Discrete dipole approximation in time domain through the Laplace transform.

Electromagnetic scattering by an arbitrary threedimensional structure, in time domain, is usually modelled using the finite difference in time domain (FDTD) method [1–3]. In the FDTD, one solves numerically the differential form of Maxwell’s equations on a grid. A constraint of the FDTD is that the entire computational domain needs to be discretized [4]. By contrast, the discrete dipole approximation (DDA), a scattering computation method, requires that only the scatterer (or its immediate neighborhood) be discretized [5–8]. In the DDA the outgoing wave condition is automatically satisfied by using dyadic field susceptibility tensors to describe the linear response of the fields. However, in its traditional formulation, the DDA is a frequency domain method, restricted to time-harmonic fields. Recently we generalized the DDA to handle arbitrary, nontime-harmonic electromagnetic waves. The method, detailed in Ref. [9], consists of solving the electromagnetic scattering in frequency domain, and performing a Fourier transform to generate the time evolution of electromagnetic quantities. Of course, with this approach, one drawback of the DDA is that we must solve a large system of linear equations to find the fields inside the scatterer [4]. This can be prohibitive in terms of computer memory requirements. A common way to circumvent this problem is to use an iterative method. However, such an approach requires us to calculate many times a large matrix-vector product (MVP), and to do so for all the frequencies required to accurately describe the time evolution of the fields. Therefore, the main bottleneck for the computation time is the total number of MVP required in order to achieve the desired level of convergence of the iterative method. One can decrease the number of MVP by choosing an efficient iterative method; for instance we use a combination of the generalized product-type methods based on biconjugate gradient (GPBICG) [10], a good initial value [9] and a preconditoner of Jacobi [11,12], but nevertheless the convergence is still slow. In this article we present a general strategy to reduce the number of MVPs by introducing complex frequencies into the problem via the Laplace transform. The outcome is a reduction of the number of MVPs, and hence a speedup of the computation. However, this is not the only benefit. This approach allows us to handle resonant scatterers, for instance a plasmon resonance, in time domain, something that the Fourier transform approach of Ref. [9] cannot do. In Sec. II we briefly present the DDA method in both its time-harmonic and time domain versions, and then, in Sec. III we present the results. Finally in Sec. IV we present our conclusion.