniometers were measured experimentally with six monocrystalline samples cut in different orientations from a large highly perfect silicon crystal with a rocking curve of less than 0.01°. Transparency functions were measured in steps of 0.02 to 0.2° in the pole-figure angles α, β. The window size Δα depends on the Bragg angle θ in the form 1/sinθ; the window size Δω is constant for each goniometer. The dominant instrumental parameter determining the long axis Δα of the pole-figure window is the axial width of the detector entrance slit. This parameter is smallest for area detectors (smaller by more than an order of magnitude compared with conventional scintillation detectors as well as one-dimensional position-sensitive detectors). The main features of the pole-figure window and their dependence on the instrumental parameters can be deduced fairly well from a simple geometrical model. The particular shapes of the transparency functions of the studied goniometers are markedly different. Particularly, they are not very well represented by Gauss functions. The two-dimensional transparency function can be fairly well characterized by its α and β profiles. The normalized profiles are virtually independent of the goniometer angles 2θ and α. The increasing size of the pole-figure window with decreasing θ puts a lower limit on the Bragg angle below which pole-figure measurement ceases to be meaningful.

Approximate description for Bessel, Bessel–Gauss, and Gaussian beams with finite aperture

An approximate analysis is derived for the propagation of Bessel, Bessel–Gauss, and Gaussian beams with a finite aperture. This treatment is based on the fact that the circ function can be expanded into an approximate sum of complex Gaussian functions, so that these three beams are typically expressed as a combination of a set of infinite-aperture Bessel–Gauss beams. Correspondingly, the evaluation of the diffracted field distribution of the beams is reduced to the summation of Bessel–Gauss functions. From analytical results, the present approach provides a good description of the diffracted beams in the region far (greater than a factor of the Fresnel distance) from the aperture. A possible extension of this method to other apertured beams is also discussed.

A new discrete fractional Fourier transform based on constrained eigendecomposition of DFT matrix by Lagrange multiplier method

This paper is concerned with the definition of the discrete fractional Fourier transform (DFRFT). First, an eigendecomposition of the discrete Fourier transform (DFT) matrix is derived by sampling the Hermite Gauss functions, which are eigenfunctions of the continuous Fourier transform and by performing a novel error-removal procedure. Then, the result of the eigendecomposition of the DFT matrix is used to define a new DFRFT. Finally, several numerical examples are illustrated to demonstrate that the proposed DFRFT is a better approximation to the continuous fractional Fourier transform than the conventional defined DFRFT.

Bloch and Wannier functions composed of Gaussian orbitals—analyticity, localization, and expectation values

We discuss Bloch and Wannier functions composed of Gaussian orbitals. The fact that the Bloch functions may be expressed in terms of θ-functions is used to discuss them in detail. We study not only their analyticity properties, but also features of expectation values of some specifically chosen Hamiltonians. We also investigate the localization properties of various choices of Wannier functions and compare them with the original Gauss functions. Likewise, we discuss in a more general context the relationship between the analyticity properties of Bloch functions and the localization properties of corresponding Wannier functions.