Finite Scatterers in Homogeneous Medium

The Mie theory [1] and the T-matrix method [4] are very efficient for (multilayered) spheres and axisymmetric particles (with moderate aspect ratios), respectively. Several methods, applicable to particles of arbitrary shapes, have been used in plasmonic simulations the boundary element method (BEM) [5, 6], the DDA [7-9], the finite-difference time-domain method (FDTD) [10, 11], the finite element method (FEM] [12,13], the finite integration technique (FIT) [14] and the null-field method with discrete sources (NFM-DS) [15,16]. There is also quasi-static approximation for spheroids [12], but it is not discussed here. 

The BEM, the DDA, the FEM, and the NFM-DS solve the Maxwell s equations in the frequency domain. The BEM and the DDA solve by discretization the corresponding surface- and volume-integral equation, respectively. In the FEM the differential form of Maxwell s 

The FDTD and the FIT solve the time-domain Maxwell s equations in the original and modified form, respectively. These two methods, as well as the FEM, need to discretize not only the particle but also some space around it. 

A systematic comparison of the methods should include the simulation of the several (the more—the better) test problems by these methods running on the same hardware. Such comparisons were performed for dielectric particles, see, e.g. Refs. [20-22], but they are not relevant for the plasmonics. On contrary, in plasmonics such comparisons are very rare. We can cite three examples, which both considered a single specific scattering problem, making it hard to generalize the conclusions. In particular, the FDTD and the FEM were compared for computation of near-field around 50-nm silver cube interacting with 600-nm plane-wave [12]. Accuracy of the FEM was worse than that of the FDTD but still satisfactory. The FEM simulation required 4 hours on a single 3.4 GHz processor, while FDTD—8 hours on 256 double-core 2.6 GHz processors. Another comparison [14] addressed the DDA and the FIT (the latter implemented in the commercial software) for simulation of refractive index sensitivity of rhombic hybrid Au-Ag nanostructure array. Both methods obtained the same value of sensitivity, but the DDA was faster (not specified how much). 

The third example [19] is the most S5 ematic one. The DDA, the FEM, and the FDTD were used for the calculation of scattering spectrum of 80-nm gold sphere with several discretizations levels for each method. Unfortunately, the accuracy and simulation times are discussed separately, so it is impossible to say which time corresponds to which accuracy (discretization level). General trend is that DDA is faster but less accurate than the FEM. Simulation time of the FDTD is comparable to that of the FEM, but its accuracy is the worst of three methods. Summarizing all three examples, truly systematic comparisons in these helds will definitely benefit the community. 

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