1. Introduction

As is well-known, there are many dark points with vanishing intensity and undefined phase in speckle fields [1] produced by the scattering of light waves from random surfaces. These points are so-called phase singularities or optical vortices [2, 3], which have attracted much interest since the seminal work of Nye and Berry in the early 1970s [2]. So far, quite a few [4, 5] researches have shown that measurement of the scattered light in near field can obtain the information on scatterers. Generally the scattering screens are divided into strong scattering screens, of which the standard deviation of surface height fluctuations is greater than the wavelength of illuminating light and weak ones with the standard deviation of surface height fluctuations less than the wavelength of illuminating light [3]. Recently, though many efforts [6–10] have been focused on the phase vortices in the speckle fields, to our knowledge, no studies was reported on the experimental study relating to the existence and properties of speckle phase vortices in the diffraction region near random surfaces.

The complex optical retrieval technologies [11–14] have been proposed and widely used in extracting the amplitude and the phase of a light field. One of them, the interference technique, was firstly applied to extract the phase vortices of a speckle field by Wang group [7]. In the early studies of phase vortices, researchers have studied qualitatively phase vortices by observing the variations of interference fringes of reference beam and object beam [15, 16]. Nowadays, CCD has been widely used in the detection of light intensity distributions, and the development of digital holographic techniques make the quantitative analysis of speckle vortices feasible and convenient.

In this paper, we make four random screen samples with different roughness values and design an optical setup to magnify speckle fields in the diffraction region near random screens. The interference intensity of reference light wave and speckle fields is recorded by CCD. Fourier transform is used to extract speckle phase vortices from the interference intensity and we present a qualitative explanation for the experimental results.

2. Experiments and extraction of the speckle wave field in the diffraction region near random screens

In order to study the existence and properties of speckle phase vortices in the diffraction region near random screens, we first make random screen samples. We grind four glass substrates of holographic plate with silicon carbide powders with sizes of 3.5$\mu m$, 5$\mu m$, 7$\mu m$ and 40$\mu m$, respectively. Correspondingly, the four samples are labeled as sample No. 1, No. 2, No. 3 and No. 4. All samples are measured with an atomic force microscope (AFM, PARK, Autoprobe CP, Contact mode, UL20 tip). For each sample, six AFM images are scanned at different areas of the sample and one of them is shown in Fig. 1 . The average roughness values of the four samples calculated from AFM image data are $w_1=0.059\pm 0.006\mu \text{m}$, $w_2=0.266\pm 0.021\mu \text{m}$, $w_3=0.368\pm 0.013\mu \text{m}$ and $w_4=0.526$ $w_4=0.526\pm 0.012\mu \text{m}$ respectively. From the results we can see that the roughness of all samples is shorter than the incident light wavelength of 0.6328$\mu m$.

 

Fig. 1 AFM images of the four random screen samples grinded with silicon carbide powders with grain sizes of (a) $3.5\mu m$, (b)$5\mu m$, (c)$7\mu m$ and (d)$40\mu m$, respectively, with scanning areas of $40\mu m\times 40\mu m$.

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We detect the speckle wave field with a Mach-Zehnder type interferometer as depicted in Fig. 2 . Vertically polarized light wave from a He-Ne laser is divided into two beams by beam -splitter BS1. The one used as reference light wave is expanded and filtered by spatial pinhole filter SPF and adjusted by lens L1 ($f=50mm$).Another is attenuated by a series of neutral filter A for matching the reference beam, and then reflected by mirror M2 and illuminates a random screen sample S mounted on a three-dimensional nanometer stage which works depending on the converse piezoelectric effect of piezoelectric ceramic and can move in three dimensions. Immediately speckle field is formed behind the sample. In order to obtain the magnified image of the speckle field, we design an imaging system consisting of a microscopic objective MO (Nikon, 40x, NA = 0.7) and a convex lens L2 ($f=50mm$).The magnified real image of the speckle field is first formed by the MO at plane P1, and then is further magnified by lens L2. The last real image is captured by a CCD (Cascade-1K, number of pixels is 1004 × 1002 and pixel size is 8$\mu m$ × 8$\mu m$) placed at the image plane. In experiment, the distance between CCD and lens L2 is twice greater than the focal length of lens L2 and fixed, so we need to find the exact object plane. We adopt a white light source to illuminate the sample and when the sample is slightly deviated from the object plane, the image will become blurry due to the short-focus MO and the extremely short coherence length of the white light. We slowly move the sample longitudinally back and forth by the aid of the nanometer stage until there is a clear image on screen of CCD and now the sample is just on the object plane. Then the laser substitutes the white light as the illuminating source, and the CCD records the intensity distribution of the speckle field on the surface of the sample. With the sample going slowly farther away from MO, speckle fields at different diffraction distances pass the object plane, and CCD records the intensity distributions of speckle fields at different distances read from the controller of nanometer stage.

