Expressions For Green Functions Research Articles

Different perspectives on predicting the thermoacoustic energy conversion response in a Rijke tube

In this work, six different perspectives on characterizing the thermoacoustic field in an open-ended Rijke tube are considered and discussed. These begin with a three-pronged approach consisting of theoretical, experimental, and numerical investigations of the Rijke tube's time-dependent field. It is followed by a discussion of effective techniques that rely on either Green's function or differential equation models. Finally, a perturbation expansion is introduced that leverages a naturally occurring small parameter in the open tube configuration. This approach is shown to produce accurate predictions of pressure modal shapes and frequencies for an arbitrary specified temperature distribution. It also leads to a set of linear partial differential equations that can be solved in conjunction with a Green's function expression for the thermoacoustic pressure, velocity, and heat oscillations. In this study, the underlying framework is presented and evaluated for the pressure disturbance only. Another fundamental result includes a similarity parameter, coined the Rijke number, which plays an essential role in driving thermoacoustic oscillations, namely, by relating heat-flux fluctuations to the acoustic velocity and pressure. In this context, we find that the peak value of the energy-flux vector modulus, which stands for the modular product of acoustic velocity and pressure, does indeed occur at the heat source location and increases with the heat power input.

A generalised approach to calculate various transport observables for a linear array of series and parallel quantum dots

A systematic generalised approach to find transport observables for a linear array of different quantum dot (QD) systems has been discussed, using non-equilibrium Green function (NEGF) formalism, in the presence of on-dot Coulomb interaction and inter-dot tunnelling. The equation of motion (EOM) method has been used to derive expressions for Green functions (GFs) within the simplest mean-field approximation to tackle the Coulomb correlation term. Starting from the mathematical structures of GFs for single, double and triple quantum dot systems, the expressions for GFs and transport observables have been generalised for the quantum dot systems containing N number of quantum dots in series as well as parallel linear array of dots. Further, the formulae so obtained have been used for numerical calculations of transmission probability and the I–V characteristics of linear arrays of quantum dots in series as well as parallel configuration containing up to three dots. The results show that, with the increase in number of dots in the scattering region, transmission probability and electron current decrease in series case, while both quantities increase in parallel configuration of dots. The inter-dot tunnelling leads to the splitting of transmission peaks in double QD system in series case whereas, it induces Fano effect in triple QD system in parallel configuration.

Consistent expressions for the free-surface Green function in finite water depth

Fast and accurate computation of the free-surface Green function is of key importance for the numerical solution of linear and second-order wave-structure interaction problems in three dimensions. Integral and series expressions for the Green function are derived for which the limiting values for zero and infinite frequency are consistent with the zero and infinite frequency Green function defined in terms of infinite series of Rankine image sources. The integral expressions presented here have the advantage that they are slowly varying with the non-dimensional wave frequency, making them more efficient to approximate compared with previous expressions.

Tuning the coupling between quantum dot and microdisk with photonic crystal nanobeam cavity.

Strong coupling between solid-state quantum emitters and microcavities paves the way for optical coherent manipulation of quantum state and provides opportunities for quantum information processing. However, it is still a challenge to realize strong coupling due to the spectral and spatial mismatch between quantum emitters and cavity modes. Here, we propose a scheme to tune the coupling between a single QD and a microdisk with 1D photonic crystal nanobeam cavity. Based on Finite-Difference Time-Domain (FDTD) method and Green's function expression for the evolution operator, we demonstrate that QDs with emission wavelengths +1.27 nm and -1.44 nm detuned from the bare microdisk mode can be coupled to the system strongly. Particularly, we observe simultaneous coupling between QD and two cavity supermodes, which enriches the optical coherent control methods of quantum states. By adjusting the distance between the two cavities, we can control the coupling between QD and photons. Furthermore, benefiting from the natural integration of nanobeam cavity to waveguide, such a system provides advantages for implementing quantum internet.

