Gérard Granet, "Modal spectral element method with modified Legendre polynomials to analyze binary crossed gratings," J. Opt. Soc. Am. A 40, 652-660 (2023)
In a previous paper, a modal spectral element method (SEM), the originality of which comes from the use of a hierarchical basis built with modified Legendre polynomials, was shown to be very powerful for the analysis of lamellar gratings. In this work, keeping the same ingredients, the method has been extended to the general case of binary crossed gratings. The geometric versatility of the SEM is illustrated with gratings whose patterns are not aligned with the boundaries of the elementary cell. The method is validated by a comparison to the Fourier modal method (FMM) in the case of anisotropic crossed gratings and with the FMM with adaptive spatial resolution in the case of a square-hole array in a silver film.
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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Effective Index of the Mode of a Circular Fiber Computed with the SEM and Two Different Pseudo-Periodic Conditions for Different Degrees of Freedom
dim
144
1.200365457
1.200366926
225
1.200992189
1.200992199
324
1.200508308
1.200508423
441
1.200501451
1.200501561
576
1.200502416
1.200502525
729
1.200502263
1.200502373
900
1.200502247
1.200502356
1089
1.200502244
1.200502354
Table 2.
Four Diffraction Efficiencies of an Anisotropic Grating Computed Using the SEMa
SEM
dim
(1,1)
(0, 0)
3
81
0.0332
0.0156
0.0528
0.3046
4
144
0.0268
0.0139
0.0620
0.2974
5
225
0.0268
0.0139
0.0620
0.2979
7
441
0.0268
0.0137
0.0620
0.2979
The grating is the one that is shown in Fig. 4. The grating is suspended in air, and the electric field is aligned along the $x$ axis with normal incidence.
Table 3.
Four Diffraction Efficiencies of an Anisotropic Grating Computed Using the FMMa
FMM
dim
(1,1)
(0,0)
4
81
0.0271
0.0138
0.613
0.2986
7
225
0.0269
0.0137
0.0617
0.2981
10
441
0.0269
0.0137
0.619
0.2980
The grating is the one that is shown in Fig. 4. The grating is suspended in air, and the electric field is aligned along the $x$ axis with normal incidence.
Table 4.
Reflected and Transmitted Efficiencies of a Membrane with Square Holes Suspended in Air Computed Using the SEM
dim
Re
Te
Rm
Tm
144
0.11546
0.88453
0.02856
0.97144
324
0.11568
0.88431
0.02858
0.97142
900
0.11578
0.88421
0.02856
0.97143
2304
0.11580
0.88419
0.02856
0.97143
Table 5.
Reflected and Transmitted Efficiencies of a Membrane with Square Holes Suspended in Air Computed Using the FMM
dim
Re
Te
Rm
Tm
153
0.11611
0.88388
0.02838
0.97161
325
0.11588
0.88411
0.02843
0.97156
561
0.11582
0.88417
0.02847
0.97152
3321
0.11580
0.88419
0.02854
0.97145
Table 6.
Reflected and Transmitted Efficiencies of Square Dielectric Pillars Suspended in Air Computed Using the FMM
dim
Re
Te
Rm
Tm
153
0.039271
0.96072
0.92855
0.071446
325
0.039813
0.96018
0.91141
0.088583
561
0.039989
0.09601
0.90570
0.094291
861
0.040066
0.95993
0.90320
0.096790
1225
0.040106
0.95989
0.90190
0.098097
1653
0.040129
0.95987
0.90114
0.098855
2145
0.040143
0.95958
0.90066
0.099334
2701
0.040153
0.95894
0.90034
0.099655
3321
0.040160
0.95983
0.90011
0.099881
4005
0.040160
0.95983
0.89999
0.100045
Table 7.
Reflected and Transmitted Efficiencies of Square Dielectric Pillars Suspended in Air Computed Using the SEM with a Tilted Elementary Cell
dim
Re
Te
Rm
Tm
144
0.040364
0.95963
0.89163
0.10836
324
0.040244
0.95975
0.89672
0.10327
576
0.040212
0.95978
0.89811
0.10188
900
0.040200
0.95979
0.89862
0.10137
1296
0.040194
0.95980
0.89886
0.10113
1764
0.040191
0.95981
0.89898
0.10103
2304
0.040189
0.95981
0.89905
0.10094
2916
0.040188
0.95981
0.89909
0.10090
3600
0.040188
0.95981
0.89912
0.10087
Table 8.