 

Fig. 2 Schematic diagram of the experimental setup for recording interference patterns.

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The size of speckle pattern is adjusted by changing the distance either between lens L2 and CCD or between lens L2 and MO. We adjust the whole optical path with special care to eliminate, as we possibly can, the influence of spherical wave resulting from the scattering of samples. Additionally, we obtain the magnification of the imaging system by observing the lateral displacement relation between a sample and the pixels. The magnification of our experiment is 167.36.

Figure 3(a) shows the interference intensity pattern of reference light and a speckle field produced by random screen sample No. 2 at distance$Z=20\mu \text{m}$. For clear indication, the white squared part in Fig. 3(a) is enlarged and its brightness is increased in Fig. 3(b). We can easily see from Fig. 3(b) that there are a few of phase dislocations such as in the region within black circles. We now discuss how to extract a speckle field from interference intensity pattern. A speckle field $U(x,y)$ and a reference wave $r(x,y)$may be expressed respectively as:

 

Fig. 3 (a) The interference intensity pattern of a reference light and a speckle field produced by random screen sample No. 2 at distance$Z=20\mu \text{m}$. (b) Local interference intensity pattern of the white squared part in Fig. 3(a). The gray-scale values take arbitrary units and the number of gray-scale level is 32.

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The intensity of their interference pattern can be represented in the form:

 

Fig. 4 (a) The gray-scale image of the spatial frequency spectrum of the image in Fig. 3(a) after Fourier transform. The gray-scale images from (b) to (e), extracted from the image in Fig. 3(a), are the distributions of the phase, the real part, the imaginary part and the intensity, respectively. (f) The same intensity distribution directly recorded by CCD.

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Speckle intensity may be obtained by the above mentioned approach. In addition, CCD may directly record speckle intensity with reference light blocked. Figure 4(f) shows the directly recorded intensity distribution of the same speckle field. Comparing the images in Fig. 4(e) and in Fig. 4(f), we can see that they resemble each other except their brightness due to different gray levels and the image in Fig. 4(f) has more background noises. Twice Fourier transforms effectively filter the inherent electrical noises of the CCD. Those noises vary as fast as almost pixel-by-pixel with rather small fluctuations. Comparatively, the signals of speckle grains vary much slower but fluctuate much greatly. Hence, such filtering can be easily done and has almost no significant influence on the phase vortices. Most important of all, the real and the imaginary parts of the speckle field are extracted.

