Vector Maxwell Equations Research Articles

Full transmission of vectorial waves through 3D multiple-scattering media.

A striking prediction from the random matrix theory (RMT) in mesoscopic physics is the existence of "open channels": waves that use multipath interference to achieve perfect transmission across an opaque disordered medium even in the multiple-scattering regime. Realization of such open channels requires a coherent control of the complete incident wavefront and has only been achieved for scalar waves in two dimensions (2D) so far. Here, we utilize a recently proposed "augmented partial factorization" full-wave simulation method to compute the polarization-resolved scattering matrix from 3D vectorial Maxwell's equations and demonstrate the existence of open channels in 3D disordered media. We examine the spatial profile of such open channels, demonstrate the existence of a bimodal transmission eigenvalue distribution, and study the effects of incomplete polarization control and finite-area illumination. The simulations provide full access to all spatiotemporal properties of the complex wave transport in 3D disordered systems, filling the gap left by experimental capabilities.

Polarization transport in optical fibers beyond Rytov's law

We consider the propagation of light in arbitrarily curved step-index optical fibers. Using a multiple-scales approximation scheme, set-up in Fermi normal coordinates, the full vectorial Maxwell equations are solved in a perturbative manner. At leading order, this provides a rigorous derivation of Rytov's law. At next order, we obtain non-trivial dynamics of the electromagnetic field, characterized by two coupling constants, the phase and the polarization curvature moments, which describe the curvature response of the light's phase and its polarization vector, respectively. The latter can be viewed as an inverse spin Hall effect of light, where the direction of propagation is constrained along the optical fiber and the polarization evolves in a frequency-dependent way.

Boundary Element Solution of Electromagnetic Fields for Non-Perfect Conductors at Low Frequencies and Thin Skin Depths

A novel boundary element formulation for solving problems involving eddy currents in the thin skin depth approximation is developed. It is assumed that the time-harmonic magnetic field outside the scatterers can be described using the quasistatic approximation. A two-term asymptotic expansion with respect to a small parameter characterizing the skin depth is derived for the magnetic and electric fields outside and inside the scatterer, which can be extended to higher order terms if needed. The introduction of a special surface operator (the inverse surface gradient) allows the reduction of the problem complexity. A method to compute this operator is developed. The obtained formulation operates only with scalar quantities and requires computation of surface operators that are usual for boundary element (method of moments) solutions to the Laplace equation. The formulation can be accelerated using the fast multipole method. The method is much faster than solving the vector Maxwell equations. The obtained solutions are compared with the Mie solution for scattering from a sphere and the error of the solution is studied. Computations for much more complex shapes of different topologies, including for magnetic and electric field cages used in testing are also performed and discussed.

3-D Modeling for Airborne EM using the Spectral-element Method

The 3-D spectral-element method (SE) based on Gauss–Lobatto–Legendre (GLL) polynomials is a very efficient and accurate solver for high-frequency computational electromagnetic (EM) modeling. Although the SE method is based on a weighted residual technique similar to the finite-element method (FE), it utilizes polynomial interpolation functions rather than linear interpolation functions. The GLL polynomials have the characteristic of exponential convergence with the order of polynomial that helps improve the modeling accuracy. We apply the SE method in the frequency range (900 Hz to 56 kHz) to airborne EM modeling. Starting from the vector Maxwell's equations, we use the SE method to establish spatial-discrete forms of 3-D vector Helmholtz equations and adopt GLL integration to calculate the matrix elements. The magnetic field is calculated using Faraday's law. To check the accuracy of the SE algorithm, we compare our results with semi-analytical solutions for a homogeneous half-space and a layered earth model. We further analyze the influence of grids, expansion of outside boundary of model domain, and the order of interpolation polynomial on EM modeling accuracy and reveal that increasing the SE polynomial order can especially improve the accuracy for complex models. Finally, we demonstrate the feasibility of our SE algorithm for modeling an airborne EM system by numerical 3-D experiments.

