Fourier Expansion Method Research Articles

Optical Properties of Dielectric and Magnetic Photonic Crystals in the Low-Frequency Limit

ABSTRACTWe consider the long-wavelength limit for two-dimensional photonic crystals - periodic arrangement of magneto-dielectric rods with dielectric and magnetic constant εa and μa embedded in a magneto-dielectric background (εb,βb). Using the Fourier expansion method in the low-frequency limit (ω → 0 ) we develop an effective medium theory and give a rigorous proof that, in this limit, a periodic medium behaves like a homogeneous one. We derive compact analytical formulas for the effective index of refraction of a 2D photonic crystal. These formulas are very general, namely the Bravais lattice, the cross-sectional form of cylinders, their filling fractions and the dielectric and magnetic constants are all arbitrary. For non-magnetic materials, μa = μb = 1, we show how to introduce index ellipsoid and demonstrate that the E-mode is an ordinary wave and the H -mode is an extraordinary wave. For magnetic materials the both modes turn out to be extraordinary. This unusual property is unknown for natural crystals.

Long-wavelength limit (homogenization) for two-dimensional photonic crystals

Using the Fourier expansion method in the low-frequency limit we develop an effective-medium theory for two-dimensional (2D) periodic composites. We give a rigorous proof that, in this limit, a periodic medium behaves like a homogeneous one and we derive compact analytical formulas for the effective dielectric constants of a 2D photonic crystal, i.e., a periodic arrangement of infinite cylinders. These formulas are very general, namely the Bravais lattice, the cross-sectional form of cylinders, their filling fractions, and the dielectric constants are all arbitrary. So is the direction of propagation of the Bloch wave---out of plane in general, with special attention paid to the limiting cases of propagation in the plane of periodicity and parallel to the cylinders. In the latter case we report a behavior that is qualitatively different from that encountered in natural crystals and in 3D photonic crystals. Namely, for propagation along the cylinder axes, the wave fronts are not plane but rippled, with the distribution of the ripples following the pattern of the 2D Bravais lattice. We also demonstrate that the other long-wavelength optical properties can be described by means of the index ellipsoid. This allows us to apply the classification used in the optics of natural crystals (``crystal optics'') to photonic crystals. Namely, we characterize the photonic crystal entirely in terms of its three ``principal'' dielectric constants. One of these is associated with the direction parallel to the cylinders, and is given simply by the spatially averaged dielectric constant. For the two in-plane principal dielectric constants we derive three representations that are equivalent in principle, however, give rise to different rates of numerical convergence, depending on whether the dielectric constant or its reciprocal have been expanded in a Fourier series (respectively, $``\ensuremath{\varepsilon}$ representation'' and ``\ensuremath{\eta} representation''). Numerical results are given for a uniaxial (biaxial) photonic crystal with square (rectangular) lattice and circular cylinders. We conclude that for dielectric cylinders in air the \ensuremath{\eta} representation leads to much better convergence than the $\ensuremath{\varepsilon}$ representation. The opposite holds for air cylinders in a dielectric. The accuracy is checked by applying Keller's theorems to conjugate structures.

Exterior stokes flows with stick-slip boundary conditions

Steady two-dimensional creeping flows induced by line singularities in the presence of an infinitely long circular cylinder with stick-slip boundary conditions are examined. The singularities considered here include a rotlet, a potential source and a stokeslet located outside a cylinder and lying in a plane containing the cylinder axis. The general exterior boundary value problem is formulated and solved in terms of a stream function by making use of the Fourier expansion method. The solutions for various singularity driven flows in the presence of a cylinder are derived from the general results. The stream function representation of the solutions involves a definite integral whose evaluation depends on a non-dimensional slip parameter λ1. For extremal values, λ1 = 0 and λ1 = 1, of the slip parameter our results reduce to solutions of boundary value problems with stick (no-slip) and perfect slip conditions, respectively.The slip parameter influences the flow patterns significantly. The plots of streamlines in each case show interesting flow patterns. In particular, in the case of a single rotlet/stokeslet (with axis along y-direction) flows, eddies are observed for various values of λ1. The flow fields for a pair of singularities located on either side of the cylinder are also presented. In these flows, eddies of different sizes and shapes exist for various values of λ1 and the singularity locations. Plots of the fluid velocity on the surface show locations of the stagnation points on the surface of the cylinder and their dependencies on λ1 and singularity locations.

