## Abstract

This paper reports experimental studies on speckles produced by the rough silver films. The speckles on the rough glass/silver surfaces are measured with a microscopic imaging system. The structures of speckle patterns have the characteristics of fractals and multi-scaled sizes. We find that with the increase of the silver film thickness, the contrast of the speckles increases, and the intensity probability density functions gradually transit to exponential decay. We calculate the global and the local correlation functions of the speckle patterns, and find that both the fractal exponent and correlation length of the small-sized speckles decrease with the thickness of the silver films. We use the mechanisms of rough dielectric interface scattering and random surface plasmon waves to give the preliminary explanations for the evolutions of the speckles.

© 2013 OSA

## 1. Introduction

Speckles are produced by the interference of light waves scattered from random media or surfaces. They are important phenomena in the optics and have found wide applications in many fields [1–3]. When the scattering objects induce random optical paths larger than a wavelength, the speckles in far-field are usually Gaussian processes and their autocorrelation functions depend on illumination aperture [4]; but speckles produced by a few scattering grains or weak scattering objects are non-Gaussian processes [5] with their intensity probability density function deviating from the negative exponential decay. Recently, the researches on speckles in deep Fresnel region and the three dimensional speckles have attracted great interest [6–8]. The distinctive properties of the speckles in these cases are that their autocorrelation functions are determined by the surface parameters or by the scattering particle sizes, which enable important applications of characterizations of scattering media by the measurement of the correlation function of speckles.

When the light waves are scattered from rough metal films deposited on random dielectric interfaces, the speckle fields involve the mechanism of the surface plasmon polaritons (SPPs) which have been one of the most animated areas in recent years [9]. At present the manipulations of the light waves by micro-device engineering [10–12] have become the most interesting aspect of surface plasmon studies. As one of the common means for surface plasmon coupling, the scattering of metal interfaces has been studied for a long time in the literature. Typical examples of the studies include the far-field scattering intensity profiles [13], the near-field speckle correlations [14] and surface enhanced Raman scattering [15].

In this paper, we report the experimental study on the speckles produced by the rough silver films in the extreme case of the deep Fresnel diffraction region [16]. The speckles on the rough glass/silver surfaces are measured with a microscopic imaging system composed of microscopic objective, lens and CCD. The structures of speckle patterns are very different from those in the traditional optical systems. The speckles have structures of obvious fractal characteristics and multi-scale sizes. By calculation with the speckle intensity data, we find that with the increase of the silver film thickness, the contrast of the speckles increases, and the intensity probability density functions gradually transit to exponential decay, i.e. the speckles approach Gaussian distribution. We calculate the global and the local correlation functions of the speckle patterns, and find that both the fractal exponent and small-scale size of the speckles decrease with the thickness of the silver films. We use the mechanisms of rough dielectric interface scattering and random surface plasmon waves to give explanations for the evolutions of the speckles.

## 2. Experiments for surface plasmon speckle acquisitions

#### 2.1 *Experimental setup*

Figure 1 schematically shows the experimental system for acquiring the speckles on the surface of the rough silver films. A He-Ne laser with the wavelength $\lambda =0.6328\mu \text{m}$ and the power $30\text{mW}$is used as the light source. Two polarizers A1 and A2 are used to adjust the light intensity with A2 keeping the polarization of the incident wave unchanged, and the pinhole with diameter about 3mm is used to filter strayed light. The sample of ground glass coated with silver film is placed on a three-dimensional piezo nanometer stage (PI E516) for accurate positioning. A microscopic objective MO (Nikon, Dry, 100X, N.A.0.9, WD 1mm) is used for collecting and imaging the speckles on the surface of sample. Lens L1 is used for convenient adjustment of the image magnification. The image is received by a CCD (Roper, Cascade 1k) with 16bit dynamic range and an array of 1004 × 1002 pixels (pixel size 8$\mu \text{m}\times \text{8}\mu \text{m}$).

