Maxwell Wave Equation Research Articles

Resonant-state expansion of dispersive open optical systems: Creating gold from sand

A resonant-state expansion (RSE) for open optical systems with a general frequency dispersion of the relative permittivity, described by a finite number of simple poles, is presented. As in the non-dispersive case, the RSE of dispersive systems converts Maxwell's wave equation into a linear matrix eigenvalue problem in the basis of unperturbed resonant states, in this way numerically exactly determining all relevant eigenmodes of the optical system. This dispersive RSE is verified by application to the analytically solvable system of a sphere in vacuum, with a dispersion of the dielectric constant described by the Drude and Drude-Lorentz models. We calculate the change of the optical modes when converting the sphere material from gold to non-dispersive silica and back to gold, and evaluate the accuracy using the exact solutions.

Structural disorder correlation examined using the Fourier–Bessel technique

Abstract The presence of structural disorder in a photonic crystal is examined through the rotational symmetry extracted from a Fourier-Bessel approach to solving Maxwell's wave equation in cylindrical space. A dielectric correlation function is proposed that relates the original structure to the disordered structure and when normalized it can be used to quantify the level of any disorder mechanism present. It is shown that the presence of disorder causes a mixing of localized and extended states and that the mixing can be directly attributed to “off diagonal” elements of the eigen-matrix and rotational symmetry breaking within the structure. The properties of disorder in an ordered structure are used to identify locations of local order in disordered structures. The Fourier–Bessel analysis of a disordered structure confirms the presence of localized light states at these sites.

Dielectric profile segmentation used to reduce computation effort of the Fourier-Bessel mode solving technique.

The Fourier-Bessel space conversion of Maxwell's wave equations into an eigenvalue formulation is a useful numerical tool for computing the steady states of cylindrically symmetric dielectric structures. The relative dielectric profile, inverse (1/εr) present in wave equations, is split into a constant offset and corresponding spatially dependent residue and greatly reduces the matrix building time (and thus computation time) when alternate dielectric configurations are considered with identical spatial distributions. Such a process significantly speeds up the theoretical analysis of numerous optical designs, such as index of refraction sensors, hole infiltration sensors and resonator tuning. The theoretical steps involved are presented along with examples of the technique applied to the well-known Bragg resonator and central defect containing hexagonal array.

Generic propagation of beams with sharp spatial boundaries.

The propagation of spatial beams with initially sharp transverse boundaries is investigated theoretically and experimentally with the paraxial wave equation (PWE). The sharp boundaries generate a universal pattern, which is a consequence of the Schrödinger-like nature of the paraxial dynamics. As a consequence, an approximate analytical expression can be derived for the longitudinal propagation dynamics of the beam. Furthermore, it is shown that the validation of the derived analytical approximation is not limited to the zone in which the PWE is valid, but it is valid in the entire space. Therefore, this solution is a good approximation for the solution of the scalar wave equation (and to the Maxwell wave equation whenever the aperture is much wider than the wavelength of light) in the entire space. Good agreement between the analytical expression and experiment results is presented.

Theoretical study of pre-formed hole geometries on femtosecond pulse energy distribution in laser drilling.

Maxwell's wave equation was solved for fs laser drilling of silicon. The pre-formed hole wall's influence on the propagation behavior of subsequent laser pulses was investigated. The laser intensity at hole bottom shows distinct profile as compared with that at hole entrance. The multi-peaks and ring structure of the laser intensity were found at hole bottom. The position of maximum laser intensity (MLI) in relation to the wall taper angle was studied. It was found that the position of the MLI point would be closer to the hole entrance with increasing taper angle. This observation provides valuable information in predicting the position of plasma plume which is a key factor influencing laser drilling process. The elliptical entrance hole shape and zonal structure at the hole bottom reported in the literatures have been reasonably explained using the laser intensity distribution obtained in the present model.

Quantum cascade structure for mid-IR four-wave mixing

The design and modeling of a quantum cascade optical amplifier (QCOA) using intra-cavity non-linear interactions to achieve wavelength conversion is proposed. The model is based on the nonlinear equation coupled with Maxwell wave equations for different emission modes. In the proposed structure, four wave mixing (FWM) output exhibits a peak as a function of pump and probe frequency if they are tuned to the energy levels of the QC structure subbands. Results demonstrate that the FWM output signal power significantly depends on how subbands are engineered and interact with optical pulses which propagate in multi layer medium. In addition, we show that by adjusting pump and probe signal frequencies, FWM output power can be tuned.