AN ANALYSIS OF GROUNDWATER FLOW IN A FINITE REGION WITH A SINUSOIDAL TOP

ABSTRACT In this paper we consider a groundwater flow problem in a semi-infinite vertical region bounded on top by a sloping sinusoidal curve dealt with by Shivakumar et al. (Shivakumar, P.N.; Williams, J.J.; Qiang Ye.; Chuanxiang, Ji. An Analysis of Groundwater Flow in an Infinite Region with Sinusoidal Top. Numer. Funct. Anal. and Optimiz. 2000, 21(1&2), 263–271). We first extend the analysis to the more realistic case of a finite region Ω. The problem is described by an elliptic equation for the hydraulic head under given boundary conditions. The obtained results are in good agreement with a classical result (Toth, J. A Theoretical Analysis of Groundwater Flow in Small Drainage Basins. J. Geophys. Res. 1963, 67, 4795–4812). We extend the problem further by allowing Dirichlet-type boundary conditions on the lower boundary. Our first goal is to obtain a semi-analytical solution. The problem is reduced to an infinite system of linear equations by using a suitable Fourier expansion method, transforming low...

Multiscale analysis of vegetation surface fluxes: from seconds to years

The variability in land surface heat (H), water vapor (LE), and CO 2 (or net ecosystem exchange, NEE) fluxes was investigated at scales ranging from fractions of seconds to years using eddy-covariance flux measurements above a pine forest. Because these fluxes significantly vary at all these time scales and because large gaps in the record are unavoidable in such experiments, standard Fourier expansion methods for computing the spectral and cospectral statistical properties were not possible. Instead, orthonormal wavelet transformations ( OWT) are proposed and used. The OWT are ideal at resolving process variability with respect to both scale and time and are able to isolate and remove the effects of missing data (or gaps) from spectral and cospectral calculations. Using the OWT spectra, we demonstrated unique aspects in three appropriate ranges of time scales: turbulent time scales (fractions of seconds to minutes), meteorological time scales (hour to weeks), and seasonal to interannual time scales corresponding to climate and vegetation dynamics. We have shown that: (1) existing turbulence theories describe the short time scales well, (2) coupled physiological and transport models (e.g. CANVEG) reproduce the wavelet spectral characteristics of all three land surface fluxes for meteorological time scales, and (3) seasonal dynamics in vegetation physiology and structure inject strong correlations between land surface fluxes and forcing variables at monthly to seasonal time scales. The broad implications of this study center on the possibility of developing low-dimensional models of land surface water, energy, and carbon exchange. If the bulk of the flux variability is dominated by a narrow band or bands of modes, and these modes “resonate” with key state and forcing variables, then low-dimensional models may relate these forcing and state variables to NEE and LE.

Interior Stokes flows with stick-slip boundary conditions

Two-dimensional Stokes flows generated by line singularities inside a circular cylinder are studied in the presence of stick-slip boundary conditions. For simplicity, line singularities are assumed to be parallel to the cylinder axis, all axes in the same plane. The interior boundary value problem associated with these flows is solved in terms of a stream function. Analytic solutions are obtained for flows induced by a rotlet, a potential-source and Stokeslets with axes radial (normal) or tangential to the cylinder by the Fourier expansion method. These solutions are used to plot streamline topologies of these flows and the flow patterns are studied as the slip parameter and the locations of the singularities are varied. Eddies of various sizes and shapes appear as the slip parameter is varied. Interesting flow patterns are observed in flows generated by a pair of rotlets. In this case, streamline patterns reveal interesting flow topologies. Some of the flow patterns observed here are similar to that of vortex mixing flows. Interior saddle points are found in these flows for certain values of the slip parameter and locations of the rotlets. The flows induced by a source and a sink and a pair of Stokeslets also exhibit interesting features. The plots of the fluid velocity on the surface of the cylinder show the locations of surface stagnation points, if they exist. A study of the movement of surface stagnation points as the slip parameter and the locations of the singularities are varied shed some light on the qualitative features of the flow patterns. The results presented may be relevant for a variety of applications including vortex mixing and journal bearing flows.