Since the system has a very small object distance (about work distance of MO 1mm) and a comparatively large image distance (several hundred mm), it is difficult for the position of the image plane to be exactly determined in the case of laser illumination. In the optical adjustment, we first use a white light source with optical fiber output to illuminate the sample, and move L1 and the CCD in direction of optical axis for the rough determination of the image plane and the magnification. We use the nanometer stage to adjust the object distance repeatedly until the image becomes the clearest. Then we remove the white light source and enable the laser beam illuminate the sample for the speckle pattern acquisition. Due to the high ability of collecting scattered light and the good resolution of the MO, we take recorded speckle patterns as the speckles on the sample surface.

#### 2.2. The sample fabrication and the fractal description of rough surfaces

We first make the ground glass screens by grinding a holographic plate glass of 2mm thickness with 320-mesh silicon carbide powder. One of them is measured by an atomic force microscope (AFM) (PARK, Auto probe CP), and one of the images is shown in Fig. 2(a). The scanning area is 80$\mu \text{m}\times \text{80}\mu \text{m}$and the datapoints are $\text{256}\times \text{256}$. It has been demonstrated that random self-affine fractal surface model [17] can be well used to describe the ground glass screens [18]. In this model, the morphological characteristics of rough surface are described by the height-height correlation function proposed by S. K. Sinha et al [19]:

This shows that in the level $\rho <<\xi $, $H\text{(}\rho \text{)}$ varies in power-law and the $\text{ln}H\text{(}\rho \text{)~}\rho $ curve is straightline. Though the strict mathematical requirement of $\rho <<\xi $ in the above Equation, curves of $\text{ln}H\text{(}\rho \text{)~}\rho $ for practical samples appear to be well linear in the level $\rho <\xi $, and in experimental practices, the fractal level or the pre-fractal of the random surfaces [20] is roughly taken as the lateral correlation length, or the average grain sizes of the screen [17].

From the AFM image in Fig. 2(a), we may see that sample surface is distributed with larger valleys and mounds, which constitutes the so-called scattering grains of the rough surface. On these valleys and mounds, there are smaller grains and fluctuations with different size scales, and they are the obvious characteristics of random fractals. With the data of the AFM images in Fig. 2(a), we calculate the height-height correlation function and fit it with that of random self-affine fractal surface model defined and given in Eq. (1). Then the parameters obtained from the fit are $w=\text{1}\text{.243}\mu \text{m}$, $\xi =\text{9}\text{.53}\pm \text{0}\text{.3375}\mu \text{m}$, $\alpha =\text{0}\text{.51477}\pm \text{0}\text{.0071}$, and ${D}_{f}=\text{2}\text{.543}$. In the following context, the ground glass sample is labeled as sample No.1.

The identical ground glass samples are carefully cleaned and silver films are deposited on their rough side by the magnetron sputtering. The thicknesses of the silver films are 20nm, 40nm, 60nm, 80nm, and 100nm, respectively, and the corresponding samples are labeled as No. 2, No. 3, No. 4, No. 5, and No. 6. Since the maximum thickness of the silver films is 100nm, which is much smaller than the fluctuations of the ground, we may neglect the influence of the silver films on the surface morphology.

The speckle patterns on the surfaces of different samples are shown in the Figs. 3(a)- 3(f) with an imaging area $95\mu \text{m}\times \text{9}5\mu \text{m}$. These patterns have a smaller magnification and larger view for a better overall observation of the speckle characteristics. For more accurate statistical analysis, we also take 5 speckle patterns with larger magnification for each sample at different positions. Each of Figs. 4(a)- 4(f) gives one of the magnified patterns for samples No. 1-No. 6, respectively. The area of the patterns is $\text{4}5\mu \text{m}\times \text{4}5\mu \text{m}$. Here the patterns are not a corresponding part of those given in Figs. 3(a)- 3(f), for the optical system and the sample positions have been changed during the adjustment of the magnification. Since the measured speckle patterns are imaged by the MO, relatively abundant evanescent waves existing in the near-field cannot be received. So the surface speckles measured in this work exclude these waves. In addition, what should be noticed is that the scale ranges of the speckles may reach as small as the order of nanometers in the Figs. 3 and Figs. 4, and speckle grains are much smaller than the ones in the ordinary optical systems.

