High-frequency switching and Kerr effect - nonlinear problems solved with nonstationary time domain techniques

A time domain method building on the concept of wave splitting is used to study direct wave propagation phenomena in weakly nonlinear media. The starting point is the linear wave equation with time-dependent coefficients. This means that the studied nonlinear medium in some sense has to be approximated with a nonstationary medium which changes while the wave passes through. For the nonstationary equation homogeneous as well as particular solutions can be obtained. Two different iterative procedures to find the nonlinear solutions are discussed. They are illustrated by two problems fetched from different research fields of current interest. In the first case, the nonlinear term is linearized using the Fr´echet derivative. This leads into a truly nonstationary, mixed initial boundary value problem with a linear equation characterized by both time-dependent coefficients and source terms. In this example a semiconductor device used for switching in high-frequency applications is considered. It can be described as a coplanar waveguide loaded with distributed resonant tunnel diodes. In the other example, wave propagation in Kerr media is considered. Then Taylor expansion transforms the nonlinear equation into a linear one with nonstationary source terms. In this case the nonlinearity does not lead to time-depending coefficients in the equation. The way to obtain the solution is a nonlinear variant of the Born approximation

Transient waves from internal sources in non-stationary media - Numerical implementation

In this paper, the focus is on numerical results from calculations of scattered direct waves, originating from internal sources in non-stationary, dispersive, stratified media. The mathematical starting point is a general, inhomogeneous, linear, first order, 2 × 2 system of equations. Particular solutions are obtained, as integrals of waves from point sources distributed inside the scattering medium. Resolvent kernels are used to construct time dependent fundamental wave functions at the location of the point source. Wave propagators, closely related to the Green functions, at all times advance these waves into the surrounding medium. Two illustrative examples are given. First waves, propagating from internal sources in a Klein-Gordon slab, are calculated with the new method. These wave solutions are compared to alternative solutions, which can be obtained from analytical fundamental waves, solving the Klein-Gordon equation in an infinite medium. It is shown, how the Klein-Gordon wave splitting, which transforms the Klein-Gordon equation into a set of uncoupled first order equations, can be used to adapt the infi- nite Klein-Gordon solutions to the boundary conditions of the Klein-Gordon slab. The second example hints at the extensive possibilities offered by the new method. The current and voltage waves, evoked on the power line after an imagined strike of lightning, are studied. The non-stationary properties are modeled by the shunt conductance, which grows exponentially in time, together with dispersion in the shunt capacitance

One-way wave operators for nonstationary dielectrics

Propagation of transient electric and magnetic (TEM) pulses in nonstationary, linear, homogeneous, and isotropic dielectric and magnetic materials is investigated using an exact wave splitting. Key intrinsic properties are the index of refraction and the relative admittance, which are both temporal integral operators with kernels that depend on two time variables. In addition, the Sommerfeld forerunners in dispersive nonstationary materials are derived. A numerical example — a single-resonance Lorentz model with time-dependent plasma frequency — is presented

Transient waves in non-stationary media

This paper treats propagation of transient waves in non-stationary media, which has many applications in e.g. electromagnetics and acoustics. The underlying hyperbolic equation is a general, homogeneous, linear, first order 2×2 system of equations. The coefficients in this system depend only on one spatial coordinate and time. Furthermore, memory effects are modeled by integral kernels, which, in addition to the spatial dependence, are functions of two different time coordinates. These integrals generalize the convolution integrals, frequently used as a model for memory effects in the medium. Specifically, the scattering problem for this system of equations is addressed. This problem is solved by a generalization of the wave splitting concept, originally developed for wave propagation in media which are invariant under time translations, and by an imbedding or a Green functions technique. More explicitly, the imbedding equation for the reflection kernel and the Green functions (propagator kernels) equations are derived. Special attention is paid to the problem of non-stationary characteristics. A few numerical examples illustrate this problem

Propagation of transient electromagnetic waves in time-varying media - Direct and inverse scattering problems

Wave propagation of transient electromagnetic waves in time-varying media is considered. The medium, which is assumed to be inhomogeneous and dispersive, lacks invariance under time translations. The spatial variation of the medium is assumed to be in the depth coordinate, i.e., it is stratified. The constitutive relations of the medium is a time integral of a generalized susceptibility kernel and the field. The generalized susceptibility kernel depends on one spatial and two time coordinates. The concept of wave splitting is introduced. The direct and inverse scattering problems are solved by the use of an imbedding or a Green functions approach. The direct and the inverse scattering problems are solved for a homogeneous semi-infinite medium. Explicit algorithms are developed. In this inverse scattering problem, a function depending on two time coordinates is reconstructed. Several numerical computations illustrate the performance of the algorithms

