Poles, physical bounds, and optimal materials predicted with approximated Mie coefficients

Claire-Hélène Guidet; Claire-Hélène Guidet; Brian Stout; Redha Abdeddaim; Nicolas Bonod

doi:10.1364/JOSAB.417078

Journal of the Optical Society of America B
Vol. 38,
Issue 3,
pp. 979-989
(2021)
•https://doi.org/10.1364/JOSAB.417078

Poles, physical bounds, and optimal materials predicted with approximated Mie coefficients

Claire-Hélène Guidet, Brian Stout, Redha Abdeddaim, and Nicolas Bonod

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Claire-Hélène Guidet, Brian Stout, Redha Abdeddaim, and Nicolas Bonod, "Poles, physical bounds, and optimal materials predicted with approximated Mie coefficients," J. Opt. Soc. Am. B 38, 979-989 (2021)

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- Surface Optics
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About this Article
History
- Original Manuscript: December 7, 2020
- Revised Manuscript: January 18, 2021
- Manuscript Accepted: February 1, 2021
- Published: February 25, 2021

Abstract

Resonant electromagnetic scattering with particles is a fundamental problem in electromagnetism that has been thoroughly investigated through the excitation of localized surface plasmon resonances (LSPR) in metallic particles or Mie resonances in high refractive index dielectrics. The interaction strength between electromagnetic waves and scatterers is limited by maximum and minimum physical bounds. Predicting the material composition of a scatterer that will maximize or minimize this interaction is an important objective, but its analytical treatment is challenged by the complexity of the functions appearing in the multipolar Mie theory. Here, we combine different kinds of expansions adapted to the different functions appearing in Mie scattering coefficients to derive simple and accurate expressions of the scattering electric and magnetic Mie coefficients in the form of rational functions. We demonstrate the accuracy of these expressions for metallic and dielectric homogeneous particles before deriving the analytical expressions of the complex eigen-frequencies (poles) for both cases. Approximate Mie coefficients can be used to derive simple but accurate expressions for determining complex dielectric permittivities that lead to poles of the dipolar Mie coefficient and ideal absorption conditions. The same expressions also predict the real dielectric permittivities that maximize (unitary limit) or minimize (anapole) electromagnetic scattering.

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Dipolar Mie coefficient	$a_{1} (z_{0}) \approx \frac{- z_{0}^{3}}{3 (z_{0} + i) e^{i z_{0}} (1 + \frac{z_{0}^{2}}{10})} \frac{ϵ (2 - \frac{z_{0}^{2}}{5}) - 2 \frac{1 - {(\frac{z_{s}}{q_{1}})}^{2}}{1 - {(\frac{z_{s}}{r_{1}})}^{2}}}{ϵ (i z_{0} - \frac{1}{1 - i z_{0}}) - 2 \frac{1 - {(\frac{z_{s}}{q_{1}})}^{2}}{1 - {(\frac{z_{s}}{r_{1}})}^{2}}}$
Poles	$ϵ_{p, e}^{2} (- i \frac{z_{0}^{3}}{r_{1}^{2}} - \frac{z_{0}^{4}}{r_{1}^{2}} + \frac{z_{0}^{2}}{r_{1}^{2}}) + ϵ_{p, e} (i z_{0} + z_{0}^{2} - 1 + \frac{2 z_{0}^{2}}{r_{0}^{2}} - 2 i \frac{z_{0}^{3}}{r_{0}^{2}}) + (2 i z_{0} - 2) = 0$
Zero (anapole)	$ϵ_{z_{0}} ≃ \frac{- (2 - \frac{z_{0}^{2}}{5} + \frac{z_{0}^{2}}{r_{0}^{2}}) + \sqrt{Δ_{z}}}{2 (\frac{- 2 z_{0}^{2}}{r_{1}^{2}} + \frac{z_{0}^{4}}{r_{1}^{2}})}$
Unitary limit	$ϵ_{U L}^{a_{1}} ≃ \frac{- (φ_{1}^{2} (z_{0}) + \frac{2 z_{0}^{2}}{r_{0}^{2}}) \pm \sqrt{Δ_{UL}}}{- \frac{2 φ_{1}^{2} (z_{0}) z_{0}^{2}}{r_{1}^{2}}}$
Ideal absorption	$ϵ_{IA}^{e} ≃ \frac{- (- i z_{0} + z_{0}^{2} - 1 + \frac{2 z_{0}^{2}}{r_{0}^{2}} + \frac{2 i z_{0}^{3}}{r_{0}^{2}}) \pm \sqrt{Δ_{IA}}}{2 (\frac{i z_{0}^{3}}{r_{1}^{2}} - \frac{z_{0}^{4}}{r_{1}^{2}} + \frac{z_{0}^{2}}{r_{1}^{2}})}$

