Scattering from fractally corrugated surfaces with use of the extended boundary condition method

S. Savaidis; P. Frangos; D. L. Jaggard; K. Hizanidis

doi:10.1364/JOSAA.14.000475

Journal of the Optical Society of America A
Vol. 14,
Issue 2,
pp. 475-485
(1997)
•https://doi.org/10.1364/JOSAA.14.000475

Scattering from fractally corrugated surfaces with use of the extended boundary condition method

S. Savaidis, P. Frangos, D. L. Jaggard, and K. Hizanidis

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S. Savaidis, P. Frangos, D. L. Jaggard, and K. Hizanidis, "Scattering from fractally corrugated surfaces with use of the extended boundary condition method," J. Opt. Soc. Am. A 14, 475-485 (1997)

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Abstract

A new method for scattering from fractally corrugated conducting surfaces is formulated with use of the extended boundary condition method. Here we expand the fields through generalized Floquet modes and obtain analytical closed-form expressions for the scattering amplitudes for both horizontal and vertical polarization of the incident optical and electromagnetic waves. The accuracy of the proposed method is checked in several ways, such as comparison with approximate methods previously presented in the literature (Rayleigh and Kirchhoff methods) and calculation of the energy-balance parameter. Finally, numerical scattering results from fractal surfaces are provided.

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N_p	χ	kσ	κ	N_b	$a = 0.1$	$a = 0.3$	$a = 0.5$	$a = 0.7$	$a = 0.9$
$M = 2$ , $θ_{i} = 30 °$
4	0.4	0.1	1	5	1.0000 (0.9999)	1.0000 (0.9999)	0.9999 (0.9999)	0.9999 (0.9999)	0.9999 (0.9998)
14		0.5	3	25	1.0000 (1.0000)	1.0000 (1.0000)	0.9999 (0.9998)	0.9999 (0.9998)	0.9998 (0.9997)
18		1.0	4	41	0.9997 (1.0001)	0.9991 (1.0004)	0.9982 (1.0016)	0.9972 (1.0025)	0.9965 (1.0026)
22		1.5	5	61	0.9958 (1.0065)	0.9918 (1.0081)	0.9878 (1.0054)	0.9873 (1.0000)	0.9888 (0.9959)
3	1.0	0.1	1	5	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)
6		0.5	3	25	1.0000 (1.0003)	1.0000 (1.0004)	1.0000 (1.0006)	1.0000 (1.0010)	1.0000 (1.0015)
7		1.0	4	41	0.9978 (1.0105)	0.9927 (1.0161)	0.9907 (1.0010)	0.9854 (1.0033)	0.9817 (1.0005)
9		1.5	5	61	1.0206 (0.9621)	0.9729 (1.3741)	0.9641 (3.2446)	0.8896 (1.5798)	0.8832 (1.7243)
$M = 4$ , $θ_{i} = 60 °$
5	0.2	0.1	1	9	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (0.9999)	1.0000 (0.9999)
18		0.5	2	41	0.9994 (0.9991)	0.9991 (0.9985)	0.9983 (0.9971)	0.9971 (0.9945)	0.9960 (0.9917)
18		0.7	2	41	0.9960 (0.9940)	0.9939 (0.9903)	0.9892 (0.9818)	0.9818 (0.9661)	0.9752 (0.9501)
4	0.8	0.1	1	9	1.0000 (0.9998)	1.0000 (0.9998)	1.0000 (0.9997)	1.0000 (0.9997)	1.0000 (0.9998)
11		0.5	2	41	1.0006 (1.0030)	1.0001 (1.0030)	0.9990 (0.9984)	0.9986 (0.9830)	1.0010 (0.9670)
1	3.0	0.1	1	9	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)
3		0.5	2	41	1.0022 (0.9998)	1.0089 (0.9997)	1.0084 (1.0027)	1.0073 (1.0019)	1.0173 (0.9960)
$M = 6$ , $θ_{i} = 45 °$
11	0.1	0.1	1	13	1.0000 (1.0000)	1.0000 (1.0000)	0.9999 (0.9999)	0.9999 (0.9999)	0.9999 (0.9999)
63		0.5	2	85	0.9992 (0.9991)	0.9985 (0.9983)	0.9969 (0.9963)	0.9937 (0.9917)	0.9903 (0.9848)
63		0.7	2	85	0.9994 (0.9938)	0.9899 (0.9887)	0.9802 (0.9769)	0.9620 (0.9510)	0.9430 (0.9148)
2	1.5	0.1	1	13	1.0000 (1.0004)	1.0001 (1.0004)	1.0001 (1.0005)	1.0002 (1.0009)	1.0003 (1.0011)
6		0.5	2	85	1.0023 (1.0115)	1.0076 (1.0108)	1.0233 (1.0151)	1.0591 (1.0258)	1.0837 (1.0170)
1	2.5	0.1	1	13	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)
3		0.5	2	85	1.0021 (0.9998)	1.0104 (0.9988)	1.0154 (0.9982)	1.0163 (0.9962)	1.0369 (0.9947)

