Model with two roughness levels for diffraction gratings: the generalized Rayleigh expansion

R. Dusséaux

doi:10.1364/JOSAA.15.002684

Journal of the Optical Society of America A
Vol. 15,
Issue 10,
pp. 2684-2697
(1998)
•https://doi.org/10.1364/JOSAA.15.002684

Model with two roughness levels for diffraction gratings: the generalized Rayleigh expansion

R. Dusséaux

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R. Dusséaux, "Model with two roughness levels for diffraction gratings: the generalized Rayleigh expansion," J. Opt. Soc. Am. A 15, 2684-2697 (1998)

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Abstract

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	Grating Number $n_{0}$
Accuracy	1	2	3	4	5
$Δ ξ_{n}^{(1)} > 1$	$M > M_{C, 1} = 4$	$M > M_{C, 2} = 8$	$M > M_{C, 3} = 16$	$M > M_{C, 4} = 20$	$M > M_{C, 5} = 25$
$\sum T_{C} = 23.3$	$T_{C_{1}, 1} = 0.2$	$T_{C_{1}, 2} = 0.4$	$T_{C_{1}, 3} = 2.7$	$T_{C_{1}, 4} = 5.6$	$T_{C_{1}, 5} = 11.7$
	$T_{C_{2}, 1} = 0.1$	$T_{C_{2}, 2} = 0.3$	$T_{C_{2}, 3} = 0.6$	$T_{C_{2}, 4} = 0.7$	$T_{C_{2}, 5} = 1.0$
	$T_{C, 1} = 0.3$	$T_{C, 2} = 0.7$	$T_{C, 3} = 3.3$	$T_{C, 4} = 6.3$	$T_{C, 5} = 12.7$
$Δ ξ_{n}^{(1)} > 2$	$M > M_{C, 1} = 7$	$M > M_{C, 2} = 13$	$M > M_{C, 3} = 19$	$M > M_{C, 4} = 28$	$M > M_{C, 5} = 32$
$\sum T_{C} = 52$	$T_{C_{1}, 1} = 0.3$	$T_{C_{1}, 2} = 1.5$	$T_{C_{1}, 3} = 4.6$	$T_{C_{1}, 4} = 16.2$	$T_{C_{1}, 5} = 25.6$
	$T_{C_{2}, 1} = 0.3$	$T_{C_{1}, 2} = 0.3$	$T_{C_{2}, 3} = 0.6$	$T_{C_{2}, 4} = 0.8$	$T_{C_{2}, 5} = 1.8$
	$T_{C, 1} = 0.6$	$T_{C, 2} = 1.8$	$T_{C, 2} = 5.2$	$T_{C, 4} = 17$	$T_{C, 5} = 27.4$
$Δ ξ_{n}^{(1)} > 3$	$M > M_{C, 1} = 8$	$M > M_{C, 2} = 17$	$M > M_{C, 3} = 25$	$M > M_{C, 4} = 36$	$M > M_{C, 5} = 45$
$\sum T_{C} = 148.4$	$T_{C_{1}, 1} = 0.4$	$T_{C_{1}, 2} = 3.0$	$T_{C_{1}, 3} = 10.7$	$T_{C_{1}, 4} = 39.6$	$T_{C_{1}, 5} = 87.4$
	$T_{C_{2}, 1} = 0.3$	$T_{C_{2}, 2} = 0.5$	$T_{C_{2}, 3} = 0.9$	$T_{C_{2}, 4} = 2$	$T_{C_{2}, 5} = 3.6$
	$T_{C, 1} = 0.7$	$T_{C, 2} = 3.5$	$T_{C, 3} = 11.6$	$T_{C, 4} = 41.6$	$T_{C, 5} = 91$
$Δ ξ_{n}^{(1)} > 4$	$M > M_{C, 1} = 10$	$M > M_{C, 2} = 19$	$M > M_{C, 3} = 30$	$M > M_{C, 4} = 40$	$M > M_{C, 5} = 50$
$\sum T_{C} = 213.6$	$T_{C_{1}, 1} = 0.8$	$T_{C_{1}, 2} = 4.6$	$T_{C_{1}, 3} = 20.6$	$T_{C_{1}, 4} = 55.3$	$T_{C_{1}, 5} = 111.5$
	$T_{C_{2}, 1} = 0.3$	$T_{C_{2}, 2} = 0.6$	$T_{C_{2}, 3} = 1.2$	$T_{C_{2}, 4} = 4.6$	$T_{C_{2}, 5} = 14.1$
	$T_{C, 1} = 1.1$	$T_{C, 2} = 5.2$	$T_{C, 3} = 21.8$	$T_{C, 4} = 59.9$	$T_{C, 5} = 125.6$
$Δ S > 3$	$M > M_{C, 1} = 8$	$M > M_{C, 2} = 15$	$M > M_{C, 3} = 24$	$M > M_{C, 4} = 32$	$M > M_{C, 5} = 36$
$\sum T_{C} = 80.7$	$T_{C_{1}, 1} = 0.4$	$T_{C_{1}, 2} = 2.3$	$T_{C_{1}, 3} = 9.7$	$T_{C_{1}, 4} = 24.6$	$T_{C_{1}, 5} = 38.4$
	$T_{C_{2}, 1} = 0.3$	$T_{C_{2}, 2} = 0.4$	$T_{C_{2}, 3} = 0.7$	$T_{C_{2}, 4} = 1.8$	$T_{C_{2}, 5} = 2.1$
	$T_{C, 1} = 0.7$	$T_{C, 2} = 2.7$	$T_{C, 3} = 10.4$	$T_{C, 4} = 26.4$	$T_{C, 5} = 40.5$

