^{1}and Cecilia Noguez

^{1,a)}

### Abstract

A spectral representation formalism in the quasistatic limit is developed to study the optical response of nanoparticles, such as nanospheres, nanospheroids, and concentric nanoshells. A transfer matrix theory is formulated for systems with an arbitrary number of shells. The spectral representation formalism allows us to analyze the optical response in terms of the interacting surface plasmons excited at the interfaces by separating the contributions of the geometry from those of the dielectric properties of each shell and surroundings. Neither numerical nor analytical methods can do this separation. These insights into the physical origin of the optical response of multishelled nanoparticles are very useful for engineering systems with desired properties for applications in different fields ranging from materials science and electronics to medicine and biochemistry.

This research was supported by UNAM (PAPIIT IN106408), Air Force Office of Scientific Research (AFOSR) (SOARD FA9550-09-1-0579), and a grant from the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Award No. DE-SC0004752.

I. INTRODUCTION

II. SPECTRAL REPRESENTATION FORMALISM

A. Spectral representation of the polarizability

B. Spectral variable

III. SPECTRAL REPRESENTATION OF NANOSPHEROIDAL PARTICLES

A. Nanospheres

B. Nanospheroids

C. High-multipolar SPRs

D. Nanocavity

IV. SPECTRAL REPRESENTATION OF SHELL NANOPARTICLES

A. Effects of geometry

B. Effects of the dielectric properties

C. Multishell nanoparticles

V. SUMMARY AND CONCLUSIONS

### Key Topics

- Nanoparticles
- 61.0
- Eigenvalues
- 54.0
- Dielectric properties
- 28.0
- Gold
- 28.0
- External field
- 23.0

## Figures

(a) Real (solid line) and imaginary (dotted line) parts of the spectral variable of nanospheres, *s*(ω), for silver in air and gold in water with *a* = 5 nm. The horizontal line at 1/3 shows the eigenvalue, *s*(*n*). (b) Corresponding absorption cross sections. Vertical arrows show the location of the resonances.

(a) Real (solid line) and imaginary (dotted line) parts of the spectral variable of nanospheres, *s*(ω), for silver in air and gold in water with *a* = 5 nm. The horizontal line at 1/3 shows the eigenvalue, *s*(*n*). (b) Corresponding absorption cross sections. Vertical arrows show the location of the resonances.

(a) Real (solid line) and imaginary (dotted line) parts of the spectral variable, *s*(ω), for gold nanospheres with *a* = 15 nm in different hosts with refraction indices , 1.33, 1.47, 1.77, and 2.79. The horizontal line at 1/3 shows the eigenvalue, *s*(*n*). (b) Corresponding absorption cross sections. Vertical arrows show the location of the resonances.

(a) Real (solid line) and imaginary (dotted line) parts of the spectral variable, *s*(ω), for gold nanospheres with *a* = 15 nm in different hosts with refraction indices , 1.33, 1.47, 1.77, and 2.79. The horizontal line at 1/3 shows the eigenvalue, *s*(*n*). (b) Corresponding absorption cross sections. Vertical arrows show the location of the resonances.

(a) Real (solid line) and imaginary (dotted line) parts of *s*(ω) for silver nanospheroids in water. Nanoprolates have equal volume, but different aspect ratios. Horizontal lines show the eigenvalues for the longitudinal modes for aspect ratios, *a*/*c*, varying from 1 to 5. Vertical arrows indicate the position of the resonances in the absorption cross sections shown in (b).

(a) Real (solid line) and imaginary (dotted line) parts of *s*(ω) for silver nanospheroids in water. Nanoprolates have equal volume, but different aspect ratios. Horizontal lines show the eigenvalues for the longitudinal modes for aspect ratios, *a*/*c*, varying from 1 to 5. Vertical arrows indicate the position of the resonances in the absorption cross sections shown in (b).

(a) Real (solid line) and imaginary (dotted line) parts of *s*(ω) for gold nanospheroids in water. Nanoprolates have equal volume, but different aspect ratios. Horizontal lines show the eigenvalues for the longitudinal modes for aspect ratios, *a*/*c*, varying from 1 to 5. Vertical arrows indicate the position of the resonances in the absorption cross sections shown in (b).

(a) Real (solid line) and imaginary (dotted line) parts of *s*(ω) for gold nanospheroids in water. Nanoprolates have equal volume, but different aspect ratios. Horizontal lines show the eigenvalues for the longitudinal modes for aspect ratios, *a*/*c*, varying from 1 to 5. Vertical arrows indicate the position of the resonances in the absorption cross sections shown in (b).

Model of the cross section of a spherical nanoshell.

Model of the cross section of a spherical nanoshell.

(a) Eigenvalues and (b) weights of the two SPRs, and , corresponding to multipolar distributions with *l* = 1, 2, 3, 4, 5, and 100 for a single nanoshell as a function of . The weights are normalized by the factor for each *l*.

(a) Eigenvalues and (b) weights of the two SPRs, and , corresponding to multipolar distributions with *l* = 1, 2, 3, 4, 5, and 100 for a single nanoshell as a function of . The weights are normalized by the factor for each *l*.

(a) Real (solid line) and imaginary (dotted line) parts of the spectral variable, *s*(ω), for silver nanoshells in water with equal outer radius nm and inner radii with (red line) and 0.8 (green line). The horizontal solid lines in the right-hand side represent the eigenvalues and weights for both nanoshells. We continue the horizontal lines with dotted lines to use as eye guides. (b) Corresponding absorption cross sections. The inset shows a zoom of the resonances at low wavelengths.

(a) Real (solid line) and imaginary (dotted line) parts of the spectral variable, *s*(ω), for silver nanoshells in water with equal outer radius nm and inner radii with (red line) and 0.8 (green line). The horizontal solid lines in the right-hand side represent the eigenvalues and weights for both nanoshells. We continue the horizontal lines with dotted lines to use as eye guides. (b) Corresponding absorption cross sections. The inset shows a zoom of the resonances at low wavelengths.

(a) The same as Fig. 7 but for gold nanoshells in water with , 0.4, 0.6, and 0.8. (b) Corresponding absorption cross sections.

(a) The same as Fig. 7 but for gold nanoshells in water with , 0.4, 0.6, and 0.8. (b) Corresponding absorption cross sections.

(a) Real (solid line) and imaginary (dotted line) parts of the spectral variable, *s*(ω), for the silver–water system shown in the model. The horizontal solid lines show the eigenvalues and their weights for *N* = 3 with nm, , and when (green line) or (red line). (b) Corresponding absorption cross sections.

(a) Real (solid line) and imaginary (dotted line) parts of the spectral variable, *s*(ω), for the silver–water system shown in the model. The horizontal solid lines show the eigenvalues and their weights for *N* = 3 with nm, , and when (green line) or (red line). (b) Corresponding absorption cross sections.

Schematic model of a concentric multishell particle.

Schematic model of a concentric multishell particle.

## Tables

Eigenvalues of the longitudinal and transversal SPRs of prolates nanospheroids of different aspect ratios, *a*/*c*.

Eigenvalues of the longitudinal and transversal SPRs of prolates nanospheroids of different aspect ratios, *a*/*c*.

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