## 3. Analysis of speckle evolution with thickness of silver film

#### 3.1 The basic characteristics of the speckles on the sample surfaces

By comparing the speckle patterns in Figs. 3 and Figs. 4, we may find that the speckles produced by rough silver film samples evolve on the basis of those by rough ground glass sample. Here we first notice that the structures of the speckle patterns in Figs. 3(a) and 4(a) produced by the ground glass sample, i.e., sample No.1, are greatly different from the traditional far field speckles. These structures mainly consist of small-sized speckle grains but also include large-sized speckles of different scales. The large-sized speckles may be divided into platform-like speckle grains, random ridge stripes and grains shivy speckles which are labeled by the red, green and white rectangular boxes in Fig. 3(a), respectively. Inside the large platform-like grains, there are small grains which are arranged densely in somewhat order and even distribution. With the increase of the silver film thickness, the borders of the platform-like grains become blurry, and the random ridges and stripes become shorter and unclear. This may be easily seen in the speckle patterns of the20nm and the 40nm samples in the Figs. 3(b) and 3(c), and these two patterns bear obvious traces of that of the ground glass sample. The patterns in the Figs. 3(d) and 3(e) for the 60nm and samples 80nm samples show that with further increase of the film thickness, the platform-like structures disappear, the number of the random ridge stripes decreases, and their lengths become much shorter. Furthermore, the ridge stripes become modulated by the speckle grains and they change into chains of speckle grains. Meanwhile, large-scaled speckles appear in the form of intensity variations. When the film thickness is 100nm, the speckle pattern in the Fig. 3(f) consists of small-sized speckle grains on large-scaled speckles (or intensity variations). All the patterns from Figs. 3(a) to 3(f) show that the small-sized speckle grains become gradually smaller, minimized to about 0.4$\mu \text{m}$for the 100nm sample, and a little bit larger than the limit of half wavelength.

In Fig. 5, we give a diagrammatic sketch for the formation of speckles on the rough dielectric-metal surfaces. There exist two mechanisms. The first one is the light scattering from the ground glass surface as shown in Fig. 5(a). The observation plane for speckles on the surface can be regarded as the one determined by the largest height of rough surface, as indicated by dash line in the Fig. 5(a). The speckle intensity at point *P* is the superposition of the waves scattered from a small area on rough surface neighboring to point *P*. The incident light illuminating the neighboring small area is indicated by three parallel shorter blue arrows, and the three smallest blue arrows on the right side of the surface indicate the scattered waves contributing to intensity at *P*. The pink arrows indicate the waves scattered to other points on the observation plane. According to the diffraction theory of Kirchhoff approximation, the scattered wave field for the ground glass sample is expressed in the form of Green’s integral:

*m*the reflective index of the sample, $B=1-{\text{(}\partial h/\partial {x}_{1}\text{)}}^{2}-{\text{(}\partial h/\partial {y}_{1}\text{)}}^{2}$,$M=\text{(}x-{x}_{1}\text{)}\partial h/\partial {x}_{1}+\text{(}y-{y}_{1}\text{)}\partial h/\partial {y}_{1}+B\text{(}h-z\text{)}$$A=\sqrt{1+{\text{(}\partial h/\partial {x}_{1}\text{)}}^{2}+{\text{(}\partial h/\partial {y}_{1}\text{)}}^{2}}$.

Though given in the above forms of the superposition of the diffracted waves, $U\text{(}x,y\text{)}$ can be considered as the overall results of both the diffraction and the geometrical effects as reflection, refraction, and even total reflection occurring at the interface. Since the observation point is very close to the surface, the diffraction is mainly induced by surface height fluctuations in the lateral scale smaller than a wavelength as describe by surface fractality, while the geometrical effects are mainly produced by height fluctuations of the larger lateral scale as described by the lateral correlation length and the roughness of the surface.