Theoretical study of the dielectric constant in porous sandstone saturated with hydrocarbon and water

Porous sandstone reservoirs generally consist of three components: sand grains, hydrocarbons, and water. This paper proposes a hybrid method to predict the effective dielectric constant of such reservoir rocks using a combination of existing formulas and a simplified Fourier expansion method. Taking into consideration fluid distribution in pore space as well as pore structure, we have obtained the relations between the effective dielectric constant of sandstone and water saturation. The calculation results indicate that the effective dielectric constant of sandstone varies as a function of the water saturation when porosity is above 10%, but it cannot predict well the hydrocarbon content in sandstone with low porosity (/spl les/10%). In this study, two different cases are discussed. In the first, water is continuous and oil is in dispersed form in pore space, and in the second, oil is continuous and water is as a dispersed phase in pore space. Very different results are obtained from the two distributions. In Case 1, the effective dielectric constant approximately linearly increases with water saturation, but in Case 2, it is seen that a typical nonlinear relationship exists between the effective dielectric constant and water saturation. Furthermore, three models with microgeometry are considered, and the calculating result shows that the influence of geometry on the effective dielectric constant is large.

Theoretical simulation of noncontact AFM images of Si(111) √3×√3-Ag surface based on Fourier expansion method

A Fourier expansion method is introduced to simulate the noncontact atomic force microscopy (NC-AFM) images in an efficient way, where three dimensional (3D) tip–surface interaction force distribution calculated by the first-principles density functional (LDA) calculations are fitted to an analytical function. As an example of application of this method, we adopt a Si(111) √3×√3 R30°-Ag surface. Force spectroscopies and 2D images for different tip height are presented.

Multipole analysis for absolute magnetic field measured by multi-probe pulsed-NMR method

For accelerator dipole magnets, the study on multipole coefficients in a local region is useful for understanding magnet characteristics and checking construction procedure. The multi-probe pulsed-NMR method is a candidate for measurement of the local multipole coefficients. For the absolute fields measured by the multi-probe pulsed-NMR method, a rigorous application of the Fourier expansion is found to yield multipole coefficients. It is made clear that the direction of the dipole field is unable to be determined by the Fourier expansion alone. For estimating alignment accuracy of pulsed-NMR probes, the relationship between errors of multipole coefficients and those of probe position is derived by the Fourier expansion method. For applying of multi-probe pulsed-NMR method to accelerator dipole magnets, we designed and fabricated a field measuring system using two probes. The probes had NMR coils wound around glass ampoules with a diameter of about 1.4 mm. The NMR signal was processed with a phase lock loop circuit to produce the NMR frequency. Dipole field uniformity and multipole coefficients were measured at 0.35 T for a 1.5 m long dipole magnet. It was confirmed that the field data by this NMR system had an accuracy of 10/sup -6/.

On the measurement of directional wave spectra at the southern Brazilian coast

The detailed reconstruction of the directional spectrum of wind waves from measurements of the wave field is an essential requirement for several applications, including the numerical modeling of wave evolution. Three reconstruction techniques that provide estimates of the directional distribution function D( f, θ), given the one-dimensional frequency spectrum, are compared using data from a coastal locality at the southern Brazilian coast. The techniques are the maximum entropy method (MEM), the Fourier Expansion Method using a cos 2 type function (FEM cos) and the Fourier Expansion Method using a sech type function (FEM sech). The main patterns of the wave climate at the study site are qualitatively assessed. Three main sea states, including swell, transition between local sea and swell, and directionally bimodal wind sea, are identified. Time series from three events associated with the main sea states provide test cases for inter comparison of the three reconstruction techniques. Maximum entropy estimates of D( f, θ) provide results that are more consistent than those obtained from the two FEM techniques in all cases considered.

