Resonances and poles of weakly absorbing spheres

Gorden Videen; J. Li; Petr Chýlek

doi:10.1364/JOSAA.12.000916

Journal of the Optical Society of America A
Vol. 12,
Issue 5,
pp. 916-921
(1995)
•https://doi.org/10.1364/JOSAA.12.000916

Resonances and poles of weakly absorbing spheres

Gorden Videen, J. Li, and Petr Chýlek

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Gorden Videen, J. Li, and Petr Chýlek, "Resonances and poles of weakly absorbing spheres," J. Opt. Soc. Am. A 12, 916-921 (1995)

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Abstract

We found that the resonance positions for a weakly absorbing sphere are much different from those of a nonabsorbing sphere. The shifts of resonance positions for Q_ext, Q_sca, and Q_abs from their original resonance positions for a real refractive index are different, and the formulas to describe these different kinds of shift are given. Calculations of 12 resonance cases show that our results are quite close to those given by exact Lorenz–Mie theory. We also derive the line shapes of the different resonances and calculate values for the resonance maxima and half-width as a function of absorption.

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Equations (67)

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	x₀	m_r	Δ₁			Δ₂
									Δ₃
			p₁	q₁	p₁/q₁	p₂	q₂	p₂/q₂	q₃
$a_{20}^{1}$	11.650466	2.2	−1.16 × 10⁻⁶	40.9	−2.82 × 10⁻⁸	−2.31 × 10⁻⁶	−8.48	2.73 × 10⁻⁷	92.7
$a_{20}^{2}$	13.683570	2.2	−2.08 × 10⁻⁴	43.3	−4.80 × 10⁻⁶	−4.15 × 10⁻⁴	−2.69	1.54 × 10⁻⁴	92.0
$a_{38}^{1}$	31.473740	1.4	3.91 × 10⁻³	318.	1.23 × 10⁻⁵	7.83 × 10⁻³	408.	1.92 × 10⁻⁵	251.
$a_{38}^{1}$	24.777621	1.8	−1.56 × 10⁻⁸	222.	−7.00 × 10^{− 11}	−3.11 × 10⁻⁸	114.	−2.74 × 10⁻¹⁰	339.
$b_{40}^{1}$	32.542541	1.4	6.38 × 10⁻³	364.	1.76 × 10⁻⁵	1.28 × 10⁻²	690.	1.85 × 10⁻⁵	71.4
$b_{40}^{2}$	36.228377	1.4	0.652	264.	2.47 × 10⁻³	1.30	845.	1.54 × 10⁻³	−279.
$a_{60}^{1}$	47.953327	1.4	1.44 × 10⁻⁵	854.	1.69 × 10⁻⁸	2.88 × 10⁻⁵	1016.	2.84 × 10⁻⁸	726.
$a_{60}^{2}$	52.081543	1.4	9.71 × 10⁻³	742.	1.31 × 10⁻⁵	1.94 × 10⁻²	1100.	1.77 × 10⁻⁵	419.
$a_{60}^{3}$	55.525274	1.4	0.564	544.	1.04 × 10⁻³	1.13	1202.	9.39 × 10⁻⁴	−80.0
$b_{60}^{1}$	47.490751	1.4	4.77 × 10⁻⁵	861.	5.54 × 10⁻⁸	9.55 × 10⁻⁵	1533.	6.23 × 10⁻⁸	238.
$b_{60}^{2}$	51.676716	1.4	1.78 × 10⁻²	757.	2.36 × 10⁻⁵	3.57 × 10⁻²	1769.	2.02 × 10⁻⁵	−202.
$b_{60}^{3}$	55.217676	1.4	0.762	567.	1.34 × 10⁻³	1.52	2118.	7.20 × 10⁻⁴	−926.