Measurement and general modeling of optical rotation in anisotropic crystal

In this paper a novel method of computing the optical rotation of a parallel plated crystal is presented. The calculation is based on Fourier optics and solves generally the vectorial Maxwell's equations for both the refraction and propagation. It does not require the paraxial approximation, instead, the incident beam may have complex shape, not only a single plane wave. The medium may be anisotropic and optically active. We compare the simulation and experimental results which agree well within the error limit of the measurement. The comparison of the results enabled us to determine the optical rotatory power. The main advantage of the method is that it computes accurately the polarization in case of arbitrary incident angle. The primary purpose of the model is to calculate accurately the electric fields for the different polarizations in case of acousto-optical cells of various orientations.

Comparison of different approaches to the numerical calculation of the LMJ focal

The beam smoothing in the focal plane of high power lasers is of particular importance to laser- plasma interaction studies in order to minimize plasma parametric and hydrodynamic instabilities on the target. Here we investigate the focal spot structure in different geometrical configurations where standard paraxial hypotheses are no longer verified. We present numerical studies in the cases of single flat top square beam, LMJ quadruplet and complete ring of quads with large azimuth angle. Different calculations are made with Fresnel diffraction propagation model in the paraxial approximation and full vector Maxwell's equations. The first model is based on Fourier transform from near to far field method. The second model uses first spherical wave decomposition in plane waves with Fourier transform and propagates them to the focal spot. These two different approaches are compared with Miro (1) modeling results using paraxial or Feit and Fleck options. The methods presented here are generic for focal spot calculations. They can be used for other complex geometric configurations and various smoothing techniques. The results will be used as boundary conditions in plasma interaction computations. where f(x,y) is the amplitude of the field with a spatial extension size of a and k0 = /c is the wave number of the field. The field expression in the k space (spatial frequencies) with one fast Fourier transform (FFT) requires a large number of points. But for a parabolic phase lens, plane wave decomposition of the electric field in the k space with double FFT allow to reduce the spatial sampling (2048 points instead of 256000 for a = 1.2m and zf = 0.8m). The plane wave decomposition with two FFT is given by:

Vector wave analysis of an electromagnetic high-order Bessel vortex beam of fractional type α: II. The cases of standing and quasi-standing waves

Stemming from the vector Maxwell's equations and Lorenz’ gauge condition, a full vector-wave derivation for the electric and magnetic fields components of a high-order Bessel vortex beam of fractional type α (HOBVB-Fα) is provided. The field corresponds to the most generalized case of quasi-standing waves that reduce to perfect (i.e. equi-amplitude) standing waves or progressive waves with appropriate choice of the quasi-standing wave coefficient ϒ. The results are of particular importance in the study of the optical/electromagnetic wave scattering, radiation force and torque in optical tweezers operating with this fractional type of non-diffracting vortex beams.

Incidence angle dependence of the enhancement factor in attenuated total reflection surface enhanced infrared absorption spectroscopy studied by numerical solution of the vectorial Maxwell equations

The finite element method (FEM) was used to solve the time-harmonic Maxwell equations in a study of the effect of the incidence angle of infrared light on the surface enhancement caused by colloidal gold particles in attenuated total reflection surface enhanced infrared absorption spectroscopy (ATR-SEIRAS). The spectral enhancement factor was obtained from computations of absorbance from a thin organic layer in the presence and absence of the metal nanostructure. For computations of an isolated particle the enhancement factor is high around the critical angle and decreases with an increase in incidence angle. This trend was also observed in experiments performed with gold particles immobilised on a silane modified silicon ATR crystal. Computations where gold particles are touching each other show low enhancement factors around the critical angle and an increase with increasing incidence angle. These two opposing trends are analysed based on the electric field distribution around the particle.