Level shift and decay dynamics of a quantum emitter around a plasmonic nanostructure

We put forward a general approach for calculating the quantum energy level shift for emitter in arbitrary nanostructures, in which the energy level shift is expressed by the sum of the real part of the scattering photon Green function (GF) and a simple integral about the imaginary part of the photon GF in the real frequency range without principle value. Compared with the method of direct principal value integral over the positive frequency axis and the method by transferring into the imaginary axis, this method avoids the principle value integral and the calculation of the scattering GF with imaginary frequency. In addition, a much narrower frequency range about the scattering photon GF in enough to get a convergent result. It is numerically demonstrated in the case for a quantum emitter (QE) located around a nanosphere and in a gap plasmonic nanocavity. Quantum dynamics of the emitter is calculated by the time domain method through solving Schr\"{o}dinger equation in the form of Volterra integral of the second kind and by the frequency domain method based on the Green's function expression for the evolution operator. It is found that the frequency domain method needs information of the scattering GF over a much narrower frequency range. In addition, reversible dynamics is observed. These findings are instructive in the fields of coherent light-matter interactions.

Breakdown of separability due to confinement

A simple system of two particles in a bidimensional configurational space S is studied. The possibility of breaking in S the time-independent Schrodinger equation of the system into two separated one-dimensional one-body Schrodinger equations is assumed. In this paper, we focus on how the latter property is countered by imposing such boundary conditions as confinement to a limited region of S and/or restrictions on the joint coordinate probability density stemming from the sign-invariance condition of the relative coordinate (an impenetrability condition). Our investigation demonstrates the reducibility of the problem under scrutiny into that of a single particle living in a limited domain of its bidimensional configurational space. These general ideas are illustrated introducing the coordinates X c and x of the center of mass of two particles and of the associated relative motion, respectively. The effects of the confinement and the impenetrability are then analyzed by studying with the help of an appropriate Green's function and the time evolution of the covariance of X c and x . Moreover, to calculate the state of a single particle constrained within a square, a rhombus, a triangle and a rectangle, the Green's function expression in terms of Jacobi θ 3 -function is applied. All the results are illustrated by examples.

Semiclassical construction of Green's functions for multidimensional systems with barrier tunneling

We develop a treatment that divides phase space into regions bounded by surfaces and forms the energy-dependent semiclassical Green's function from short classical trajectory segments confined to the individual regions. We use this to generalize Bogomolny's quantization method to systems with barrier tunneling. Combined with an appropriate treatment of tunneling amplitudes, this yields a unified semiclassical framework that uses ordinary, real trajectories in classically allowed regions and imaginary-time trajectories in forbidden regions to obtain energies, decay rates, and wave functions for arbitrary states of multidimensional systems undergoing tunneling. Because the trajectory segments in the allowed region are short, components in the Green's function expression can be numerically evaluated using semiclassical initial value methods, yielding a promising approach for practical calculations.

Local spin excitations in the rectangular ferromagnetic semiconductor nanowires

A Green function analysis is used to study spin-waves excitations in a ferromagnetic semiconductor nanowires. The expressions of Green function for different spins of ferromagnetic nanowires are derived. The nanowire is modeled as having a cubic cross section. The results are illustrated numerically for a particular choice of parameters

Mathematic Model of Green Function with Two-Dimensional Free Water Surface

Adopting complex number theory, a mathematic model of Green function is built for two dimension free water surface, and an analytic expression of Green function is obtained by introducing two parameters. The intrinsic properties of Green function are discussed on vertical line and horizontal line. At last, the derivation expression of Green function is obtained from the formula of Green function.

Inherent Properties of Two Dimension Green Function with Linear Boundary Condition of Free Water Surface

A mathematic model of Green function is build for two dimension free water surface. The analytic expression of Green function is obtained by introducing a parameter of complex number. The intrinsic properties of Green function are discussed for the special parameter values. The real and imaginary parts of H function are shown in the paper.

Efficient Closed Form Cut-Off Planes and Propagation Planes Characteristics for Dielectric Slab Loaded Boundary Value Problems

The geometrical relationship between the cut-off and propagating planes of any waveguide system is a prerequisite for any design process. The characterization of cut-off planes and optimisation are challenging for numerical methods, closed-form solutions are always preferred. In this paper Maxwells coupled field equations are used to characterise twin E-plane and H-plane slab loaded boundary value problems. The single mode bandwidths and dispersion characteristics of these structures are presented and critically compared. The impact of slab mobility upon cut-off and propagations planes has been envisaged. The presented formulation has been extended further to derive a vectorized Green function expression linking electric and magnetic fields for the characterization of planar waveguide structures.