Mie coefficients	$b_{1} (z_{0}) ≃ \frac{- z_{0}^{3}}{3 (z_{0} + i) e^{i z_{0}} (1 + \frac{z_{0}^{2}}{10})} \frac{3 - \frac{z_{0}^{2}}{5} + r_{0} (z_{s} - r_{0})}{(i z_{0} - \frac{1}{1 - i z_{0}}) + 1 + r_{0} (z_{s} - r_{0})}$
Poles	$ϵ_{p, h} ≃ {(\frac{- z_{0}^{2} + r_{0}^{2} - i r_{0}^{2} z_{0}}{r_{0} z_{0} - i r_{0} z_{0}^{2}})}^{2}$
Unitary limit	$ϵ_{UL}^{b_{1}} ≃ {(\frac{r_{0}^{2} - φ_{1}^{2} (z_{0}) - 1}{r_{0} z_{0}})}^{2}$
Ideal absorption	$ϵ_{IA}^{h} ≃ \frac{{(- r_{0}^{2} + z_{0}^{2} - i r_{0}^{2} z_{0})}^{2}}{{(r_{0} z_{0} + i r_{0} z_{0}^{2})}^{2}}$

Dipolar Mie coefficient	$a_{1} (z_{0}) \approx \frac{- z_{0}^{3}}{3 (z_{0} + i) e^{i z_{0}} (1 + \frac{z_{0}^{2}}{10})} \frac{ϵ (2 - \frac{z_{0}^{2}}{5}) - 2 \frac{1 - {(\frac{z_{s}}{q_{1}})}^{2}}{1 - {(\frac{z_{s}}{r_{1}})}^{2}}}{ϵ (i z_{0} - \frac{1}{1 - i z_{0}}) - 2 \frac{1 - {(\frac{z_{s}}{q_{1}})}^{2}}{1 - {(\frac{z_{s}}{r_{1}})}^{2}}}$
Poles	$ϵ_{p, e}^{2} (- i \frac{z_{0}^{3}}{r_{1}^{2}} - \frac{z_{0}^{4}}{r_{1}^{2}} + \frac{z_{0}^{2}}{r_{1}^{2}}) + ϵ_{p, e} (i z_{0} + z_{0}^{2} - 1 + \frac{2 z_{0}^{2}}{r_{0}^{2}} - 2 i \frac{z_{0}^{3}}{r_{0}^{2}}) + (2 i z_{0} - 2) = 0$
Zero (anapole)	$ϵ_{z_{0}} ≃ \frac{- (2 - \frac{z_{0}^{2}}{5} + \frac{z_{0}^{2}}{r_{0}^{2}}) + \sqrt{Δ_{z}}}{2 (\frac{- 2 z_{0}^{2}}{r_{1}^{2}} + \frac{z_{0}^{4}}{r_{1}^{2}})}$
Unitary limit	$ϵ_{U L}^{a_{1}} ≃ \frac{- (φ_{1}^{2} (z_{0}) + \frac{2 z_{0}^{2}}{r_{0}^{2}}) \pm \sqrt{Δ_{UL}}}{- \frac{2 φ_{1}^{2} (z_{0}) z_{0}^{2}}{r_{1}^{2}}}$
Ideal absorption	$ϵ_{IA}^{e} ≃ \frac{- (- i z_{0} + z_{0}^{2} - 1 + \frac{2 z_{0}^{2}}{r_{0}^{2}} + \frac{2 i z_{0}^{3}}{r_{0}^{2}}) \pm \sqrt{Δ_{IA}}}{2 (\frac{i z_{0}^{3}}{r_{1}^{2}} - \frac{z_{0}^{4}}{r_{1}^{2}} + \frac{z_{0}^{2}}{r_{1}^{2}})}$

Mie coefficients	$b_{1} (z_{0}) ≃ \frac{- z_{0}^{3}}{3 (z_{0} + i) e^{i z_{0}} (1 + \frac{z_{0}^{2}}{10})} \frac{3 - \frac{z_{0}^{2}}{5} + r_{0} (z_{s} - r_{0})}{(i z_{0} - \frac{1}{1 - i z_{0}}) + 1 + r_{0} (z_{s} - r_{0})}$
Poles	$ϵ_{p, h} ≃ {(\frac{- z_{0}^{2} + r_{0}^{2} - i r_{0}^{2} z_{0}}{r_{0} z_{0} - i r_{0} z_{0}^{2}})}^{2}$
Unitary limit	$ϵ_{UL}^{b_{1}} ≃ {(\frac{r_{0}^{2} - φ_{1}^{2} (z_{0}) - 1}{r_{0} z_{0}})}^{2}$
Ideal absorption	$ϵ_{IA}^{h} ≃ \frac{{(- r_{0}^{2} + z_{0}^{2} - i r_{0}^{2} z_{0})}^{2}}{{(r_{0} z_{0} + i r_{0} z_{0}^{2})}^{2}}$

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Journal of the Optical Society of America B