Frequency-domain Formulation Research Articles

Frequency-domain formulation of photonic crystals using sources and gain

We present a formulation to analyze photonic periodic structures from viewpoints of sources and gain. The approach is based on a generalized eigenvalue problem and mode expansions of sources which sustain optical fields with phase boundary conditions. Using this scheme, we calculate power spectra, dispersion relations, and quality factors of Bloch modes in one-dimensional periodic structures consisting of dielectrics or metals. We also compare the results calculated from this scheme with those from the complex-frequency method. The outcomes of these two approaches generally agree well and only deviate slightly in the regime of low quality factors.

PREDICTION OF ACOUSTIC WAVE PROPAGATION IN A SHALLOW WATER CONFIGURATION USING THE METHOD OF FUNDAMENTAL SOLUTIONS

An efficient and accurate numerical frequency domain formulation is proposed to investigate the 2D acoustic wave propagation within a shallow water region with a rigid bottom and a free surface. The proposed configuration combines different regions that have either a sloping or a flat rigid bottom.The numerical approach used here is based on the method of fundamental solutions (MFS). In this model only the vertical interface between different regions is discretized, as the model incorporates Green's functions that take into account the free water surface and the presence of either a horizontal or sloping rigid bottom.

Comparison of signal to noise ratios from spatial and frequency domain formulations of nonprewhitening model observers in digital mammography

Image quality indices based upon model observers are promising alternatives to laborious human readings of contrast-detail images. This is especially appealing in digital mammography as limiting values for contrast thresholds determine, according to some international protocols, the acceptability of these systems in the radiological practice. The objective of the present study was to compare the signal to noise ratios (SNR) obtained with two nonprewhitening matched filter model observer approaches, one in the spatial domain and the other in the frequency domain, and with both of them worked out for disks as present in the CDMAM phantom. The analysis was performed using images acquired with the Siemens Novation and Inspiration digital mammography systems. The spatial domain formulation uses a series of high dose CDMAM images as the signal and a routine exposure of two flood images to calculate the covariance matrix. The frequency domain approach uses the mathematical description of a disk and modulation transfer function (MTF) and noise power spectrum (NPS) calculated from images. For both systems most of the SNR values calculated in the frequency domain were in very good agreement with the SNR values calculated in the spatial domain. Both the formulations in the frequency domain and in the spatial domain show a linear relationship between SNR and the diameter of the CDMAM discs. The results suggest that both formulations of the model observer lead to very similar figures of merit. This is a step forward in the adoption of figures of merit based on NPS and MTF for the acceptance testing of mammography systems.

Frequency-domain method for discrete frequency noise prediction of rotors in arbitrary steady motion

A novel frequency-domain formulation for the prediction of the tonal noise emitted by rotors in arbitrary steady motion is presented. It is derived from Farassat's ‘Formulation 1A’, that is a time-domain boundary integral representation for the solution of the Ffowcs-Williams and Hawkings equation, and represents noise as harmonic response to body kinematics and aerodynamic loads via frequency-response-function matrices. The proposed frequency-domain solver is applicable to rotor configurations for which sound pressure levels of discrete tones are much higher than those of broadband noise. The numerical investigation concerns the analysis of noise produced by an advancing helicopter rotor in blade–vortex interaction conditions, as well as the examination of pressure disturbances radiated by the interaction of a marine propeller with a non-uniform inflow.

