Measurement of surface parameters from autocorrelation function of speckles in deep Fresnel region with microscopic imaging system

Chunxiang Liu; Qingrui Dong; Haixia Li; Zhenhua Li; Xing Li; Chuanfu Cheng

doi:10.1364/OE.22.001302

1. Introduction

The study of random surfaces has been paid great attention in many scientific and technological fields such as the growth fronts of thin films [1–3], surfaces of fabricated optical devices [4,5], the engineering of optical elements [6], and so on. In random surface characterizations, scattering technique is one of the most powerful tools and has been widely applied to the surfaces due to the advantages of non-contact and non- destructiveness. The methods of speckles essentially make use of the dependence of the statistical properties of the speckles fields on those of the rough surfaces, and the roughness and lateral correlation length are determined from the measurement of certain statistical quantities of the speckle patterns. The structures of the speckles are statistically described by the intensity autocorrelation function of the scattered light field with a far-field form given by the Van Cittert-Zernike theorem [7]. However, little useful information about the physical properties of the surface samples can be obtained [8] from this function. Recently, the investigations [8–13] on speckles in the region near the scattering object have attracted great interest, in which the average scattering grain sizes of the objects are successfully extracted. For the characterizations of random surfaces, more parameter such as the roughness, the lateral correlation length (or average scattering grain size) and the roughness exponent are needed. The later research [12] has a guide significance to extract the surface parameters from the correlation function of speckles, however, the experimental measurements have been rarely reported. In reference [13], the extraction of the surface parameters in the deep Fresnel region with a scanning fibre-optic probe has been realized, in which the acquisition process of the light signals is complicated and the data processing is very time-consuming.

In this paper, we propose a simple and convenient method for measuring the surface parameters with a microscopic imaging system. Based on the light scattering theory of Kirchoff approximation, we derive the two-dimensional expressions of the normalized autocorrelation function of speckles in the deep Fresnel region. The theoretical analysis shows that three surface parameters have great influence on the distribution of the scattered profile, and then determine the width of the normalized autocorrelation function of the speckles, i.e., the average size of the speckle grains. In the experiment, we setup a microscopic imaging system which can collect a wider range of the spatial spectrum components scattered from the random surface, and the speckle patterns in the deep Fresnel region are received by a CCD. A multi-scale behavior of the speckles has been identified, which is compatible with fractal character, and the structures of speckle patterns are related to the scattering properties of the surface topography. We calculate the autocorrelation function of the speckles and obtain the roughness $w$ , the lateral correlation length $ξ$ and the roughness exponent $α$ by fitting the formulated expression to the autocorrelation function data of the speckles. The extracted results by our method agree well with those by an atomic force microscope (AFM), which indicates that our method has a satisfying accuracy.

2. Theory

In the typical light scattering diagram shown in Fig. 1(a), a parallel laser beam with wavelength $λ$ illuminates a rough scattering surface. The observation plane at distance $z$ from the random surface is in the deep Fresnel region, which refers to the diffraction region from a few wavelengths near the surface to the traditional Fresnel region. We suppose that $z$ is small and the diameter D of the illuminating beam is large, so that the scattered waves will spread and then enough scattering elements on the screen may contribute to the light field at an observation point. Figure 1(b) gives the schematic diagram of the superposition of the light waves scattered from an equivalent scattering aperture contributing to the intensity of a point on the observation plane, which represents the one-dimensional case shown in Fig. 1(a) with the red dashed line. From the scattering theory, we know that the light intensity scattered onto the observation plane from a small adjacent region of a point $x_{0}$ on the random surface is distributed in the form of a decaying profile, which are shown in Fig. 1(b) with five small bell-shaped curves whose centers are at the conjugate positions a, b, c, d, e, respectively. The profile broadens with the increase of the distance $z$ . Correspondingly, the light intensity at an arbitrary point $x$ on the observation plane is the superposition of the light waves scattered from an equivalent aperture on the random surface with its transmittance variation mirroring the decaying profile and its center at the conjugate point $x_{0} = x$ . The big bell-shaped curve in Fig. 1(b) represents the equivalent scattering aperture with the center at $x_{0} = 0$ on the object plane and hence is that contributing to the intensity of point $x = 0$ on the observation plane.