On the Lie point symmetries of the vector Maxwell equations in non-linear optics

The vector Maxwell equations of non-linear optics coupled to a single Lorentz oscillator is analyzed by using Lie symmetries. The symmetry are used to obtain classical similarity solutions of the equations to reduce to ordinary differential equations for the electric field intensity. Also, nth dimensional classification of Lie point symmetry subalgebras are given. Finally we obtain the conservation laws of symmetries.

Field equations for massless bosons from a Dirac–Weinberg formalism

Dirac’s first-order differential equation for relativistic spin 1/2 fermions can be extended to massless bosons using a formalism developed by Steven Weinberg. Weinberg’s theory is based heavily on details of the Lorentz and Poincaré groups and may not be accessible to the average reader. Although Weinberg’s theory was formulated over 40 years ago and his equations are easy to write down, few physicists seem to be familiar with them, and they are not discussed in standard quantum texts. In this paper we demonstrate the utility of his methods. Weinberg’s equations, which are valid for arbitrary spin, can be put in a form analogous to the Dirac equation for massless particles. Due to transversality conditions imposed on the theory so that it satisfies the second-order Klein–Gordon equation, we recast the theory in terms of a two-component Dirac–Pauli spinor. The theory is presented in some detail for j=1, corresponding to the photon and the vector Maxwell equations, and for j=2, corresponding to a graviton-like particle and a tensor version of Maxwell’s equations. These latter equations determine the polarization of gravitational waves in the weak field approximation. Our results correspond to those from the traceless-transverse gauge of general relativity.

A Boundary Perturbation Method for Vector Electromagnetic Scattering from Families of Doubly Periodic Gratings

In this paper the authors extend their previous work on Boundary Perturbation methods for scattering calculations from families of diffraction gratings to three dimensions and the full vector electromagnetic Maxwell equations. This extension is non-trivial in both its algorithmic implementation (not only are new terms added to the recursions, but also the full, coupled, vector Maxwell equations must be simulated) and in the size of the relevant computer simulations. Not only do we give details of the implementation of the method, but also provide results of numerical simulations.

Dynamical model for coherent optical manipulation of a single spin state in a charged quantum dot

AbstractThe optically‐induced coherent spin dynamics of a single spin confined in a charged quantum dot (QD) is theoretically studied employing coupled vector Maxwell‐pseudospin formalism. Generalized pseudospin master equation is derived for description of the time evolution of spin coherences and spin populations including spin population transfer and dissipation in the system through spin relaxation processes. The equation is solved in the time domain self‐consistently with the vector Maxwell equations for the optical wave propagation coupled to it via macroscopic medium polarisation. Using the model the long‐lived electron spin coherence left behind a single resonant ultrashort optical excitation of the electron‐trion transition in a charged QD is simulated in the low‐ and high‐intensity Rabi oscillations regime. Signatures of the polarised photoluminescence (PL), predicted by the model, such as the appearance of a second echo pulse after the excitation and characteristic PL trace shape, are discussed for realization of high‐fidelity schemes for coherent readout of a single spin polarisation state. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Model for the coherent optical manipulation of a single spin state in a charged quantum dot

The optically driven coherent dynamics associated with the single-shot initialization and readout of a localized spin in a charged semiconductor quantum dot embedded in a realistic structure is theoretically studied using a new Maxwell-pseudospin model. Generalized pseudospin master equation is derived for description of the time evolution of spin coherences and spin populations in terms of the real state pseudospin coherence vector including dissipation in the system through spin-relaxation processes. The equation is solved in the time-domain self-consistently with the vector Maxwell equations for the optical wave propagation coupled to it via macroscopic medium polarization. Using the model, the long-lived electron spin coherence left behind a single resonant ultrashort optical excitation of the electron-trion transition in a charged quantum dot is simulated in the low- and high-intensity Rabi oscillation regime. Signatures of the polarized photoluminescence PL resulting from the numerical simulations, such as the appearance of a second echo pulse following the excitation and a characteristic nonmonotonic PL trace shape, specific for initial spin-up orientation, are discussed for realization of high-fidelity schemes for coherent readout of a single spin polarization state.