Semi-Analytical Methods for Different Problems of Diffraction-Radiation by Vertical Circular Cylinders

As in the other fields of mechanics, analytical methods represent an important analysis tool in marine hydrodynamics. The analytical approach is interesting for different reasons : it gives reference results for numerical codes verification, it gives physical insight into some complicated problems, it can be used as a simplified predesign tool, etc. This approach is of course limited to some simplified geometries (cylinders, spheres, ...), and only the case of one or more cylinders, truncated or not, will be considered here. Presented methods are basically eigenfunction expansions whose complexity depends on the boundary conditions. The hydrodynamic boundary value problem (BVP) is formulated within the usual assumptions of potential flow and is additionally simplified by the perturbation method. By using this approach, the highly nonlinear problem decomposes into its linear part and the higher order (second, third, ...) corrections. Also, periodicity is assumed so that the time dependence can be factorized i.e. the frequency domain formulation is adopted. As far as free surface flows are concerned, only cases without or with small forward speed are sufficiently simple to be solved semi-analytically. The problem of the floating body advancing in waves with arbitrary forward speed is far more complicated. These remarks are also valid for the general numerical methods where the case of arbitrary forward speed, even linearized, is still too difficult from numerical point of view, and "it is fair to say that there exists at present no general practical numerical method for the wave resistance problem" [9], and even less for the general seakeeping problem. We note also that, in the case of bluff bodies like cylinders, the assumptions of the potential flow are justified only if the forward speed is less than the product of wave amplitude with wave frequency.

Frequency Domain Analysis of Superregenerative Receivers in the Linear and the Logarithmic Modes

Despite their simplicity, analysis and design of superregenerative oscillators is often still based on approximations which provide limited insight into its behavior. In this paper we investigate the superregenerative oscillator in the linear and logarithmic operation modes without any approximations. A frequency-domain formulation allows us to efficiently compute the relevant waveforms both in the time and in the frequency domains. One of the main results is the ability to efficiently predict the exact envelope and the instantaneous phase and frequency of the generated waveforms for sinusoidal input signals. The formulation allows taking into account different amplitude-limiting nonlinearities that are responsible for the logarithmic response. We analyze different superregenerative oscillator designs under various operating conditions. We also provide a comparison of the results with those obtained with other techniques.

Generalized Frequency Domain Formulation of the Switching Frequency for Hysteresis Current Controlled VSI Used for Load Compensation

In this paper, a generalized frequency domain method is proposed to obtain the switching frequency formulation for hysteresis current controlled voltage source inverter (VSI)-based shunt compensator. The formulation obtained from the high-frequency model based on the Tsypkin's method explicitly shows the relation between the maximum switching frequency with the system and design parameters. The shunt compensator has been used for the load compensation in a distribution system for both weak and strong feeder supplying a nonlinear load. The maximum switching frequency has been shown related with hysteresis bandwidth and parameters of the VSI, feeder, and load of the distribution system. It is shown that the feeder and load reactance has a significant effect in determining the maximum switching frequency for the weak feeder distribution systems. However, the maximum switching frequency is mainly dependent upon the shunt compensator parameters for strong feeder distribution systems. The minimum switching frequency in general depends upon the modulation depth of the VSI. The results are verified using as power systems CAD (PSCAD) simulation studies for single-phase load compensation. A laboratory model distribution system has been used for the experimental verification.

Transmission of cochlear distortion products as slow waves: A comparison of experimental and model data

There is a long-lasting question of how distortion products (DPs) arising from nonlinear amplification processes in the cochlea are transmitted from their generation sites to the stapes. Two hypotheses have been proposed: (1) the slow-wave hypothesis whereby transmission is via the transverse pressure difference across the cochlear partition and (2) the fast-wave hypothesis proposing transmission via longitudinal compression waves. Ren with co-workers have addressed this topic experimentally by measuring the spatial vibration pattern of the basilar membrane (BM) in response to two tones of frequency f(1) and f(2). They interpreted the observed negative phase slopes of the stationary BM vibrations at the cubic distortion frequency f(DP) = 2f(1) - f(2) as evidence for the fast-wave hypothesis. Here, using a physically based model, it is shown that their phase data is actually in accordance with the slow-wave hypothesis. The analysis is based on a frequency-domain formulation of the two-dimensional motion equation of a nonlinear hydrodynamic cochlea model. Application of the analysis to their experimental data suggests that the measurement sites of negative phase slope were located at or apical to the DP generation sites. Therefore, current experimental and theoretical evidence supports the slow-wave hypothesis. Nevertheless, the analysis does not allow rejection of the fast-wave hypothesis.

