Randomized exponential integrators for modulated nonlinear Schr\"odinger equations

We consider the nonlinear Schr\"odinger equation with dispersion modulated by a (formal) derivative of a time-dependent function with fractional Sobolev regularity of class $W^{\alpha,2}$ for some $\alpha\in (0,1)$. Due to the loss of smoothness in the problem classical numerical methods face severe order reduction. In this work, we develop and analyze a new randomized exponential integrator based on a stratified Monte Carlo approximation. The new discretization technique averages the high oscillations in the solution allowing for improved convergence rates of order $\alpha+1/2$. In addition, the new approach allows us to treat a far more general class of modulations than the available literature. Numerical results underline our theoretical findings and show the favorable error behavior of our new scheme compared to classical methods

Electromagnetic scattering from thin tubular objects and an application in electromagnetic chirality

Asymptotic perturbation formulas characterize the effective behavior of waves as the volume of the scattering object tends to zero. In this work, wave propagation is described by time-harmonic Maxwell\u27s equations in free space and the corresponding scattering objects are thin tubular objects that feature a different electric permittivity and a different magnetic permeability than their surrounding medium. For this setting, we derive an asymptotic representation of the scattered electric field away from the thin tubular object and use the corresponding leading order term in a shape identification problem and for designing highly electromagnetically chiral objects. In inverse problems, the leading order term may be used to find the center curve of a thin wire that is supposed to emit a scattered field, which is reasonably close to a given measured field. For the optimal design of electromagnetically chiral structures, the representation formula provides an explicit formula for the leading order term of an asymptotic far field operator expansion. A chirality measure, usually requiring the far field operator, will now map aforementioned leading order term to a value between $0$ and $1$ dependent on the level of electromagnetic chirality of the thin tubular scatterer. This approximation greatly simplifies the challenge to maximize the chirality measure with respect to thin tubular objects. The fact that neither the evaluation of the leading order term nor the calculation of corresponding derivatives require a Maxwell system to be solved implies that the shape optimization scheme is highly efficient compared to shape optimization algorithms that use e.g. domain derivatives. In the visible range, the metallic nanowires obtained by our optimization scheme attain high values of electromagnetic chirality and even exceed those attained by traditional metallic helices

Inverse medium scattering for a nonlinear Helmholtz equation

We discuss a time-harmonic inverse scattering problem for a nonlinear Helmholtz equation with compactly supported inhomogeneous scattering objects that are described by a nonlinear refractive index in unbounded free space. Assuming the knowledge of a nonlinear far field operator, which maps Herglotz incident waves to the far field patterns of corresponding solutions of the nonlinear scattering problem, we show that the nonlinear index of refraction is uniquely determined. We also generalize two reconstruction methods, a factorization method and a monotonicity method, to recover the support of such nonlinear scattering objects. Numerical results illustrate our theoretical findings

A Fourier integrator for the cubic nonlinear Schr\"{o}dinger equation with rough initial data

Standard numerical integrators suffer from an order reduction when applied to nonlinear Schr\"{o}dinger equations with low-regularity initial data. For example, standard Strang splitting requires the boundedness of the solution in $H^{r+4}$ in order to be second-order convergent in $H^r$, i.e., it requires the boundedness of four additional derivatives of the solution. We present a new type of integrator that is based on the variation-of-constants formula and makes use of certain resonance based approximations in Fourier space. The latter can be efficiently evaluated by fast Fourier methods. For second-order convergence, the new integrator requires two additional derivatives of the solution in one space dimension, and three derivatives in higher space dimensions. Numerical examples illustrating our convergence results are included. These examples demonstrate the clear advantage of the Fourier integrator over standard Strang splitting for initial data with low regularity

Maximizing the electromagnetic chirality of thin dielectric tubes

Any time-harmonic electromagnetic wave can be uniquely decomposed into a left and a right circularly polarized component. The concept of electromagnetic chirality (em-chirality) describes differences in the interaction of these two components with a scattering object or medium. Such differences can be quantified by means of em-chirality measures. These measures attain their minimal value zero for em-achiral objects or media that interact essentially in the same way with left and right circularly polarized waves. Scattering objects or media with positive em-chirality measure interact qualitatively different with left and right circularly polarized waves, and maximally em-chiral scattering objects or media would not interact with fields of either positive or negative helicity at all. This paper examines a shape optimization problem, where the goal is to determine thin tubular structures consisting of dielectric isotropic materials that exhibit large measures of em-chirality at a given frequency. We develop a gradient based optimization scheme that uses an asymptotic representation formula for scattered waves due to thin tubular scattering objects. Numerical examples suggest that thin helical structures are at least locally optimal among this class of scattering objects

