Sam Campbell, Lindsay C. Botten, Ross C. McPhedran, and C. Martijn de Sterke, "Modal method for classical diffraction by slanted lamellar gratings," J. Opt. Soc. Am. A 25, 2415-2426 (2008)
We consider lamellar gratings made of dielectric or lossy materials used in classical diffraction mounts. We show how the modal diffraction formulation may be generalized to deal with slanted lamellar gratings and illustrate the accuracy and versatility of the new method through study of highly slanted gratings in a homogenization limit. We also comment on the completeness of the eigenmode basis and present tests enabling this completeness to be verified numerically.
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Truncation order of the lamellar mode basis necessary to keep the maximum error in the completeness matrix of Eq. (18) below as a function of the imaginary part of the dielectric permittivity of the metal. Six different truncation orders of the plane wave basis are shown. Data: period ; ; width of air region, , ; width of metal region, ; normal incidence ; polarization.
Mode 1 is a propagating mode in the limit as , mode 2 is an evanescent mode, and mode 3 is an “anomalous” mode. Data as in Table 1.
Table 3
Comparison of Efficiencies Calculated Using the DMM and the C Methoda
Case
Polarization
Polarization
C Method
DMM
C Method
DMM
A
10
5
10
5
25
10
25
Reflected orders
0.0179
0.0179
0.023
0.0231
0
0.0137
0.0137
0.001
0.0011
Transmitted orders
0.0399
0.0398
0.0232
0.0227
0
0.9286
0.9286
0.9528
0.9531
B
40
15
40
15
75
40
85
Reflected Orders
0.4191
0.4179
0.2742
0.275
0
0.0562
0.0562
0.2361
0.237
Transmitted orders
0.0232
0.0229
0.096
0.097
0
0.5015
0.5030
0.3938
0.391
C
10
10
20
10
40
50
Reflected orders
0.2357
0.2359
0.2214
0.2247
0
0.4269
0.4267
0.3061
0.307
Transmitted orders
0.1645
0.1646
0.2066
0.2075
0
0.1558
0.1557
0.2408
0.241
For the C method is the truncation parameter of the Floquet harmonics ( in total). For the DMM is the truncation parameter of plane waves ( in total), and is the truncation parameter for the modal basis. Parameters: , , , , , , , . Case (A) , ; (B) , ; (C) , .
Truncation order of the lamellar mode basis necessary to keep the maximum error in the completeness matrix of Eq. (18) below as a function of the imaginary part of the dielectric permittivity of the metal. Six different truncation orders of the plane wave basis are shown. Data: period ; ; width of air region, , ; width of metal region, ; normal incidence ; polarization.
Mode 1 is a propagating mode in the limit as , mode 2 is an evanescent mode, and mode 3 is an “anomalous” mode. Data as in Table 1.
Table 3
Comparison of Efficiencies Calculated Using the DMM and the C Methoda
Case
Polarization
Polarization
C Method
DMM
C Method
DMM
A
10
5
10
5
25
10
25
Reflected orders
0.0179
0.0179
0.023
0.0231
0
0.0137
0.0137
0.001
0.0011
Transmitted orders
0.0399
0.0398
0.0232
0.0227
0
0.9286
0.9286
0.9528
0.9531
B
40
15
40
15
75
40
85
Reflected Orders
0.4191
0.4179
0.2742
0.275
0
0.0562
0.0562
0.2361
0.237
Transmitted orders
0.0232
0.0229
0.096
0.097
0
0.5015
0.5030
0.3938
0.391
C
10
10
20
10
40
50
Reflected orders
0.2357
0.2359
0.2214
0.2247
0
0.4269
0.4267
0.3061
0.307
Transmitted orders
0.1645
0.1646
0.2066
0.2075
0
0.1558
0.1557
0.2408
0.241
For the C method is the truncation parameter of the Floquet harmonics ( in total). For the DMM is the truncation parameter of plane waves ( in total), and is the truncation parameter for the modal basis. Parameters: , , , , , , , . Case (A) , ; (B) , ; (C) , .