Benchmark for convergence test of circular cylinder gratings

[Simulator-CEM]

Hi Dr Schubert,

This issue is cut from the previous discussion about "Asymmetric results for symmetric structure? #86", I think the contents are a little disordered there, so I just rearrange the contents about the convergence of the circular cylinder grating and open a new issue, and put the discussion about the divergence problem of single precision calculation in another new issue. Please forgive the ugly formatting.

I would recommend three kinds of circular cylinder gratings for the convergence test: metallic gratings, semiconductor gratings, and all-dielectric gratings, all with tabulated diffraction efficiencies for benchmark.

In my experience, the metal cylinder gratings might be very challenging, the semi-conductor cylinder grating should be easier, and the all-dielectric grating can be piece of cake, unless the nonorthogonal unit cell is considered.

############################################# Metallic gratings ###############################

(1)
The first metallic grating is from Ref. [1] and [2] (note that there is typo of radius, it should be diameter)

                                 Reference :
                                            PMM
                                            T=0.004205996990137
                                            R=0.896084684513263

I have checked the parameters, and confirm that the "radius of 0.5 microns" should be a typo, and the correct statement should be "diameter of 0.5 microns". I verified this example by FEM (COMSOL) with very fine mesh (5 nm in the metal cylinder), and the results read

                                       T=0.004191844836374037
                                       R=0.897337165028805

I also check the structure by FMM (RCWA-2D) with ASR_G=0.05, discretization=1024,truncation=“parallelogrammic”, formulation='tangent', but the convergence seems not very well. The result are below

                             N             T                                 R 
                            121       0.0095640011                       0.8546378911 
                          **441       0.0040046979                       0.9040339125**
                            961       0.0216957240                       0.7857508857
                           1681       0.0067179374                       0.8839969118
                           2601       0.0097550629                       0.8456889720

############################################################################################
(2)
The second metallic cylinder grating is from Ref. [1], which is free-standing in air. A exponent factor "2" is missing in the original paper, and I add it in red here.

                            Reference:
                            PMM (PMM-MC):
                                      T=0.002999846668958
                                      R=0.900969507356113

I also checked this example with other methods, and the results are as follows:

Reticolo (Standard FMM with zigzag mesh and Li's inverse rule without NVM):
Diverge, see the table in the lower-right

S4 (FMM):
Seems not converge, see the table in the lower-right

RCWA-2D (FMM, No ASR (ASR helps little), discretization=1024,truncation=“parallelogrammic”, formulation='tangent'**):
Seems not converge, similar to results from S4

                             N              T                                 R 
                            441        0.0016166222                       0.9202642683
                           1681       0.0041357704                       0.8757414320
                           3721       0.0046738549                       0.8731557801
                       **6561       0.0027819274                        0.9067833647**
                          10201      0.0038960030                       0.8872613159

COMSOL (FEM):

                                      T=0.002838156332494414
                                      R=0.9016793641445542

CST (FEM):

                                      T=0.0025015345411362
                                      R=0.9019416460147 

######################################### Semi-conductor grating ##############################

The semi-conductor grating example is from Ref. [3-5],

                        reference：
                                 FEM:
                                 R_00=0.24415

######################################### All-dielectric grating ##############################

The all-dielectric grating is from Ref. [6], this unit cell is slant rather than orthogonal, and the algorithm is just based on zigzag mesh and common inverse rule without NVM.

                        reference：
                                 FMM-Li:
                                 T_00=0.05929
                                 R_00=0.00394

I have verified this result by FEM (COMSOL), and the result is:

                                 T_00=0.05969414193466224
                                 R_00=0.00393598283403028

[1] Edee K, Plumey J P, Moreau A, et al. Matched coordinates in the framework of polynomial modal methods for complex metasurface modeling. Journal of the Optical Society of America A, 35(4): 608-615 (2018).
[2] S. Essig and K. Busch, “Generation of adaptive coordinates and their use in the Fourier modal method,” Opt. Express 18, 25258–23274 (2010).
[3] Demésy, Guillaume, et al. "Finite element method." Gratings: theory and numeric applications (2012): 5-1.
[4] T. Schuster, J. Ruoff, N. Kerwien, S. Rafler and W. Osten, “Normal vector method for convergence improvement using the rcwa for crossed gratings,” J. Opt. Soc. Am. A 24, 2880–2890 (2007).
[5] Popov, Evgeni. "Differential method for periodic structures." Gratings: Theory and Numeric Applications, Second Revisited Edition (2014): 7-1.
[6] Li L. New formulation of the Fourier modal method for crossed surface-relief gratings. Journal of the Optical Society of America A, 1997, 14(10): 2758-2767.