Real Incidence: $sin θ$ and $p = 0$	$n_{0} - 1 Additional Incidences :$ $sin θ + p λ / D, p \neq 0$
Spectrum of $4 M + 2$ elements	$n_{0} - 1$ spectra of $4 M$ elements
$ψ_{0, n}^{(j)} (x, u) = ϕ_{0, n}^{(j)} (x) exp ({jk}^{(j)} r_{0, n}^{(j)} u)$	$ψ_{p, n}^{(j)} (x, u) = ϕ_{p, n}^{(j)} (x) exp ({jk}^{(j)} r_{p, n}^{(j)} u)$
$2 M + 1$ outgoing waves	$2 M$ outgoing waves
$2 M + 1$ incoming waves	$2 M$ incoming waves

Accuracy	Grating Number $n_{0}$
Accuracy	1	2	3	4	5
$Δ ξ_{n}^{(1)} > 1$	$M > M_{R, 1} = 4$	$M > M_{R, 2} = 6$	$M > M_{R, 3} = 5$	$M > M_{R, 4} = 11$	$M > M_{R, 5} = 10$
$\sum T_{R} = 17.9$	$T_{R_{1}, 1} = 0.2$	$T_{R_{1}, 2} = 0.7$	$T_{R_{1}, 3} = 0.7$	$T_{R_{1}, 4} = 4.4$	$T_{R_{1}, 5} = 4.4$
	$T_{R_{2}, 1} = 0.2$	$T_{R_{2}, 2} = 0.5$	$T_{R_{2}, 3} = 0.5$	$T_{R_{2}, 4} = 5.1$	$T_{R_{2}, 5} = 7.2$
	$T_{R, 1} = 0.4$	$T_{R, 2} = 1.2$	$T_{R, 3} = 1.2$	$T_{R, 4} = 9.5$	$T_{R, 5} = 11.6$
$Δ ξ_{n}^{(1)} > 2$	$M > M_{R, 1} = 7$	$M > M_{R, 2} = 8$	$M > M_{R, 3} = 7$	$M > M_{R, 4} = 11$	$M > M_{R, 5} = 14$
$\sum T_{R} = 37.7$	$T_{R_{1}, 1} = 0.3$	$T_{R_{1}, 2} = 0.9$	$T_{R_{1}, 3} = 2.1$	$T_{R_{1}, 4} = 4.4$	$T_{R_{1}, 5} = 11$
	$T_{R_{2}, 1} = 0.3$	$T_{R_{2}, 2} = 0.7$	$T_{R_{2}, 3} = 1.0$	$T_{R_{2}, 4} = 5.2$	$T_{R_{2}, 5} = 19.5$
	$T_{R, 1} = 0.7$	$T_{R, 2} = 1.6$	$T_{R, 3} = 3.1$	$T_{R, 4} = 9.6$	$T_{R, 5} = 30.5$
$Δ ξ_{n}^{(1)} > 3$	$M > M_{R, 1} = 8$	$M > M_{R, 2} = 11$	$M > M_{R, 3} = 10$	$M > M_{R, 4} = 16$	$M > M_{R, 5} = 18$
