Fluctuation theory for radiative transfer in random media

We consider the effect of small scale random fluctuations of the constitutive coefficients on boundary measurements of solutions to radiative transfer equations. As the correlation length of the random oscillations tends to zero, the transport solution is well approximated by a deterministic, averaged, solution. In this paper, we analyze the random fluctuations to the averaged solution, which may be interpreted as a central limit correction to homogenization.With the inverse transport problem in mind, we characterize the random structure of the singular components of the transport measurement operator. In regimes of moderate scattering, such components provide stable reconstructions of the constitutive parameters in the transport equation. We show that the random fluctuations strongly depend on the decorrelation properties of the random medium.     

Self-consistent simulations of quantum cascade laser structures for frequency comb generation

Maxwell–Bloch equations are widely used to model the dynamics due to coherent light-matter interaction in quantum cascade laser (QCL) structures, which plays an essential role especially for the generation of frequency combs and mode-locked pulses. While the modest numerical complexity of the Maxwell–Bloch system allows for a full spatiotemporal treatment, its main disadvantage is the inclusion of dissipation by empirical dephasing rates and electron lifetimes. We present a self-consistent multi-domain approach which couples the Maxwell–Bloch equations to advanced carrier transport simulations based on a density matrix Monte Carlo technique, yielding the scattering and dephasing rates. In this way, the compact spatiotemporal modeling of the carrier-light dynamics by the Maxwell–Bloch system can be combined with the versatility and reliability of self-consistent carrier transport approaches. Simulation results are shown for a QCL-based terahertz frequency comb source, and good agreement with experiment is obtained.     

Numerical test of the theory of pseudo-diffusive transmission at the Dirac point of a photonic band structure

It has recently been predicted that a conical singularity (=Dirac point) in the band structure of a photonic crystal produces an unusual 1/L scaling of the photon flux transmitted through a slab of thickness L. This inverse-linear scaling is unusual, because it is characteristic of radiative transport via diffusion modes through a disordered medium - while here it appears for propagation of Bloch modes in an ideal crystal without any disorder. We present a quantitative numerical test of the predicted scaling, by calculating the scattering of transverse-electric (TE) modes by a two-dimensional triangular lattice of dielectric rods in air. We verify the 1/L scaling and show that the slope differs by less than 10% from the value predicted for maximal coupling of the Bloch modes in the photonic crystal to the plane waves in free space.     

Numerical method for hydrodynamic modulation equations describing Bloch oscillations in semiconductor superlattices

We present a finite difference method to solve a new type of nonlocal hydrodynamic equations that arise in the theory of spatially inhomogeneous Bloch oscillations in semiconductor superlattices. The hydrodynamic equations describe the evolution of the electron density, electric field and the complex amplitude of the Bloch oscillations for the electron current density and the mean energy density. These equations contain averages over the Bloch phase which are integrals of the unknown electric field and are derived by singular perturbation methods. Among the solutions of the hydrodynamic equations, at a 70 K lattice temperature, there are spatially inhomogeneous Bloch oscillations coexisting with moving electric field domains and Gunn-type oscillations of the current. At higher temperature (300 K) only Bloch oscillations remain. These novel solutions are found for restitution coefficients in a narrow interval below their critical values and disappear for larger values. We use an efficient numerical method based on an implicit second-order finite difference scheme for both the electric field equation (of drift-diffusion type) and the parabolic equation for the complex amplitude. Double integrals appearing in the nonlocal hydrodynamic equations are calculated by means of expansions in modified Bessel functions. We use numerical simulations to ascertain the convergence of the method. If the complex amplitude equation is solved using a first order scheme for restitution coefficients near their critical values, a spurious convection arises that annihilates the complex amplitude in the part of the superlattice that is closer to the cathode. This numerical artifact disappears if the space step is appropriately reduced or we use the second-order numerical scheme.     

Stochastic radiative transfer in multilayer broken clouds. Part I: Markovian approach

A two-part paper describes the statistical treatment of solar radiative transfer in multilayer broken clouds. The proposed approach is a logical development of the statistical ones originally suggested for a single-layer broken clouds. This first part introduces a new statistically inhomogeneous Markovian model that allows one to properly account for different combinations of the random and maximum overlap of broken clouds at distinct vertical layers. The statistically inhomogeneous Markovian model and the stochastic radiative transfer equation have been used to derive equations for the mean radiance of solar radiation. It was demonstrated that in extreme cases the obtained equations agree with corresponding equations previously derived for (i) the statistically homogeneous broken clouds and (ii) the vertically inhomogeneous overcast clouds.     