3. The existence and properties of speckle phase vortices in the diffraction region near random screens

In Fig. 5 , the red solid and the black dash lines give the zero lines of the real and the imaginary parts of the speckle fields, respectively, produced by different samples at different distances with the image size$8000\mu m\times 8000\mu m$. We see that those lines for different samples obviously have different distributions. From Figs. 5(a1)–5(a3) we may see that the zero lines of the sample No. 1 are the most rarefied; the lines for the real and the imaginary parts appear alternatively and they do not intersect. This indicates that for a very weak scattering screen the phase vortices do not appear in the region near the random screen. In the experiment, we have conducted the measurement to the extent of distances of the order of several millimeters, and no phase vortex is observed for this sample. From Figs. 5(d1)–5(d3) we can see that for sample No. 4 with the largest roughness, the zero lines are closest and the zero line densities are most uniformly distributed. Figures 5(c1)–5(c3) show that for sample No. 3 with a comparatively large roughness, the zero lines go a little farther apart than those for sample No. 4, and a view limited to some local areas the zero line densities may appear non-uniformly distributed. From Figs. 5(b1)–5(b3) we may see that for sample No. 2 with moderately smaller roughness, though the zero lines of the real and the imaginary parts do not appear alternatively as in the case of sample No. 1, their distributions are so highly non-uniform that in some local areas, the lines of the real part are denser and the imaginary part denser in the some other areas. Anyway, this non-uniform distributions lead to some intersection points and phase vortices are formed. On the whole, the results in Fig. 5(b1), Fig. 5(c1) and Fig. 5(d1) show that phase vortices can be formed on the surfaces of the random screens with enough large roughness. This is obviously different from conclusions in references [15] and [16].

 

Fig. 5 Distributions of zero lines of the real and the imaginary parts. The red solid lines and the black dashed lines are respectively the zero lines of the real parts and the imaginary parts. Figures (a1)-(a3), (b1)-(b3), (c1)-(c3) and (d1)-(d3) are for sample No.1, No.2, No.3 and No.4, respectively. Dimensions of each image are$8000\mu m\times 8000\text{μm}$and distances are given under each image.

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It would be interesting to know how many phase vortices in each image of Fig. 5, which reflect the vortex density in each case. We design a program to count the vortices and to calculate the average core eccentricity of the vortices, and the results are shown in Table 1 . The numbers of the vortices therein quantitatively demonstrate that the vortex densities increase with the roughness of the screens at the same distance, and increase with distances for the same screen with large enough roughness. For clarity, Fig. 6 shows enlarged color phase maps in correspondence with the images in Fig. 5, where the dimensions of each image are$2000\mu m\times 2000\mu m$ with the phase ranging from$-\pi$ to $\pi$ in eight color levels. We can see from Figs. 6(a1)–6(a3) that in each image not all the eight colors appear, and some colors occupy most part of it, and other colors fill in a few and small regions. This indicates that the small phase fluctuations for sample No. 1 cannot form phase vortices. In phase maps of Figs. 6(b1)–6(b3) for sample No. 2, all colors appear but the sizes of the regions of different colors are evidently different; a few phase vortex pairs may be discerned in some areas as labeled by the red circles in Figs. 6(b1)–6(b2) though they are small. In Figs. 6(c1)–6(c3) for sample No. 3 the proportion of the areas of different colors becomes more uniform and more vortex pairs appear. From Figs. 6(d1)–6(d3) we can see that more phase vortices appear in groups and they become uniformly distributed within a distance of several tens of micrometers. From all the maps of Fig. 6, expect for screen No. 1, which has so small roughness that no vortices are formed at any distance, the obvious increase in densities of phase vortices with the increase of distance is clearly demonstrated. It is interesting that the average core eccentricities of the phase vortices shown in the Table 1 are about 0.9 for samples No. 2, No. 3 and No. 4, which is a little bit larger than the value 0.87 given by Berry’s theoretical estimation for the far field speckle fields [17]. This may be correlated with comparatively rarefied existence of phase vortex pairs with very small separation between the two vortices of each pair, as labeled by red circles in Figs. 6(b1) and (b2). Phenomenologically, such structure of phase distribution would elongate the closed intensity contour around each vortex, and this leads to higher eccentricity values.

Table 1. The Number and the Average Core Eccentricity of Phase Vortices of
Different Samples at Different Distances

View Table

 

Fig. 6 Local phase distributions one-to-one corresponding to the images in Fig. 5. Dimensions of each image are$2000\mu m\times 2000\mu m$.

