1. Introduction

Many researchers have focused their attention on the speckle phenomenon since long ago [1,2]. From a pure theoretical point of view, a typical speckle pattern appears when a very large number of “effective” scattering centers contribute coherently to the scattered intensity at a given point. In this case, the total scattered field behaves as a circular complex Gaussian random variable and the statistical fluctuations of the intensity follow an exponential law whose moment of order m is given by < I^m > / < I >^m = m!. When this happens, the spatial fluctuations of the scattered light are commonly referred to as “Gaussian speckle” [1]. However, when the “effective” number of scattering centers is very low, the statistics of the scattered radiation presents clear differences with respect to the Gaussian case and it is common to find in the literature the terms “non-Gaussian” speckle or “non-Gaussian” statistics [3,4]. The non-Gaussian regime has been extensively analyzed in the past, especially when there is no multiple scattering and the total scattered field is calculated by the coherent addition of the fields scattered by the isolated “effective” scattering centers. In most of these cases, theoretical models were developed and experiments were carried out for scattering systems like random phase screens [5–9]. More recently, a lot of research has been done about wave propagation through random media when multiple scattering is important. The statistical analysis of speckle fluctuations produced when light propagates through disordered media has constituted an important part of that research. It includes “Gaussian” and “non-Gaussian” regimes for scattering systems like isolated particles [10], bulk diffusers [11–14], surface corrugated waveguides [15,16], rough surfaces [17] and surfaces contaminated with particles [18,19].

Surfaces with particles constitute an interesting electromagnetic problem [20] for both basic (as a model to solve more complicated surfaces) and applied research (for instance, for monitoring surface contamination in optical and microelectronic components). Several authors have analyzed theoretically and experimentally the intensity and polarization fluctuations of such a system but have not considered the possible multiple scattering effects between the particles [18,19]. From the applied research point of view, the purpose of this paper is to show how to obtain information about the surface density of scattering particles by means of a theoretical and experimental analysis of the statistics of the fluctuations of the scattered intensity. The system considered here is a surface contaminated with small particles, multiple scattering effects being included in the calculations. As a consequence of these effects, different polarizations appear in the light scattered within the plane of incidence showing a non-zero component in the direction perpendicular to that of the incident one (it is the so called cross-polarized component). For both, Gaussian and non-Gaussian regimes, we study the intensity fluctuations of both the co- and the cross-polarized components of the scattered light from this kind of surfaces. We focus on the cross-polarized component because it is more sensitive to multiple scattering, and therefore, it is easier to obtain information about particle surface density. In particular, we introduce a method to determine this density based on the experimental measurement of the probability of detecting zeros in the cross-polarized scattered intensity, P(I_cross=0).

2. Theoretical model

We have developed a very simple model for obtaining an analytical expression for P(I_cross=0) as a function of both particle surface density (which we consider constant) and the illuminated area. The geometry of the problem consists on a flat surface that separates two semi-infinite media of dielectric constants ε₁ and ε₂. The substrate is seeded with particles of spherical symmetry. The particle distribution obeys a Poisson distribution where the particle density, ρ, is constant over the substrate. The surface is illuminated by a top-hat profile beam (only for simplicity in the calculation process) which produces either a circular illumination area of radius w for normal incidence or an elliptical one of semiaxes a and b for other incidences. It is assumed that the detection system has a threshold intensity I₀, which is the minimum scattered intensity it can detect (this is in fact the experimental case) and it can be considered zero for calculation purposes. Our theoretical model is based on the previous assumptions and the following hypothesis: If a particle scatters a crosspolarized intensity I₀ (or less) when its nearest neighbor is at a distance L, the probability of detecting zero for the scattered cross-polarized intensity is the probability of having the nearest neighbor at a distance l larger than L and is therefore given by

 

Fig. 1. Experimental set-up. The output polarization of the laser is perpendicular to the plane of the drawing (S polarization).

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where d is the mean distance between particles, d = ρ ^-1/2. Therefore, the probability of detecting zero cross-polarized scattered intensity when the mean number of illuminated particles is N̅ (= either πw² ρ, or πabρ) is

For normal incidence, a=b=w. As we will see in the following, from the measurement of P(I_cross=0) the parameter γ can be determined and it can be used to estimate the parameter d if L is known and vice-versa.