$\sum T_{R} = 73$	$T_{R_{1}, 1} = 0.3$	$T_{R_{1}, 2} = 3$	$T_{R_{1}, 3} = 2.4$	$T_{R_{1}, 4} = 12.5$	$T_{R_{1}, 5} = 19.8$
	$T_{R_{2}, 1} = 0.3$	$T_{R_{2}, 2} = 1.1$	$T_{R_{2}, 3} = 1.8$	$T_{R_{2}, 4} = 15.1$	$T_{R_{2}, 5} = 34.9$
	$T_{R, 1} = 0.6$	$T_{R, 2} = 4.1$	$T_{R, 3} = 4.2$	$T_{R, 4} = 27.6$	$T_{R, 5} = 54.7$
$Δ ξ_{n}^{(1)} > 4$	$M > M_{R, 1} = 10$	$M > M_{R, 2} = 14$	$M > M_{R, 3} = 13$	$M > M_{R, 4} = 16$	$M > M_{R, 5} = 21$
$\sum T_{R} = 99.6$	$T_{R_{1}, 1} = 0.9$	$T_{R_{1}, 2} = 3.8$	$T_{R_{1}, 3} = 5.1$	$T_{R_{1}, 4} = 12.5$	$T_{R_{1}, 5} = 35.2$
	$T_{R_{2}, 1} = 0.8$	$T_{R_{2}, 2} = 3.5$	$T_{R_{2}, 3} = 3.8$	$T_{R_{2}, 4} = 15.1$	$T_{R_{2}, 5} = 41.2$
	$T_{R, 1} = 1.7$	$T_{R, 2} = 7.3$	$T_{R, 3} = 8.9$	$T_{R, 4} = 27.6$	$T_{R, 5} = 76.4$

Norm Value	Amplitude
Norm Value	$h_{1}$ $= 0.02 λ$	$h_{1}$ $= 0.04 λ$	$h_{1}$ $= 0.08 λ$	$h_{1}$ $= 0.12 λ$
$σ_{c}$	0.1331	0.2661	0.5024	0.6892
$σ_{d}$	0.1351	0.2662	0.5026	0.6896
Class of Results	1 in $E ‖$	1 in $E ‖$	1 in $E ‖$	1 in $E ‖$
	1 in $H ‖$	1 in $H ‖$	2 in $H ‖$	2 in $H ‖$

Diffraction Order	$E ‖$ Polarization			$H ‖$ Polarization
Diffraction Order	$θ = 20 °$	$θ = 50 °$	$θ = 80 °$	$θ = 20 °$	$θ = 50 °$	$θ = 80 °$
-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]	—	—
$S = \sum_{n} ξ_{n}^{(1)}$	0.880	0.895	0.971	0.0708	0.174	0.592

Abstract

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Equations (78)

Journal of the Optical Society of America A