Vector Potential Green's Functions Research Articles

The evaluation of the interaction potential of two non-polar molecules in Green function approach

In this research, by assuming that the vacuum electric field fluctuations can convert a molecule (or an atom) into an oscillating electric dipole, the interaction potential of the two atoms or two non-polar molecules is calculated in terms of the molecular polarizabilities and the distance between them. Considering the net electric field at the position of a molecule as a sum of the vacuum electric field and the electric field due to the induced electric dipole of the other molecule, and substituting in the quadratic stark shift formula, the interaction potential of the two molecules is related to the vacuum electric field correlation function. By writing the interaction potential in terms of the imaginary parts of vector potential Green function tensor components (via the fluctuation- dissipation theorem and Kubo’s formula in statistical mechanics) and computing the required Green function components, the interaction potential between the two molecules is evaluated. The small and large distances limits of the general formula investigated and the consistencies with the previous works are shown.‎

The spontaneous emission rate determination of a molecule near a perfect conductive wall

In this paper, the formalism describing the spontaneous emission rate of an excited molecule (or atom) with definite transition electric dipole moment, in vicinity of a perfect conductive wall is developed. Separately, for molecules (or atoms) with the electric dipole moment parallel and perpendicular to the wall, the spontaneous emission rate of the molecule is calculated in two ways: (1) by using the explicit form of the vector potential operator in the Fermi' golden rule and taking the expectation value in vacuum state and (2) by considering the relation between the spontaneous emission rate and the vector potential Green function, and calculating the required Green function components of the space at the presence of a perfect conductive wall. The two methods yield the same expression for the spontaneous emission rate. The agreement of the results with some previous works is shown and the variations of the spontaneous emission rate for each dipole moment orientation are demonstrated.

The calculation of the atomic spontaneous emission rate between two dielectric slabs

In this paper, the spontaneous emission rate of an excited atom between the two dispersive and dissipative dielectric slabs is determined. It is assumed that the atom has a definite transition electric dipole moment, and the dielectric function of the slabs is an arbitrarily complex function of frequency satisfying Kramers–Kronig relations. In this research, by noting to the connection between the spontaneous emission rate and the imaginary part of the vector potential Green function, only by evaluating the vector potential Green function tensor of the system of the two slabs, an expression for the spontaneous emission rate is introduced. In this formalism, it is needless to derive the vacuum electric field operator. The emission rate is obtained in terms of the dielectric function, the thickness and the distance between the slabs. By considering a Lorentz model for the dielectric function of the slabs, the typical spatial variations of the spontaneous emission rate between the two slabs are demonstrated.

The Interaction Potential between an Atom and a Conductive Wall

In this work, the interaction potential between an atom (with definite Static Polarizability) and a perfect neutral conductive wall is evaluated in two approaches. First, using, the explicit form of the singlet and doublet parts of the vacuum vector potential operator in atomic quadratic stark shift formula and taking the expectation value in vacuum state. Second, writing, the quadratic stark shift formula in terms of electric field correlation function, relating the correlation function to the imaginary parts of the vector potential Green function tensor components (via fluctuation dissipation theorem in statistical mechanics) and the evaluation of the required vector potential Green function tensor components. The consistency of the two methods is shown.

The Casimir force between an ideal metal plate and a dissipative dielectric slab

In this research, a general formula for the Casimir force between ideal metal plate and a dissipative dielectric slab has been obtained. The dielectric function of the slab is assumed to be an arbitrary complex function of frequency satisfying Kramers–Kronig relations. A classical expression for the radiation pressure of the vacuum fields on the slab is presented by using the Maxwell stress tensor. With the transition to the quantum domain and using the fluctuation dissipation theorem and Kubo's formula, the resulting expression is written in terms of the imaginary part of the vector potential Green functions components of the system. Finally, by computing the Green function, the Casimir force on the slab is obtained. This formalism enables us to calculate the Casimir force without resorting to the explicit form of the field operators. The general expression is confirmed by limiting and comparing with one of the previous works.

Fast High-Frequency Impedance Extraction of Horizontal Interconnects and Inductors in 3-D ICs With Multiple Substrates

We present a high-frequency impedance extraction method for horizontal interconnects as needed in 3-D integrated circuits (ICs), where the horizontal interconnects are sandwiched between substrate layers of possibly different electromagnetic parameters. In particular, for the first time, we develop an extension of the discrete complex images method based on a 2D, or alternatively, 3D magneto-quasi-static (MQS) vector potential Green's functions to extract analytical solutions to the series impedance (resistance and inductance) matrix elements for wire filaments. We then follow standard methods to extract the port impedance. Using the 2D approach, the series impedance per unit length of horizontal wire loops is obtained, which shows excellent accuracy (<; 1% error to Maxwell SV) and significantly improved computational cost (two orders faster than Maxwell SV). Using our 3D approach and combining the series impedance matrix from the MQS extraction engine with the capacitance matrix from an electrostatic extraction engine, we produce an electro-magneto-quasi-static impedance matrix extraction engine, which is used to extract the input impedance of a spiral inductor. In the frequency range spanning near dc to a high frequency cutoff given by four times the frequency of the maximum in the quality factor, we show that our results agree to within less than 5% and 11% deviation to the full-wave simulator HFSS, for the self and mutual loop impedance, respectively. The CPU time using our approach is 18-25t faster than HFSS. These results provide a reasonable foundation for circuit block-level impedance extraction for interconnects and inductors in 3-D integrated systems.