Fig. 1 (a) Schematic diagram for generating and observing the speckle fields in deep Fresnel region. (b) Diagram of the equivalent aperture contributing to the intensity of the point x = 0.

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By use of Kirchhoff approximation and with the terms of surface height derivatives neglected, the scattered light field on the observation plane $(x, y)$ can be written as [14]:

U (x, y) = \iint exp[i k (n - 1) h (x_{0}, y_{0})] \frac{exp(i k r)}{r} \cos θ d x_{0} d y_{0},

where

h (x_{0}, y_{0})

is the height function of the surface, n is the refractive index of the screen,

cos θ

is the inclination factor,

r = {[z^{2} + {(x - x_{0})}^{2} + {(y - y_{0})}^{2}]}^{1 / 2}

is the distance from the object point to the observation point, and

k = 2 π / λ

is the magnitude of the scattered light wave

k

.

I (x, y) = < U (x, y) U^{*} (x, y) >

is the scattered intensity at point

(x, y)

and the autocorrelation function of the scattered intensity is given by:

R_{I} (x_{1}, y_{1}; x_{2}, y_{2}) = < I (x_{1}, y_{1}) I (x_{2}, y_{2}) >,

where

〈 〉

represents the ensemble average. The normalized autocorrelation function of the intensity fluctuation is defined by [15]:

γ_{I} (x_{1}, y_{1}; x_{2}, y_{2}) = [R_{I} (x_{1}, y_{1}; x_{2}, y_{2}) - < I >^{2}] / < I >^{2},

where

< I > = < I (x_{1}, y_{1}) > = < I (x_{2}, y_{2}) >

is the average intensity of the speckle ðelds.

γ_{I} (x_{1}, y_{1}; x_{2}, y_{2})

may also be called autocovariance, and it tends to zero when the two points are moved away from each other. Compared with the autocorrelation function

R_{I} (x_{1}, y_{1}; x_{2}, y_{2})

, the width of

γ_{I} (x_{1}, y_{1}; x_{2}, y_{2})

can be well-defined. When

U (x, y)

is the circular complex Gaussian ðeld, the autocorrelation function of the speckle intensity takes the form

R_{I} (x_{1}, y_{1}; x_{2}, y_{2}) = < I >^{2} + | < U (x_{1}, y_{1}) U^{*} (x_{2}, y_{2}) > |^{2},

Thus, the correlation function

γ_{I} (x_{1}, y_{1}; x_{2}, y_{2})

assumes form:

γ_{I} (x_{1}, y_{1}; x_{2}, y_{2}) = {| < U (x_{1}, y_{1}) U^{*} (x_{2}, y_{2}) > |}^{2} / < I >^{2},

The problem of calculating

γ_{I} (x_{1}, y_{1}; x_{2}, y_{2})

is thus reduced to that of calculating the autocorrelation of the fields.

R_{U} (x_{1}, y_{1}; x_{2}, y_{2}) = < U (x_{1}, y_{1}) U^{*} (x_{2}, y_{2}) >,

Subsitituting Eq. (1) to Eq. (6) and using the following statistical principle about the average calculation [16]:

\begin{array}{l} < exp{i k (n - 1)[h (x_{01}, y_{01})- h (x_{02}, y_{02})]} > \\ = \iint \iint p_{j} [h (x_{01}, y_{01}), h (x_{02}, y_{02})]exp{i k (n - 1)[h (x_{01}, y_{01})- h (x_{02}, y_{02})]} d x_{01} d y_{01} d x_{02} d y_{02}, \end{array}

where

p_{j} [h (x_{01}, y_{01}), h (x_{02}, y_{02})]

is the joint-probability density function of surface height functions. After some calculations, we derive the following expression:

