Laboratoire des Sciences et Matériaux pour l’Electronique et d’Automatique, Unité Mixte de Recherche, Centre National de la Recherche Scientifique No. 6602, Université Blaise Pascal, Les Cézeaux Aubière Cedex, France
Gérard Granet, "Analysis of diffraction by surface-relief crossed gratings with use of the Chandezon method: application to multilayer crossed gratings," J. Opt. Soc. Am. A 15, 1121-1131 (1998)
A new formulation of the Chandezon method for crossed gratings is presented. In the nonorthogonal translation coordinate system, an arbitrary field in a homogeneous source-free region can be expressed as the sum of a TE field and a TM field. It is shown that the whole solution can be derived from the eigensolutions of an operator independent of the polarization. In addition, use is made of the -matrix formalism to include multilayer coated crossed gratings with parallel faces. Numerical results are given for sinusoidal crossed gratings and pyramidal gratings.
Gérard Granet, "Analysis of diffraction by surface-relief crossed gratings with use of the Chandezon method: application to multilayer crossed gratings: errata," J. Opt. Soc. Am. A 15, 2444-2444 (1998) https://opg.optica.org/josaa/abstract.cfm?uri=josaa-15-9-2444
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Comparison of the Computational Speed of the Codes Based on Two Formulations of the C Methoda
Present Formulation
Previous Formulation
∊
Time (s)
∊
Time (s)
5
0.18318308
0.04441311
0.54480306
31
0.18317513
0.04441197
0.54482602
103
6
0.18318536
0.04441384
0.54480149
81
0.18318483
0.04441374
0.544803
274
7
0.18318539
0.04441385
0.54480151
207
0.18318535
0.04441385
0.5448016
673
8
0.18318538
0.04441385
0.54480152
472
0.18318538
0.04441385
0.54480153
1225
The grating is the perfectly conducting sinusoidal crossed grating of Eq. (71) under normal incidence with a wavelength-to-period ratio and a height-to-period ratio
Table 2
Comparison of the Numerical Results of the Present Study with the Results of Bruno and Reitich (Ref. 2) for the Perfectly Conducting Sinusoidal Crossed Grating of Eq. (71) under Normal Incidence with a Wavelength-to-Period Ratio
Convergence Study of the Zero eflected Order for the Perfectly Conducting Sinusoidal Crossed Grating of Eq. (71), with Various Height-to-Period Ratios, under Normal Incidence with a Wavelength-to-Period Ratio λ/d = 0.83a
Previous Formulation
∊
∊
∊
∊
1
0.544826
0.544803
0.544802
0.544802
1.1
0.41056
0.410519
0.410516
0.410516
1.2
0.32145
0.321359
0.321354
0.321354
1.3
0.24448
0.244219
0.244207
0.244208
1.4
0.15582
0.155217
0.15519
0.155114
1.5
0.77793
0.77119
0.77093
0.077096
1.6
0.05719
0.0575
0.05754
0.05755
1.7
0.09636
0.09851
0.09871
0.098734
1.8
0.16541
0.169798
0.170278
0.17033
1.9
0.248207
0.255549
0.256564
0.256695
2
0.350388
0.362702
0.364821
0.365035
Present Formulation
∊
∊
∊
∊
0.544803
0.544801
0.544802
0.544802
0.410517
0.410516
0.410516
0.410516
0.321369
0.321354
0.321354
0.321355
0.244292
0.24421
0.244208
0.244208
0.155435
0.155201
0.155195
0.155195
0.077402
0.077106
0.077099
0.077099
0.0575
0.057538
0.057553
0.057554
0.097901
0.098667
0.098738
0.098742
0.168347
0.170154
0.170337
0.170344
0.252754
0.256267
0.256667
0.256695
0.357264
0.364129
0.36501
0.36508
It should be noted that an error has occurred in Ref. 9. In Table 2 on p. 787, the value of for should read 0.36482 instead of 0.036482.
