On the validity of the integral localized approximation for Bessel beams and associated radiation pressure forces

Leonardo A. Ambrosio; Jiajie Wang; Gérard Gouesbet

doi:10.1364/AO.56.005377

Applied Optics
Vol. 56,
Issue 19,
pp. 5377-5387
(2017)
•https://doi.org/10.1364/AO.56.005377

On the validity of the integral localized approximation for Bessel beams and associated radiation pressure forces

Leonardo A. Ambrosio, Jiajie Wang, and Gérard Gouesbet

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Leonardo A. Ambrosio, Jiajie Wang, and Gérard Gouesbet, "On the validity of the integral localized approximation for Bessel beams and associated radiation pressure forces," Appl. Opt. 56, 5377-5387 (2017)

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Abstract

In this paper we investigate the integral version of the localized approximation (ILA)—a powerful technique for evaluating the beam shape coefficients in the framework of the generalized Lorenz–Mie theory—as applied to ideal scalar Bessel beams (BBs). Originally conceived for arbitrary shaped beams with a propagating factor $\exp (\pm i k z)$ , it has recently been shown that care must be taken when applying the ILA for the case of ideal scalar BBs, since they carry a propagating factor $\exp (\pm i k z \cos α)$ , with $α$ being the axicon angle, which cannot be smoothly accommodated into its mathematical formalism. Comparisons are established between the beam shape coefficients calculated from both ILA and exact approaches, assuming paraxial approximation and both on- and off-axis beams. Particular simulations of radiation pressure forces are provided based on the existing data in the literature. This work helps us in elucidating that ILA provides adequate beam shape coefficients and descriptions of ideal scalar BBs up to certain limits and, even when it fails to do so, reliable information on the physical optical properties of interest can still be inferred, depending on specific geometric and electromagnetic aspects of the scatterer.

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	$α = 1 °$					$α = 5 °$
$n$	$g_{n}^{exa}$	$g_{n}^{loc}$	$δ ϵ (g_{n}^{loc})$	$g_{n, vec}^{loc}$	$δ ϵ (g_{n, vec}^{loc})$	$g_{n}^{exa}$	$g_{n}^{loc}$	$δ ϵ (g_{n}^{loc})$	$g_{n, vec}^{loc}$	$δ ϵ (g_{n, vec}^{loc})$
1	0.999848	0.999829	$1.90 \times 10^{- 3}$	0.999676	$1.71 \times 10^{- 2}$	0.996198	0.995732	$4.68 \times 10^{- 2}$	0.99195	$4.26 \times 10^{- 1}$
2	0.999543	0.999524	$1.90 \times 10^{- 3}$	0.999372	$1.71 \times 10^{- 2}$	0.988617	0.988166	$4.56 \times 10^{- 2}$	0.984421	$4.24 \times 10^{- 1}$
5	0.997717	0.997698	$1.89 \times 10^{- 3}$	0.997546	$1.71 \times 10^{- 2}$	0.94373	0.943374	$3.77 \times 10^{- 2}$	0.939841	$4.12 \times 10^{- 1}$
10	0.991641	0.991622	$1.85 \times 10^{- 3}$	0.991472	$1.70 \times 10^{- 2}$	0.801412	0.801339	$9.13 \times 10^{- 3}$	0.798478	$3.66 \times 10^{- 1}$
25	0.951110	0.951095	$1.57 \times 10^{- 3}$	0.950952	$1.66 \times 10^{- 2}$	0.0970896	0.0979056	$8.40 \times 10^{- 1}$	0.098292	1.24
50	0.815038	0.815034	$4.91 \times 10^{- 4}$	0.814917	$1.49 \times 10^{- 2}$	−0.340569	−0.34198	$4.14 \times 10^{- 1}$	−0.340206	$1.07 \times 10^{- 1}$
100	0.366707	0.366731	$6.57 \times 10^{- 3}$	0.366697	$2.53 \times 10^{- 3}$	−0.0312103	−0.0283921	9.03	−0.0281146	9.92
250	−0.347796	−0.347851	$1.57 \times 10^{- 2}$	−0.347778	$5.21 \times 10^{- 3}$	−0.102946	−0.0993501	3.49	−0.0987599	4.07
500	−0.0220289	−0.0219155	$5.15 \times 10^{- 1}$	−0.0219058	$5.59 \times 10^{- 1}$	0.0553595	0.0494443	10.69	0.0491529	11.21