LOCALIZATION OF NONLOCAL LAGRANGIANS AND MASS GENERATION FOR NON-ABELIAN GAUGE FIELDS

We introduce and study the four-dimensional analogue of a mass generation mechanism for non-Abelian gauge fields suggested in the paper, Phys. Lett. B403, 297 (1997), in the case of three-dimensional space–time. The construction of the corresponding quantized theory is based on the fact that some nonlocal actions may generate local expressions for Green functions. An example of such a theory is the ordinary Yang–Mills field where the contribution of the Faddeev–Popov determinant to the Green functions can be made local by introducing additional ghost fields. We show that the quantized Hamiltonian for our theory unitarily acts in a Hilbert space of states and prove that the theory is renormalizable to all orders of perturbation theory. One-loop coupling constant and mass renormalizations are also calculated.

Exact kinetic transport equation solutions in the particle propagation theory in the scattering medium

Charged particle kinetics in an inhomogeneous medium (stochastic magnetic field) is investigated. Exact analytic expressions for Green function of kinetic equation in relaxation-time approximation are derived in one and three dimensions with arbitrary particle absorption. We separately consider the case of isotropic particle injection as well as the case of unidirectional instantaneous particle injection. The new way of solution makes it possible to get off any Cauchy-Principal Value integrals in some solutions which arise in the inverse Fourier–Laplace transform. Weak scattering regime and diffusion approximation is considered, and particle density is derived in three dimensions and arbitrary particle source.

Green function method for ferromagnetic superlattice

Abstract The expression of Green function for different layers in a ferromagnetic superlattice is derived by the recurrence relations technique. The elementary unit cell of the superlattice under consideration consists of alternating layers of two simple-cubic Heisenberg ferromagnets. The results are illustrated numerically for a particular choice of parameters.

Closed-form Expressions for Layered Media Green's Functions That are Reliable Both in the Near Field and in the Far Field

The authors have developed an algorithm for the determination of accurate closed- form expressions of Green's functions for multilayered media. The algorithm is based on the fltting of the spectral domain Green's functions in terms of an asymptotic term plus a ratio of two polynomials. The fltting is carried out via the method of total least squares, which is a non-iterative method demanding very few computational resources. The resulting closed-form expressions for the Green's functions consist of a quasistatic term plus a sum of cylindrical waves. The quasistatic term provides the near-fleld behavior and the cylindrical waves provide the farfleld behavior. As a consequence of this, the closed-form expressions are very accurate irrespective of the distance between the source point and the observation point. DOI: 10.2529/PIERS060807124124 The application of the method of moments (MoM) to the solution of mixed potential integral equations (MPIE) has proven to be an e-cient numerical tool for the analysis of planar circuits and antennas (1). In fact, several commercial software products that are currently used in the design of planar structures (such as \Ansoft Ensemble, \Zeland IE3D, and \Agilent Momentum) are based on the solution of MPIE by means of the MoM. In order to solve the MPIE arising from the analysis of planar structures, it is necessary to calculate the Green's functions (GF) for the scalar and vector potentials in multilayered media. These GF can be determined via numerical computation of inflnite integrals that are commonly known as Sommerfeld integrals (SI). However, the highly-oscillatory nature of the integrands involved makes the numerical computation of SI cumbersome and time consuming (1). Some researches have developed speciflc techniques that make it possible to accelerate the nu- merical computation of SI (2,3). Unfortunately, these techniques have the drawback that they have to be repeatedly used every time the distance between the source and observation points changes. One technique for the computation of multilayered media GFs that has received much attention in the last flfteen years is the discrete complex image method (DCIM) (4). This technique makes it possible to obtain closed-form expressions of the GFs without requiring numerical integration. However, the closed-form expressions have been recently found to produce unpredictable errors in the far-fleld computation of the GFs (5). Although it is possible to raise the threshold value of the distance between source and observation points that marks the onset of numerical errors, this is achieved at the expense of a considerable CPU time consumption (5). The sensitivity of the DCIM to far-fleld numerical errors seems to decrease when a surface waves term is included in the closed-form expression of the GFs (4,6). Unfortunately, the surface-waves term contains poles of the spectral domain GFs as well as residues of the spectral domain GFs at these poles, and the accurate determination of both the poles and the residues requires time-consuming algorithms (6). Very recently, the rational function fltting method (RFFM) has been proposed as an alternative technique to DCIM for the determination of closed-form expressions of multilayered media GFs (7). In the new technique the GFs are expressed in terms of a quasistatic term plus a linear com- bination of cylindrical waves. The amplitudes and propagation constants of the cylindrical waves are obtained in the spectral domain by means of a vector fltting algorithm, which is an iterative algorithm. The main drawback of the RFFM is that the closed-form expressions for the GFs in- troduce unacceptable errors in the near-fleld (7). These errors are caused by the Hankel functions representing the cylindrical waves, and in particular, by the non-physical near-fleld singularities present in these Hankel functions. In this paper, we present a revised version of the RFFM which substantially improves the original approach published in (7). In one hand, the determination of

