Under conditions differing from those subjected for central limit theorem, the spatial autocorrelation function of speckle pattern resulting from illuminated rough surface is investigated. Its dependence on different illuminating apertures and the average of the roughness heights is presented theoretically and experimentally. The experiments were carried out using a set of circular and square apertures having different sizes. The results indicate that, increasing the size of the illuminating aperture leads to a decrease in the width of the main lobe of the spatial autocorrelation function.
As a result of illuminating a rough surface with coherent or partially coherent light, interference between reflected or transmitted waves occurs. Due to the random roughness heights of the surface, the interfering beams are of random phases. This leads to intensities of random values with random spatial distribution called speckle patterns [
The majority of the publications in this respect rely on the central limit theorem which considers limited conditions on the statistics of the roughness. Under such conditions, the speckle correlation and contrast do not depend on the statistical characteristics of the roughness heights. The present work deals with studying the theoretical and experimental dependence of the statistical properties of laser speckle mainly the autocorrelation function of the speckle fields on the aperture illuminating shape and its dimension as well as the roughness root mean square beyond the conditions of the central limit theorem.
The autocorrelation function between two intensities of speckle fields located at positions r1 and r2 on the observation plane is given from the ensemble average of I(r1) and I(r2) as [
R I ( r 1 , r 2 ) = 〈 I ( r 1 ) I ( r 2 ) 〉 , (1)
where 〈 ⋯ 〉 denotes the ensemble average.
Under the statistical conditions of the central limit theorem, the autocorrelation function according to Goodman [
R I ( r 1 , r 2 ) = 〈 I ( r 1 ) I ( r 2 ) 〉 = 〈 I s 〉 2 + | 〈 A ( r 1 ) A * ( r 2 ) 〉 | 2 . (2)
where A ( r 1 ) and A ( r 2 ) are the complex field amplitudes at the points r1 and r2, respectively. The asterisk denotes the conjugate complex. Considering
〈 I ( r 1 ) 〉 = 〈 I ( r 2 ) 〉 = 〈 I s 〉 and μ A ( r 1 , r 2 ) = 〈 A ( r 1 ) A * ( r 2 ) 〉 〈 I s 〉 , (3)
where μ A ( r 1 , r 2 ) is the complex coherence factor of the speckle field, then Equation (2) can be written as [
R I ( r 1 , r 2 ) = 〈 I s 〉 2 ( 1 + | μ A ( r 1 , r 2 ) | 2 ) . (4)
In fact μ A is the Fourier transform of the intensity distribution of the object and 〈 I s 〉 is the average intensity of the speckle pattern.
It is worth noting that 〈 I s 〉 in Equation (4) obeys according to Goodman [
Maack et al. [
To achieve the aim of the present work, the dependence of RI on the root mean square of the roughness heights, we have to consider a formula for 〈 I s 〉 depending on such statistical parameters. According to [
〈 I s 〉 = N + N ( N − 1 ) ( x 2 + y 2 ) γ , (5)
x = 〈 cos ( φ ) 〉 = ∫ − ∞ ∞ cos ( φ ) P ( φ ) d φ = exp ( − σ φ 2 / 2 ) ,
and y = 〈 sin ( φ ) 〉 = ∫ − ∞ ∞ sin ( φ ) P ( φ ) d φ = 0 .
where N is the number of the interfering beams and γ is the mutual degree of the temporal coherence between the interfering beams. γ tends to one when using monochromatic light. P ( φ ) is a Gaussian phase probability density distribution with zero mean value and σ φ is the root mean square of the random phase ( φ ). It is to be noted that, the statistical conditions of formula (5) do not obey the limited statistical conditions of the central limit theorem. For formula (4) to be still valid, formula (5) of 〈 I s 〉 is modified such that it verifies as nearly as possible the statistical conditions of the central limit theorem. It follows:
1) 〈 I s 〉 is normalized through dividing it by N2.
2) let N tends to infinity which is one of the conditions of the central limit theorem.
3) consider σ φ to be of large values which is a second condition of the central limit theorem.
Noting that a Gaussian distribution of large σ φ approaches to a uniform distribution which is one of the limited conditions. Under such modification we get:
〈 I s 〉 = exp ( − σ φ 2 ) .
The speckle contrast reaches one as σ φ takes large values and thus the central limit theorem is reached [
R I ( r 1 , r 2 ) = exp ( − 2 σ φ 2 ) ( 1 + | μ A ( r 1 , r 2 ) | 2 ) (6)
Equation (6) holds only for large values of σ φ . It indicates that, the correlation of the speckle field decreases with increasing σ φ .
To investigate the effect of the illuminating point spread function, circular and square apertures of various dimensions are chosen. The correlation ( R I ) of the speckle field produced when using a circular aperture is given by [
R I ( r 1 , r 2 ) = 〈 I s 〉 2 ( 1 + | 2 J 1 ( k a r / f ) k a r / f | 2 ) . (7)
where r = x 2 + y 2 and a is the radius of the aperture, and J 1 is the Bessel function of the first kind of order one.