< exp{i k (n - 1)[h (x_{01}, y_{01})- h (x_{02}, y_{02})]} > = exp{-[k (n - {1)]}^{2} [w^{2} - R_{h} (x_{01}, y_{01}; x_{02}, y_{02})]},

where

R_{h} (x_{01}, y_{01}; x_{02}, y_{02})

is the correlation function of surface height. Combinning with Eqs. (6)-(8) and continuing calculations, we get:

\begin{array}{l} R_{U} (x_{1}, y_{1}; x_{2}, y_{2}) = R_{U} (Δ x, Δ y) = \frac{λ}{z^{2}} \iint \iint exp{-[k (n - 1 {)]}^{2} [w^{2} - R_{h} (Δ x_{0}, Δ y_{0})]} \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} \end{matrix} \times exp[i 2 π k_{p} (Δ x_{0} -Δ x)]exp[i 2 π k_{q} (Δ y_{0} -Δ y)] d Δ x_{0} d Δ y_{0} d k_{p} d k, \end{array}

Where

Δ x = x_{2} - x_{1}

,

Δ y = y_{2} - y_{1}

,

Δ x_{0} = x_{02} - x_{01}

,

Δ y_{0} = y_{02} - y_{01}

and

k_{p} = sin θ cos γ / λ

,

k_{q} = sin θ sin γ / λ

are the components of wave vector

k

on the object plane, respectively,

θ

and

γ

are the azimuths angles of

k

. The integration of

Δ x_{0}

and

Δ y_{0}

in Eq. (9) is the duplex Fourier transform of the function

F (Δ x_{0}, Δ y_{0}) = exp{-[k (n - 1 {)]}^{2} [w^{2} - R_{h} (Δ x_{0}, Δ y_{0})]}

, furthermore,

F T^{- 1} [F (Δ x_{0}, Δ y_{0})]

is actually the scattered intensity profile:

I (k_{| |})= \iint exp{- {[k (n - 1)]}^{2} [w^{2} - R_{h} (ρ)]} \cdot exp(i k_{| |} \cdot ρ) d^{2} ρ = F T^{- 1} [F (Δ x_{0}, Δ y_{0})],

where

k_{| |} = 2 π (k_{p} i + k_{q} j)

is the parallel vector of

k

,

i

and

j

are the unit vectors, and

ρ =

| ρ | = \sqrt{{(Δ x_{0})}^{2} + {(Δ y_{0})}^{2}}

is the distance between the two points on the object plane.

Obviously, in the deep Fresnel region the scattered intensity profile takes the place of the scattering aperture in the Van Cittert-Zernike theorem of the far-field form. Afterward, substituting Eqs. (9) and (10) into Eq. (5), we finally obtain

γ_{I} (Δ x, Δ y) = exp{-2[2 π (n - 1) / λ]^{2} [w^{2} - R_{h} (Δ x, Δ y)]},

The above equation reveals the quantitative relation of the normalized autocorrelation function

γ_{I} (Δ x, Δ y)

and the autocorrelation function of the random surface

R_{h} (Δ x, Δ y)

. It has been demonstrated that random self-affine fractal surface model [17] can be well used to describe the ground glass screens [18] and its height autocorrelation function is given by [19]:

R_{h} (ρ) = < h (r_{0}) h (r_{0} + ρ) > = w^{2} exp[- {(ρ / ξ)}^{2 α}],

where

w

is root-mean-square deviation roughness,

ξ

is the lateral correlation length, and

α

is roughness exponent of the surface.

