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Two-dimensional arrays of particles are of great interest because of their very characteristic optical properties and numerous potential applications. Although a variety of theoretical approaches are available for the description of their properties, methods that are accurate and convenient for computational procedures are always sought. In this work, a new technique to study the diffraction of a monochromatic electromagnetic field by a two-dimensional lattice of spheres is presented. The method, based on Fourier series, can take into account an arbitrary number of terms in the multipole expansion of the field scattered by each sphere. This method has the advantage of leading to simple formulas that can be readily programmed and used as a powerful tool for nanostructure characterization.

The selective absorption in the visible spectral region observed in granular noble-metal films or nanoparticles has been the subject of numerous investigations in the past [

To characterize the two-dimensional distribution of nanoparticles on a substrate, it was frequent to use an effective-medium theory that approximated the granular films to an effective film having an effective optical thickness and effective optical properties [

In a previous work, we introduced a new way to solve the problem of particle arrays consistently and efficiently, when only the electric dipole radiation of the particles was considered [

In the following, we will make use of the basis of vector harmonics [

The problem at hand is the diffraction of a monochromatic electromagnetic field of angular frequency

We will call

In view of (

Finally, we have for the scattered field the following:

There is a fundamental difficulty involved in the approach just described: although the series in (

Our approach distinguishes itself from the traditional scheme in two fashions: first, we will use Fourier series, instead of (

The calculation of the Fourier series of

Let us call

Our task is to compute the scattered field in terms of the multipole moments of

Next, we develop the vector potential and the scattered field in Fourier series, with respective Fourier coefficients

The last step consists in expressing (

Using a point source, like we just did, is not suitable to compute

We will now show how to compute the multipole coefficients

Assuming that

The task is a bit more complicated for

In the following, we will show a few examples of calculations for illustration purpose. In particular, we would like to compare calculations based uniquely on the electric dipole terms and those obtained by including multipolar terms for better accuracy. The spherical particles are distributed in a two-dimensional hexagonal array as described previously and the metal composing the particles is gold. As particle deposits are usually done on glass, the optical refractive index supposed in the wavelength range considered (from 400 nm to 800 nm) is 1.5. The optical constants for gold have been determined by ellipsometric studies using thin films of this metal (we acknowledge with thanks Professor Georges Bader of Université de Moncton for kindly providing us with the gold optical constants).

From the calculations performed with small particles, such as those with a diameter smaller than 50 nm, it is observed that the dipolar approximation with retardation effects is quite similar to more accurate calculations using higher multipolar terms. Figure

Reflectance of a two-dimensional array of 50 nm gold nanoparticles deposited on glass separated by an interparticle distance of 60 nm. Solid line: dipolar approximation, dotted line: multipolar calculations.

For larger particles, the difference between the dipolar and multipolar calculations becomes considerable. Figure

Reflectance of a two-dimensional array of 200 nm gold nanoparticles deposited on glass separated by an inter-particle distance of 400 nm. Solid line: dipolar approximation, dotted line: multipolar calculations.

Reflectance of a two-dimensional array of 200 nm gold nanoparticles deposited on glass separated by an inter-particle distance of 300 nm. Solid line: dipolar approximation, dotted line: multipolar calculations.

It should be noted that calculations done for other metals such as silver and copper will give similar qualitative features.

The formulas we have obtained are relatively simple, considering the complexity of the problem encountered. When the spheres are suspended in vacuo, all that remains to do, to completely solve the problem, is to calculate explicitly the Fourier coefficients of the scattered field. This task might be a little tedious, but the results will involve nothing more elaborate than the complementary error function. From a numerical perspective, this means that in terms of special functions, the only procedures needed are those involving the computation of the erfc(

As for the case where a substrate has to be taken into account, a technique analogous to the one introduced in [

When solving Maxwell's equations, the electrical field

Financial support from Faculté des Études Supérieures at Université de Moncton and the Natural Science and Engineering Research Council of Canada (NSERC) is gratefully acknowledged.

_{2}cermet films: a comparison of effective-medium theories