A Fully Analytical Model for the Series Impedance of Through-Silicon Vias With Consideration of Substrate Effects and Coupling With Horizontal Interconnects

This paper introduces a fully analytical and physical model capable of extracting high-frequency series impedance of through-silicon vias (TSVs) in 3-D integrated chips with consideration of the eddy currents in the surrounding Si substrate and coupling with horizontal interconnects. The model employs the vertical-to-vertical and horizontal-to-vertical 3D vector potential Green's function in layered media and is concise and sufficiently accurate in the entire range of interest for both the frequency and the center-to-center distance between TSVs. Along with the series impedances between horizontal wires, which are extracted from the discrete complex image method, as well as the TSV and horizontal wire capacitance values, the total loop impedance can be obtained. Our approach is verified against a full-wave finite- element-method electromagnetic solver High Frequency Structure Simulator, and it shows good accuracy (<; 7% error) in the entire frequency range examined (up to 100 GHz). Given the fact that the formulated TSV series impedance model is purely analytical, the model could be efficiently used for system-level interconnect impedance extraction in emerging 3-D integrated systems.

High-Frequency Behavior of Graphene-Based Interconnects—Part II: Impedance Analysis and Implications for Inductor Design

This paper provides the first detailed insights into the ultrahigh-frequency behavior of graphene ribbons (GRs) and analyzes their consequences in designing interconnects and low-loss on-chip inductors. In the companion paper (part I), an accurate impedance modeling methodology has been developed based on the Boltzmann equation with the magnetic vector potential Green's function approach incorporating the dependency of current on the nonlocal electric field. Based on the developed methodology, this paper for the first time embarks on the rigorous investigation of the intricate processes occurring at high frequencies in GRs, such as anomalous skin effect (ASE), high-frequency resistance and inductance saturation, intercoupled relation between edge specularity and ASE, and the influence of the linear dimensions on impedance. A comparative study of the high-frequency response of GRs with that of carbon nanotubes (CNTs) and Cu is made to highlight the potential of GR interconnects for high-frequency applications. Subsequently, the high-frequency performance of GR inductors is analyzed, and it is shown that they can achieve 32% and 50% improvements in maximum Q-factor compared to Cu and single-walled CNT inductors with 1/3 metallic fraction, respectively.

Computation of Electromagnetic Whispering Gallery Mode Field in a Dielectric Microring Fabricated on a Metal Wire

Closed-form formulas to calculate electromagnetic whispering gallery mode laser fields in a circular microring are developed. Such a microring is typically made from an optically active polymer, which is fabricated on a metallic wire. Because of the operation frequency, the microring is modeled as an infinite dielectric cylinder with a metallic inner boundary. Vector potential Green functions, from which the whispering gallery mode fields are computed, are presented, and a simplified method of computing fields is introduced. Computed results show that the whispering gallery mode field in the dielectric cylinder is detached from its inner boundary and the phenomena can be achieved by increasing the thickness of the dielectric layer.

Casimir force between two dielectric slabs

We examine the Casimir effect between two dispersive dissipative slabs in three dimensions. The dielectric function of the slabs is assumed to be an arbitrary complex function of frequency satisfying Kramers-Kronig relations. The Maxwell stress tensor is used to evaluate the vacuum radiation pressure of the electromagnetic field on each slab in terms of the field correlation functions. These correlations are simply related to the imaginary part of the vector potential Green function by using the fluctuation dissipation theorem and Kubo's formula. The response function is obtained by employing the standard method for the present configuration. The formalism enables us to calculate the attractive Casimir force between a pair of dielectric slabs without resorting to the electromagnetic field quantization. Special attention is paid to the various limits of the general expression and the results are compared with previous work whenever possible. The Lorentz model of the dielectric function is also used to show the typical behavior of this force in terms of the slab thickness. We also consider the effect of finite temperature on the Casimir force between two dielectric slabs.

Radiative properties of an atom in the vicinity of a mirror

The decay rate of an excited atom is described using Fermi's golden rule. Welton's interpretation of the Lamb shift is extended by introducing a damping term in the Heisenberg equation of motion associated with the fluctuation in the position of the electron. These expressions are related to the imaginary part of the vector potential Green function through the fluctuation dissipation theorem and Kubo's formula. The results are applied to the calculation of the radiative properties of an atom in the vicinity of a perfect mirror.

Electromagnetic Field and Conduction Current Equivalence in the Lorentz and Coulomb Gauges

ABSTRACT Coulomb and Lorentz gauge equivalence of the electric and magnetic fields and the conduction current in the time and spectral domains is demonstrated. The analysis relies entirely on the Green's functions for both scalar and vector potentials in the two gauges, It is shown that the non-causal portion of the Coulomb gauge electric field expression is removed by the Coulomb gauge vector potential Green's function. The fact that the current is gauge invarient has important implications in the verification of electromagnetic field numerical codes based on integral equation methods.

Vector potential green's function for a radially directed line dipole in open space

A vector potential Green's function for a radially directed line dipole (Figure 1) is derived. Methods for obtaining two-dimensional Green's functions are available for the z-directed line current and for the OS-directed line dipole. With the results of this article available, the set of Green's functions for two-dimensional sources in cylindrical coordinates is complete.

A coulomb gauge analysis of a wire scatterer

The Lorentz gauge is the preferred gauge when problems in electromagnetic scattering and diffraction are solved by a classical Maxwell equation potential function approach. The free space magnetic vector and scalar potentials subject to the Coulomb gauge are obtained in the paper. It is shown that the Coulomb gauge vector potential dyadic Green's function can be extracted from the well known Lorentz gauge free space vector potential Green's function. The electric field mixed potential integral equation (MPIE) for a straight, circular, finite-length, narrow conducting cylinder scatterer is formulated and solved in both the Lorentz and Coulomb gauges and results are presented in terms of the cylinder surface current. The merits and drawbacks of the Coulomb gauge formulation are discussed.