Table 4
Convergence Study of the Efficiencies for a Perfectly Conducting Pyramidal Crossed Grating with and
0.027249
0.02727
0.027377
0.027393
0.027422
0.027428
0.037875
0.037428
0.037375
0.037287
0.037247
0.03721
0.028982
0.028708
0.028571
0.028477
0.02842
0.02838
0.005062
0.005115
0.005104
0.005115
0.005115
0.005118
0.152897
0.154705
0.15546
0.155939
0.156184
0.156363
0.036213
0.035908
0.035951
0.035922
0.035925
0.035916
0.139854
0.139041
0.138317
0.138054
0.137882
0.137798
0.019859
0.020346
0.020814
0.021003
0.021148
0.021229
∊
0.001291
0.001121
0.000768
0.000649
0.000549
0.000493
Table 5
Comparison among Different Methods for a Pyramidal Dielectric Grating with and
Reflected Efficiencies of a 22-Layer Coated Grating
TE Polarization
0.00371816
0.00379195
0.00379255
0.0174459
0.01763833
0.01763991
0.79512396
0.79463043
0.79462637
∊
TM Polarization
0.03900172
0.03888695
0.0388858
0.02094925
0.02104628
0.02104708
0.05357425
0.05366466
0.05366555
∊
Tables (6)
Table 1
Comparison of the Computational Speed of the Codes Based on Two Formulations of the C Methoda
Present Formulation
Previous Formulation
∊
Time (s)
∊
Time (s)
5
0.18318308
0.04441311
0.54480306
31
0.18317513
0.04441197
0.54482602
103
6
0.18318536
0.04441384
0.54480149
81
0.18318483
0.04441374
0.544803
274
7
0.18318539
0.04441385
0.54480151
207
0.18318535
0.04441385
0.5448016
673
8
0.18318538
0.04441385
0.54480152
472
0.18318538
0.04441385
0.54480153
1225
The grating is the perfectly conducting sinusoidal crossed grating of Eq. (71) under normal incidence with a wavelength-to-period ratio and a height-to-period ratio
Table 2
Comparison of the Numerical Results of the Present Study with the Results of Bruno and Reitich (Ref. 2) for the Perfectly Conducting Sinusoidal Crossed Grating of Eq. (71) under Normal Incidence with a Wavelength-to-Period Ratio
Convergence Study of the Zero eflected Order for the Perfectly Conducting Sinusoidal Crossed Grating of Eq. (71), with Various Height-to-Period Ratios, under Normal Incidence with a Wavelength-to-Period Ratio λ/d = 0.83a
Previous Formulation
∊
∊
∊
∊
1
0.544826
0.544803
0.544802
0.544802
1.1
0.41056
0.410519
0.410516
0.410516
1.2
0.32145
0.321359
0.321354
0.321354
1.3
0.24448
0.244219
0.244207
0.244208
1.4
0.15582
0.155217
0.15519
0.155114
1.5
0.77793
0.77119
0.77093
0.077096
1.6
0.05719
0.0575
0.05754
0.05755
1.7
0.09636
0.09851
0.09871
0.098734
1.8
0.16541
0.169798
0.170278
0.17033
1.9
0.248207
0.255549
0.256564
0.256695
2
0.350388
0.362702
0.364821
0.365035
Present Formulation
∊
∊
∊
∊
0.544803
0.544801
0.544802
0.544802
0.410517
0.410516
0.410516
0.410516
0.321369
0.321354
0.321354
0.321355
0.244292
0.24421
0.244208
0.244208
0.155435
0.155201
0.155195
0.155195
0.077402
0.077106
0.077099
0.077099
0.0575
0.057538
0.057553
0.057554
0.097901
0.098667
0.098738
0.098742
0.168347
0.170154
0.170337
0.170344
0.252754
0.256267
0.256667
0.256695
0.357264
0.364129
0.36501
0.36508
It should be noted that an error has occurred in Ref. 9. In Table 2 on p. 787, the value of for should read 0.36482 instead of 0.036482.
Table 4
Convergence Study of the Efficiencies for a Perfectly Conducting Pyramidal Crossed Grating with and
0.027249
0.02727
0.027377
0.027393
0.027422
0.027428
0.037875
0.037428
0.037375
0.037287
0.037247
0.03721
0.028982
0.028708
0.028571
0.028477
0.02842
0.02838
0.005062
0.005115
0.005104
0.005115
0.005115
0.005118
0.152897
0.154705
0.15546
0.155939
0.156184
0.156363
0.036213
0.035908
0.035951
0.035922
0.035925
0.035916
0.139854
0.139041
0.138317
0.138054
0.137882
0.137798
0.019859
0.020346
0.020814
0.021003
0.021148
0.021229
∊
0.001291
0.001121
0.000768
0.000649
0.000549
0.000493
Table 5
Comparison among Different Methods for a Pyramidal Dielectric Grating with and