fixed as ⌬t ϭ 0.5⌬z/c. Then the 2 nd -order Liao's T2 ABC is applied in the lower boundary where the material is air and the 2 nd -order Liao's T3 ABC is applied in the upper boundary where the material is glass, with a refractive index of 1.5.The stability criterion for the hybrid algorithm is given by Eq. (7), but Liao's ABC requires ⌬t ϭ 0.5⌬z/c. In general, if we assume ⌬x ϭ ⌬y, then we can obtain the following stability criterion for using Liao's ABC: ⌬x ϭ ⌬y Ն ⌬z ͱ6 Ϸ 1.283⌬z.This is generally satisfied since, in the x and y directions, a lower sampling rate is usually used to take the advantage of the PSTD algorithm.Because we use Fourier PSTD with a uniform grid in the x and y directions, the hybrid algorithm in this paper requires that the material contrast in the x and y directions is not too large. Otherwise, the severe Gibbs phenomenon will contaminate the simulation results. In our numerical experiments, we found that when skin depth of the absorbing layer is very small, in which the case Gibbs phenomenon is sure to appear, there are fairly large differences in the field values obtained via the hybrid method and those obtained via FDTD with smaller grid size (see Fig. 7). Here we propose several possible remedies for high-contrast materials and good conductors, if they are included in the computation domain: (i) If the object geometry is rectangular on the mask surface, we can use the mapped PSTD along the x and y directions to help reduce the magnitude of the Gibbs phenomenon in the mapped space [12,14]; (ii) if the object geometry is not rectangular but can be handled with multidomain PSTD, we suggest using multidomain PSTD in the x and y directions; (iii) use regular Fourier PSTD for the x and y directions and then apply a low-pass filter to the final steady-state results contaminated by the Gibbs phenomenon. This may help restore the global exponential convergence property of the PSTD algorithm. We expect to see more progress in these areas in the near future.
CONCLUSIONIn this paper, we have applied a hybrid PSTD-FDTD algorithm for electrically large thin-plate problems. We adopted FDTD along the vertical direction of the thin plate and PSTD along the horizontal directions. This arrangement can combine the advantages of both the FDTD and PSTD algorithms. As an example application, we studied the near-field scattering of a photo mask, and the hybrid algorithm was shown to offer large savings in memory usage and computation time, as compared with the FDTD method.