맥스웰-슈뢰딩거 방정식을 이용한 다이폴 안테나 해석

본 논문은 양자역학 관점으로 본 다이폴 안테나의 정량적 분석과 그것의 특성에 관한 논문이다. 분석 방법으로 현존하는 안테나 전파 방정식에 이용되는 맥스웰 방정식을 사용한다. 이는 안테나의 길이와 주파수에 관한 함수로, 방사저항, 입력 리액턴스, 안테나 효율을 포함한다. 본 논문의 주요 관심사는 슈뢰딩거 방정식이다. 또한 본 논문은 맥스웰과 슈뢰딩거 방정식을 결합하여 다이폴 안테나의 전계와 자계를 해석한다. 현존하는 맥스웰 방정식과 슈뢰딩거 방정식을 비교함으로써, 단일 맥스웰 방정식을 썼을 때 보다 양자-전기 이동의 정확성이 향상됨을 보인다. We present a quantitative analysis of a dipole antenna and its characteristics from the viewpoint of quantum mechanics. The method makes use of a Maxwell equation used in an existing antenna propagation formula. This includes radiation resistance, input reactance, and antenna efficiency as functions of frequency and antenna length. Particular attention is paid to the Schr<TEX>$\ddot{o}$</TEX>odinger equation. We accomplish E-field and H-field analyses of a dipole antenna by combining the Maxwell and Schr<TEX>$\ddot{o}$</TEX>odinger wave equations. When comparing the existing Maxwell wave equation with the Schr<TEX>$\ddot{o}$</TEX>odinger wave equation, quantum-electric movement is more accurate than using the Maxwell wave equation alone.

Phase-Controlled Transparent Superluminal Light Propagation in a Doppler-Broadened Four-Level N-Type System

The propagation of a weak probe field in a four-level N-type quantum system in the presence of spontaneously generated coherence (SGC) is theoretically investigated. The optical properties of the system are studied and it is shown that the group velocity of light pulse can be controlled by relative phase of applied fields. By changing the relative phase of applied fields, the group velocity of light pulse changes from transparent subluminal to the transparent superluminal light propagation. Thus, the phase-controlled absorption-free superluminal light propagation is obtained without applying an incoherent laser fields to the system. The propagation of a weak probe light pulse is studied by solving the Maxwell's wave equation on numerical grid in space and time. Moreover, we study the third order self- and cross-Kerr susceptibility of probe field and calculate the nonlinear cross-phase shift for different values of intensity of applied fields. In addition, we take into account the effect of Doppler broadening on the light pulse propagation and it is found that a suitable choice of laser propagation directions allows us to preserve our results even in the presence of Doppler effect. It is demonstrated that by increasing the Doppler width of distribution to the room temperature, the dispersion changes from transparent subluminal to transparent superluminal light propagation which is our major motivation for this work.

Simplified skin‐effect formulation for power transmission lines

This study presents a simplified formulation to calculate the skin effect on transmission lines wires. The methodology is developed from the solution of the time-domain Maxwell's wave equations whose frequency-domain solutions are obtained using Fourier transforms. Results obtained from the proposed formulation are compared with those calculated using the conventional method based on the approach by Bessel functions. The proposed skin-effect formulation is also validated based on electromagnetic transients simulations in the time domain considering a conventional 69-kV transmission system.

Intense attosecond pulse generated from a molecular harmonic plateau of H2+ in mid-infrared laser fields

High-order harmonic generation from the molecular ion H2+ exposed to intense laser fields is investigated by the time-dependent quantum wave packet method. Molecular and atomic plateaus of harmonic spectra are effectively distinguished at large internuclear distances, where the harmonic efficiency of the molecular plateau is several orders of magnitude higher than that of the latter. We report on a physical model of the origin of the molecular supercontinua and reveal that the creation of this plateau directly results from the interference of the intramolecular electronic wave packet localized in two potential wells following the laser field. This is our first effort in utilizing the efficient molecular plateau to generate intense isolated attosecond pulses by controlling the dynamics of the nucleus and electrons with a mid-infrared laser. Further, we show that the harmonic plateau is enhanced at the macroscopic level by solving the Maxwell wave equation coupled with the Schrödinger equation.

