Standard numerical integrators suffer from an order reduction when applied to nonlinear Schrödinger 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.

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.

Designing objects with predefined optical properties is a task of fundamental importance for nanophotonics, and chirality is a prototypical example of such a property, with applications ranging from photochemistry to nonlinear photonics. A measure of electromagnetic chirality with a well-defined upper bound has recently been proposed. Here, we optimize the shape of silver helices at discrete frequencies ranging from the far-infrared to the optical band. Gaussian process optimization, taking into account also shape derivative information on the helices scattering response, is used to maximize the electromagnetic chirality. We show that the theoretical designs achieve more than 90% of the upper bound of em-chirality for wavelenghts 3 μm or larger, while their performance decreases toward the optical band. We fabricate and characterize helices for operation at 800 nm and identify some of the imperfections that affect the performance. Our work motivates further research both on the theoretical and fabrication sides to unlock potential applications of objects with large electromagnetic chirality at optical frequencies, such as helicity filtering glasses. We show that, at 3 μm, a thin slab of randomly oriented helices can absorb 99% of the light of one helicity while absorbing only 10% of the opposite helicity.

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.

We consider the nonlinear Schrödinger equation with dispersion modulated by a (formal) derivative of a time-dependent function with fractional Sobolev regularity of class W α,2 for some α ∈ (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 α + 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.