J. Phys.: Condens. Matter l(1989) 8035-8044. Printed in the UK
Optical non-linear response of small metal particles:
a complete self-consistent calculation including exchange and
correlation
F Hache and D Ricard
Laboratoire d’Optique Quantique du CNRS, Ecole Polytechnique, 91128 Palaiseau
CCdex. France
Received 10 February 1989, in final form 12 May 1989
Abstract. We report a fully self-consistent calculation of the third-order Kerr polarisation in
small metal spheres embedded in a dielectric medium. The random phase approximation
with exchange is used leading to a fully symmetric expression accounting for the non-locality
of the electronic response as well as exchange and correlation. This treatment generalises
the results reported in a previous paper by Hache et al. The expression of the octupolar
component of the charge density is also given.
1. Introduction
The linear optical properties of small metal particles have been extensively studied [ 11.
The connection between the absorption spectrum of a metal colloid and the surface
plasma resonance [2-41, as well as the broadening of the absorption band when the size
of the particles is reduced [5-71 have given rise to a large number of publications.
The calculations were first performed assuming independent electrons in the local
approximation [8-101 but later papers gave improved treatments taking into account
the non-local character of the electronic response [ll-131. A more general situation,
including exchange and correlation effects in the calculation of the dynamical polaris-
ability, has also been considered [14].
On the other hand, and more recently, we have performed the first experimental
studies of non-linear optical properties (the optical Kerr effect) of gold and silver
colloids [15-171 using optical phase conjugation in the degenerate four-wave mixing
configuration. A resonant enhancement of the non-linearity was observed in the vicinity
of the surface plasma resonance. The enhancement was first interpreted as a local field
effect [ 151, the local field correction being derived from an effective dielectric constant
formalism [3] from which the four factors fi(o), wherefl is the ratio between the field
inside and outside the sphere, usually entering the expression of the third-order Kerr
non-linear source polarisation PNLs (w) are easily recovered. Of course, such a treatment
was obtained assuming a local response of the electrons, the metal sphere being charac-
terised by a dielectric constant ~~(0).
We have recently published [ 181 an improved treatment taking into account the non-
locality of the electronic res onse, at least in its most important aspects, obtaining a self-
(3)
consistent expression for PNLs K (U). In the present paper, we will refer to [18] as paper I.
0953-8984/89/438035 + 10 $02.50 @ 1989 IOP Publishing Ltd 8035