The Dynamic Gravitation of Photons from the Perspective of Maxwell’s Wave Equations

Although the gravitational constant (G) does not explicitly occur in the Maxwell Wave Equations, this paper will show that G is indeed implicitly contained in them. The logical consequence hereby is that electromagnetic radiation is associated with dynamic gravitation and not—as assumed in Einstein’s Special Theory of Relativity—with “static” gravitation, dynamic gravitation being at the time unknown. According to the Maxwell Wave Equations, gravitation experiences the same dynamic (speed of light c) as electromagnetic radiation and must therefore also be of a quantum nature. There must exist an equal number of gravitational quanta as there are photons. Since photons do not possess a baryonic rest mass but only a relativistic mass, this mass must be nonbaryonic in nature—precisely as their dynamic gravitation.

Laser-imposed phase in resonant absorption of an isolated attosecond pulse

We calculate resonant absorption line shapes in helium atoms subject to the combination of an attosecond pulse of XUV radiation and a delayed, few-cycle infrared (IR) laser pulse. We are in particular interested in how the resonant features evolve with the delay between the fields and with increasing IR intensity. We first compare the single-atom absorption calculated by a nonperturbative solution of the time-dependent Schr\"odinger equation (TDSE) to a simple model in which the resonant absorption is modified only by a laser-imposed phase (LIP) that originates in the instantaneous ac Stark shift of the resonant state. We find that the LIP model can explain many features of the resonant absorption, including the direction of the resonance energy shift and the onset of dispersive line shapes with increasing IR intensity. We also find, however, that the LIP model predicts the energy shifts and the line shapes only qualitatively when the resonant states are strongly IR coupled to nearby states. We then use a coupled solution of the Maxwell wave equation and the TDSE to examine how the line shapes are modified by propagation effects at helium densities such that the resonant absorption is strong. We find that when the resonant absorption saturates, their line shapes are strongly modified due to reshaping of the attosecond pulse time profile during propagation. This in turn alters the ultrafast absorption and emission dynamics.

Correlation Approach to Propagation of Terahertz Pulse in Dielectrics for Material Characterization

With Maxwell's wave equation and an auxiliary equation, we devise a novel approach to simulate terahertz (THz) pulse propagation in a typical THz time-domain spectrometer for material diagnostics. This is based on a correlation analysis of experimentally measured THz wave from propagated through the sample and that of a simulated one with material properties as adjustable parameters. The Lorentz model for dielectrics is assumed. Using fused quartz as an example, we find this approach can yield reliable static relative permittivity and optical constants beyond the bandwidth of a THz pulse.

Dispersion of effective refractive indices of mid-infrared quantum cascade lasers

We point out the difference between the two mostly used methods for calculating the effective refractive index of mid-infrared quantum cascade lasers, namely by solving the Maxwell's wave equation and by analyzing the frequency spacing of the longitudinal modes of the Fabry-Perot cavity. The effective refractive indices obtained by these methods are shown to be different, as one refers to the phase effective refractive index while the other refers to the group effective refractive index, respectively. Dispersion relationships for these two effective refractive indices are deduced for mid-infrared quantum cascade lasers, which show an increase in group refractive index and a decrease in phase refractive index with wavelength. Experiments are conducted to obtain the group effective refractive indices of quantum cascade lasers emitting at various wavelengths, and good agreement has been achieved between the experimental data and theory.

SPIREs: A Finite-Difference Frequency-Domain electromagnetic solver for inhomogeneous magnetized plasma cylinders

We present SPIREs (plaSma Padova Inhomogeneous Radial Electromagnetic solver), a Finite-Difference Frequency-Domain (FDFD) electromagnetic solver in one dimension for the rapid calculation of the electromagnetic fields and the deposited power of a large variety of cylindrical plasma problems. The two Maxwell wave equations have been discretized using a staggered Yee mesh along the radial direction of the cylinder, and Fourier transformed along the other two dimensions and in time. By means of this kind of discretization, we have found that mode-coupling of fast and slow branches can be fully resolved without singularity issues that flawed other well-established methods in the past. Fields are forced by an antenna placed at a given distance from the plasma. The plasma can be inhomogeneous, finite-temperature, collisional, magnetized and multi-species. Finite-temperature Maxwellian effects, comprising Landau and cyclotron damping, have been included by means of the plasma Z dispersion function. Finite Larmor radius effects have been neglected. Radial variations of the plasma parameters are taken into account, thus extending the range of applications to a large variety of inhomogeneous plasma systems. The method proved to be fast and reliable, with accuracy depending on the spatial grid size. Two physical examples are reported: fields in a forced vacuum waveguide with the antenna inside, and forced plasma oscillations in the helicon radiofrequency range.

