3.4.1 Rectangular-Groove Gratings
In this subsection accuracy and convergence are investigated again with the same parameters as those in preceding section except for TM polarization.
Note that here the method of discretization in each layer is distributing grid points according to the ratio of width of each region inside that layer. This process could avoid too large difference of distances from sample points to discontinuities between points on each side of discontinuities and will be discussed in the last part of this subsection.
A. Dielectric
Consider a dielectric grating with εg1 = εtra = (3)2, and the results are shown in Fig. 3.27 and Fig. 3.28. It could be found that although the propagation constants β of five-point FDMM without GD and methods with even higher order converge faster than RCWA, the convergence of diffraction efficiencies of FDMM are slower than RCWA, and the results of every FDMM are on the same degree of accuracy except for three-point FDMM.
B. Lossless metal
A lossless grating with εg1 = εtra = (−10i)2 is tested, and the results is shown in Fig. 3.29 and Fig. 3.30. In this case, RCWA’s results are quite unstable and do not converge to a satisfying degree of accuracy. However, such kind of situation will not happen in FDMM. Therefore, although not
all convergence lines of β of FDMM converge faster than that of RCWA, the convergence of diffraction efficiencies of FDMM is superior to RCWA even for three-point FDMM without GD.
C. Lossy metal I
A lossy metallic grating with εg1 = εtra = (3.18 − 4.41i)2 is considered.
Like dielectric gratings with TM polarization, the diffraction efficiencies of FDMM converge slower than RCWA. The results are shown in Fig. 3.31 and Fig. 3.32.
D. Lossy metal II
If the parameter εg1 of lossy gratings is replaced by highly conductive one, FDMM will become superior to RCWA. To verify this statement, con-sider a lossy metallic grating with εg1 = εtra = (0.22 − 6.71i)2 which is the permittivity of gold at λ = 1000nm, and the results are shown in Fig. 3.33 and Fig. 3.34.
E. Lossy metal with high conductivity
Finally, we test FDMM in an even higher conductive grating with εg1 = εtra = (1 − 40i)2. The results are shown in Fig. 3.35 and Fig. 3.36. Although the results of FDMM will be more correct than RCWA until exceeding 200 grid’s number, FDMM can still be seen as a better choice than RCWA be-cause the result of RCWA converges quite slowly in this case. Note that the FDMM inside each layer can not be taken non-uniform discretization under
TM polarization.
Figure 3.27: Accuracy and convergence of β of fundamental mode. A dielec-tric grating with εr,g1 = 32 under TM incidence. Grids are distributed by ratio of width.
Figure 3.28: Convergence of the zeroth-order reflection. A dielectric grating with εr,g1 = 32 under TM incidence. Grids are distributed by ratio of width.
Figure 3.29: Accuracy and convergence of β of fundamental mode. A loss-less metallic grating with εr,g1 = (−10i)2 under TM incidence. Grids are distributed by ratio of width.
Figure 3.30: Convergence of the zeroth-order reflection. A lossless metallic grating with εr,g1 = (−10i)2 under TM incidence. Grids are distributed by ratio of width.
Figure 3.31: Accuracy and convergence of β of fundamental mode. A lossy grating with εr,g1= (3.18−4.41i)2under TM incidence. Grids are distributed by ratio of width.
Figure 3.32: Convergence of the zeroth-order reflection. A lossy grating with εr,g1 = (3.18 − 4.41i)2 under TM incidence. Grids are distributed by ratio of width.
Figure 3.33: Accuracy and convergence of β of fundamental mode. A lossy grating with εg1 = (0.22 − 6.71i)2 under TM incidence. Grids are distributed by ratio of width.
Figure 3.34: Convergence of the zeroth-order reflection. A lossy grating with εg1 = (0.22 − 6.71i)2 under TM incidence. Grids are distributed by ratio of width.
Figure 3.35: Accuracy and convergence of β of fundamental mode. A highly conductive grating with εg1 = (1 − 40i)2 under TM incidence. Grids are distributed by ratio of width.
Figure 3.36: Convergence of the zeroth-order reflection. A highly conductive grating with εg1 = (1 − 40i)2 under TM incidence. Grids are distributed by ratio of width.
At the beginning of this subsection, we noted a proper way for TM po-larization to dicretize. It is found that the ratio of distance between dis-continuities and their adjacent grid points affect the convergence properties, especially for the convergence of diffraction efficiencies. This problem may be due to ignoring some boundary conditions while constructing the sparse ma-trix and solving eigen-problem in each layer. And this omission will cause the contradiction of continuity of fields as matching boundary condition between every layers.
Figure 3.37: Continuity of Ex fields for TM polarization.