Resonance	$C_{n}^{r m} / A_{n}^{r o}$	$A_{n}^{r o} / C_{n}^{i δ}$
1	4.4840 × 10⁷	−1.144 × 10⁻⁷
2	2.3451 × 10⁵	−2.550 × 10⁻⁵
3	1.7181 × 10⁴	−1.197 × 10⁻³
4	7.5080 × 10⁹	−1.782 × 10⁻⁹
5	4.5780 × 10⁴	−4.818 × 10⁻⁴
6	8.3340 × 10²	−2.875 × 10⁻²
7	8.7961 × 10⁶	−3.714 × 10⁻⁶
8	3.3220 × 10⁴	−1.049 × 10⁻³
9	1.1049 × 10³	−3.252 × 10⁻²
10	1.3046 × 10⁷	−2.522 × 10⁻⁶
11	5.3740 × 10⁴	−6.598 × 10⁻⁴
12	1.9562 × 10³	−1.903 × 10⁻²

	x₀	m_r	Δ₁			Δ₂
									Δ₃
			p₁	q₁	p₁/q₁	p₂	q₂	p₂/q₂	q₃
$a_{20}^{1}$	11.650466	2.2	−1.16 × 10⁻⁶	40.9	−2.82 × 10⁻⁸	−2.31 × 10⁻⁶	−8.48	2.73 × 10⁻⁷	92.7
$a_{20}^{2}$	13.683570	2.2	−2.08 × 10⁻⁴	43.3	−4.80 × 10⁻⁶	−4.15 × 10⁻⁴	−2.69	1.54 × 10⁻⁴	92.0
$a_{38}^{1}$	31.473740	1.4	3.91 × 10⁻³	318.	1.23 × 10⁻⁵	7.83 × 10⁻³	408.	1.92 × 10⁻⁵	251.
$a_{38}^{1}$	24.777621	1.8	−1.56 × 10⁻⁸	222.	−7.00 × 10^{− 11}	−3.11 × 10⁻⁸	114.	−2.74 × 10⁻¹⁰	339.
$b_{40}^{1}$	32.542541	1.4	6.38 × 10⁻³	364.	1.76 × 10⁻⁵	1.28 × 10⁻²	690.	1.85 × 10⁻⁵	71.4
$b_{40}^{2}$	36.228377	1.4	0.652	264.	2.47 × 10⁻³	1.30	845.	1.54 × 10⁻³	−279.
$a_{60}^{1}$	47.953327	1.4	1.44 × 10⁻⁵	854.	1.69 × 10⁻⁸	2.88 × 10⁻⁵	1016.	2.84 × 10⁻⁸	726.
$a_{60}^{2}$	52.081543	1.4	9.71 × 10⁻³	742.	1.31 × 10⁻⁵	1.94 × 10⁻²	1100.	1.77 × 10⁻⁵	419.
$a_{60}^{3}$	55.525274	1.4	0.564	544.	1.04 × 10⁻³	1.13	1202.	9.39 × 10⁻⁴	−80.0
$b_{60}^{1}$	47.490751	1.4	4.77 × 10⁻⁵	861.	5.54 × 10⁻⁸	9.55 × 10⁻⁵	1533.	6.23 × 10⁻⁸	238.
$b_{60}^{2}$	51.676716	1.4	1.78 × 10⁻²	757.	2.36 × 10⁻⁵	3.57 × 10⁻²	1769.	2.02 × 10⁻⁵	−202.
$b_{60}^{3}$	55.217676	1.4	0.762	567.	1.34 × 10⁻³	1.52	2118.	7.20 × 10⁻⁴	−926.

Resonance	$C_{n}^{r m} / A_{n}^{r o}$	$A_{n}^{r o} / C_{n}^{i δ}$
1	4.4840 × 10⁷	−1.144 × 10⁻⁷
2	2.3451 × 10⁵	−2.550 × 10⁻⁵
3	1.7181 × 10⁴	−1.197 × 10⁻³
4	7.5080 × 10⁹	−1.782 × 10⁻⁹
5	4.5780 × 10⁴	−4.818 × 10⁻⁴
6	8.3340 × 10²	−2.875 × 10⁻²
7	8.7961 × 10⁶	−3.714 × 10⁻⁶
8	3.3220 × 10⁴	−1.049 × 10⁻³
9	1.1049 × 10³	−3.252 × 10⁻²
10	1.3046 × 10⁷	−2.522 × 10⁻⁶
11	5.3740 × 10⁴	−6.598 × 10⁻⁴
12	1.9562 × 10³	−1.903 × 10⁻²

Abstract

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Equations (67)

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