Influence of gas pressure on high-order-harmonic generation of Ar and Ne

We study the effect of gas pressure on the generation of high-order harmonics where harmonics due to individual atoms are calculated using the recently developed quantitative rescattering theory, and the propagation of the laser and harmonics in the medium is calculated by solving the Maxwell's wave equation. We illustrate that the simulated spectra are very sensitive to the laser focusing conditions at high laser intensity and high pressure since the fundamental laser field is severely reshaped during the propagation. By comparing the simulated results with several experiments we show that the pressure dependence can be qualitatively explained. The lack of quantitative agreement is tentatively attributed to the failure of the complete knowledge of the experimental conditions.

Adiabatic transfer of light in a double cavity and the optical Landau-Zener problem

We analyze the evolution of an electromagnetic field inside a double cavity when the difference in length between the two cavities is changed, e.g., by translating the common mirror. We find that this allows photons to be moved deterministically from one cavity to the other. We are able to obtain the conditions for adiabatic transfer by first mapping the Maxwell wave equation for the electric field onto a Schr\"odinger-like wave equation and then using the Landau-Zener result for the transition probability at an avoided crossing. Our analysis reveals that this mapping only rigorously holds when the two cavities are weakly coupled (i.e., in the regime of a highly reflective common mirror) and that, generally speaking, care is required when attempting a Hamiltonian description of cavity electrodynamics with time-dependent boundary conditions.

On the dipole, velocity and acceleration forms in high-order harmonic generation from a single atom or molecule

We consider high-order harmonic generation from a single atom or molecule and show that the generated harmonic field is proportional to the dipole velocity. We derive this result by solving Maxwell's wave equation for an infinitely thin gas. Hence, the dipole velocity form is the one that relates directly to the harmonic field, and it should be used as a reference when performing calculations with the dipole or acceleration forms.

Influence of phase matching on the Cooper minimum in Ar high-order harmonic spectra

We study the influence of phase matching on interference minima in high harmonic spectra. We concentrate on structures in atoms due to interference of different angular momentum channels during recombination. We use the Cooper minimum (CM) in argon at 47 eV as a marker in the harmonic spectrum. We measure 2d harmonic spectra in argon as a function of wavelength and angular divergence. While we identify a clear CM in the spectrum when the target gas jet is placed after the laser focus, we find that the appearance of the CM varies with angular divergence and can even be completely washed out when the gas jet is placed closer to the focus. We also show that the argon CM appears at different wavelengths in harmonic and photo-absorption spectra measured under conditions independent of any wavelength calibration. We model the experiment with a simulation based on coupled solutions of the time-dependent Schr\"odinger equation and the Maxwell wave equation, including both the single atom response and macroscopic effects of propagation. The single atom calculations confirm that the ground state of argon can be represented by its field free $p$ symmetry, despite the strong laser field used in high harmonic generation. Because of this, the CM structure in the harmonic spectrum can be described as the interference of continuum $s$ and $d$ channels, whose relative phase jumps by $\pi$ at the CM energy, resulting in a minimum shifted from the photoionization result. We also show that the full calculations reproduce the dependence of the CM on the macroscopic conditions. We calculate simple phase matching factors as a function of harmonic order and explain our experimental and theoretical observation in terms of the effect of phase matching on the shape of the harmonic spectrum. Phase matching must be taken into account to fully understand spectral features related to HHG spectroscopy.

Effect of photoconductivity and oscillation frequency shift on the signal beam intensity in two beam coupling in photorefractive materials

A general theory of the two-beam coupling between a pump beam and a signal beam in photorefractive materials is presented. The coupled wave equations describing the non-linear two beam coupling are derived, based on Maxwell's wave equation. The coupled equation for the intensities of the two beams in the photorefractive crystals with the absorbing properties have been derived analytically. The intensity of the signal beam increases with the increasing crystal thickness, reaches a maximum and then decreases. The influence of energy beam coupling coefficient, oscillation frequency shift, crystal thickness, absorption coefficient and the input beam intensity ratio on the signal beam intensity have been studied in details. The effect of the photoconductivity of the materials on the intensities of the two beams in both the co-directional as well as contra-directional two beam coupling cases have been studied.