Frequency shift between near- and far-field scattering resonances in dielectric particles

Alex J. Yuffa; Yael Gutierrez; Juan M. Sanz; Rodrigo Alcaraz de la Osa; José M. Saiz; Francisco González; Fernando Moreno; Gorden Videen

doi:10.1364/JOSAA.32.001638

Journal of the Optical Society of America A
Vol. 32,
Issue 9,
pp. 1638-1642
(2015)
•https://doi.org/10.1364/JOSAA.32.001638

Frequency shift between near- and far-field scattering resonances in dielectric particles

Alex J. Yuffa, Yael Gutierrez, Juan M. Sanz, Rodrigo Alcaraz de la Osa, José M. Saiz, Francisco González, Fernando Moreno, and Gorden Videen

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Alex J. Yuffa, Yael Gutierrez, Juan M. Sanz, Rodrigo Alcaraz de la Osa, José M. Saiz, Francisco González, Fernando Moreno, and Gorden Videen, "Frequency shift between near- and far-field scattering resonances in dielectric particles," J. Opt. Soc. Am. A 32, 1638-1642 (2015)

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Abstract

The near-field electromagnetic scattering intensity resonances are redshifted in frequency with respect to their far-field counterparts. We derive simple, approximate, analytical formulas for this shift in the case of a plane wave interacting with a dielectric sphere. Numerical results comparing the approximate formulas to the numerically exact solutions show that the two are in good agreement. We also consider the Rayleigh limit of the formulas to gain more insight into the phenomenon.

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Equations (40)

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Symbol	Description
$j_{n}$	Spherical Bessel function of the 1st kind
$y_{n}$	Spherical Bessel function of the 2nd kind
$h_{n}$	Spherical Hankel function of the 1st kind
$ψ_{n} (η) = η j_{n} (η)$	Riccati–Bessel function
$χ_{n} (η) = - η y_{n} (η)$	Riccati–Bessel function
$η (η) = η h_{n} (η)$	Riccati–Bessel function
$P_{n}^{m}$	Associated Legendre polynomial with the Condon–Shortley phase
$c$	Speed of light in vacuum
$ϵ_{1}$	Permittivity of the host space (Region 1)
$ϵ_{2}$	Permittivity of the sphere (Region 2)
$k_{1} = \sqrt{ϵ_{1}} \frac{ω}{c}$	Wavenumber in Region 1
$k_{2} = \sqrt{ϵ_{2}} \frac{ω}{c}$	Wavenumber in Region 2

$n$	$ℓ$	$x_{ℓ}^{e}$	Rel. Error in %
1	1	0.44404161231126	$3.40 \times 10^{- 2}$
1	2	0.76111381942357	$9.14 \times 10^{- 1}$
1	3	1.0730776730363	4.45
1	4	1.3888710632302	5.07
1	5	1.7085740733775	2.12
1	6	2.0284063829955	$6.33 \times 10^{- 1}$
1	7	2.3472206484602	$1.66 \times 10^{- 1}$
1	8	2.6652685984771	$1.99 \times 10^{- 2}$
1	9	2.9830223110202	$3.88 \times 10^{- 2}$
2	10	0.57329872365611	$2.89 \times 10^{- 5}$
2	11	0.90414933943808	$2.59 \times 10^{- 3}$
2	12	1.2237952354462	$4.89 \times 10^{- 2}$
2	13	1.5386833689535	$3.99 \times 10^{- 1}$
2	14	1.8515469192830	1.61
2	15	2.1657001377980	3.07
2	16	2.4832719779298	2.73
2	17	2.8029458320520	1.41

$n$	$ℓ$	$x_{ℓ}^{m}$	Rel. Error in %
1	1	0.31130447515856	$7.94 \times 10^{- 3}$
1	2	0.62398570479241	$6.50 \times 10^{- 2}$
1	3	0.93834713028058	$1.54 \times 10^{- 1}$
1	4	1.2540335431079	$2.42 \times 10^{- 1}$
1	5	1.5707963267949	$3.53 \times 10^{- 1}$
1	6	1.8889337144859	$5.93 \times 10^{- 1}$
1	7	2.2102197037432	1.48
2	8	0.44775047977432	$2.55 \times 10^{- 5}$
2	9	0.76961156442652	$1.21 \times 10^{- 3}$
2	10	1.0862284580243	$1.09 \times 10^{- 2}$
2	11	1.4015575650554	$4.14 \times 10^{- 2}$
2	12	1.7168078189283	$9.43 \times 10^{- 2}$
2	13	2.0324277448812	$1.61 \times 10^{- 1}$
2	14	2.3485738549284	$2.44 \times 10^{- 1}$
2	15	2.6653903766980	$3.72 \times 10^{- 1}$
2	16	2.9832885136309	$6.52 \times 10^{- 1}$

Symbol	Description
$j_{n}$	Spherical Bessel function of the 1st kind
$y_{n}$	Spherical Bessel function of the 2nd kind
$h_{n}$	Spherical Hankel function of the 1st kind
$ψ_{n} (η) = η j_{n} (η)$	Riccati–Bessel function
$χ_{n} (η) = - η y_{n} (η)$	Riccati–Bessel function
$η (η) = η h_{n} (η)$	Riccati–Bessel function
$P_{n}^{m}$	Associated Legendre polynomial with the Condon–Shortley phase
$c$	Speed of light in vacuum
$ϵ_{1}$	Permittivity of the host space (Region 1)
$ϵ_{2}$	Permittivity of the sphere (Region 2)
$k_{1} = \sqrt{ϵ_{1}} \frac{ω}{c}$	Wavenumber in Region 1
$k_{2} = \sqrt{ϵ_{2}} \frac{ω}{c}$	Wavenumber in Region 2

$n$	$ℓ$	$x_{ℓ}^{e}$	Rel. Error in %
1	1	0.44404161231126	$3.40 \times 10^{- 2}$
1	2	0.76111381942357	$9.14 \times 10^{- 1}$
1	3	1.0730776730363	4.45
1	4	1.3888710632302	5.07
1	5	1.7085740733775	2.12
1	6	2.0284063829955	$6.33 \times 10^{- 1}$
1	7	2.3472206484602	$1.66 \times 10^{- 1}$
1	8	2.6652685984771	$1.99 \times 10^{- 2}$
1	9	2.9830223110202	$3.88 \times 10^{- 2}$
2	10	0.57329872365611	$2.89 \times 10^{- 5}$
2	11	0.90414933943808	$2.59 \times 10^{- 3}$
2	12	1.2237952354462	$4.89 \times 10^{- 2}$
2	13	1.5386833689535	$3.99 \times 10^{- 1}$
2	14	1.8515469192830	1.61
2	15	2.1657001377980	3.07
2	16	2.4832719779298	2.73
2	17	2.8029458320520	1.41

Abstract

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Figures (4)

Tables (3)

Equations (40)

Journal of the Optical Society of America A