Noise-amplitude dependence of the invariant density for noisy, fully chaotic one-dimensional maps.

We present some analytic, nonperturbative results for the invariant density rho(x) for noisy one-dimensional maps at fully developed chaos. Under periodic boundary conditions, the Fourier expansion method is used to show precisely how noise makes rho(x) absolutely continuous and smooths it out. Simple solvable models are used to illustrate the explicit dependence of rho(x) on the amplitude eta of the noise distribution, all the way from the case of zero noise (eta-->0) to the completely noise-dominated limit (eta=1).

Form-birefringence limits of Fourier-expansion methods in grating theory: errata

Form-birefringence limits of Fourier-expansion methods in grating theory: errata

Form-birefringence limits of Fourier-expansion methods in grating theory: arbitrary angle of incidence

In a recent publication [J. Opt. Soc. Am. A13, 1013 (1996)] a modified formulation of a rigorous Fourier-expansion eigenmode method was presented in TM polarization, which produces the correct effective refractive indices of form-birefringent stratified media in the lowest-order approximation. In this communication it is shown that a further refinement is consistent, for any angle of incidence, with the extraordinary nature of the TM-polarized wave in the quasi-static limit.

Azimuthal anisotropy in S + Au reactions at 200 A GeV

Abstract Azimuthal correlations of photons produced at mid-rapidity in 200 A GeV S + Au collisions have been studied using a preshower photon multiplicity detector in the WA93 experiment. The Fourier expansion method has been employd to estimate the event plane via the anisotropy of the event as a function of centrality. The event plane correlation technique has been used to determine the true event anisotropy, beyond the anisotropy which arises due to finite multiplicity. The VENUS event generator with rescattering and proper simulation of the detector response can explain only a portion of the observed anisotropy. The residual anisotropy is found to be of the order of 5% for semi-central collisions. This suggests that directed collective flow of the produced particles is present at SPS energies.

Formula for the calculation of integral cross sections in a Fourier expansion method

A method has been developed to calculate the integral cross sections from the measured generalized oscillator strengths that are fitted by a Fourier expansion. The method has been applied to the e-Xe and e-N{sub 2} scattering problems. Excellent agreement has been obtained with existing measured values for the transitions to 5p{sup 5}({sup 2}P{sub 3/2})6s and 5p{sup 5}({sup 2}P{sub 1/2})6s of Xe at 100 and 500 eV. For the vibrational states, v=1{endash}4, of the b{sup 1}{Pi}{sub u} electronic state of N{sub 2} at 300 eV, good agreement with the data calculated by the Lassettre expansion has also been achieved. {copyright} {ital 1997} {ital The American Physical Society}

Optical Gain Calculation for Strained Quantum Well Lasers by the Fourier Expansion Method

In this paper, we extend the Fourier expansion method to calculate the optical gain of strained quantum well (QW) lasers within the 6×6 Luttinger-Kohn Hamiltonian. The optical matrix elements, in a Fourier expansion framework, can be derived in a simple form by averaging around the axis normal to the well plane. As a numerical example, we calculate the band structure and the optical gain of InxGa1-xAs/InGaAsP/InP QW lasers. We also study the effect of the spin-orbit split-off band coupling on the optical gain. It is shown that the calculated results are in reasonable agreement with those obtained by the conventional finite difference method.

Form-birefringence limits of Fourier-expansion methods in grating theory

Standard formulations of rigorous Fourier-expansion analysis methods of lamellar gratings reduce, irrespective of the state of polarization, to the TE result of the lowest-order theory of form birefringence when only the zeroth-order terms are retained in the field and permittivity expansions. A reformulation is presented that reduces to the correct form-birefringence result also in the TM case. Expressions are given for the effective relative permittivities ∊eff of a subwavelength-period grating with an arbitrary relative-permittivity profile ∊r(x): the lowest-order approximation for ∊eff is given by the zeroth-order Fourier coefficient of ∊r(x) in TE polarization and of 1/∊r(x) in TM polarization.

