Université de Versailles et de Saint-Quentin en Yvelines, Centre d’ étude des Environnements Terrestre et Planétaires, 10-12 Avenue de l’Europe, 78140 Vélizy, France
A model with two roughness levels for the diffraction of a plane wave by a metallic grating with periodic imperfections is presented. The grating surface is the sum of a reference profile and a perturbation profile. First, the diffraction by the reference grating is treated. At this stage the Chandezon method is used. This method leads to the resolution of eigenvalue systems. Each eigensolution defines an elementary wave function that characterizes a propagating or an evanescent wave. Second, the periodic errors are taken into account and a Rayleigh hypothesis is expressed: Everywhere in space the diffracted fields can be written as a linear combination of reference wave functions. The boundary conditions on the perturbed grating allow the diffraction amplitudes to be determined and therefore lead to the energetic magnitudes (efficiencies). The domain of analytical validity of this hypothesis is not defined. In fact, this method is considered to be an approximation. The proposed numerical study leads to some utilization rules. With a plane as the reference surface, the electromagnetic fields are given by classical Rayleigh expansions. Here the reference profile is a grating, hence the term generalized Rayleigh expansion.
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Accuracy in Efficiencies versus Grating Period with the C Methoda
Accuracy
Grating Number
1
2
3
4
5
Here with and The accuracy is obtained from The CPU times in seconds correspond to computing diffraction efficiencies with the matlab program on a Power-Mac 8500. The values take into account only the polarization. For the case with the same accuracy, the truncation order and the computation times are different.
Table 2
Spectra of Elementary Wave Functions Associated with Incidences
Real Incidence: and
Spectrum of elements
spectra of elements
outgoing waves
outgoing waves
incoming waves
incoming waves
Table 3
Accuracy in Efficiencies versus Grating Period with the Generalized Rayleigh Expansiona
Accuracy
Grating Number
1
2
3
4
5
Parameters and comments are as for Table 1. It should be observed that the ideal case is not the general case.
Table 4
Norm Values and for the Gratings Studied in Subsection 5.A
Norm Value
Amplitude
0.1331
0.2661
0.5024
0.6892
0.1351
0.2662
0.5026
0.6896
Class of Results
1 in
1 in
1 in
1 in
1 in
1 in
2 in
2 in
Table 5
Truncation Order Range Ensuring the Exploitation of Diffraction Efficiencies for the Metallic Grating Referenced in Subsection 5.D
Diffraction Order
Polarization
Polarization
-7
—
[8, 32]
[8, 33]
—
[15, 28]
[14, 30]
-6
—
[8, 32]
[8, 33]
—
[15, 27]
[14, 30]
-5
[8, 29]
[8, 34]
[8, 38]
[13, 25]
[13, 27]
[11, 30]
-4
[<8, 32]
[<8, 34]
[<8, >39]
[13, 26]
[13, 29]
[12, 32]
-3
[8, 29]
[8, 32]
[8, 37]
[13, 25]
[14, 28]
[14, 30]
-2
[8, 31]
[8, 32]
[8, 33]
[15, 25]
[15, 29]
[12, 30]
-1
[8, 32]
[8, 32]
[8, 33]
[14, 24]
[14, 28]
[13, 30]
0
[<8, 31]
[<8, 33]
[8, >39]
[13, 26]
[11, 29]
[10, 30]
1
[9, 26]
—
—
[13, 26]
—
—
2
[9, 26]
—
—
[14, 25]
—
—
0.880
0.895
0.971
0.0708
0.174
0.592
Tables (5)
Table 1
Accuracy in Efficiencies versus Grating Period with the C Methoda
Accuracy
Grating Number
1
2
3
4
5
Here with and The accuracy is obtained from The CPU times in seconds correspond to computing diffraction efficiencies with the matlab program on a Power-Mac 8500. The values take into account only the polarization. For the case with the same accuracy, the truncation order and the computation times are different.
Table 2
Spectra of Elementary Wave Functions Associated with Incidences
Real Incidence: and
Spectrum of elements
spectra of elements
outgoing waves
outgoing waves
incoming waves
incoming waves
Table 3
Accuracy in Efficiencies versus Grating Period with the Generalized Rayleigh Expansiona
Accuracy
Grating Number
1
2
3
4
5
Parameters and comments are as for Table 1. It should be observed that the ideal case is not the general case.
Table 4
Norm Values and for the Gratings Studied in Subsection 5.A
Norm Value
Amplitude
0.1331
0.2661
0.5024
0.6892
0.1351
0.2662
0.5026
0.6896
Class of Results
1 in
1 in
1 in
1 in
1 in
1 in
2 in
2 in
Table 5
Truncation Order Range Ensuring the Exploitation of Diffraction Efficiencies for the Metallic Grating Referenced in Subsection 5.D