Rigorous theory of scattering by perfectly conducting periodic surfaces with trapezoidal height profile. TE and TM polarization

G.M. Whitman; D. M. Leskiw; F. Schwering

doi:10.1364/JOSA.70.001495

Journal of the Optical Society of America
Vol. 70,
Issue 12,
pp. 1495-1503
(1980)
•https://doi.org/10.1364/JOSA.70.001495

Rigorous theory of scattering by perfectly conducting periodic surfaces with trapezoidal height profile. TE and TM polarization

G.M. Whitman, D. M. Leskiw, and F. Schwering

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G.M. Whitman, D. M. Leskiw, and F. Schwering, "Rigorous theory of scattering by perfectly conducting periodic surfaces with trapezoidal height profile. TE and TM polarization," J. Opt. Soc. Am. 70, 1495-1503 (1980)

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Abstract

A rigorous method for analyzing plane-wave scattering from perfectly conducting periodic surfaces is examined and applied to trapezoidal profiles. Both TE and TM polarizations of the incident plane wave are considered. An integral equation for the unknown current distribution in the scatter surface is formulated by invoking the extended boundary condition. Upon expressing the current density in terms of its physical optics approximation multiplied by a Fourier series, the integral equation reduces to a linear system of equations. For the case of a piecewise linear surface profile, the coefficient matrix of this system is amenable to efficient computer evaluation, which furnishes the Fourier coefficients of the current distribution. The method is applied to trapezoidal scatterers for which little data is available in the literature, and, by using appropriate limiting procedures, to triangular and rectangular profiles. Scatter fields and surface current densities are calculated. The accuracy of the method, its range, and its limitations, are investigated and comparisons are made with the results of others. The method has given accurate results for surface groove depths of less than half a wavelength and for surface periods of greater than a wavelength at minimal computational cost.

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Trapezoidal parameters: d/λ = 3.0 h/λ = 0.15 a/b = 0.9
Rectangular parameters: d/λ = 3.0 h/λ =.15
θ	m	θ_m	TRAP P_mt	TE REC P_mr	$∊_{w}^{TE}$	TRAP P_mt	TM REC P_mr	$∊_{w}^{TM}$
30.°	−4	−56.4	0.00550	0.00542	0.01	0.00552	0.00585	0.03
	−3	−30.0	0.02651	0.02212	0.53	0.06576	0.07787	1.02
	−2	−9.6	0.00788	0.01099	0.22	0.00160	0.00167	0.01
	−1	9.6	0.23715	0.23339	0.38	0.26317	0.27422	1.06
	0	30.0	0.57451	0.57889	0.43	0.44845	0.40367	4.97
	1	56.4	0.14449	0.14909	0.45	0.23290	0.23721	0.42
Relative power error (∊ %)			0.396	0.000		1.741	0.000
85.0°	−5	−42.1	0.00096	0.00154	0.04	0.37229	0.34373	3.09
	−4	−19.8	0.0082	0.00115	0.02	0.00662	0.01125	0.27
	−3	−0.2	0.00252	0.00312	0.05	0.23383	0.22983	0.41
	−2	19.2	0.00109	0.00171	0.04	0.00742	0.01076	0.23
	−1	41.5	0.01920	0.02332	0.34	0.38022	0.34949	3.34
	0	85.0	0.97473	0.96916	0.56	0.00572	0.05495	0.51
Relative power error (∊ %)			0.069	0.000		0.610	0.000

	h/λ	Method	\|E₀\|²	\|E₁\|² = \|E₁\|²	\|E₂\|²	Total power
TE	0.172	Present	0.698	0.163	0.009	1.000
		KN	0.699	0.163	0.008	1.000
		P	0.697	0.163	0.008	0.999
	0.345	Present	0.186	0.392	0.103	0.991
		KN	0.182	0.397	0.105	0.999
	1.145	P Present	0.179	0.397	0.107	0.999
		KN	0.472	0.110	0.334	0.994
		P	0.457	0.114	0.343	1.044
TM	0.172	Present	0.644	0.184	0.025	1.000
		KN	0.646	0.184	0.023	1.001
		P	0.642	0.184	0.015	0.997
	0.345	Present	0.132	0.328	0.279	0.994
		KN	0.133	0.335	0.277	0.997
		P	0.133	0.329	0.281	0.994
	1.145	Present	0.659	0.053	0.270	1.016
		KN	0.792	0.049	0.130	1.006
		P

