## Optical response of a single spherical particle in a tightly focused light beam: application to the spatial modulation spectroscopy technique

JOSA A, Vol. 25, Issue 2, pp. 493-514 (2008)

http://dx.doi.org/10.1364/JOSAA.25.000493

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### Abstract

We develop a new and numerically efficient formalism to describe the general problem of the scattering and absorption of light by a spherical metal or dielectric particle illuminated by a tightly focused beam. The theory is based on (i) the generalized Mie theory equations, (ii) the plane-wave decomposition of the converging light beam, and (iii) the expansion of a plane wave in terms of vector spherical harmonics. The predictions of the model are illustrated in the case of silver nanoparticles. The results are compared with the Mie theory in the local approximation. Finally, some effects related to the convergence of the beam are analyzed in the context of experiments based on the spatial modulation spectroscopy technique.

© 2008 Optical Society of America

**OCIS Codes**

(180.0180) Microscopy : Microscopy

(260.0260) Physical optics : Physical optics

(290.0290) Scattering : Scattering

(290.2200) Scattering : Extinction

(290.4020) Scattering : Mie theory

(290.5850) Scattering : Scattering, particles

(300.0300) Spectroscopy : Spectroscopy

**ToC Category:**

Scattering

**History**

Original Manuscript: June 5, 2007

Revised Manuscript: September 20, 2007

Manuscript Accepted: September 25, 2007

Published: January 30, 2008

**Virtual Issues**

Vol. 3, Iss. 3 *Virtual Journal for Biomedical Optics*

**Citation**

Jean Lermé, Guillaume Bachelier, Pierre Billaud, Christophe Bonnet, Michel Broyer, Emmanuel Cottancin, Salem Marhaba, and Michel Pellarin, "Optical response of a single spherical particle in a tightly focused light beam: application to the spatial modulation spectroscopy technique," J. Opt. Soc. Am. A **25**, 493-514 (2008)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-25-2-493

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- All the calculations have been performed with a standard PC (Intel Pentium 4 processor with 3.2 GHz clock speed, computational code written in the C language). Typically the spectra or error parameter curves (as a function of λ) displayed in the manuscript involve roughly Nλ=280 wavelengths and a maximum order Nmax in the expansion of the fields equal to 15 [n=1,2,3,...,Nmax in Eqs. ]. Actually this large Nmax value is not necessary for particles of radius equal to or smaller than 100 nm. Besides the Lorenz-Mie coefficients [Eqs. ] the present theory requires the Bessel functions Jm(x=kρPsin(θ)) and Legendre polynomials Pnm(θ) to be computed for each θ value in the integral of Eq. . Typically 800-1200 θ values are retained, depending on the parameter kρP in the argument of the Bessel function Jm (and also on kzP in the exponential factor), though this quite large number is unnecessary in most cases of interest. Thanks to efficient recurrence relations obeyed by the Bessel functions and the Legendre polynomials , a spectrum is obtained within 3-4 s for an arbitrary particle location (ρP≠0; zP≠0). This time obviously depends on the Nλ, Nmax, and θ step values that are selected. This high speed is rooted in the simple mathematical ingredients (standard analytical functions) involved in the theory.

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