Magnetic field nonuniformities and NMR of protons diffusing in a porous medium

Magnetic field inhomogeneity can arise either because of an externally applied field gradient or because of spatial variations in magnetic susceptibility. The latter are most important when the solid matrix includes paramagnetic substances and when the uniform applied field, and, consequently, also the Larmor precession frequency are very large. Both types of field inhomogeneity add extra phase shifts to the precessing spins. These phase shifts vary with time and position in a complex and random fashion as a result of the diffusive motion of the spins. We have studied these effects by performing detailed calculations for the case of a fluid filled porous medium with a periodic microstructure. Special attention was devoted to the question of whether the statistical distribution of the phase shifts encountered in a Hahn spin echo experiment or in a Carr-Purcell-Meiboom-Gill (CPMG) spin-echo train can be approximated as a Gaussian. The mean square phase shift is measured in such experiments as an enhanced relaxation rate of the precessing transverse magnetization. We determine this mean square phase shift for periodic composites from the diffusion eigenstates, which were calculated using a previously developed Fourier expansion method. 1 The enhanced relaxation rate depends on the echo spacing time τ in a way that can be correlated with important length scales of the porous microstructure. Those correlations can be extended also to disordered microstructures, like the ones that are found in natural rocks. We compare these theoretically predicted correlations with CPMG measurements performed on protons in laboratory samples of brine saturated sandstone.

Active Flatness Control of a Large-Scale Silicon-Wafer Slicer Cutting a Crystal Ingot. Effect on the Deflection Control of a Air Film Stiffness between the Blade and the Ingot.

The response of an ID sawblade slicing a crystal ingot is studied analytically with the effect of the air film stiffness between the ingot and the blade taken into account. The rotating blade is clamped and stressed initially in the radial direction at the outer periphery, while the inner edge is subjected to both the in-plane stationary and the lateral dynamic slicing loads from the workpiece. The solution is obtained by using the Fourier expansion method, and also the Galerkin method on the governing equations of the system. The distributed air film between the blade and the ingot is approximated by a set of parallel translational springs. Numerical results are presented for an actual SUS 301 blade cutting a 6-diameter silicon ingot at a speed of 1550 rpm. Both the steady-state and the transient response of the blade are obtained. Effects of the spring stiffness and the location on the blade deflection are examined. The results obtained show that the blade deflection is greatly affected by the air film stiffness, and that the flatness of the rotating blade is improved effectively by controlling the air flow rate and increasing the air film stiffness appropriately between the blade and the ingot.

Fitting of the Derivative Voigt ESR Line under Conditions of Modulation Broadening

An analysis is made of the effect of the magnetic-field-modulation amplitude on the Voigt lineshape in electron spin resonance spectroscopy. The problem is solved numerically using the Fourier expansion method. Only the coefficient of the first harmonic in the Fourier expansion, corresponding to the output of the phase-sensitive detector, is analyzed. The modulation-broadened first-derivative Voigt ESR line, here after referred to as the derivative Voigt, is fitted with the sum of a first-derivative Gaussian function and a first-derivative Lorentzian function of equal linewidths to extract the Lorentzian contribution of the line. Fitting to the sum is applicable as long as the amplitude of modulation is less than the true peak-to-peak linewidth of ESR line. Modulation broadening substantially contributes only to the derivative Gaussian component of the derivative Voigt line. The same effect is observed for the modulation-broadened derivative Lorentzian lineshape, which also can be fitted to the sum function. The linewidth of the Lorentzian component of the modulation-broadened derivative Lorentzian is constant if the modulation amplitude is less than the peak-to-peak linewidth. For the modulation-broadened derivative Voigt line, the goodness of the fit depends on the ratio of the Gaussian component to the Lorentzian component of the derivative Voigt line. When the Lorentzian component of the derivative Voigt is dominant, the linewidth of the Lorentzian component can be extracted for a broader range of the modulation amplitudes than when the Gaussian component is dominant.