Scattering by Two Rayleigh-Debye Spheres

G. O. Olaofe

doi:10.1364/AO.9.000429

Applied Optics
Vol. 9,
Issue 2,
pp. 429-437
(1970)
•https://doi.org/10.1364/AO.9.000429

Scattering by Two Rayleigh-Debye Spheres

G. O. Olaofe

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G. O. Olaofe, "Scattering by Two Rayleigh-Debye Spheres," Appl. Opt. 9, 429-437 (1970)

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Abstract

The Rayleigh-Gans approximation theory is applied to the problem of scattering by two neighboring spherical particles. For simplicity, we have examined in detail the case of two identical uniform spheres. Differential intensities for any arbitrary location of the particle pair relative to the incident wave, mean intensities, and mean scattering cross sections for randomly oriented particles are considered. The results show good agreement with the exact theory. Some numerical results are presented for the particular case of touching spheres (dumbbells), which are either at random, or are at specific, orientation relative to the direction of polarization of the incident wave. It is observed that for small spheres at random orientation, the scattering cross section is four times the value for that of a single sphere. This factor is two for large particles. We also observe that both the intensity and dissymmetry patterns for two spheres are totally different from the single particle ones.

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x	Single sphere	Two spheres					Random orientation (or mean)
		End illumination α = 0		Broadside illumination α = π/2			Random orientation (or mean)
		η = 2	η = 3	η = 4	All η	η = 2	η = 3	η = 4
0.1	1.0052	1.0317	1.0667	1.1304	Same as for single sphere	1.0222	1.0222	1.0455
0.2	1.0229	1.1481	1.3359	1.6765		1.0623	1.1123	1.1850
0.3	1.0525	1.3745	2.4317	3.7774		1.1444	1.2659	1.4435
0.4	1.0957	1.7956	3.9322	24.988		1.2692	1.5027	1.8237
0.5	1.1543	2.6148	13.405	57.300		1.4463	1.8322	2.2244
0.6	1.2312	4.4187	970.36	5.3141		1.6886	2.2312	2.3979
0.7	1.3303	9.4652	24.999	2.0963		2.0108	2.6103	2.2398
0.8	1.4568	33.224	6.0748	1.3789		2.4248	2.8372	1.9563
0.9	1.6183	1275.2	3.0731	1.2144		2.9328	2.8584	1.7450
1.0	1.8255	90.617	2.1397	1.3666		3.5178	2.7556	1.6709
1.1	2.0934	20.622	1.8276	2.0030		4.1415	2.6525	1.7473
1.2	2.4445	10.1935	1.8344	4.2705		4.7609	2.6359	1.9886
1.3	2.9122	6.8814	2.1351	20.850		5.3599	2.7591	2.4140
1.4	3.5484	5.5918	2.9007	363.84		5.9745	3.0736	3.0279
1.5	4.4369	5.2005	4.7359	11.4285		6.6974	3.6589	3.8209
1.6	5.7182	5.4126	9.9895	4.3100		7.6787	4.6518	4.8476
1.7	7.6444	6.2510	32.872	2.8696		9.1519	6.2903	6.3491
1.8	10.6857	8.2187	498.0	2.6549		11.5225	8.9959	8.8833
1.9	15.8395	11.5395	307.91	3.2263		15.6025	13.5822	13.6475
2.0	25.4324	19.0394				23.2792	21.9016	23.4992
Comparison of dissymetry for a single sphere with (i) dissymmetry for two spheres for the cases of end and broadside illumination; (ii) dissymmetry for the case of two spheres randomly positioned relative to incident light. [Separation = ηx, Z₄₅ = intensity (45°)/intensity (135°)].

Concentric spheres			Dumbbells
x = 1, x₀ = 0.25, m = 1.05, n = 1.0125			x = 0.5,	m = 1.05
sca angle	vpi	hpi	vpi	hpi
0°	0.000077	0.000077	0.000073	0.000073
15°	0.000076	0.000071	0.000072	0.000067
30°	0.000074	0.000055	0.000070	0.000053
45°	0.000070	0.000034	0.000067	0.000034
60°	0.000064	0.000016	0.000064	0.000016
75°	0.000059	0.000004	0.000060	0.000004
90°	0.000052	0.0	0.000056	0.0
105°	0.000047	0.000003	0.000052	0.000004
120°	0.000043	0.000011	0.000049	0.000012
135°	0.000039	0.000019	0.000047	0.000023
150°	0.000036	0.000027	0.000045	0.000034
165°	0.000035	0.000033	0.000044	0.000041
180°	0.000034	0.000034	0.000043	0.000043