Trapezoidal parameters: d/λ = 3.0 h/λ = 0.20 a/b = 0.27815
Sinusoidal parameters: d/λ = 3.0 h_s/λ = 0.2254
θ	m	θ_m	TRAP P_mt	SINE P_ms	TE ∊	TRAP P_mt	TM SINE P_ms	TM ∊_w
0.1°	−3	−86.6	0.0003	0.00007		0.02860	0.00307
	−2	−41.7	0.02840	0.03395	0.46	0.01262	0.03431	7.98
	−1	−19.4	0.29929	0.29633	0.29	0.33197	0.32063	1.17
	0	0.1	0.34465	0.33571	0.92	0.26822	0.28534	1.61
	1	19.6	0.29876	0.29798	0.08	0.31554	0.31645	0.09
	2	41.9	0.02857	0.03395	0.45	0.04333	0.04020	3.37
Relative power error, ∊(%)			0.003	0.000		0.028	0.000
60°	−5	−53.2	0.00020	0.00000		0.00752	0.00000
	−4	−27.9	0.00070	0.00002		0.00542	0.00003
	−3	−7.7	0.00287	0.00088	0.65	0.00761	0.00169	2.67
	−2	11.5	0.01498	0.01825	0.27	0.02565	0.03345	0.60
	−1	32.2	0.16710	0.16995	0.28	0.25818	0.26241	0.42
	0	60.0	0.81414	0.81090	0.33	0.69569	0.70242	0.67
Relative power error, ∊(%)			0.001	0.000		0.008	0.000

Trapezoidal parameters: d/λ = 3.0 h/λ = 0.15 a/b = 0.9
Rectangular parameters: d/λ = 3.0 h/λ =.15
θ	m	θ_m	TRAP P_mt	TE REC P_mr	$∊_{w}^{TE}$	TRAP P_mt	TM REC P_mr	$∊_{w}^{TM}$
30.°	−4	−56.4	0.00550	0.00542	0.01	0.00552	0.00585	0.03
	−3	−30.0	0.02651	0.02212	0.53	0.06576	0.07787	1.02
	−2	−9.6	0.00788	0.01099	0.22	0.00160	0.00167	0.01
	−1	9.6	0.23715	0.23339	0.38	0.26317	0.27422	1.06
	0	30.0	0.57451	0.57889	0.43	0.44845	0.40367	4.97
	1	56.4	0.14449	0.14909	0.45	0.23290	0.23721	0.42
Relative power error (∊ %)			0.396	0.000		1.741	0.000
85.0°	−5	−42.1	0.00096	0.00154	0.04	0.37229	0.34373	3.09
	−4	−19.8	0.0082	0.00115	0.02	0.00662	0.01125	0.27
	−3	−0.2	0.00252	0.00312	0.05	0.23383	0.22983	0.41
	−2	19.2	0.00109	0.00171	0.04	0.00742	0.01076	0.23
	−1	41.5	0.01920	0.02332	0.34	0.38022	0.34949	3.34
	0	85.0	0.97473	0.96916	0.56	0.00572	0.05495	0.51
Relative power error (∊ %)			0.069	0.000		0.610	0.000

	h/λ	Method	\|E₀\|²	\|E₁\|² = \|E₁\|²	\|E₂\|²	Total power
TE	0.172	Present	0.698	0.163	0.009	1.000
		KN	0.699	0.163	0.008	1.000
		P	0.697	0.163	0.008	0.999
	0.345	Present	0.186	0.392	0.103	0.991
		KN	0.182	0.397	0.105	0.999
	1.145	P Present	0.179	0.397	0.107	0.999
		KN	0.472	0.110	0.334	0.994
		P	0.457	0.114	0.343	1.044
TM	0.172	Present	0.644	0.184	0.025	1.000
		KN	0.646	0.184	0.023	1.001
		P	0.642	0.184	0.015	0.997
	0.345	Present	0.132	0.328	0.279	0.994
		KN	0.133	0.335	0.277	0.997
		P	0.133	0.329	0.281	0.994
	1.145	Present	0.659	0.053	0.270	1.016
		KN	0.792	0.049	0.130	1.006
		P

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Journal of the Optical Society of America