Frequency-domain bridging method for the analysis of wave propagation in damaged structures

This paper proposes a frequency-domain formulation of the bridging multiscale method specifically developed to study the interaction of elastic waves with heterogeneities and defects. The adopted approach utilizes a coarse-scale discretization that captures the global wave behavior, while a localized fine-scale mesh resolves the portion of the domain around the discontinuity. The occurrence of spurious waves at the interface is avoided by imposing proper dynamic compatibility conditions between the two domains. The formulation of such conditions is based on the two scale bridging. The frequency domain implementation of the bridging presented herein simplifies the dynamic compatibility operators between the discretizations which constitutes the major source of computational costs when implemented as part of commonly used time marching schemes. The potentials of the proposed technique are presented in the analysis of propagating elastic waves in several kinds of one- and two-dimensional structures with localized imperfections.

Plate with decentralised velocity feedback loops: Power absorption and kinetic energy considerations

This paper is focused on the vibration effects produced by an array of decentralised velocity feedback loops that are evenly distributed over a rectangular thin plate to minimise its flexural response. The velocity feedback loops are formed by collocated ideal velocity sensor and point force actuator pairs, which are unconditionally stable and produce ‘sky-hook’ damping on the plate. The study compares how the overall flexural vibration of the plate and the local absorption of vibration power by the feedback loops vary with the control gains. The analysis is carried out both considering a typical frequency-domain formulation based on kinetic energy and structural power physical quantities, which is normally used to study vibration and noise problems, and a time-domain formulation also based on kinetic energy and structural power, which is usually implemented to investigate control problems. The time-domain formulation shows to be much more computationally efficient and robust with reference to truncation errors. Thus it has been used to perform a parametric study to assess if, and under which conditions, the minimum of the kinetic energy and the maximum of the absorbed power cost functions match with reference to: (a) the number of feedback control loops, (b) the structural damping in the plate, (c) the mutual distance of a pair of control loops and (d) the mutual gains implemented in a pair of feedback loops.

Time and Frequency Domain Finite Element Models for Axial Wave Analysis in Hyperelastic Rods

The propagation of axial waves in hyperelastic rods is studied using both time and frequency domain Finite Element models. The nonlinearity is introduced using the Murnaghan strain energy function and the equations governing the dynamics of the rod are derived assuming linear kinematics. In the time domain, the standard Galerkin Finite Element Method, Spectral Element Method and Taylor-Galerkin Finite Element Method are considered. A frequency domain formulation based on the Fourier Spectral Method is also developed. It is found that the time domain Spectral Element Method provides the most efficient numerical tool for the problem considered.

Boundary element method for surface nonlinear optics of nanoparticles

We present the frequency-domain boundary element formulation for solving surface second-harmonic generation from nanoparticles of virtually arbitrary shape and material. We use the Rao-Wilton-Glisson basis functions and Galerkin's testing, which leads to very accurate solutions for both near and far fields. This is verified by a comparison to a solution obtained via multipole expansion for the case of a spherical particle. The frequency-domain formulation allows the use of experimentally measured linear and nonlinear material parameters or the use of parameters obtained using ab-initio principles. As an example, the method is applied to a non-centrosymmetric L-shaped gold nanoparticle to illustrate the formation of surface nonlinear polarization and the second-harmonic radiation properties of the particle. This method provides a theoretically well-founded approach for modelling nonlinear optical phenomena in nanoparticles.

Full frequency-domain approach to reciprocal microlasers and nanolasers–perspective from Lorentz reciprocity

We develop a frequency-domain formulation in the form of generalized eigenvalue problems for reciprocal microlasers and nanolasers. While the goal is to explore the resonance properties of dispersive cavities, the starting point of our approach is the mode expansion of arbitrary current sources inside the active regions of lasers. Due to the Lorentz reciprocity, a mode orthogonality relation is present and serves as the basis to distinguish various cavity modes. This scheme can also incorporate the asymmetric Fano lineshape into the emission spectra of cavities. We show how to obtain the important parameters of laser cavities based on this formulation. The proposed approach could be an alternative to other computation schemes such as the finite-difference-time-domain method for reciprocal cavities.