θ_i	χ	kσ	κ	EBCM (%)			Rayleigh (%)
θ_i	χ	kσ	κ	$a = 0.1$	$a = 0.5$	$a = 0.9$	$a = 0.1$	$a = 0.5$	$a = 0.9$
$M = 2$
60°	0.2	1.90	6	$7.27 \times 10^{- 2}$	0.29	0.31	0.27	1.98	2.12
		2.00		$9.78 \times 10^{- 2}$	0.38	0.40	0.59	4.35	4.57
		2.10		0.17	0.64	0.66	1.22	9.23	9.56
45°	2.0	0.50		$1.91 \times 10^{- 4}$	$7.38 \times 10^{- 3}$	$8.46 \times 10^{- 3}$	$2.21 \times 10^{- 4}$	$1.21 \times 10^{- 2}$	$5.08 \times 10^{- 3}$
		0.55		$9.06 \times 10^{- 4}$	$4.58 \times 10^{- 3}$	$8.58 \times 10^{- 4}$	$3.40 \times 10^{- 4}$	$7.40 \times 10^{- 3}$	$6.73 \times 10^{- 3}$
		0.65		$1.12 \times 10^{- 2}$	$4.69 \times 10^{- 2}$	0.25	$1.61 \times 10^{- 2}$	0.12	9.16
30°	3.0	0.40		$7.81 \times 10^{- 4}$	$3.05 \times 10^{- 3}$	$1.67 \times 10^{- 2}$	$2.33 \times 10^{- 2}$	$1.80 \times 10^{- 2}$	$5.17 \times 10^{- 2}$
		0.45		$4.42 \times 10^{- 3}$	$6.24 \times 10^{- 3}$	$5.47 \times 10^{- 2}$	0.14	$5.97 \times 10^{- 2}$	0.22
		0.50		$1.67 \times 10^{- 2}$	$3.12 \times 10^{- 2}$	0.37	0.66	0.36	0.76
$M = 4$
80°	0.3	0.50	2	$1.99 \times 10^{- 2}$	$5.52 \times 10^{- 2}$	0.12	$2.21 \times 10^{- 2}$	$6.72 \times 10^{- 2}$	0.15
		0.60		$5.71 \times 10^{- 2}$	0.15	0.34	$6.66 \times 10^{- 2}$	0.21	0.47
		0.70		0.14	0.36	0.78	0.17	0.54	1.23
$M = 6$
15°	0.1	0.30	2	$1.90 \times 10^{- 2}$	$7.82 \times 10^{- 2}$	0.19	$1.96 \times 10^{- 2}$	$8.43 \times 10^{- 2}$	0.22
		0.35		$4.68 \times 10^{- 2}$	0.19	0.46	$4.9 \times 10^{- 2}$	0.21	0.54
		0.40		0.10	0.41	0.97	0.11	0.47	1.21