3. Experiment

In order to verify the proposed model we have used the experimental set-up shown in Fig. 1. This consists on a heterodyne detection system with a Mach-Zender configuration [21]. The light source, a polarized He-Ne laser, is incident on a beam splitter (BS1). The transmitted beam acts as a local oscillator (LO) and the reflected one, which acts as the reference beam (REF), illuminates the sample at an incident angle of a few degrees in order to be sure that the scattered light comes from the first copolarized backscattered lobe close to its maximum. The REF beam goes through an acousto-optic modulator (AOM) which shifts its frequency by 80 MHz. This beam is focused on the sample by means of a lens whose focal length is varied in order to change the size of the illuminating spot on the sample. The local oscillator interferes with the component of the backscattered light from the sample with parallel polarization to that of the local oscillator. A λ/2 plate is introduced in its path so that either the co- or the cross-polarized component of the scattered intensity can be detected. The sample is rotating at 1 revolution per second producing a change in the effective target, and therefore intensity fluctuations. The detected signal is sent to a spectrum analyzer where the interference term is selected and translated to the temporal domain. The analog signal is digitalized by an oscilloscope and processed and stored in a personal computer.

The samples consist on PMMA spherical particles (1.1 μm in diameter) deposited on a flat substrate (usually a microscope slide). The samples are gold coated by using a typical sputtering process. The manufacturing and characterization details can be found in Ref [22]. In Fig. 2 (a) we show the probability density function (PDF) of the copolarized backscattered intensity obtained for a sample with d≅7.7 μm (as obtained form electron microscope photographs) when the illuminated area has a width w≅7.4 μm. As the number of illuminated particles is quite small, deviations appear from the negative exponential law predicted for the Gaussian regime. It has been shown that, for independent scatterers, those deviations can serve to obtain the number of scattering particles in the illuminated area [4] (and then the particle surface density) by means of the measurement of the second moment of the PDF of the intensity fluctuations. When the scatterers are not independent (i.e. there is multiple scattering) it is interesting to pay attention to the PDF of the fluctuations of the cross-polarized backscattered intensity, that is shown in Fig. 2 (b). Its temporal evolution is quite different to the one observed for the co-polarized component, with many values falling to zero or close to zero and with a few very strong peaks that reach very large values compared to the mean scattered intensity. In order to extract the required information, we propose the measurement of particular values of the PDF, like P(I_cross=0). At that point the function can be determined with the smallest relative error. For spherical scatterers this parameter can be understood as a measurement of particle interaction (it decreases as interaction increases and vice-versa). The evolution of the measured values of P(I_cross=0) with the radius of the illuminated area, w, is shown in Fig.3 for six different illuminating spot sizes (w≅ 7.4, 11, 15, 16.5, 17, 26.6 μm). The circles represent the experimental data, with their corresponding error bars. The fit to Eq.(2) is plotted as a continuous line. As can be seen, the agreement is quite good.

 

Fig. 2. a) Experimental PDF of the copolarized backscattered intensity. The straight line would correspond to the pure Gaussian regime. b) Experimental PDF of the crosspolarized backscattered intensity.

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Fig. 3. Experimental values of P(I_cross=0) (marked with dots) and their corresponding experimental errors, as a function of the size of the illuminated area w (see text for details). The continuous line corresponds to the fitted function given by Eq. 2.

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From this fitting, a value for γ can be obtained (γ= 8.6 10^-3 μm^-2), so we can obtain d if L is known and vice-versa.. In our case, we can calculate the mean interparticle distance by two different ways: 1) From the photographs taken with the electronic microscope, d is 7.7 μm approximately. This corresponds to a value of L≅1.25 μm. 2) From the second normalized moment of the fluctuations of the copolarized scattered intensity [4], neglecting multiple scattering

and from the experimental result shown in Fig 2a, < I ² > / < I >² = 2.43 ± 0.11. Then d≅8.6 μm, and L ≅1.74 μm. Both values of L are larger than the particle diameter, so they are consistent with the physical meaning of L. Furthermore, they agree with the experimental work presented by González et al. [23] for the same kind of samples. In this work, the authors show that the amount of cross-polarized scattered light decreases sharply when the mean interparticle separation is larger than values similar to those obtained here for L.

Once the illumination and detection conditions are fixed (including the detection threshold level), the parameter L only depends on the characteristics of the particles (size, shape and optical properties) and flatness and optical properties of the substrate. This means that the value calculated from one sample can be used in Eq.(2) to determine the particle surface density of similar samples contaminated by the same kind of particles.

4. Conclusions

We have made an analysis of the statistics of the fluctuations of the scattered light from flat substrates contaminated with particles including multiple scattering effects. A simple model for obtaining an analytical expression of the probability density of obtaining zeros in the cross-polarized component, P(I_cross=0), has been introduced. It has been shown that this function decreases exponentially, with a factor that depends on i) the area of the illuminating spot, ii) the particle surface density and iii) a parameter we have denoted by L. This can be physically interpreted as an effective minimum interparticle distance necessary to observe non-zero cross-polarized intensity, and therefore multiple scattering. An experiment based on a heterodyne detection system has been performed in order to confirm the proposed model. From the discussion of the experimental results we can conclude that the parameter L/d³ can be assessed from the experimental measurement of P(I_cross=0). This can be used either for determining particle surface density of samples, all with the same individual scatterer characteristics (same L) or for determining the interacting level of different samples with well known particle surface density.