The correlation at the speckle field produced when using a square aperture is given by [
R I ( r 1 , r 2 ) = 〈 I s 〉 2 ( 1 + ( sin α α ) 2 ( sin β β ) 2 ) , (8)
where α = k L x 2 f and β = k L y 2 f , L is the side length of the square aperture.
| 2 J 1 ( k a r / f ) k a r / f | 2 and ( sin α α ) 2 ( sin β β ) 2 arise from the diffraction phenomena of a circular and square apertures, respectively. They are called the intensity point spread functions of the illuminating apertures.
Figures 1(a)-(d) and Figures 2(a)-(d) show the behavior of the speckle correlation with the speckle separated distance Δx for circular and square illuminating apertures, respectively. The figures are evaluated numerically, for different values of σ φ , through the theoretical formula given by Equation (6). The figures indicate that the speckle correlation function decreases with increasing both of Δx and σ φ which is directly proportional to the square root of the average of the square heights of the roughness. This is due to the fact that by increasing σ φ the phase randomization of the scattered interfering waves increases leading to a decrease in the spatial autocorrelation function of the speckle. Figures 3(a)-(d) show the decaying of the speckle correlation at arbitrary point x = 33 μm on the speckle pattern plane, with increasing σ φ for circular and square illuminating apertures, respectively.
Schematic diagram of the optical experimental setup used to generate speckle patterns is shown in
A He-Ne laser was expanded and collimated by the use of a spatial filter (SF) and a lens (L1). The object consists of a combination of an aperture and different transparent diffusers. Circular and square apertures having different dimensions are used. To obtain an optical Fourier transform, the object has to be located at the front focal plane of the imaging lens (L2) and the CCD Camera is located at the back focal plane of (L2). Speckle patterns are recorded on the CCD Camera (1920 × 2560 pixels, 2.2 μm pixel size, 8 bit).
With the aid of the optical setup shown in
Figures 9(a)-(d) and Figures 10(a)-(d) represent a section profile of the normalized autocorrelation function maps that numerically evaluated through the Fourier transform of the recorded intensities of the speckle patterns when using circular and square apertures, respectively. Each figure is for particular roughness with the dimension of the illuminating aperture as a parameter. The considered average roughness heights are 1.98 μm, 3.8 μm, 5 μm and 7.6 μm and the apertures dimensions are 0.4 mm, 0.6 mm, 0.8 mm and 1mm.
The figures show as expected that the speckle size decrease as the dimension of the aperture increases.
The apparent crowding of the R I curves as the aperture tends to 1 mm is due to the nonlinear behavior of the Bessel function as well as the Sinc function. Moreover, it is observed that the crowding density increases with increasing the roughness by keeping the dimension of the aperture unchanged. This behavior is due to the relatively greater aperture area that leads to shrinking the illuminating point spread function and thus to R I . This behavior is more pronounced as the square aperture is considered because of the greater area of the square aperture in comparison with the circular aperture of the same dimension.
As a result, the behavior of the speckle correlation function obtained experimentally agrees with that numerically obtained through the theoretical formula given by Equation (6). An important general result indicated theoretically and experimentally is the speckle correlation decreases with increasing the speckle spacing (x).
To have a general overview of the parameters affecting the normalized autocorrelation functions,
Speckle pattern is a result of two optical phenomena namely interference between scattered waves acquired random phases from a rough surface and diffraction due to the illuminating aperture. Thus the speckle intensity arises from spatial intensity of interference pattern modulated by spatial intensity of diffraction pattern.
The correlation between the speckle intensities is basically attributed to those two optical phenomena. The characteristics of the produced random spatial interference pattern are referred to the statistical physical conditions of the rough surface, while the modulated factor of the diffraction is attributed to the conditions of the imaging system including the shape and dimensions of the illuminating aperture. These physical bases of arising speckle correlation lead us to consider the speckle correlation function as a convolution of the two spatial intensity distributions of interference and diffraction phenomena. The effect of one of them on the behavior of the speckle correlation function can overcome the effect of the second one depending on the following factors:
1) The phase probability density distribution which is acquired by the scattering waves.
2) The phase correlation function of the roughness and its correlation length.
3) The shape and dimensions of the illuminating aperture and in turn the resulting illuminating point spread function of the imaging system.
In general, the correlation between the speckle patterns mainly arises from the correlation between the interfering waves which causing the speckle pattern.
In conclusion, we have presented the theoretical and experimental dependence of the spatial autocorrelation function of the speckle intensity on the aperture shape, the aperture size and the roughness of transparent diffusers. A significant effect of the aperture size on the spatial normalized autocorrelation function is observed considering both the circular and square apertures. Increasing the size of the illuminating aperture permits more number of waves to be incident on the scattering surface and in turn more number of scattered waves of random phases to produce speckle pattern. This leads to decrease the spatial autocorrelation between the speckle patterns. These explanations confirm the experimental and theoretical results presented in Figures 1-3 and
The authors declare no conflicts of interest regarding the publication of this paper.
El-Nicklawy, M.M., Hassan, A.F., Elmahdy, A.E., Abd-Rabou, A.M. and Eesee, E.A. (2019) Effect of the Illuminating Point Spread Function and Roughness on the Speckle Correlation Field. Optics and Photonics Journal, 9, 165-177. https://doi.org/10.4236/opj.2019.911015