In order to deeply understand the influence of the above surface parameters on the speckle patterns, we will firstly analyse their effects on the scattered profiles which determine the distribution of correlation function $γ_{I} (Δ x, Δ y)$ . We have simulated the scattered profile produced by random surfaces with different parameters. In the simulation, we generate 2000 one-dimensional self-affine surface samples and take them as one surface ensemble [20]. Based on Eq. (10), we calculate the corresponding light field $U (k_{| |})$ and the ensemble average intensity $I (k_{| |}) = < I_{u} (k_{| |}) > = < U (k_{| |}) U^{*} (k_{| |}) >$ . Figures 2(a)-2(c) give the simulated profiles scattered by surface samples with different parameters. Through careful observation, we find that the distributions of the scattered profiles with the change of surface parameters are obviously different. The increase of the roughness $w$ , the decrease of the lateral correlation length $ξ$ or the roughness exponent $α$ will broaden the width of scattered profiles. The average intensity $I (k_{| |})$ drops slowly with the increase of $k_{| |}$ as shown in Figs. 2(a) or 2(b) but drops rapidly shown in Fig. 2(c). It may be difficult to see clearly the difference of the shape of the curves in Figs. 2(a)-2(c) by naked eyes, but more quantitative description can be fulfilled by the full-width at half maximum, whose approximate expression is written as $W_{p} = 2 ξ^{- 1} {ln \frac{Ω}{ln[exp(Ω -1) + 1-1/ e]}}^{-1/2 α}$ with $Ω = {[2 π (n-1)/ λ]}^{2} w^{2}$ for transmissive random surface samples [20]. This expression may depict to some extent the sensitivity of the surface parameters influencing the scattered profiles. Additionally, for the broadest curves in Figs. 2(a) and 2(b) drop so slowly that truncations seemingly exist. Such truncations may really happen and can be overcome by using an objective of high numerical aperture to collect large scattering-angle waves. Overall, the differences of the scattered profiles as an equivalent scattering aperture will lead to the change of the distribution of correlation function $γ_{I} (Δ x, Δ y)$ . Therefore, we can extract the surface parameters through these changes of $γ_{I} (Δ x, Δ y)$ , i.e., we can firstly calculate $γ_{I} (Δ x, Δ y)$ with the data of the measured speckle intensities, and then extract the parameters of the random surface by fitting the curve of $γ_{I} (Δ x, Δ y)$ using Eqs. (11) and (12).

Fig. 2 The scattered intensity profiles produced by random surfaces with different parameters.

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3. Experimental study and discussion

3.1 Setup of the optical system and measurement of random surface samples with AFM

Figure 3 schematically shows the experimental system for acquiring the speckle field in deep Fresnel region. We use a microscope objective (MO) with high numerical aperture (Nikon, Dry, 100x, N.A.0.9, WD 1mm) to collect a wider range of the spatial spectrum components scattered from the random surface. As is well known, the higher spatial spectrum components corresponds to the resolution for local tiny structure in the surface morphology, therefore, using this microscope objective to collect the speckle intensity will lead to the increase in the measurement resolution of surface characteristics. In the experiment, a laser beam with a wavelength of 532 nm is filtered and expanded with a spatial filter, and is paralleled to be a laser beam again by a collimating lens 1, and then it illuminates the random surface sample. The diameter D of the incident beam is 6cm and the distance $z$ from the microscope objective to the random surface is about 1cm,so that the aperture contributing to an observation point is within the area of the illumination beam. For accurate and large-scale positioning, the sample of random surface is placed on a three-dimensional piezo nanometer stage (PI E516) which has been installed on a two-dimensional precise mobile platform. Lens 2 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 8μm × 8μm). 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 firstly use a white light source with optical fiber output to illuminate the sample, and move L2 and the CCD in the direction of optical axis for the rough determination of the image plane and the magnification. We use the piezo stage to adjust the object distance repeatedly to obtain the clearest image, and then move the mobile platform until the distance between microscope objective and the image plane is 1cm. Then we remove the white light source and use laser beam to illuminate the sample for acquiring the speckle pattern. Due to the high ability of collecting scattered light and the good resolution of the MO, the light intensity distributions in observation planes in the deep Fresnel region can be recorded. The imaging range of CCD for collecting the speckle field is 34 × 34μm². We need not a beam stop in the setup to dispose the non-diffused beam as in Ref [8] for it disappears since the roughness values of the samples are large enough.

Fig. 3 Diagram of the optical setup for collecting the speckle fields in deep Fresnel region.