Computation of resonances in photonic crystal nanocavities

[1] The spatial confinement of light is an important prerequisite for the excitation of light-matter interactions and nonlinear optics. Photonic crystal nanocavities can have very high quality factors that lead to the desired confinement. We study the feasibility of a hybrid time and frequency domain formulation for the computation of the resonance frequencies and the external quality factors of photonic crystal nanocavities. The time domain approach yields estimates for quality factor and resonance frequency by a special postprocessing technique as well as an approximate eigenvector. In the eigenvalue computation the approximate results from time domain analysis are used for the accelerated solution of the eigenvalue problem.

Generalized Partial Element Equivalent Circuit (PEEC) Modeling With Radiation Effect

In this paper, a new frequency-domain formulation of the partial element equivalent circuit (PEEC) model incorporating the concept of generalized complex partial inductance and pure real capacitance is introduced for modeling of 3-D structures, to which the radiation effect is not negligible. Unlike conventional PEEC-based models, the proposed formulation accounts for the radiation effect by introducing physically meaningful complex-valued inductors and pure real-valued capacitors. In essence, the imaginary part of such an inductor represents a frequency-dependent radiation resistance. Having introduced the complex inductance, there is no inversion of the complex matrix of coefficients of potential, which is not physically meaningful and inevitably creates negative resistance. It is proven in this paper that the imaginary part of the generalized complex inductance for a short dipole exactly reflects the radiation resistance of the dipole. Several numerical examples are given to validate the proposed theory. The results obtained are in good agreement with those from commercial full-wave EM solvers, showing the potential of this technique for analyzing and designing high-frequency and high-speed electronic devices.

Polarized atomic radiative emission in electric and magnetic fields1This article is part of a Special Issue on the 10th International Colloquium on Atomic Spectra and Oscillator Strengths for Astrophysical and Laboratory Plasmas.

A reduced density matrix approach is employed to provide a general theoretical description of polarized radiative emission during single-photon transitions from bound and auto-ionizing states of many-electron atomic systems in the presence of a general arrangement of static (or quasi-static) electric and magnetic fields. Polarized radiative emission from partially ionized atomic systems can occur as a result of the excitation of the radiating atomic states by electrons or ions with an anisotropic velocity distribution, which can be produced in an electron or ion beam experiment, and in a non-equilibrium plasma environment. Polarized radiative emission can also be produced or modified during the excitation of the atomic system in the presence of electric and magnetic fields, and electromagnetic fields. In electric and magnetic fields, the normally overlapping angular momentum projection components of atomic spectral lines can be substantially shifted from their field-free positions and split into spectroscopically resolvable (and inherently polarized) features. Because of the breakdown of the field-free angular momentum and parity selection rules, otherwise forbidden components of atomic spectral lines can be generated. Using a representation based on the field-free many-electron atomic states, the Stark–Zeeman patterns can be determined by a diagonalization of the atomic Hamiltonian in the presence of electric and magnetic fields. In the density operator approach, account can be taken of the coherent excitation of a particular subspace of the initial atomic bound or auto-ionizing states. A general expression for the matrix elements of the detected-photon density operator is obtained and provides a unified framework for the analysis of the spectral intensity, angular distribution, and polarization of the Stark–Zeeman patterns. From a unified development of time-domain (equation-of-motion) and frequency-domain (resolvent-operator) formulations of the more comprehensive reduced density matrix approach, the non-equilibrium atomic state kinetics and the homogeneous spectral line shapes can be systematically and self-consistently described.