An asymptotic representation formula for scattering by thin tubular structures and an application in inverse scattering

We consider the scattering of time-harmonic electromagnetic waves by a penetrable thin tubular scattering object in three-dimensional free space. We establish an asymptotic representation formula for the scattered wave away from the thin tubular scatterer as the radius of its cross-section tends to zero. The shape, the relative electric permeability and the relative magnetic permittivity of the scattering object enter this asymptotic representation formula by means of the center curve of the thin tubular scatterer and two electric and magnetic polarization tensors. We give an explicit characterization of these two three-dimensional polarization tensors in terms of the center curve and of the two two-dimensional polarization tensor for the cross-section of the scattering object. As an application we demonstrate how this formula may be used to evaluate the residual and the shape derivative in an efficient iterative reconstruction algorithm for an inverse scattering problem with thin tubular scattering objects. We present numerical results to illustrate our theoretical findings. Mathematics subject classifications (MSC2010): 35C20, (65N21, 78A46

Randomized exponential integrators for modulated nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation with dispersion modulated by a (formal) derivative of a $\alpha$-Hölder continuous time-dependent function. Due to the highly oscillatory nature of the problem classical numerical methods face severe order reduction in non-smooth regimes $\alpha < 1$. In this work, we develop a new randomized exponential integrator based on a stratified Monte Carlo approximation which allows us to average the high oscillations in the problem and obtain improved error bounds of order $\alpha + 1/2$. In addition, the new approach allows us to treat a far more general class of modulations than the available literature. Numerical results underline our theoretical findings and show the favorable error behavior of our new scheme compared to classical methods

An asymptotic representation formula for scattering by thin tubular structures and an application in inverse scattering

We consider the scattering of time-harmonic electromagnetic waves by a penetrable thin tubular scattering object in three-dimensional free space. We establish an asymptotic representation formula for the scattered wave away from the thin tubular scatterer as the radius of its cross-section tends to zero. The shape, the relative electric permeability and the relative magnetic permittivity of the scattering object enter this asymptotic representation formula by means of the center curve of the thin tubular scatterer and two electric and magnetic polarization tensors. We give an explicit characterization of these two three-dimensional polarization tensors in terms of the center curve and of the two two-dimensional polarization tensor for the cross-section of the scattering object. As an application we demonstrate how this formula may be used to evaluate the residual and the shape derivative in an efficient iterative reconstruction algorithm for an inverse scattering problem with thin tubular scattering objects. We present numerical results to illustrate our theoretical findings

Maximizing the electromagnetic chirality of thin metallic nanowires at optical frequencies

Electromagnetic waves impinging on three-dimensional helical metallic metamaterials have been shown to exhibit chiral effects of large magnitude both theoretically and in experimental realizations. Chirality here describes different responses of scatterers, materials, or metamaterials to left and right circularly polarized electromagnetic waves. These differences can be quantified in terms of electromagnetic chirality measures. In this work we consider the optimal design of thin metallic free-form nanowires that possess measures of electromagnetic chirality as large as fundamentally possible. We focus on optical frequencies and use a gradient based optimization scheme to determine the optimal shape of highly chiral thin silver and gold nanowires. The electromagnetic chirality measures of our optimized nanowires exceed that of traditional metallic helices. Therefore, these should be well suited as building blocks of novel metamaterials with an increased chiral response. We discuss a series of numerical examples, and we evaluate the performance of different optimized designs

Maximizing the electromagnetic chirality of thin metallic nanowires at optical frequencies

Electromagnetic waves impinging on three-dimensional helical metallic metamaterials have been shown to exhibit chiral effects of large magnitude both theoretically and in experimental realizations. Chirality here describes different responses of scatterers, materials, or metamaterials to left and right circularly polarized electromagnetic waves. These differences can be quantified in terms of electromagnetic chirality measures. In this work we consider the optimal design of thin metallic free-form nanowires that possess measures of electromagnetic chirality as large as fundamentally possible. We focus on optical frequencies and use a gradient based optimization scheme to determine the optimal shape of highly chiral thin silver and gold nanowires. The electromagnetic chirality measures of our optimized nanowires exceed that of traditional metallic helices. Therefore, these should be well suited as building blocks of novel metamaterials with an increased chiral response. We discuss a series of numerical examples, and we evaluate the performance of different optimized designs