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In the derivation of Eq. (5), the speckle fields in the region near the scattered screen are assumed to be circular Gaussian. For this to be satisfied, the roughness values of the surface samples need to be large enough so that the non-diffused component in the scattered waves, which results in partially developed speckles deviating from Gaussianity, should be negligible compared with the diffused component. In the experiment, we select holographic plate as glass substrate due to its higher flatness and grind three glass substrates with silicon carbide powders with sizes of 10µm, 14µm, and 20µm, respectively. Correspondingly, the three samples are labelled as sample No.1, No.2 and No.3. All samples are measured with an atomic force microscope (AFM, PARK, Autoprobe CP, Contact mode, UL20 tip). The scanned area is 80μm × 80μm and the data points are 256 × 256. For each sample, five AFM images are scanned in different parts of the sample and one of them is shown in Figs. 4(a)-4(c). From these images, the height data $h (x_{0}, y_{0})$ as having appeared in Eq. (1) in the scanned area of each sample can be obtained. It is seen that sample surfaces are 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. The root-mean-square roughness and numerical height-height correlation are obtained by averaging those quantities calculated from the data of each image. The average roughness values of the samples No.1, No. 2 and No.3 calculated from AFM image are $w_{1} = 0.409 \pm 0.0013 μm$ , $w_{2} = 0.458 \pm 0.0012 μm$ and $w_{3} = 0.563 \pm 0.1559 μm$ , respectively. These roughness values are large enough for the non-diffused component in the scattered waves to be neglected. Our experience is that for the samples we have fabricated in the above manner, the non-diffused component disappears when the roughness value is greater than about 0.3μm. We calculate the numerical distributions of the autocorrelation of the three samples and fit it with Eq. (12), respectively. Thus, the lateral correlation length and the roughness exponent of three samples can be obtained by fitting. The results are $ξ_{1} = 4.147 \pm 0.212 μm$ and $α_{1} = 0.789 \pm 0.017$ for No.1; $ξ_{2} = 4.855 \pm 0.1829 μm$ and $α_{2} = 0.8056 \pm 0.0125$ for No.2; $ξ_{3} = 5.859 \pm 0.156 μm$ and $α_{3} = 0.8225 \pm 0.0112$ for No.3, respectively.

Fig. 4 AFM images of three random surfaces labeled as No.1, No. 2 and No. 3, respectively.

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3.2 Measurement of surface parameters with the speckle method

With the optical system we constructed above, the images of speckle intensity distributions in the deep Fresnel region are obtained and shown in Figs. 5(a)-5(c) for surface samples No.1, No.2 and No.3, respectively. Here we notice that the structures of the speckle patterns are different from the traditional far field speckles. In each of the patterns speckle grains have no obvious specific size and they appear the distribution in size of certain range from small to large scales. Moreover, the large-sized speckles appear intensity fluctuations consisted of many small-sized speckle grains and this differs from the smooth local intensity in the traditional far field speckles. These features indicate that the structures of speckle patterns in the deep Fresnel region have the characteristics of fractals and are distributed in multi-scaled sizes, and they are the reflection of the scattering properties of the surface topography.

Fig. 5 Speckle patterns for the samples No.1, No.2 and No.3, respectively.

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Through careful observation of the speckle patterns in Figs. 5(a)-5(c), we may see that the speckle pattern in Fig. 5(a) has more small-sized grains and less large-sized grains, while in Fig. 5(c) there appears some very large grains as shown in the red circle except for many small-sized grains with the same scales as those in Fig. 5(a), This indicates that the average size of the speckles in Fig. 5(c) should be larger than that in Fig. 5(a). The speckle grain distribution in Fig. 5(b) is in-between Figs. 5(a) and 5(c). These qualitative characteristics may be quantitatively described by the autocorrelation function of the speckle intensity. As analysed in Section 2, the correlation function $γ_{I} (Δ x, Δ y)$ is exactly determined by the scattered profile, and the detailed features of speckle grains are jointly determined by the factors such as the width of scattered profiles and the descendant shape of $I (k_{| |})$ with the increase of $k_{| |}$ . Larger roughness $w$ , smaller lateral correlation length $ξ$ and smaller roughness exponent $α$ will broaden the scattered profile, and then decrease the width of autocorrelation function $γ_{I} (Δ x, Δ y)$ , i.e., speckle grains. More definitely, the structures of the speckle grain described by $γ_{I} (Δ x, Δ y)$ are the reflection of the surface topography.

