Faculté des Sciences et Techniques, Laboratoire d’Optique Electromagnétique, Equipe de Recherche Associée du Centre National de la Recherche Scientifique no. 597, Centre de St.-Jérôme, 13397 Marseille Cedex 13, France.
J. P. Hugonin, R. Petit, and M. Cadilhac, "Plane-wave expansions used to describe the field diffracted by a grating," J. Opt. Soc. Am. 71, 593-598 (1981)
The properties of plane waves are well known, and it is probably because they are that we often try to represent any unknown field as a combination of such waves. Is such a representation always efficient or even correct? We try to answer this question for a simple problem arising in the electromagnetic study of gratings. We first summarize in a didactic way some important theoretical results that are probably not well known to those working in optics. Thereafter we report on recently performed numerical experiments. Plane-wave field representations indeed permit us to obtain quickly reliable results for some particular profiles. Nevertheless, the methods leading to the solution of an integral equation are the most useful because they are applicable to a larger class of groove profiles.
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If, when using the LSAM, one replaces dx by dl, one obtains different values for the Rn(N), but, in both cases, Rn(N) tends to the same value when N tends to infinity.
Table 2
Elements of Matrices S and ϕ as Defined in Terms of Bn(u)
FSM
LSAM
Sn
ϕn,m
Table 3
Numerical Experiment Using the Integral Method for a Symmetrical Triangular Profile (h/d = 1/2)a
N
∊
t
e0
e1
e2
10
8.29 10−3
4.58 10−2
0.08
3.69 10−1
1.27 10−1
1.84 10−1
20
8.43 10−5
3.35 10−4
0.48
4.86 10−1
9.74 10−2
1.60 10−1
30
1.39 10−5
6.09 10−5
1.48
4.72 10−1
1.00 10−1
1.63 10−1
40
2.06 10−6
3.57 10−6
3.22
4.75 10−1
9.93 10−2
1.63 10−1
60
2.52 10−7
3.02 10−7
10.22
4.76 10−1
9.92 10−2
1.63 10−1
80
6.07 10−8
2.42 10−8
23.98
4.76 10−1
9.92 10−2
1.63 10−1
100
2.10 10−8
0
46.77
4.76 10−1
9.92 10−2
1.63 10−1
The computation time is given in seconds. N, ∊, and en are defined in the text. Only three figures have been retained to give the values of the efficiencies en.
Table 4
Comparison of Three Methods for a Sinusoidal Profile (h/d = 0.40)a
FSM
LSAM
IM
N
∊
t
∊
t
∊
t
5
1.41 10−2
1.73 10−4
0.02
2.84 10−1
5.75 10−2
0.03
3.89 10−1
8.24 10−1
0.01
10
1.10 10−5
2.50 10−8
0.07
6.80 10−2
4.26 10−3
0.07
4.38 10−3
7.12 10−3
0.08
15
5.19 10−10
2.68 10−8
0.17
1.72 10−2
3.58 10−4
0.17
2.49 10−4
4.28 10−4
0.21
20
2.83 10−8
2.67 10−8
0.31
8.63 10−3
9.89 10−5
0.31
1.44 10−5
2.33 10−5
0.84
Again t is the computation time in seconds.
Table 5
Comparison of Three Methods for a Symmetrical Triangular Profile (h/d = 0.29)
If, when using the LSAM, one replaces dx by dl, one obtains different values for the Rn(N), but, in both cases, Rn(N) tends to the same value when N tends to infinity.
Table 2
Elements of Matrices S and ϕ as Defined in Terms of Bn(u)
FSM
LSAM
Sn
ϕn,m
Table 3
Numerical Experiment Using the Integral Method for a Symmetrical Triangular Profile (h/d = 1/2)a
N
∊
t
e0
e1
e2
10
8.29 10−3
4.58 10−2
0.08
3.69 10−1
1.27 10−1
1.84 10−1
20
8.43 10−5
3.35 10−4
0.48
4.86 10−1
9.74 10−2
1.60 10−1
30
1.39 10−5
6.09 10−5
1.48
4.72 10−1
1.00 10−1
1.63 10−1
40
2.06 10−6
3.57 10−6
3.22
4.75 10−1
9.93 10−2
1.63 10−1
60
2.52 10−7
3.02 10−7
10.22
4.76 10−1
9.92 10−2
1.63 10−1
80
6.07 10−8
2.42 10−8
23.98
4.76 10−1
9.92 10−2
1.63 10−1
100
2.10 10−8
0
46.77
4.76 10−1
9.92 10−2
1.63 10−1
The computation time is given in seconds. N, ∊, and en are defined in the text. Only three figures have been retained to give the values of the efficiencies en.
Table 4
Comparison of Three Methods for a Sinusoidal Profile (h/d = 0.40)a
FSM
LSAM
IM
N
∊
t
∊
t
∊
t
5
1.41 10−2
1.73 10−4
0.02
2.84 10−1
5.75 10−2
0.03
3.89 10−1
8.24 10−1
0.01
10
1.10 10−5
2.50 10−8
0.07
6.80 10−2
4.26 10−3
0.07
4.38 10−3
7.12 10−3
0.08
15
5.19 10−10
2.68 10−8
0.17
1.72 10−2
3.58 10−4
0.17
2.49 10−4
4.28 10−4
0.21
20
2.83 10−8
2.67 10−8
0.31
8.63 10−3
9.89 10−5
0.31
1.44 10−5
2.33 10−5
0.84
Again t is the computation time in seconds.
Table 5
Comparison of Three Methods for a Symmetrical Triangular Profile (h/d = 0.29)
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