Traditional and new simulation techniques for nanoscale optics and photonics

Several classes of computational methods are available for computer simulation of electromagnetic wave propagation and scattering at optical frequencies: Discrete Dipole Approximation, the T-matrix - Extended Boundary Condition methods, the Multiple Multipole Method, Finite Difference (FD) and Finite Element (FE) methods in the time and frequency domain, and others. The paper briefly reviews the relative advantages and disadvantages of these simulation tools and contributes to the development of FD methods. One powerful tool - FE analysis - is applied to optimization of plasmon-enhanced AFM tips in apertureless near-field optical microscopy. Another tool is a new FD calculus of "Flexible Local Approximation MEthods" (FLAME). In this calculus, any desirable local approximations (e.g. scalar and vector spherical harmonics, Bessel functions, plane waves, etc.) are seamlessly incorporated into FD schemes. The notorious 'staircase' effect for slanted and curved boundaries on a Cartesian grid is in many cases eliminated - not because the boundary is approximated geometrically on a fine grid but because the solution is approximated algebraically by suitable basis functions. Illustrative examples include problems with plasmon nanoparticles and a photonic crystal with a waveguide bend; FLAME achieves orders of magnitude higher accuracy than the standard FD methods, and even than FEM.