Application of the multiscale singular perturbation method to nonparaxial beam propagations in free space

Starting from the vector Maxwell's equations and applying the multiscale singular perturbation method, the nonparaxial beam propagation is studied in free space. Two new equations have been derived for transverse and longitudinal electric fields of an arbitrary polarized electromagnetic wave even in the case of tightly focused nonparaxial laser beams. By using the analogy of the optical beam in the space domain and the optical pulse in the time domain, the higher-order diffraction term is introduced. For strongly nonparaxial beams that are characterized by large values of the perturbative parameter, our correction solutions yield an accurate description of the field in the near-field region and are consistent with all other correction results obtained by others in the far-field region. For weakly nonparaxial beams, our correction solutions can be expressed in a very simple form that is proved to be exactly consistent with the solutions obtained by Cao and Deng [J. Opt. Soc. Am. A15, 1144 (1998)]. In addition, the lowest-order correction to the paraxial approximation can be found to be in good agreement with the results of Lax et al. [Phys. Rev. A11, 1365 (1975)] and Seshadri [J. Opt. Soc. Am. A19, 2134 (2002)].

All-optical coherent control of spin dynamics in semiconductor quantum dots

A new model for rigorous theoretical description of circularly (elliptically) polarized ultrashort optical pulse interactions with singly charged quantum dots embedded in semiconductor optical waveguides and microcavities is proposed. The method is based on self-consistent solution in the time domain of the vector Maxwell equations coupled via microscopic polarization to the coherent time-evolution equations of a discrete N-level quantum system in terms of the real pseudospin (coherence) vector. The model is applied to a 4-level system describing the optical dipole transitions of the trion state in a singly charged quantum dot. Selective optical excitation of specific spin states by predefined helicity of the optical field and an onset of self-induced transparency and polarized soliton formation in the nonlinear regime are numerically demonstrated

Spin‐dependent dynamics of ultrafast polarised optical pulse propagation in coherent semiconductor quantum systems

AbstractA new model for rigorous theoretical description of circularly (elliptically) polarised ultrashort optical pulse interactions with the resonant nonlinearities in semiconductor optical waveguides is proposed. The method is based on self‐consistent solution in the time domain of the vector Maxwell equations coupled via microscopic polarisation to the coherent time‐evolution equations of a discrete N‐level quantum system in terms of the real pseudospin (coherence) vector. The model is initially applied to a generic two‐level quantum system and subsequently to a 4‐level system describing the heavy‐hole excitonic transitions in low‐dimensional semiconductor systems, such as quantum wells and quantum dots. Selective optical excitation of specific spin states by predefined helicity of the optical field in the linear regime and an onset of self‐induced transparency and polarised soliton formation in the nonlinear regime are numerically demonstrated in both discrete‐level systems. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A 3-D precise integration time-domain method without the restraints of the courant-friedrich-levy stability condition for the numerical solution of Maxwell's equations

In this paper, a new three-dimensional time-domain method for solving vector Maxwell's equations, called the precise-integration time-domain (PITD) algorithm, is proposed in order to eliminate the Courant-Friedrich-Levy (CFL) condition restraint. The new algorithm is based on the precise-integration technique. It is shown that this method is quite stable even when the CFL condition is not satisfied. Although the memory requirement of the PITD method is much larger than that of the finite-difference time-domain (FDTD) method, this new algorithm is very appealing since the time step used in the simulation is no longer restricted by stability. As a result, computation speed can be improved. Therefore, if the minimum cell size in the computational domain is required to be much smaller than the wavelength, this new algorithm is more efficient than the FDTD scheme. Theoretical proof of the unconditional stability is shown and numerical results are presented to demonstrate the effectiveness and efficiency of the method. It is found that the accuracy of the PITD is independent of the time-step size.