Transient waves in nonstationary media

This paper treats propagation of transient waves in nonstationary media, which has many applications in, for example, electromagnetics and acoustics. The underlying hyperbolic equation is a general, homogeneous, linear, first-order 2×2 system of equations. The coefficients in this system depend on one spatial coordinate and time. Furthermore, memory effects are modeled by integral kernels, which, in addition to the spatial dependence, are functions of two different time coordinates. These integrals generalize the convolution integrals, frequently used as a model for memory effects in the medium. Specifically, the scattering problem for this system of equations is addressed. This problem is solved by a generalization of the wave splitting concept, originally developed for wave propagation in media which are invariant under time translations, and by an imbedding or a Green's functions technique. More explicitly, the imbedding equation for the reflection kernel and the Green's functions (propagator kernels) equations are derived. Special attention is paid to the problem of nonstationary characteristics. A few numerical examples illustrate this problem

Transient waves in nonstationary media : with applications to lightning and nonlinear media

Propagation of transient waves in nonstationary, inhomogeneous, dispersive, stratified media is considered. Waves originating from sources exterior to the scatterer as well as from internal sources are treated. Algorithms are developed and illustrated by computations of wave phenomena in stationary, nonstationary and weakly nonlinear media. In the theoretical part, the underlying hyperbolic equation is a general, homogeneous, linear, first order 2x2 system of equations. The coefficients depend on one spatial coordinate and time. Memory effects are modeled by integral kernels, which are functions of two different time coordinates. The analysis builds on generalization of the wave splitting concept, originally developed for time-invariant media. Imbedding and Green functions (propagator kernels) equations are derived for the external source problem. The wave propagators of the internal source problem are based on generalized Green functions equations. Special attention is paid to characteristic curves and discontinuities. Particular solutions are obtained as integrals of fundamental waves from distributed point sources. Resolvent kernels and wave propagators are essential. Direct and inverse computational algorithms are developed for the nonstationary, homogeneous semi-infinite medium. Generalized susceptibility kernels with one spatial and two time coordinates are used. A function depending on two time coordinates is reconstructed. Furthermore, direct scattering algorithms for internal sources are implemented. Waves in a Klein-Gordon slab are calculated and compared to alternative solutions obtained from analytical fundamental waves of an infinite Klein-Gordon medium. In a second example, the current and voltage waves, evoked on the power line after an imagined strike of lightning, are studied. The nonstationary properties are modeled by the shunt conductance, together with dispersion in the shunt capacitance. The nonstationary theory is used to study direct wave propagation phenomena in weakly nonlinear media by linearization. Two different iterative procedures to find the nonlinear solutions are discussed. One leads into a truly nonstationary, mixed initial boundary value problem with a linear equation characterized by time-dependent coefficients and source terms. This procedure is applied to a pulse generator for high-frequency switching. The alternative time-invariant procedure, which is a variation of the nonlinear Born approximation, is used to calculate wave propagation in Kerr media

Transient waves from internal sources in non-stationary media - Numerical implementation

In this paper, the focus is on numerical results from calculations of scattered direct waves, originating from internal sources in non-stationary, dispersive, stratified media. The mathematical starting point is a general, inhomogeneous, linear, first order, 2 × 2 system of equations. Particular solutions are obtained, as integrals of waves from point sources distributed inside the scattering medium. Resolvent kernels are used to construct time-dependent fundamental wave functions at the location of the point source. Wave propagators, closely related to the Green functions, at all times advance these waves into the surrounding medium. Two illustrative examples are given. First waves, propagating from internal sources in a Klein-Gordon slab, are calculated with the new method. These wave solutions are compared to alternative solutions, which can be obtained from analytical fundamental waves, solving the Klein-Gordon equation in an infinite medium. It is shown, how the Klein-Gordon wave splitting, which transforms the Klein-Gordon equation into a set of uncoupled first order equations, can be used to adapt the infinite Klein-Gordon solutions to the boundary conditions of the Klein-Gordon slab. The second example hints at the extensive possibilities offered by the new method. The current and voltage waves, evoked on the power line after an imagined strike of lightning, are studied. The non-stationary properties are modeled by the shunt conductance, which grows exponentially in time, together with dispersion in the shunt capacitance

"The source problem" - Transient waves propagating from internal sources in non-stationary media

Direct scattering of propagating transient waves originating from internal sources in non-stationary, inhomogeneous, dispersive, stratified media, is investigated. The starting point is a general, inhomogeneous, linear, first order, 2 × 2 system of equations. Particular solutions are obtained, as integrals of fundamental waves from point sources distributed throughout the medium. First, resolvent kernels are used to construct time dependent fundamental wave functions at the location of the point source. Wave propagators, closely related to the Green functions, at all times advance these time dependent waves into the surrounding medium. The propagator equations and the propagation of propagator kernel discontinuities along the characteristics of these equations are essential in the distributional proof, which is outlined. As an illustration, three special problems are studied; the inhomogeneous, second order wave equation, and source problems in homogeneous and time invariant media