The algorithm for the calculation of speckle intensity produced by the ground glass sample is the direct numerical calculation of the integral of Eq. (3), in which the data of the AFM image in Fig. 2(a) used surface height $h\text{(}{x}_{0},{y}_{0}\text{)}$. The calculated speckle intensity pattern is shown in Fig. 2(b). Comparing the structures of speckle pattern with the AFM image, we may deduce that the ridge stripes in the speckle pattern are caused by the refraction of surface grain slopes, and the platform-like structures correspond to the valleys on the surface, indicating they are caused by the refraction of the valleys. Here the accuracy algorithm based on Kirchhoff’s approximation may not influence much on our qualitative comparison of the speckle intensity pattern with the corresponding surface image. Other algorithms based on more accurate theories such as Rayleigh-Sommerfeld diffraction may also used for the calculation [20].

The other mechanism for speckle formation is the excitation and scattering of surface plasmon polarizations in silver films, as shown in Fig. 5(b). When the light wave is scattered at glass/silver interface, some components of the scattered waves with special wave vectors may satisfy the disperse relation ${k}_{\text{sp}}={k}_{0}{\text{[}\epsilon {\text{'}}_{m}\epsilon /\text{(}\epsilon {\text{'}}_{m}+\epsilon \text{)]}}^{\text{1/2}}$, and surface plasmon waves may be excited propagating along the rough silver film. Here, $\epsilon {\text{'}}_{m}$ and ${\epsilon}_{}$are permittivities of the silver film and the dielectric, respectively. As sketched in Fig. 5(b), the red arrows indicate the propagation of this surface plasmon waves, the blue dash arrows indicate the waves scattered from the rough glass surface transmitting through the film. It has been shown that the surface plasmon waves may propagate a distance up to a few teens of microns in the silver film. These waves may be coupled by the rough silver/air interface, and radiate into the air-side space, as represented in short red arrows in Fig. 5(b). These scattering-coupled radiations of surface plasmons also contribute to the intensity of the speckles at point *P*^{’}.

The speckle field on the silver is the superposition of waves produced by these two mechanisms. For thinner the silver film sample, the main part of the scattered waves from the rough surface penetrates through silver film, and the radiated waves of surface plasmons are comparatively small. Then the superposed speckle field is influenced mainly by the scattering waves from the rough glass surface. While the silver film is thicker, absorbance of the scattered waves by the silver film becomes significant, and influence of the surface plasmon waves becomes relatively greater. The speckle field is mainly contributed from plasmon waves, causing greater difference of speckle characteristics from those of the ground glass samples. In addition, the radiation of surface plasmon waves enables waves from all directions to contribute the speckle field at the point on the observation. This is equivalent to an almost indefinite aperture in the previous far field and deep Fresnel region, so the surface plasmon mechanism produces speckles of very small grains. This is the probable cause for the decrease of small-sized speckle grains with the film thickness.

#### 3.2 The contrast and the intensity probability density of speckles

We now perform the qualitative analysis for the speckle properties. We use data of intensity patterns in Figs. 4(a)- 4(f) with the larger magnification and then better resolution for the calculation. For each sample, the data of the five patterns in different position are used for the averages of calculated speckle parameters.

The speckle contrast is defined by [4]:

We use the above equation to calculate the contrast of the speckles and the results are given in blue triangle curve in Fig. 6(a). We see that the contrast values are increased from 0.65 to 0.90 with the increase of the film thickness, except for the slight decrease the sample for No.2. As is well known, the contrast is unity for Gaussian speckle, while partially developed speckles have contrast less than unity due to the existence of the directly transmitted wave and they are non-Gaussian speckles [4]. The speckles produced by a few scatterers [21] may also be non-Gaussian with the contrast less than unity [5].

We then turn to the intensity probability density function $p\text{(}I\text{)}$ of speckles. Since the probability for all intensity values is unity in the whole range of intensity $I$, then one has:

For the fully developed Gaussian speckle, $p\text{(}I\text{)}$ is a negative exponent function of intensity:

For the calculation of $p\text{(}I\text{)}$, we first normalize intensity data with the average intensity $<I>$, and we use the histogram counts for each intensity pattern and perform the normalization according to Eq. (5). By averaging for the 5 speckle patterns of the same samples, we obtain intensity probability density function $p\text{(}I\text{)}$. The results for all the samples are shown in Fig. 6(b), and in the inset, the curve for the 40nm is also given in square dot for a better view. From Fig. 6(b), we see that in the small intensity region, the values of the probability density curves are small, almost approaching 0. This means that probability for zero intensity value is very small, but this is not the reflection of the characteristics of speckles. In fact, in the detection of the intensity with the CCD, the electrical noise background with small amplitude is inevitable. Hence for small speckle intensity, this noise greatly influences the accuracy [22]. Our experience is that when the value of $I/<I>$ is less than 0.3, i.e., the parts of the curves left to the dashed vertical line, is invalid for the analysis. Then we see that for samples of smaller film thickness, probability density functions deviate from the exponential decay, indicating the speckles are non-Guassian; while for samples of larger film thickness, the probability density curves approach the negative exponential decay, and this means speckles produced by thicker silver films will approach Gaussian speckles.

For the partially developed speckles, the wave fields may be simplified as the superposition of a transmitted and a diffused wave field, and the probability density of the speckle intensity is written as [4]:

In above equation, the value of $r$ determines to what extent the speckle field approach the Gaussian one. We use the Eq. (7) to fit the curves in the Fig. 6(b) and obtain the values of $r$ for the samples. In the inset of Fig. 6(b), the fit curve of $p\text{(}I\text{)}$ to the experimental curve for the 40nm sample is given as the example. The $r$ curve versus the film thickness for the samples is also given in the Fig. 6(a), with values of $r$ labeled by the right margin marks. We may see that $r$ decreases with the thickness of silver film, indicating speckles evolves to be Gaussian. The speckle contrast is derived as:

Then with the above equation and the fit value $r$ given in Fig. 6(a), we can also obtain the speckle contrast. The contrast values thus obtained for the samples are given in the green curve in Fig. 6(a), and they roughly consistent with those previously calculated from the intensity data.

The observation plane is so close to the scattering point on the sample surface that for samples with smaller film thickness or without the film, the geometrical optical effect may play the obvious role in the speckle field. Then some components in scattered wave are not able to be diffused to a large spatial range, which equivalently plays a role of the transmitted waves. This causes a larger value of $r$, and correspondingly the speckle fields deviate from Gaussian ones. With the increase of the film thickness, the surface plasmon waves play main roles in the speckle fields as discussed in the former context, causing the decrease of $r$ and the increase of the speckle contrast. Meanwhile, the probability density function approaches the exponential decay. On the whole, the contrast and its comparison with the experimental values can be conveniently obtained from the probability density function we use above. However, with the increase of film thickness, there might be influences from the transition from film scattering to bulk scattering, which has a probability density function of following a Gamma law of order N [23]. Obviously, due to the complexity of mechanisms for speckles on the rough silver films, the properties of the probability density function need more detailed theoretical studies.

## 4. Evolution of the autocorrelation function

The autocorrelation function of the intensities for any two points $\rho =\rho \text{(}x\text{,}y\text{)}$ and $\rho \text{(}x\text{,}y\text{)}+d$ in the plane $Oxy$ is defined as:

where < > represents the ensemble average and $\rho =\rho \text{(}x\text{,}y\text{)}$ the position vector in the observation plane, and $I\text{(}\rho \text{)}=U\text{(}\rho \text{)}{U}^{*}\text{(}\rho \text{)}$, with $U\text{(}\rho \text{)}$ the complex amplitude of speckle field. The full width at half maximum of ${R}_{I}\text{(}\rho \text{;}\rho +d\text{)}$ may be regarded as the average size of the speckle grains. For Gaussian speckles, ${R}_{I}\text{(}\rho \text{;}\rho +d\text{)}$ defined above may be simplified with the Gaussian moment theorem. In the far field or in the deep Frensel region, it is determined by the illumination aperture or scattering grains of the object, respectively [4,7].But now for the speckles on the surfaces of the rough screens with silver films, the case may be complicated. We define from Eq. (10) the normalized autocorrelation function:

where we have assumed that the speckle field is isotropic and $d=\left|d\right|$ is the correlation separation. For a stationary random process, the ensemble average may be substituted by the average over the image area in the calculation of $\gamma \text{(}d\text{)}$.#### 4.1 The global autocorrelation function

To calculate $\gamma \text{(}d\text{)}$ for the speckles of a sample, we first use the data of one image and obtain $\gamma \text{(}d\text{)}$ of the image. Then we average the $\gamma \text{(}d\text{)}$ data for the 5 images of the sample at the same separation value *d* and finally obtain the $\gamma \text{(}d\text{)}$ curve versus *d*. Since the calculation is performed over a whole image, we call $\gamma \text{(}d\text{)}$ the global autocorrelation function to distinguish it from the local autocorrelation function that will be defined in the next context.