$n$	$α = 10 °$					$α = 20 °$
$n$	$g_{n}^{exa}$	$g_{n}^{loc}$	$δ ϵ (g_{n}^{loc})$	$g_{n, vec}^{loc}$	$δ ϵ (g_{n, vec}^{loc})$	$g_{n}^{exa}$	$g_{n}^{loc}$	$δ ϵ (g_{n}^{loc})$	$g_{n, vec}^{loc}$	$δ ϵ (g_{n, vec}^{loc})$
1	0.984865	0.98311	$1.78 \times 10^{- 1}$	0.968295	1.68	0.940602	0.935275	$5.66 \times 10^{- 1}$	0.880662	6.37
2	0.954941	0.953437	$1.57 \times 10^{- 1}$	0.939182	1.65	0.827151	0.825406	$2.11 \times 10^{- 1}$	0.778892	5.83
5	0.784796	0.784638	$2.00 \times 10^{- 2}$	0.773559	1.43	0.281627	0.2928	3.98	0.284927	1.17
10	0.323824	0.32643	$8.05 \times 10^{- 1}$	0.32384	$4.72 \times 10^{- 3}$	−0.388016	−0.390914	$7.47 \times 10^{- 1}$	−0.354648	8.60
25	−0.330623	−0.336458	1.76	−0.329548	$3.25 \times 10^{- 1}$	−0.0626394	−0.0183149	70.76	−0.0148893	76.23
50	−0.0422676	−0.0310775	26.47	−0.0299154	29.22	−0.0804123	−0.137608	71.13	−0.125877	56.54
100	−0.0960508	−0.1109	15.46	−0.108523	12.99	−0.127309	−0.0765319	39.88	−0.0695583	45.36
250	0.0596963	0.0356454	40.29	0.0347973	41.71	0.0214699	−0.0860111	300.61	−0.0783911	265.12
500	0.0146158	−0.0228053	56.03	−0.0223026	52.59	−0.0250512	0.0446645	78.29	0.0407195	62.55

	$α = 8.88 °$ , $ρ_{0} = 4.38 μm$
$\| m \|$	$\| g_{n, TM}^{m, exa} \|$	$\| g_{n, TM}^{m, loc} \|$	$δ ϵ$ (%)
0	$9.86143 \times 10^{- 1}$	$9.86413 \times 10^{- 1}$	0.0273562
1	$3.09428 \times 10^{- 1}$	$3.09497 \times 10^{- 1}$	0.0224999
2	$1.13530 \times 10^{- 2}$	$1.12837 \times 10^{- 2}$	0.61119
3	$6.84498 \times 10^{- 5}$	$6.75893 \times 10^{- 5}$	1.25715
4	$1.12107 \times 10^{- 5}$	$1.10110 \times 10^{- 5}$	1.78188
5	$2.88748 \times 10^{- 7}$	$2.81894 \times 10^{- 7}$	2.374
6	$3.38528 \times 10^{- 9}$	$3.28508 \times 10^{- 9}$	2.95993

	$α = 8.88 °$ , $ρ_{0} = 4.38 μm$
$\| m \|$	$\| g_{n, TM}^{m, exa} \|$	$\| g_{n, TM}^{m, loc} \|$	$δ ϵ$ (%)
0	$2.77639 \times 10^{0}$	$2.79563 \times 10^{0}$	0.692896
1	$1.26028 \times 10^{- 2}$	$1.27086 \times 10^{- 2}$	0.839318
2	$3.90318 \times 10^{- 3}$	$3.90536 \times 10^{- 3}$	0.0557908
3	$1.10489 \times 10^{- 6}$	$9.87599 \times 10^{- 7}$	10.6163
4	$8.77310 \times 10^{- 6}$	$8.63177 \times 10^{- 6}$	1.61091
5	$2.30193 \times 10^{- 7}$	$2.25052 \times 10^{- 7}$	2.23305
6	$2.78664 \times 10^{- 9}$	$2.70741 \times 10^{- 9}$	2.84302

	$α = 8.88 °$ , $ρ_{0} = 4.38 μm$
$\| m \|$	$\| g_{n, TM}^{m, exa} \|$	$\| g_{n, TM}^{m, loc} \|$	$δ ϵ$ (%)
0	$2.99054 \times 10^{0}$	$2.96272 \times 10^{0}$	0.930215
1	$5.34937 \times 10^{- 3}$	$5.35749 \times 10^{- 3}$	0.151735
2	$3.60255 \times 10^{- 3}$	$3.60657 \times 10^{- 3}$	0111498
3	$7.52630 \times 10^{- 5}$	$7.48755 \times 10^{- 5}$	0.514915
4	$3.36897 \times 10^{- 6}$	$3.33982 \times 10^{- 6}$	0.865197
5	$9.68067 \times 10^{- 8}$	$9.52050 \times 10^{- 8}$	1.6545
6	$1.34867 \times 10^{- 9}$	$1.31648 \times 10^{- 9}$	2.38678

Abstract

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Equations (7)

Applied Optics