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We now take the case of $z=60\mu m$as the example to discuss the evolution of phase vortices with roughness. Figures 7(a) –7(d) give respectively the color phase maps for all samples. In Fig. 7(a), we may see that no phase vortices exist as depicted above. In Fig. 7(b) for sample No. 2, though the map is filled with all colors, we still can see that the green area dominates the whole image, which indicates that there is a larger probability for phase to take the values around zero. For sample No. 3 as shown in Fig. 7(c), all colors fill smaller areas of the map and mix with each other, demonstrating more uniform distribution of the phase values. In Fig. 7(d) for sample No. 4, we can see that all the colors distribute randomly on the whole map with each color taking almost uniform proportion. The variation of phase distributions shown in Fig. 7 demonstrates that the phase fluctuations become stronger with increasing roughness. For quantitative analysis of the phase distribution, we give in Fig. 8 the phase probability density curves calculated from the data of phase maps in Figs. 7(a)–7(d). We see that there are maxima in the curves for sample No. 1 and No. 2, and the maximum for No. 1 is evidently larger than that for sample No. 2. For sample No. 3 and sample No. 4 the maxima are not apparent, and the curves are roughly flat though some smaller fluctuations exist. These curves further demonstrate that the phase distributions tend to be uniform with the increase of roughness.

 

Fig. 7 Phase distributions of (a) sample No. 1, (b) sample No. 2, (c) sample No. 3 and (d) sample No. 4 with distance $z=60\mu \text{m}$$z=60\mu \text{m}$and dimensions of each image are$8000\mu m\times 8000\mu \text{m}$.

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Fig. 8 Phase probability density distributions of four samples when$z=60\mu m$.

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Generally for a weak-scattering screen, the scattered wave can be regarded as a specular component and a diffuse component [1]. In our experiment, the non-diffused specular component is clearly observed for the two screens with smaller roughness, and it does not exist for two screens with larger roughness. In the speckle field of the screen with smallest roughness (sample No. 1), the specular component is far stronger than the diffuse scattered component, so that, as is shown in Fig. 9 , the phase distribution of this sample is quite similar to that of the incident light though many weak random phase fluctuations appear in its speckle field. With an increase of roughness, as we can see from Figs. 7(b)–7(d), phase fluctuations become stronger and the phase vortex density increases due to the specular component and the scattered component growing weaker and stronger, respectively.

 

Fig. 9 Phase distributions of (a) incident light and (b) sample No. 1.

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We now take the phase distributions of sample No. 2 for example to discuss phase vortex evolution with distance. Figures 10(a) –10(c) show the color phase images for sample No. 2 at distances (a) $z=0\mu \text{m}$, (b) $z=30\mu \text{m}$ and (c) $z=60\mu \text{m}$, respectively. From Fig. 10 we can easily see that the phase fluctuations in Fig. 10(b) are evidently stronger than that in Fig. 10(a) and a little weaker than that in Fig. 10(c). This phase fluctuation evolution with distance agrees well with the phase vortex number changes shown in Table. 1. For clear indication, we give in Fig. 11 the phase probability density distributions of the images in Fig. 10. It can be seen from Fig. 11 that the curve maxima decrease gradually with the increase of distance. This variation mainly results from the increase of the scattered components meeting at any point of speckle field.

 

Fig. 10 Phase distributions of sample No. 2 when (a)$z=0\mu \text{m}$ (b) $z=30\mu \text{m}$and (c) $z=60\mu \text{m}$ and dimensions of each image are$8000\mu m\times 8000\mu \text{m}$.

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Fig. 11 Phase probability density distributions at different distances of sample No. 2.

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It should be noted that in the experiments of References [15] and [16] the authors did not report the existence of phase vortices immediately behind a random screen. That might be due to their experimental conditions were different from those in this paper. This indicates that proper conditions such as change of roughness of the screens and choice of imaging lens can make phase vortices near the screens detectable.

4. Conclusions

An optical setup was designed to extract phase vortices. Experimental results show that the phase vortices cannot be produced unless the screen is rough enough. It is very interesting that on condition that a screen is rough enough, phase vortices may appear on its surface and its density of phase vortices would become larger with an increase of distance in the region near the screen. In addition, with the distance from the screen certain, the density of phase vortices will become larger with the increase of roughness.