The global autocorrelation curves for all the samples are given in Fig. 7(a). We may see that the curves can be divided into two parts with the division point at about $d={d}_{0}\approx 1\mu m$. In the region $d<1\mu m$, the curves descend quickly with the increase of *d*; the thicker the silver film of sample is, the more quickly the curve descends. This corresponds to facts that small-sized speckle grains are smaller for thicker silver film samples, as shown in Figs. 3 and Figs. 4. While $d>1\mu m$, the curves drop slowly and finally tends to somewhat straight horizontal lines. This means the autocorrelation function approach a limit at the large separation, and this limit decreases with the increase of the film thickness. Furthermore, for the samples of thicker films, the curves drop more slowly in this region, and for the 100nm sample, the curve approaches its limit at about $d$ = $15\mu m$.

These properties of the autocorrelation function are related to the speckle structures in Figs. 3 and Figs. 4. For the ground glass sample, the small-sized grains in the speckle pattern therein are the largest compared with other samples, and the large-scale grains such as the platform-like structures have relatively distinct borders. With increase of the film thickness, the size of the small-sized grains decrease, and the range of the larger-sized intensity fluctuations expands due to the smoothing of border of the large-sized structures. This leads to the increase the range of speckle size for the samples of the thicker films, with their autocorrelation curves dropping more quickly at small *d* region, and more slowly at larger *d* region.

According to fractal theory and the self-affine fractal model given in Section 2.2, speckles with grain sizes of different scales may have the fractal properties. The normalized autocorrelation function of speckle intensity also takes form:

By fitting the Eq. (12) to the correlation curves in Fig. 7(a), we obtain the correlation length $\delta $ and fractal exponent ${\alpha}_{I}$ for speckles of all samples, and the results are given in Table 1. The experimental correlation data and the fit curve for the 100nm sample are shown in the inset of Fig. 7(a) as the example.

From the data in Table 1, we can see that the fractal exponent of the speckle intensity descends from 0.4972 to 0.2626 with the increase of the film thickness. This means that the speckle fields produced by thicker film samples have larger fractal dimension ${D}_{f}$ and more obvious fractality. This may be judged from speckle patterns in Figs. 3 and Figs. 4. In addition, Eq. (12) can be simplified as $\gamma \text{(}d\text{)}\approx \text{1-(1-}\beta \text{)(}d/\delta {\text{)}}^{\text{2}\alpha}$ for $d<<\delta $. Since the power spectrum is defined as the Fourier transform of the autocorrelation function $\gamma \text{(}d\text{)}$ [4], at larger spectrum region, it is the contribution of $\gamma \text{(}d\text{)}$ at smaller $d$ region. We may derive that power spectrum decreases in the large spectrum region in negative power law with the power value $-2{\alpha}_{I}-2$ [18]. Then the fractals in the speckles are related to the straightness of the tangent to the power spectrum in log-log scale. The conformation of the fit curve with the experimental data in the inset of Fig. 7(a) is an indirect verification of this property.

The case for fit values of the correlation length $\delta $ is different. For the thinner film samples, as the samples of 0nm, 20nm, and 40nm, the $\delta $ values decrease with film thickness. This is consistent with intuitive vision for patterns in Figs. 3 and Figs. 4. For the samples of thicker films, as samples of 60nm, 80nm, and 100nm, the $\delta $ values increase with the film thickness. This seems to contradictory to the intuitive judge from the speckle patterns. By scrutinizing the features of the speckle patterns, we may notice that this is the results of the increase of speckle size range, which causes the global autocorrelation function to vary with the increase of both the fractal dimension ${D}_{f}$ and the correlation length $\delta $. Anyhow we need to find quantitative ways for the description of the small-sized speckles.