Concentric spheres			Dumbbells
x = 1, x₀ = 0.25, m = 1.10, n = 1.025			x = 0.5, m = 1.0
sca. angle	vpi	hpi	vpi	hpi
0°	0.000313	0.000313	0.000306	0.000306
15°	0.000309	0.000289	0.000303	0.000283
30°	0.000298	0.000223	0.000296	0.000222
45°	0.000280	0.000140	0.000283	0.000142
60°	0.000258	0.000065	0.000268	0.000067
75°	0.000235	0.000016	0.000252	0.000017
90°	0.000212	0.0	0.000235	0.0
105°	0.000191	0.000013	0.000220	0.000015
120°	0.000173	0.000043	0.000207	0.000052
135°	0.000158	0.000079	0.000196	0.000098
150°	0.000148	0.000111	0.000188	0.000141
165°	0.000142	0.000132	0.000183	0.000171
180°	0.000141	0.000141	0.000182	0.0001822

Concentric spheres			Dumbbells
x = 2, x₀ = 0.5, m = 1.05, n = 1.025			x = 1, m = 1.05
x = 2, x₀ = 0.5, m = 1.05, n = 1.025			x = 1, m = 1.05		sca. angle	vpi	hpi	vpi	hpi
0°	0.00494	0.00494	0.00467	0.00467
15°	0.00469	0.00438	0.00450	0.00420
30°	0.00402	0.00301	0.00405	0.00304
45°	0.00311	0.00156	0.00343	0.00171
60°	0.00220	0.00055	0.00277	0.00069
75°	0.00144	0.00010	0.00218	0.00015
90°	0.00088	0.0	0.00711	0.00
105°	0.00052	0.000035	0.00137	0.000092
120°	0.00030	0.000075	0.00113	0.00028
135°	0.00018	0.000090	0.00097	0.00049
150°	0.00012	0.000087	0.00088	0.00066
165°	0.000085	0.000080	0.00082	0.00077
180°	0.000077	0.000077	0.00081	0.00081

m	x	Single sphere 4C_sca	Two spheres (dumbbells) 〈C_sca〉_αβ
1.05	0.5	0.0005536	0.0004764
	1.0	0.026702	0.017676
	1.5	0.200983	0.122133
	2.0	0.726057	0.437101
1.10	0.5	0.0023239	0.0019998
	1.0	0.112082	0.074195
	1.5	0.843627	0.512656
	2.0	3.047626	1.834736
1.15	0.5	0.0054807	0.0047163
	1.0	0.264335	0.174984
	1.5	1.989626	1.209058
	2.0	7.187579	4.327075
1.20	0.5	0.0102019	0.008779
	1.0	0.0492040	0.325719
	1.5	3.703544	2.250573
	2.0	13.379151	8.054533
1.35	0.5	0.031154	0.026808
	1.0	1.502571	0.994667
	1.5	11.309721	6.872701
	2.0	40.856667	24.5965680
1.44	0.5	0.060738	0.052266
	1.0	2.929411	1.939203
	1.5	22.049420	13.399011
	2.0	79.654115	47.953467

	m	η = 2	η = 3	η = 4
x = 0.5	1.05	0.000476	0.000406	0.000345
	1.10	0.001999	0.001707	0.001452
	1.15	0.004716	0.004026	0.003424
	1.20	0.008778	0.007495	0.006374
	1.25	0.014347	0.012250	0.010418
	1.30	0.021589	0.018433	0.015676
	1.33	0.026808	0.022890	0.019466
	1.44	0.052266	0.044627	0.037951
	1.50	0.070852	0.060497	0.051448
	1.55	0.089195	0.076158	0.064767
x = 1.0	1.05	0.017676	0.014879	0.014577
	1.10	0.074195	0.062458	0.061187
	1.15	0.174983	0.147302	0.144304
	1.20	0.325719	0.274193	0.268612
	1.25	0.532332	0.448122	0.439001
	1.30	0.801007	0.674295	0.660570
	1.33	0.994667	0.837320	0.820277
	1.44	1.939203	1.632438	1.599212
	1.50	2.628804	2.212951	2.167909
	1.55	3.309359	2.785848	2.729146
x = 1.5	1.05	0.122133	0.111037	0.104930
	1.10	0.512656	0.466081	0.440444
	1.15	1.209058	1.099214	1.038752
	1.20	2.250573	2.046107	1.933561
	1.25	3.367178	3.344014	3.16007
	1.30	5.534596	5.031775	4.755003
	1.33	6.872701	6.248312	5.904625
	1.44	13.3990	12.1817	11.5117
	1.50	18.1638	16.5137	15.6053
	1.55	22.8662	20.7888	19.6453
x = 2.0	1.05	0.437101	0.391701
	1.10	1.834736	1.644165
	1.15	4.327075	3.877630
	1.20	8.054533	7.217924
	1.25	13.163763	11.796469
	1.30	19.807661	17.750278
	1.33	24.596580	22.041782
	1.44	47.953467	42.972636
	1.50	65.006239	58.254169
	1.55	81.835314	73.335242

Abstract

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Applied Optics