Nonlinear finite-difference time-domain method for the simulation of anisotropic, /spl chi//sup (2)/, and /spl chi//sup (3)/ optical effects

A two-dimensional (2-D) finite-difference time-domain (FDTD) code for the study of nonlinear optical phenomena, in which both the slowly varying and the rapidly varying components of the electromagnetic fields are considered, has been developed. The algorithm solves vectorial Maxwell's equations for all field components and uses the nonlinear constitutive relation in matrix form as the equations required to describe the nonlinear system. The stability of the code is discussed and its effectiveness is demonstrated through the simulations of self-phase modulation (SPM) and second-harmonic generation (SHG). The authors also show that the combination of nonlinear effects with PCs can result in a significant improvement in device size and integrability, using the example of a Mach-Zehnder interferometer (MZI).

Dynamical model of coherent circularly polarized optical pulse interactions with two-level quantum systems

We propose and develop a method for theoretical description of circularly (elliptically) polarized optical pulse resonant coherent interactions with two-level atoms. The method is based on the time-evolution equations of a two-level quantum system in the presence of a time-dependent dipole perturbation for electric dipole transitions between states with total angular-momentum projection difference $(\ensuremath{\Delta}{J}_{z}=\ifmmode\pm\else\textpm\fi{}1)$ excited by a circularly polarized electromagnetic field [Feynman et al., J. Appl. Phys. 28, 49 (1957)]. The adopted real-vector representation approach allows for coupling with the vectorial Maxwell's equations for the optical wave propagation and thus the resulting Maxwell pseudospin equations can be numerically solved in the time domain without any approximations. The model permits a more exact study of the ultrafast coherent pulse propagation effects taking into account the vector nature of the electromagnetic field and hence the polarization state of the optical excitation. We demonstrate self-induced transparency effects and formation of polarized solitons. The model represents a qualitative extension of the well-known optical Maxwell-Bloch equations valid for linearly polarized light and a tool for studying coherent quantum control mechanisms.

Variational principles, Lie point symmetries, and similarity solutions of the vector Maxwell equations in non-linear optics

The vector Maxwell equations of non-linear optics coupled to a single Lorentz oscillator and with instantaneous Kerr non-linearity are investigated by using Lie symmetry group methods. Lagrangian and Hamiltonian formulations of the equations are obtained. The aim of the analysis is to explore the properties of Maxwell’s equations in non-linear optics, without resorting to the commonly used non-linear Schrödinger (NLS) equation approximation in which a high frequency carrier wave is modulated on long length and time scales due to non-linear sideband wave interactions. This is important in femto-second pulse propagation in which the NLS approximation is expected to break down. The canonical Hamiltonian description of the equations involves the solution of a polynomial equation for the electric field E, in terms of the canonical variables, with possible multiple real roots for E. In order to circumvent this problem, non-canonical Poisson bracket formulations of the equations are obtained in which the electric field is one of the non-canonical variables. Noether’s theorem, and the Lie point symmetries admitted by the equations are used to obtain four conservation laws, including the electromagnetic momentum and energy conservation laws, corresponding to the space and time translation invariance symmetries. The symmetries are used to obtain classical similarity solutions of the equations. The traveling wave similarity solutions for the case of a cubic Kerr non-linearity, are shown to reduce to a single ordinary differential equation for the variable y= E 2, where E is the electric field intensity. The differential equation has solutions y= y( ξ), where ξ= z− st is the traveling wave variable and s is the velocity of the wave. These solutions exhibit new phenomena not obtainable by the NLS approximation. The characteristics of the solutions depends on the values of the wave velocity s and the energy integration constant ϵ. Both smooth periodic traveling waves and non-smooth solutions in which the electric field gradient diverges (i.e. solutions in which | E ξ |→∞ at specific values of E, but where | E| is bounded) are obtained. The traveling wave solutions also include a kink-type solution, with possible important applications in femto-second technology.