1Department of Physics, Cleveland State University, Cleveland, Ohio 44115 USA
2Laboratoire d’Energétique des Systèmes and Procédés, Unité de Recherche Associée au Centre National de la Recherche Scientifique, No. 230, Institut National des Sciences Appliquées de Rouen, B.P. 08, 76131 Mont-Saint-Aignan Cedex, France
James A. Lock and Gérard Gouesbet, "Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. I. On-axis beams," J. Opt. Soc. Am. A 11, 2503-2515 (1994)
Generalized Lorenz–Mie theory describes electromagnetic scattering of an arbitrary light beam by a spherical particle. The computationally most expensive feature of the theory is the evaluation of the beam-shape coefficients, which give the decomposition of the incident light beam into partial waves. The so-called localized approximation to these coefficients for a focused Gaussian beam is an analytical function whose use greatly simplifies Gaussian-beam scattering calculations. A mathematical justification and physical interpretation of the localized approximation is presented for on-axis beams.
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Terms in the Series Expansion of gn in Powers of s That Are Independent of R for zf = 0
Beam Shape Coefficients
k = 1 First Order
k = 3 Third Order
k = 5 Fifth Order
gnMCk
s0
s0s2
s0s2s4
gnLk
s0s2
s0s2s4
s0s2s4s6
gnBk
s0s2
s0s2s4s6
s0s2s4s6s8s10
Table 2
Average of the Magnitude of the Deviation of the Ratio Filoc/Fis from Unity in Parts per 106 for i = 1, 2, 3a
s
|F1loc/F1s − 1|ave
|F2loc/F2s − 1|ave
|F3loc/F3s − 1|ave
0.001
1
3
2
0.0033
13
34
20
0.01
117
306
178
0.033
1322
3409
1994
0.084
9366
21,954
13,270
0.1
14,167
31,399
19,382
= 1.4%
= 3.1%
= 1.9%
0.15
4.8%
7.6%
5.5%
The average extends over 0 ≤ R ≤ 2.625/s or 1.0 ≥ exp(−s2R2) ≥ 0.001. Fis for the S beam is obtained from Eqs. (90)–(93) with use of a 52-term sum for gns. loc, localized beam.
Table 3
Average of the Magnitude of the Deviation of the Ratio Fimloc/Fis From Unity in Parts per 106 for i = 1, 2, 3a
s
|F1mloc/F1s − 1|ave
|F2mloc/F2s − 1|ave
|F3mloc/F3s − 1|ave
0.001
2
3
2
0.0033
17
31
21
0.01
157
277
186
0.033
1754
3085
2068
0.084
11,737
20,095
13,208
0.1
17,301
28,858
18,943
= 1.7%
= 2.9%
= 1.9%
0.15
5.4%
6.9%
5.2%
The average extends over 0 ≤ R ≤ 2.625/s or 1.0 ≥ exp(−s2R2) ≥ 0.001. Fis for the S beam is obtained from Eqs. (90)–(93) with use of a 52-term sum for gns. mloc, modified localized beam.
Table 4
Actual rms Half-Width of the Focal Waist of the Localized Beam, the Modified Localized Beam, and the Barton Symmetrized Fifth-Order Beam Approximation
w0 (μm)
s
(w0rms)loc(μm)
(w0rms)mloc (μm)
(w0rms)B5 (μm)
1000
0.001
999.999
999.999
999.999
300
0.003333
299.997
300.000
300.000
100
0.01
99.992
99.992
99.990
30
0.03333
29.972
29.972
29.967
10
0.1
9.917
9.917
9.899
5
0.2
4.922
4.922
4.824
4
0.25
4.432
4.432
3.876
3
0.3333
5.497
5.497
3.121
2
0.5
5.283
5.283
2.622
1
1.0
—
—
1.655
Tables (4)
Table 1
Terms in the Series Expansion of gn in Powers of s That Are Independent of R for zf = 0
Beam Shape Coefficients
k = 1 First Order
k = 3 Third Order
k = 5 Fifth Order
gnMCk
s0
s0s2
s0s2s4
gnLk
s0s2
s0s2s4
s0s2s4s6
gnBk
s0s2
s0s2s4s6
s0s2s4s6s8s10
Table 2
Average of the Magnitude of the Deviation of the Ratio Filoc/Fis from Unity in Parts per 106 for i = 1, 2, 3a
s
|F1loc/F1s − 1|ave
|F2loc/F2s − 1|ave
|F3loc/F3s − 1|ave
0.001
1
3
2
0.0033
13
34
20
0.01
117
306
178
0.033
1322
3409
1994
0.084
9366
21,954
13,270
0.1
14,167
31,399
19,382
= 1.4%
= 3.1%
= 1.9%
0.15
4.8%
7.6%
5.5%
The average extends over 0 ≤ R ≤ 2.625/s or 1.0 ≥ exp(−s2R2) ≥ 0.001. Fis for the S beam is obtained from Eqs. (90)–(93) with use of a 52-term sum for gns. loc, localized beam.
Table 3
Average of the Magnitude of the Deviation of the Ratio Fimloc/Fis From Unity in Parts per 106 for i = 1, 2, 3a
s
|F1mloc/F1s − 1|ave
|F2mloc/F2s − 1|ave
|F3mloc/F3s − 1|ave
0.001
2
3
2
0.0033
17
31
21
0.01
157
277
186
0.033
1754
3085
2068
0.084
11,737
20,095
13,208
0.1
17,301
28,858
18,943
= 1.7%
= 2.9%
= 1.9%
0.15
5.4%
6.9%
5.2%
The average extends over 0 ≤ R ≤ 2.625/s or 1.0 ≥ exp(−s2R2) ≥ 0.001. Fis for the S beam is obtained from Eqs. (90)–(93) with use of a 52-term sum for gns. mloc, modified localized beam.
Table 4
Actual rms Half-Width of the Focal Waist of the Localized Beam, the Modified Localized Beam, and the Barton Symmetrized Fifth-Order Beam Approximation
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