According to the relationship between the height autocorrelation function and the height-height correlation functions of the random surface [14,17], the normalized intensity -intensity correlation functions of speckle intensity may also take the following form:

H_{I} (Δ x,Δ y) = 2 σ_{I}^{2} [1 - γ_{I} (Δ x,Δ y)],

where

σ_{I}^{2} = < {[I (x, y) - < I >]}^{2} >

is the root mean square speckle intensity fluctuation. Substituting Eq. (11) into the above equation and considering the case of

ρ \to 0

, we finally obtain

\begin{array}{l} H_{I} (Δ x,Δ y)= 2 σ_{I}^{2} {1 - exp{-2[2 π (n -1) / λ]^{2} [w^{2} - R_{h} (Δ x, Δ y)]}} \\ \begin{matrix} = 4 σ_{I}^{2} {[2 π (n -1) / λ]}^{2} w^{2} {(ρ / ξ)}^{2 α} \end{matrix}, \end{array}

it can be seen that the approximation

H_{I} (ρ) \propto ρ^{2 α}

holds in the range of

ρ \to 0

. From the speckle intensities measured with CCD shown in Figs. 5(a)-5(c), the autocorrelation function

R_{I} (Δ x, Δ y)

and the average intensity

< I >

are numerically calculated, and then

γ_{I} (Δ x, Δ y)

and

H_{I} (Δ x, Δ y)

can be calculated based on Eqs. (3) and (13). Afterward, the linear fitting in the log-log scale will give the fractal exponent

α

of the surface samples. The results are as following

α_{1} = 0.7832 \pm 0.02745

for No.1;

α_{2} = 0.7987 \pm 0.02228

for No.2 and

α_{3} = 0.802 \pm 0.0117

for No.3. With the values of

α

, we obtain the other two surface parameters

w

and

ξ

by fitting the curve of

γ_{I} (ρ)

vs. $ρ$ with Eqs. (11) and (12). Figures 6(a)-6(c) show the data of normalized autocorrelation functions calculated from the speckle intensities and the fit curves in solid lines for samples No.1, No.2 and No.3, respectively. The extracted results of samples are

w_{1} = 0.41758 \pm 0.02854 μm

and

ξ_{1} = 4.22302 \pm 0.6738 μm

for No.1;

w_{2} = 0.46495 \pm 0.06494 μm

and

ξ_{2} = 4.87313 \pm 0.4159 μm

for No.2;

w_{3} = 0.57597 \pm 0.0358 μm

and

ξ_{3} = 5.55361 \pm 0.63384 μm

for No.3, respectively. Comparing the results extracted from the autocorrelation function of speckles with those measured by AFM, we find that our method is of good accuracy.

Fig. 6 The curves of the normalized autocorrelation function produced by the samples No.1, No.2 and No.3, respectively, and the extracted values of surface parameters.

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As previously explained, the speckle patterns have the characteristics of fractals and are distributed in multi-scaled sizes. In order to deeply study the fractal features of speckles and make clear the fractal relationship between the speckles and the random surface, we can imitate the definition of the height autocorrelation function $R_{h} (ρ)$ in Eq. (12) to define the autocorrelation function of speckle intensity as the following form:

γ_{I} (ρ) = exp[- {(ρ / ξ_{I}^{})}^{2 α_{I}}],

where

ξ_{I}

is the correlation length of speckles, and

α_{I}

is the fractal exponent of the speckles. We note that the curves of

1 - γ_{I} (ρ)

are linear in the range of

ρ \to 0

in the log-log scale, and the linear fitting will give the slope values of

2 α_{I}

. The fitting results are shown in Figs. 7(a)-7(c) with the red lines, and the extracted fractal exponent of speckle patterns are

α_{I 1} = 0.7352 \pm 0.00436

,

α_{I 2} = 0.7402 \pm 0.00449

and

α_{I 3} = 0.7507 \pm 0.00522

for No.1-No.3, respectively. With the fitting values of

α_{I}

, we obtain the correlation length

ξ_{I}

of speckles for three surface samples by fitting the correlation curves of

γ_{I} (ρ)

vs. $ρ$ in Figs. 6(a)-6(c) with Eq. (15). The corresponding results of three samples are respectively

ξ_{I 1} = 0.59087 \pm 0.0039 μm

,

ξ_{I 2} = 0.59386 \pm 0.0052 μm

,

ξ_{I 3} = 0.59557 \pm 0.00412 μm

.