Modified spontaneous emission from a two-dimensional photonic crystal

Decay kinetic properties of a two-level atom near the band edges of a two-dimensional photonic crystal are studied based on the Green's function expression for the evolution operator. Our calculations show that the decay kinetics can be controlled by changing the atomic parameters. We also find that there is a quite wide lifetime distribution in the spontaneous emission process.

Quantum pumping and dissipation: From closed to open systems

Current can be pumped through a closed system by changing parameters (or fields) in time. The Kubo formula allows to distinguish between dissipative and non-dissipative contributions to the current. We obtain a Green function expression and an $S$ matrix formula for the associated terms in the generalized conductance matrix: the "geometric magnetism" term that corresponds to adiabatic transport; and the "Fermi golden rule" term which is responsible to the irreversible absorption of energy. We explain the subtle limit of an infinite system, and demonstrate the consistency with the formulas by Landauer and Buttiker, Pretre and Thomas. We also discuss the generalization of the fluctuation-dissipation relation, and the implications of the Onsager reciprocity.

Decay kinetic properties of atoms in photonic crystals with absolute gaps.

Decay kinetic properties of a two-level atom near the band edges of photonic crystals (PCs) with absolute gaps are studied based on the Green's function expression for the evolution operator. The local coupling strength between the photons and an atom is evaluated by an exact numerical method. It is found that the decay behavior of an excited atom can be fundamentally changed by the variation of the atomic position: Weisskopf-Wigner and non-Weisskopf-Wigner decay phenomena occur at different atomic positions in the PCs as a result of a significant difference in the local coupling strength. Our finding implies that it is possible to engineer the luminescence spectrum by controlling the atomic position.

Dislocation inside a piezoelectric media with an elliptic inhomogeneity

In this paper, Green functions for an infinite piezoelectric media with an elliptic piezoelectric inhomogeneity are given for a generalized electro-mechanical force and a generalized electro-mechanical line dislocation that may be located outside, inside or on the interface of elliptic boundary of inhomogeneity. Expressions of Green functions are identical when the generalized force and the generalized dislocation approach to the elliptic boundary from points outside or inside the inhomogeneity. The interaction electric enthalpy between the inhomogeneity and the dislocation is given, which can be used to find the interaction force on the dislocation owing to the existence of the inhomogeneity. Numerical illustrations for the interaction forces are also given and some discussions on the effects of the material mismatches on the interaction forces are made.

ESTIMATION METHOD FOR HYDRODYNAMIC PRESSURES ON OFFSHORE STRUCTURES DURING EARTHQUAKES

The three dimensional source distribution method is a popular technic to estimate hydrodynamic pressures on offshore structures during earthquakes. However, because only series-form expression of proper Green function was available, it takes long CPU time for accurate calculation. In this paper, an alternative integral-form expression of Green function is derived and its efficient way to calculation is presented. Resultant acting forces on circular and submerged rectangular cylinders vibrating horizontally under the effective dimensions of cylinders are also presented.