Reflected and Transmitted Efficiencies of Square Dielectric Pillars Suspended in Air Computed Using the SEM in Cartesian Coordinates
dim
Re
Te
Rm
Tm
240
0.040301
0.95956
0.89090
0.10950
540
0.040237
0.95972
0.89639
0.10367
960
0.040209
0.95977
0.89795
0.10206
1500
0.040198
0.95979
0.89854
0.10146
2160
0.040193
0.95980
0.89881
0.10118
2940
0.040190
0.95980
0.89895
0.101045
3840
0.040189
0.95980
0.89903
0.10096
Tables (8)
Table 1.
Effective Index of the Mode of a Circular Fiber Computed with the SEM and Two Different Pseudo-Periodic Conditions for Different Degrees of Freedom
dim
144
1.200365457
1.200366926
225
1.200992189
1.200992199
324
1.200508308
1.200508423
441
1.200501451
1.200501561
576
1.200502416
1.200502525
729
1.200502263
1.200502373
900
1.200502247
1.200502356
1089
1.200502244
1.200502354
Table 2.
Four Diffraction Efficiencies of an Anisotropic Grating Computed Using the SEMa
SEM
dim
(1,1)
(0, 0)
3
81
0.0332
0.0156
0.0528
0.3046
4
144
0.0268
0.0139
0.0620
0.2974
5
225
0.0268
0.0139
0.0620
0.2979
7
441
0.0268
0.0137
0.0620
0.2979
The grating is the one that is shown in Fig. 4. The grating is suspended in air, and the electric field is aligned along the $x$ axis with normal incidence.
Table 3.
Four Diffraction Efficiencies of an Anisotropic Grating Computed Using the FMMa
FMM
dim
(1,1)
(0,0)
4
81
0.0271
0.0138
0.613
0.2986
7
225
0.0269
0.0137
0.0617
0.2981
10
441
0.0269
0.0137
0.619
0.2980
The grating is the one that is shown in Fig. 4. The grating is suspended in air, and the electric field is aligned along the $x$ axis with normal incidence.
Table 4.
Reflected and Transmitted Efficiencies of a Membrane with Square Holes Suspended in Air Computed Using the SEM
dim
Re
Te
Rm
Tm
144
0.11546
0.88453
0.02856
0.97144
324
0.11568
0.88431
0.02858
0.97142
900
0.11578
0.88421
0.02856
0.97143
2304
0.11580
0.88419
0.02856
0.97143
Table 5.
Reflected and Transmitted Efficiencies of a Membrane with Square Holes Suspended in Air Computed Using the FMM
dim
Re
Te
Rm
Tm
153
0.11611
0.88388
0.02838
0.97161
325
0.11588
0.88411
0.02843
0.97156
561
0.11582
0.88417
0.02847
0.97152
3321
0.11580
0.88419
0.02854
0.97145
Table 6.
Reflected and Transmitted Efficiencies of Square Dielectric Pillars Suspended in Air Computed Using the FMM
dim
Re
Te
Rm
Tm
153
0.039271
0.96072
0.92855
0.071446
325
0.039813
0.96018
0.91141
0.088583
561
0.039989
0.09601
0.90570
0.094291
861
0.040066
0.95993
0.90320
0.096790
1225
0.040106
0.95989
0.90190
0.098097
1653
0.040129
0.95987
0.90114
0.098855
2145
0.040143
0.95958
0.90066
0.099334
2701
0.040153
0.95894
0.90034
0.099655
3321
0.040160
0.95983
0.90011
0.099881
4005
0.040160
0.95983
0.89999
0.100045
Table 7.
Reflected and Transmitted Efficiencies of Square Dielectric Pillars Suspended in Air Computed Using the SEM with a Tilted Elementary Cell
dim
Re
Te
Rm
Tm
144
0.040364
0.95963
0.89163
0.10836
324
0.040244
0.95975
0.89672
0.10327
576
0.040212
0.95978
0.89811
0.10188
900
0.040200
0.95979
0.89862
0.10137
1296
0.040194
0.95980
0.89886
0.10113
1764
0.040191
0.95981
0.89898
0.10103
2304
0.040189
0.95981
0.89905
0.10094
2916
0.040188
0.95981
0.89909
0.10090
3600
0.040188
0.95981
0.89912
0.10087
Table 8.
Reflected and Transmitted Efficiencies of Square Dielectric Pillars Suspended in Air Computed Using the SEM in Cartesian Coordinates
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