Another method to solve the problem of RCWA for highly conductive and lossless metallic gratings under TM polarization is using two-dimensional finite difference (2DFD) to solve scattering problems directly. By comparing the duty cycle variation of graded-index 2DFD with RCWA and FDMM, it could be found that the graded-index approximation is appropriate for small index difference. Fig. 3.42 shows the duty cycle dependence of the minus-first order reflection, and the its parameters are: εr,inc = εr,g2 = 1, εr,g1 = εr,tra = 32, f from 0.98 to 0.08, Λ = 500nm, λ = 632.8nm, tg = 500nm and θinc = 30◦. There is a little deviation from the results of RCWA and FDMM because the index difference between 3 and 1 is not really small.
Figure 3.42: Duty cycle variation of the minus-first-order reflection. A di-electric grating with εg1 = 32 under TM incidence. 2DFD with averaging permittivity is used.
However, for large index difference and nearly abrupt interfaces, the graded-index approximation has difficulty in accurately modeling the field behaviors. As in Fig. 3.43, which has the same parameters as in Fig. 3.12, it is found that the line of graded-index 2DFD deviates from the appropriated results but does not suffer large instability like RCWA.
Figure 3.43: Duty cycle variation of the minus-first-order reflection. A loss-less metallic grating with εg1 = (−10i)2 under TM incidence. 2DFD with averaging permittivity is used.
For the structure with abrupt interfaces or large index difference, it is better to consider the interface conditions in 2DFD, and this method is demonstrated in section 2.3. Using step-index 2DFD method to approach the same problem as Fig. 3.43, the results are shown in Fig. 3.44. It is found that the result of step-index 2DFD matches the results of FDMM well and does not suffer any instabilities.
Figure 3.44: Duty cycle variation of the minus-first-order reflection. A loss-less metallic grating with εg1 = (−10i)2 under TM incidence. 2DFD with considering boundary condition is used.
The fields diagrams obtained from four methods which have been pre-sented in the thesis are shown in Fig. 3.45-Fig. 3.48. Fig. 3.45 and Fig.
3.46 have the same parameters as Fig. 3.44 with 302nm groove width. And Fig. 3.47 and Fig. 3.48 are the results for 250nm groove width. In Fig.
3.45 and Fig. 3.47, it is found that the fields calculated by RCWA do not vanish totally inside the lossless metallic gratings. This phenomenon may be explained by the spurious modes mentioned by Lyndin et al. [19] that the spurious mode resonance causes the field enhancement in the grating region.
Figure 3.45: Field diagram of Fig. 3.12 at groove width= 302nm (Left:
RCWA, Right: FDMM)
Figure 3.46: Field diagram of Fig. 3.12 at groove width= 302nm (Left:
Graded-index 2DFD, Right: Step-index 2DFD)
Figure 3.47: Field diagram of Fig. 3.12 at groove width= 250nm (Left:
RCWA, Right: FDMM)
Figure 3.48: Field diagram of Fig. 3.12 at groove width= 250nm (Left:
Graded-index 2DFD, Right: Step-index 2DFD)
Chapter 4 Conclusion
The arbitrary-order finite-difference modal method (FDMM) with step-index formulation for the analysis of gratings with one-dimensional periodicity has been presented. The correctness of FDMM is testified by comparing with some papers, and the accuracy and convergence of this method were com-pared with the Fourier modal method (FMM), also named rigorous coupled-wave analysis (RCWA). It is found that the accuracy of three-point FDMM with generalized Douglas scheme and even higher order methods are better than FMM for TE polarization in almost all cases. In addition, using nonuni-form discretization with increased resolution near the discontinuities could accelerate the convergence. For TM polarization, the accuracy of FDMM could be superior to FMM for high conductive and lossless metallic grat-ings, and even for usual lossy metallic gratings as simulating gratings with arbitrary profiles by multi-layer approximation.
However, numerical results of rectangular-groove gratings indicates that
a proper discretization for FDMM is important under TM incidence. This problem may be attributed to the corners existing in calculated configura-tion and could be explained by disregard of some boundary condiconfigura-tions of the fields normal to the vertical interfaces as constructing the sparse matrix of each layer which causes contradiction of continuity of the fields tangential to horizontal interfaces as matching boundary conditions between every lay-ers. Therefore, for arbitrary profile gratings under TM incidence, a proper discretization might be achieved by making the distances between disconti-nuities and their adjacent grid points on both sides of them equal, and the results will avoid instabilities and converge to the correct answer smoothly.
Moreover, this technique could also be used for TE polarization to make results more stable.
Besides using FDMM to solve the serious problem of FMM as simu-lating lossless metallic gratings for TM polarization, the two-dimensional finite-difference methods of both graded-index and step-index formulation are presented and shown that the results are correct and stable, especially for step-index formulation.
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