Real-time dynamics of the acoustically driven electron–hole transport in GaAs quantum wires

We have investigated the dynamics of photogenerated carriers in GaAs quantum wires during their transport by surface acoustic waves using spatially and time-resolved photoluminescence. This technique allows us to map in real time the spatial carrier distribution during the transport as well as to study the nature of radiative defect sites in the transport path.     

Two-dimensional array of room-temperature nanophotonic logic gates using InAs quantum dots in mesa structures

We present a detailed study of the dynamics of light in passive nonlinear resonators with shallow and deep intracavity periodic modulation of the refractive index in both longitudinal and transverse directions of the resonator. We investigate solutions localized in the transverse direction (so-called Bloch cavity solitons) by means of envelope equations for underlying linear Bloch modes and solving Maxwell’s equations directly. Using a round-trip model for forward and backward propagating waves we review different types of Bloch cavity solitons supported by both focusing (at normal diffraction) and defocussing (at anomalous diffraction) nonlinearities in a cavity with a weak-contrast modulation of the refractive index. Moreover, we identify Bloch cavity solitons in a Kerr-nonlinear all-photonic crystal resonator solving Maxwell’s equations directly. In order to analyze the properties of Bloch cavity solitons and to obtain analytical access we develop a modified mean-field model and prove its validity. In particular, we demonstrate a substantial narrowing of Bloch cavity solitons near the zero-diffraction regime. Adjusting the quality factor and resonance frequencies of the resonator optimal Bloch cavity solitons in terms of width and pump energy are identified.     

Exact Non-Singular Waves in the Anti-de Sitter Universe

A class of radiative solutions of Einstein's field equations with a negative cosmological constant and a pure radiation is investigated. The space-times, which generalize the Defrise solution, represent exact gravitational waves which interact with null matter and propagate in the anti–de Sitter universe. Interestingly, these solutions have homogeneous and non-singular wave-fronts for all freely moving observers. We also study properties of sandwich and impulsive waves which can be constructed in this class of space-times.     

Gauge treatment of the intra-collisional-field-effect in electron transport theory

We study the evolution equations of the distribution functions of hot electrons. We show that a convenient choice of the gauge describing the applied uniform electric and magnetic fields considerably simplifies the explicit calculations. The main advantage of our method lies in the possibility of treating with the same simplicity free electrons (with or without a magnetic field) and Bloch electrons (without a magnetic field). We discuss the influence of the electric field on the collision term of the different transport equations we derive.     

A boundary integral method to compute Green's functions for the radiative transport equation

The Green's function for the time-independent radiative transport equation in the whole space can be computed as an expansion in plane wave solutions. Plane wave solutions are a general class of solutions for the radiative transport equation. Because plane wave solutions are not known analytically in general, we calculate them numerically using the discrete ordinate method. We use the whole space Green's function to derive boundary integral equations. Through the solution of the boundary integral equations, we compute the Green's function for bounded domains. In particular we compute the Green's function for the half space, the slab, and the two-layered half space. The boundary conditions used here are in their most general form. Hence, this theory can be applied to boundaries with any kind of reflection and transmission law.     

LTE computation of axisymmetric arc-flow interaction incircuit-breakers

A numerical procedure for the computation of arc-flow interaction in gas-blast circuit breakers is presented. In the proposed approach the flow is obtained via the solution of the Euler equations and the coupling with the arc by a source term in the energy equation. This source term is composed of an ohmic energy production part and a radiative transport part. The equations are solved by a time-marching procedure using a finite-volume discretization. The model is applied to the computation of an axisymmetric arc in a model breaker. The results reproduce qualitatively the major features of arc-flow interaction and show the presence of a localized form of choking of the flow around the arc boundary     

Ultrafast carrier dynamics in semiconductor nanostructures: interplay between coherence and relaxation

The strong coupling between coherent and incoherent ultrafast phenomena in the electro-optical response of semiconductor nanostructures is discussed theoretically within a density matrix formalism. In particular, the problem of scattering-induced damping of Bloch oscillations in superlattices is reviewed. Moreover, a generalization to ‘open systems’ of the conventional semiconductor Bloch equations is discussed. The presence of spatial boundary conditions manifests itself through self-energy corrections and additional source terms in the kinetic equations. As an example, some simulated experiments of quantum transport phenomena through double-barrier structures are reviewed.     