As in Fig. 3.37, while matching boundary of tangential field Ex for TM polarization, the fields 1∂H∂zy are continuous at the interface between the upper and the lower layer. But, inside the lower layer of Fig. 3.37, the field 1
L
∂Hy
∂z
on the left side and 1
R
∂Hy
∂z on the right side are not continuous at the vertical
interface. Therefore, it causes the contradiction of continuity of fields 1∂H∂zy at the vertical interface in the lower layer of Fig. 3.37. In addition, while constructing the sparse matrix for TM polarization in FDMM, the boundary condition of the fields 1∂H∂zy or the boundary condition of fields Ex is not considered, and what we considered are only tangential fields Hy and Ez. This effect of contradiction does not happen in TE polarization because the fields ∂E∂zy (which means Bx) used in matching boundary condition between every layers are always continuous at the vertical interface of Fig. 3.37.
As using nonuniform discretization to increasing the spatial resolution near the interfaces for TM polarization, or distributing grid points without considering the ratio of width of each different region inside one layer, such kind of contradiction may impact the stability and the convergence more drastically. Hence, this is my speculation on the failure of FDMM under TM polarization for nonuniform discretization and the dependance on the ratio of distances between the interface and its adjacent grid points. In addition, the similar problem of continuity at corners has been discussed by Chiou et al. [31].
However, this problem does not happen in FMM (or RCWA) as well, even for TM polarization. It could be explained by using 1
r
for a specific order. The boundary conditions of fields at the interfaces are matched by use of the same Fourier order terms in FMM (or RCWA). And it could be found in (3.4) that this field of the specific order is continuous at the vertical interface of Fig. 3.37 naturally because the x-dependance of the fields is a continuous function e−jkxix. Therefore, this is my speculation on why RCWA does not have such contradictory problem. Although RCWA does not suffer from the problem of contradiction of field’s continuity as matching boundary condition between every adjacent layers, it has difficulty in effec-tively describing structures with abrupt and large change in permittivity by Fourier bases because such bases are continuous functions.
In a word, the difficulty of FMM (or RCWA) is inside one layer to de-scribe the abrupt change material by continuous functions, and the problem of FDMM is contradiction of continuity of fields as matching boundary con-dition around corners.
3.4.2 Arbitrary Profiles Gratings A. Dielectric
From preceding subsection, it has been known that one needs to use proper discretization to avoid the instabilities for TM polarization. How-ever, as the gratings cut into many layer for approximating arbitrary shape of gratings, such proper discretization for every interfaces would be achieved more difficultly. But if the proper discretization is not considered, the serious problem of instability will occur, as in Fig. 3.38, which shows transmittance for TM case with respect to the grid’s number for uniform grids. The pa-rameters of the grating’s structure are: tg = 1000nm, Λ = 1000nm, θtilt = arctan(2tg/Λ), number of layers=15 εr,tra = εr,g1 = 32 and εr,inc = εr,g2 = 1.
The results of FDMM in Fig. 3.38 are inferior to that of RCWA and suffer from obvious instabilities. And these instabilities could be avoided by plac-ing grid points as in Fig. 3.22. After usplac-ing such proper discretization, the results will become quite stable, as in Fig. 3.39.
Although convergence of FDMM is a little worse than that of RCWA in Fig. 3.39, the results of FDMM is correct and stable. Therefore, it could be expected that results of FDMM become better than RCWA as simulating structures that RCWA will suffer from instabilities or slower convergence.
B. Lossless metal
A lossless metal grating with εr,g1 = εr,tra = (−10i)2 is considered, and its
results are shown in Fig. 3.40. In this case, results of RCWA are very unstable and incorrect. However, the results of FDMM with proper discretization could avoid such instability and failure. Notice that three-point FDMM without GD is superior to other higher order FDMM unexpectedly.
C. Lossy metal
Finally, a lossy metal grating with εr,g1 = εr,tra = (3.18 − 4.41i)2 is tested. Fig. 3.41 shows that the results of FDMM converge faster than that of RCWA even for three-point formulation without GD. Note that this is different from the result of FDMM for rectangular-groove grating with the same εr,g1 and εr,tra, which shows that all of the formulations of FDMM are inferior to RCWA.
Figure 3.38: Convergence of the zeroth-order transmittance. A dielectric triangular grating with εg1= 32 under TM incidence.
Figure 3.39: Convergence of the zeroth-order transmittance with proper dis-cretization. A dielectric triangular grating with εg1= 32under TM incidence.
Figure 3.40: Convergence of the zeroth-order reflection with proper dis-cretization. A lossless metal triangular grating with εg1 = (−10i)2 under TM incidence.
Figure 3.41: Convergence of the zeroth-order reflection with proper dis-cretization. A lossy triangular grating with εg1 = (3.18 − 4.41i)2 under TM incidence.