Fig. 7 The curves of $1 - γ_{I} (ρ)$ vs. $ρ$ produced by the samples No.1, No.2 and No.3, respectively, and the extracted fractal exponent $α_{I}$ of the speckle patterns. The insets show the probability density function curves of speckles for three samples.

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Comparing the fractal exponent $α_{I}$ of the speckle patterns with $α$ of the random surfaces obtained from Eq. (14), we find that the fractal characteristic of speckle patterns is the same as that of the random surfaces, which can also be drawn from the expressions of Eqs. (14) and (15).

From these results, we see that differences among the values of roughness exponent $α$ , measured either by AFM or by our method, are small. Furthermore, the obtained values of $ξ$ have a tendency to overlap in consideration of the uncertainty bars. This may be related to the adoption of the statistical model of the surface. Though the fit to the curves of the autocorrelation functions is satisfying, another statistical model instead of Eq. (12) might also give good fit. Therefore, sensitivity of the measurement depending on a change in the statistical model is an issue that is worthy to be further investigated.

Theoretically, in the derivation of Eq. (4) the circular Gaussian character of the speckle field is required. This condition simplifies the Eq. (2) to Eq. (4) by reducing the fourth order momentum of speckle field to second order momentum with the so-called Reed theorem [15,16]. In the case of non-circular Gaussian speckles, the momentum-order reduction may be further tried with the momentum theorem in statistics despite theoretical complicatedness [16]. This case is usually related in the experiment to the existence of transmitted fraction of the illuminating beam due to the non-complete scattering when the roughness of the surface sample is small.

To demonstrate the circular Gaussian speckles are formed in our experimental condition, we have calculated the probability density function $p (I / < I >)$ numerically with the intensity data of speckle patterns in Figs. 5(a)-5(c). The data of $p (I / < I >)$ are shown in insets of Figs. 7(a)-7(c) in scattered square dot, and exponential decay fits are given in the red curves. We may see that the probability density curves in the small intensity region on the left of the blue dash line deviate from the exponential decay. This is induced by the background electrical noise with small amplitude in the detection of the intensity with the CCD [21]. On the whole, the probability density curves approach the negative exponential decay, and this indicates that the speckle fields could be taken as circular Gaussian speckles.

4. Discussion and conclusion

Based on the derived two-dimensional autocorrelation function of speckles in the deep Fresnel region, a good statistical fit with one plausible statistical model has been obtained and that parametric fit works well. The influences of scattered profiles as an equivalent aperture on the autocorrelation function are discussed and it is found that the autocorrelation function is related to the scattering properties of the surface topography. In the experiment, the speckle patterns are measured and a multi-scale behavior of the speckles has been identified, which is compatible with fractal character. We calculate the normalized autocorrelation function of the speckles and obtain the roughness $w$ , the lateral correlation length $ξ$ and the roughness exponent $α$ by fitting the formulated expression to the correlation function data. The extracted results with our method are compared with those by an atomic force microscope. The results of this work show that the speckles in deep Fresnel region may be used in acquiring the surface information. The normalized autocorrelation function of the intensity fluctuation, or the autocovariance function, of speckles used here is also precisely the quantity that can be accessed in the far field from the speckle intensity average curve versus scattering angle [15]. This speckle intensity average curve is related to the far field scattering profile [15,17], and as has been pointed out in Ref. [15], it is a broad function of “coarsely varying distribution”. In this sense, the speckles both in far-field and in deep Fresnel field bear the same information about the surfaces, but the technique here for information acquisition is more convenient by recording the speckle patterns instead of the far field intensity measurement with scattering system construction. We expect that our method is useful in the investigation of the characteristics of speckle fields in the deep Fresnel region.

Acknowledgments

National Natural Science Foundation of China (Grant No. 10974122) and Shandong Provincial Natural Science Foundation, China (Grant No. ZR2013AM010)are gratefully acknowledged.

References and links

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Measurement of surface parameters from autocorrelation function of speckles in deep Fresnel region with microscopic imaging system

Abstract

1. Introduction

2. Theory

3. Experimental study and discussion

3.1 Setup of the optical system and measurement of random surface samples with AFM

3.2 Measurement of surface parameters with the speckle method

4. Discussion and conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Equations (15)

Optics Express