#### 4.2 The local autocorrelation function

To describe the average size of the small-sized speckle grains, we define the local normalized autocorrelation function ${\gamma}_{loc}\text{(}{d}_{i}\text{)}$of the speckle intensities. We first notice that autocorrelation function $\gamma \text{(}d\text{)}$ defined in Eq. (11) is normalized by the average of the intensity squared $<{I}^{2}>$ over a whole speckle pattern. If we divide the pattern into $M\times M$ sub-patterns, each of which includes enough number of the small-sized speckle grains but is smaller than or at most at the same order of large-sized speckle grains, the influence of the large-sized speckle grains may be eliminated by the local normalization factor $<{I}^{2}{>}_{loc}$. We suppose a sub-pattern consists $m\times m$ pixels, and we use ${S}_{\text{(}n1\text{,}n2\text{)}}$ to represent the sub-pattern $\text{(}n1\text{,}n2\text{)}$ in the all $\text{(}M,M\text{)}$ patterns. Then we have the expression for the numerical calculation for the local autocorrelation function of ${S}_{\text{(}n1\text{,}n2\text{)}}$:

The above defined ${\gamma}_{loc}\text{(}{d}_{i}\text{)}$ can reflect the properties of the small-sized speckles. In the actual calculations, number of the pixels in each sub-pattern takes $m\times m=$$50\times 50$, and each speckle pattern is divided into $M\times M=$$20\times 20$ sub-patterns. The local autocorrelation function ${\gamma}_{loc}\text{(}{d}_{i}\text{)}$ for each sample is also finally obtained by averaging ${\gamma}_{loc}\text{(}{d}_{i}\text{)}$ of the five speckle patterns. In Fig. 7(b), the local autocorrelation curves for all samples are shown. We fit the data of each curve with the Gaussian function, and obtain the correlation length ${\delta}_{s}$ of the small-sized speckles. The values of ${\delta}_{s}$ are given in Table 2. The calculated autocorrelation curve and the fit curve for the 60nm sample are given in the inset of Fig. 7(b). From the data in Table 2, we can see that the size of the small-scale speckles monotonously decreases with the silver film thickness. This is consistent with the intuitive vision of the patterns in Figs. 3 and Figs. 4. We also notice that the smallest average size, i.e., the correlation length, is 0.4049$\mu \text{m}$for the 100nm sample, which is only a little larger than half the wavelength of the illuminating light. It should be noted here that the limit values of ${\gamma}_{loc}\text{(}{d}_{i}\text{)}$ do not reflect the speckle contrast, as those of the global autocorrelation function do, for the local normalization factor make this limit meaningless.

## 5. Discussion and conclusion

To summarize the evolution of the speckles with thickness of the rough silver films, we say that for the pure ground glass, the speckle field on the sample surface deviates from Gaussian speckles and its small-sized speckles have relatively larger size due to the incomplete scattering in the extremely deep Fresnel region. For samples of thinner silver films, the speckle fields are the combined contribution of the speckles produced ground glass scattering and the random surface plasmon waves. With the increase of the silver film thickness, contributions of the random surface plasmon waves become greater; the contrast increase, the intensity probability density functions gradually transit to exponential decay; the average sizes of small-scale speckles decrease and the fractal exponents become smaller and the fractality more and more serious. These properties of the speckles and their evolutions are very different from those in the traditional optical system. The mechanism of the random surface plasmon waves propagating along the films and their scattering coupled radiation plays specific roles for these properties.

In conclusion, we present a systematical experimental study for the speckles produced by the random silver films. We analyze contrast, the intensity probability density function and the global and local autocorrelation functions of the speckles, discuss the evolutions of fractality, speckle grain size distributions. We expect that the work would be beneficial for the further studies of the speckles produced by surface plasmon waves.

## Acknowledgment

We gratefully acknowledge the sponsorship of the National Natural Science Foundation of China under Grant No. 10947122.

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