N_p	χ	kσ	κ	N_b	$a = 0.1$	$a = 0.3$	$a = 0.5$	$a = 0.7$	$a = 0.9$
$M = 2$ , $θ_{i} = 30 °$
4	0.4	0.1	1	5	1.0000 (0.9999)	1.0000 (0.9999)	0.9999 (0.9999)	0.9999 (0.9999)	0.9999 (0.9998)
14		0.5	3	25	1.0000 (1.0000)	1.0000 (1.0000)	0.9999 (0.9998)	0.9999 (0.9998)	0.9998 (0.9997)
18		1.0	4	41	0.9997 (1.0001)	0.9991 (1.0004)	0.9982 (1.0016)	0.9972 (1.0025)	0.9965 (1.0026)
22		1.5	5	61	0.9958 (1.0065)	0.9918 (1.0081)	0.9878 (1.0054)	0.9873 (1.0000)	0.9888 (0.9959)
3	1.0	0.1	1	5	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)
6		0.5	3	25	1.0000 (1.0003)	1.0000 (1.0004)	1.0000 (1.0006)	1.0000 (1.0010)	1.0000 (1.0015)
7		1.0	4	41	0.9978 (1.0105)	0.9927 (1.0161)	0.9907 (1.0010)	0.9854 (1.0033)	0.9817 (1.0005)
9		1.5	5	61	1.0206 (0.9621)	0.9729 (1.3741)	0.9641 (3.2446)	0.8896 (1.5798)	0.8832 (1.7243)
$M = 4$ , $θ_{i} = 60 °$
5	0.2	0.1	1	9	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (0.9999)	1.0000 (0.9999)
18		0.5	2	41	0.9994 (0.9991)	0.9991 (0.9985)	0.9983 (0.9971)	0.9971 (0.9945)	0.9960 (0.9917)
18		0.7	2	41	0.9960 (0.9940)	0.9939 (0.9903)	0.9892 (0.9818)	0.9818 (0.9661)	0.9752 (0.9501)
4	0.8	0.1	1	9	1.0000 (0.9998)	1.0000 (0.9998)	1.0000 (0.9997)	1.0000 (0.9997)	1.0000 (0.9998)
11		0.5	2	41	1.0006 (1.0030)	1.0001 (1.0030)	0.9990 (0.9984)	0.9986 (0.9830)	1.0010 (0.9670)
1	3.0	0.1	1	9	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)
3		0.5	2	41	1.0022 (0.9998)	1.0089 (0.9997)	1.0084 (1.0027)	1.0073 (1.0019)	1.0173 (0.9960)
$M = 6$ , $θ_{i} = 45 °$
11	0.1	0.1	1	13	1.0000 (1.0000)	1.0000 (1.0000)	0.9999 (0.9999)	0.9999 (0.9999)	0.9999 (0.9999)
63		0.5	2	85	0.9992 (0.9991)	0.9985 (0.9983)	0.9969 (0.9963)	0.9937 (0.9917)	0.9903 (0.9848)
63		0.7	2	85	0.9994 (0.9938)	0.9899 (0.9887)	0.9802 (0.9769)	0.9620 (0.9510)	0.9430 (0.9148)
2	1.5	0.1	1	13	1.0000 (1.0004)	1.0001 (1.0004)	1.0001 (1.0005)	1.0002 (1.0009)	1.0003 (1.0011)
6		0.5	2	85	1.0023 (1.0115)	1.0076 (1.0108)	1.0233 (1.0151)	1.0591 (1.0258)	1.0837 (1.0170)
1	2.5	0.1	1	13	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)	1.0000 (1.0000)
3		0.5	2	85	1.0021 (0.9998)	1.0104 (0.9988)	1.0154 (0.9982)	1.0163 (0.9962)	1.0369 (0.9947)

θ_i	χ	kσ	κ	EBCM (%)			Rayleigh (%)
θ_i	χ	kσ	κ	$a = 0.1$	$a = 0.5$	$a = 0.9$	$a = 0.1$	$a = 0.5$	$a = 0.9$
$M = 2$
60°	0.2	1.90	6	$7.27 \times 10^{- 2}$	0.29	0.31	0.27	1.98	2.12
		2.00		$9.78 \times 10^{- 2}$	0.38	0.40	0.59	4.35	4.57
		2.10		0.17	0.64	0.66	1.22	9.23	9.56
45°	2.0	0.50		$1.91 \times 10^{- 4}$	$7.38 \times 10^{- 3}$	$8.46 \times 10^{- 3}$	$2.21 \times 10^{- 4}$	$1.21 \times 10^{- 2}$	$5.08 \times 10^{- 3}$
		0.55		$9.06 \times 10^{- 4}$	$4.58 \times 10^{- 3}$	$8.58 \times 10^{- 4}$	$3.40 \times 10^{- 4}$	$7.40 \times 10^{- 3}$	$6.73 \times 10^{- 3}$
		0.65		$1.12 \times 10^{- 2}$	$4.69 \times 10^{- 2}$	0.25	$1.61 \times 10^{- 2}$	0.12	9.16
30°	3.0	0.40		$7.81 \times 10^{- 4}$	$3.05 \times 10^{- 3}$	$1.67 \times 10^{- 2}$	$2.33 \times 10^{- 2}$	$1.80 \times 10^{- 2}$	$5.17 \times 10^{- 2}$
		0.45		$4.42 \times 10^{- 3}$	$6.24 \times 10^{- 3}$	$5.47 \times 10^{- 2}$	0.14	$5.97 \times 10^{- 2}$	0.22
		0.50		$1.67 \times 10^{- 2}$	$3.12 \times 10^{- 2}$	0.37	0.66	0.36	0.76
$M = 4$
80°	0.3	0.50	2	$1.99 \times 10^{- 2}$	$5.52 \times 10^{- 2}$	0.12	$2.21 \times 10^{- 2}$	$6.72 \times 10^{- 2}$	0.15
		0.60		$5.71 \times 10^{- 2}$	0.15	0.34	$6.66 \times 10^{- 2}$	0.21	0.47
		0.70		0.14	0.36	0.78	0.17	0.54	1.23
$M = 6$
15°	0.1	0.30	2	$1.90 \times 10^{- 2}$	$7.82 \times 10^{- 2}$	0.19	$1.96 \times 10^{- 2}$	$8.43 \times 10^{- 2}$	0.22
		0.35		$4.68 \times 10^{- 2}$	0.19	0.46	$4.9 \times 10^{- 2}$	0.21	0.54
		0.40		0.10	0.41	0.97	0.11	0.47	1.21

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Journal of the Optical Society of America A