Sensitivity to initial conditions of the wave-packet dynamics in diluted Anderson chains

We study the one-electron wave-packet dynamics in the one-dimensional diluted Anderson model which is composed of two interpenetrating chains with pure and random on-site potentials, respectively. This model presents extended states at a particular resonance energy. Starting with one electron fully localized at the site closer to the chain center, we solve the set of coupled motion equations and calculate the time evolution of the wave-packet width. We report on a long-time memory effect which is reflected by distinct asymptotic dynamics governing the wave-function spread for electrons initially localized at random or pure sites. This anomalous behavior is discussed under the light of the Bloch character of the extended resonant state.     

Numerical solutions to 2D Maxwell–Bloch equations

Rare-earth-doped crystals contain inhomogeneously broadened two-level atoms. Optical propagation and nonlinear interaction in the crystals can be described by the Maxwell–Bloch equations. We show a consistent numerical approach that solves Maxwell’s equations by using the FFT-finite difference beam propagation method and the Bloch equations by using the finite difference method. Numerical simulation results are given for an off-axis 3-pulse photon echo.     

Finite-size effects in random energy models and in the problem of polymers in a random medium

By the use of traveling wave equations we calculate the finite-size corrections to the free energy of random energy models in their low-temperature phases and in the neighborhood of the transition temperature. We find that although the extensive part of the free energy does not show any critical behavior when the temperature approaches its transition value, the finite-size corrections signal the transition by becoming singular. We obtain a scaling form for these finite-size corrections valid in the limitN andTT _c . By considering a generalized random energy model in the limit of a very large number of steps, we obtain results for the finite-size corrections in the problem of a polymer in a random medium.     

A boundary integral method to compute Green's functions for the radiative transport equation

The Green's function for the time-independent radiative transport equation in the whole space can be computed as an expansion in plane wave solutions. Plane wave solutions are a general class of solutions for the radiative transport equation. Because plane wave solutions are not known analytically in general, we calculate them numerically using the discrete ordinate method. We use the whole space Green's function to derive boundary integral equations. Through the solution of the boundary integral equations, we compute the Green's function for bounded domains. In particular we compute the Green's function for the half space, the slab, and the two-layered half space. The boundary conditions used here are in their most general form. Hence, this theory can be applied to boundaries with any kind of reflection and transmission law.     

On the standard form of the Bloch equation

The requirement is often made in non-equilibrium statistical mechanics that a transport equation should be derived as that which governs the subdynamics relative to a (small) part of a (large) conservative dynamical system close to equilibrium. We show that such a requirement on the Markovian relaxation of a 1/2-spin imposes that this process be described by a Bloch equation of a very specific form, which we call standard. We show that this reduced dynamics is quasi-free if, and only if, the relaxation time is maximally anisotropic.Research supported in part by NSF grant MCS 76-07286     

Laboratory simulations of astrophysical blast waves with high energy lasers

We review recent efforts to simulate aspects of supernova remnants in laboratory experiments by creating energetic explosions with high energy lasers. High energy pulsed lasers are uniquely suited for these kinds of studies. By focusing a laser with pulse energy of a few joules to many hundreds of joules onto a solid target or into a dense gas target, explosive shock waves of very high Mach number can be created. With a well chosen set of laser and target parameters it has been shown by a number of groups that radiative blast waves can be produced. Such blast waves have dynamics dominated by radiation transport and exhibit unusual characteristics, the most important of which include hydrodynamic instabilities which may play an important role in the structure of the interstellar medium. As a result there are now prospects for gaining new insights into astronomical observations of supernova remnants by studying laboratory laser driven plasma systems.     

Determination of band structure in one-dimensional complex photonic crystals: A bandedge approach

We present a numerical stable method to easily compute bandgaps of one-dimensional complex basis photonic crystals using novel bandedge equations rather than the traditional dispersion equation. The bandedge equations are derived by the concept of scattering matrix and concisely expressed by the transport coefficients with operations of multiplication and addition only. It is not required to calculate the global transfer matrix or the cosine of Bloch phase to avoid numerical instability. Moreover, we present closed-form expressions to calculate the global scattering matrix without using recursive computation. Finally, numerical examples show that use of the derived bandedge equation has better numerical stability than using the traditional dispersion equation for the band structure analysis.