Efficient numerical models for the prediction of acoustic wave propagation in the vicinity of a wedge coastal region

In this paper, numerical frequency domain formulations are developed to simulate the 2D acoustic wave propagation in the vicinity of an underwater configuration which combines two sub-regions: the first one consists of a wedge with rigid seabed and free surface, and the second one is assumed to have a rigid flat bottom and a free flat surface. The problem is solved using two different numerical methods: the Boundary Element Method (BEM) and the Method of Fundamental Solutions (MFS). Two models are developed by using a sub-region technique, where only the vertical interface between sub-regions of different geometries has to be discretized. These formulations incorporate Green's functions that take into account the presence of flat rigid and free surfaces and of a wedge. Green's functions are defined using two approaches: the image source method is used to model the rigid flat bottom and free flat interface, whereas the response provided by the wedge sub-region is based on a normal mode solution. Additionally, a MFS and a BEM model are also implemented which require the discretization of the sloping rigid seabed of the wedge, therefore making use of Green's functions for a rigid flat bottom and a free surface (using the image source method). A detailed discussion on the performance of these formulations is performed, with the aim of finding an efficient formulation to solve the problem. It is found that the model based on the MFS and on the sub-region technique has a significantly lower computational cost and is stable, therefore being the most suitable for the analysis of acoustic wave propagation in the studied configurations.

A stress invariant based spectral method to estimate fatigue life under multiaxial random loading

The present paper aims to propose a frequency-domain formulation of a stress invariant based multiaxial fatigue criterion, called “Projection by Projection” (PbP) approach, suitable for estimating fatigue life in presence of complex multiaxial loadings. The new proposed spectral method is first validated with experimental data from the literature, concerning bending-torsion random loading, showing a general accuracy. Numerical simulations are next used to generate a wide set of bending-torsion random loading with different correlation degrees, variance ratios and frequency content, in order to further validate the new proposed method. Simulations confirm a general quite good accuracy for PbP spectral method and also emphasize its better precision compared to another spectral method, namely the equivalent von Mises stress approach, developed by Preumont and co-workers.

Reconstruction of aperiodic FRI signals and estimation of the rate of innovation based on the state space method

In the absence of observation noise, it is known that it is possible to develop exact sampling schemes for a large class of parametric non-bandlimited signals, namely, certain signals of finite rate of innovation (FRI signals), either periodic or aperiodic, such as streams of Diracs, nonuniform splines or piecewise polynomials. A common feature of such signals is that they have a finite number of degrees of freedom per unit of time and they can be reconstructed from a finite number of uniform samples of the filtered signal. Unfortunately, the accuracy of such reconstruction substantially degrades when the samples are distorted by noise. For the case of periodic FRI signals, good algorithms based on the state space method have been proposed which are robust against noise. However, in the case of aperiodic signals, these algorithms may still fail to accurately reconstruct the signals due to ill-conditioning problems that often arise in the matrices involved. This paper proposes a new reconstruction method for aperiodic FRI signals that is also based on the state space method but has a considerably better numerical conditioning than previous reconstruction algorithms. This advantage is achieved by using a frequency domain formulation.

A high-accuracy pseudospectral full-vectorial leaky optical waveguide mode solver with carefully implemented UPML absorbing boundary conditions

The previously developed full-vectorial optical waveguide eigenmode solvers using pseudospectral frequency-domain (PSFD) formulations for optical waveguides with arbitrary step-index profile is further implemented with the uniaxial perfectly matched layer (UPML) absorption boundary conditions for treating leaky waveguides and calculating their complex modal effective indices. The role of the UPML reflection coefficient in achieving high-accuracy mode solution results is particularly investigated. A six-air-hole microstructured fiber is analyzed as an example to compare with published high-accuracy multipole method results for both the real and imaginary parts of the effective indices. It is shown that by setting the UPML reflection coefficient values as small as on the order of 10(-40) ∼ 10(-70), relative errors in the calculated complex effective indices can be as small as on the order of 10(-12).