1. Introduction

Speckle patterns appear in different contexts whenever highly coherent waves reflect from or transmit through a rough surface or a turbid medium [1]. In optics, speckle may also arise from the interaction of multiple high order eigenmodes in multimode fibers [2]. At first, this phenomenon was seen only as a drawback in imaging systems employing coherent sources, and speckle noise is still an issue in the case of LIDAR technology, ultrasound imaging and optical coherence tomography [3–5]. However, laser speckle has found interesting applications in areas like optical micromanipulation [2,6], and studies on the connection between disorder and localization [7], for instance. Moreover, speckle metrology [8,9], dynamic speckle analysis [10], and some methods of high-resolution astronomical and microscopic imaging [11,12] are powerful techniques that have exploited this phenomenon in its favor.

From the viewpoint of wavefront topology and singular optics, speckle has also attracted considerable attention, since it has been recognized that its zeros of intensity correspond to optical vortices [13–15], whose distribution exhibit intriguing statistical properties [16,17], and whose evolution can also be used in high-resolution metrology [18]. An optical vortex is a phase singularity or screw wavefront dislocation [19], meaning that the phase of the light field changes by an integer multiple m of 2π-cycles along a closed loop around the vortex core, where the amplitude vanishes. The integer m is known as the topological charge or the singularity strength, and its sign defines the rotation direction or vortex helicity.

Many theoretical and experimental studies have been performed on the optical vortices in speckle patterns [13–16,20] and random optical fields [21,22]. In contrast with the vortices carried by Laguerre-Gaussian laser modes or high order Bessel beams for example, those arising in speckle are typically anisotropic point vortices, which means that the phase does not increase linearly with the azimuthal angle around the singularity [14–17,23,24]. Although, in principle, it is not impossible to obtain a higher order vortex (|m| > 1) in a random pattern, it would be extremely rare and unstable on propagation, so in general, vortices in a speckle have unitary topological charge [16].

Interactions between pairs of vortices are mainly determined by their topological charges, the separation distance and the conservation of the total angular momentum. While two vortices of the same helicity tend to repeal each other, a pair with opposite helicities may attract each other and annihilate along propagation or even a new pair can be created [21,25,26]. In this way, phase singularities may form curves and knots in space [23]. In random fields, it has been demonstrated that there is a strong anticorrelation in the sign of the vortices between nearest neighbors [17,27].

In this work we explore a new path to create a random optical field. Namely, we use a phase Spatial Light Modulator (SLM) to imprint arrays of vortices in a Gaussian carrier beam and investigate the conditions leading to a speckle pattern. Ordered and disordered arrays of unitary charge vortices with the same sign, random signs and anticorrelated signs are analyzed, varying the number of vortices and the pitch or mean separation among them. After reflection on the SLM, the field is propagated in free space for a given distance and then focused, with the aim of forcing the vortices to interact. The resulting field is characterized in a transverse plane for which the carrier beam would reach the same size as its initial condition in the absence of vortices. The role of the different control parameters is analyzed in order to determine the requirements to obtain either an ordered or a random pattern. We complement our experiments with numerical simulations, from which we are able to do an analysis of the phase and the vortices in the region of interest, as well as statistical ensemble studies as those typically used for characterizing speckle.

2. Ordered lattices and disordered arrays of vortices

Consider a monochromatic optical field consisting of an array of N point vortices of unitary topological charge embedded in a Gaussian beam of waist radius w₀, whose complex amplitude is given by

Our experimental setup is depicted in Fig. 1(a). A collimated laser beam of wavelength λ = 532nm and a measured radius of w₀ = 1688μm, impinges at a small angle (< 5°) on a reflection phase-SLM (Hamamatsu LCOS-SLM X10468), where a phase mask containing an array of vortices is displayed. The total area of the SLM in pixels is 800 × 600, each square pixel having a side length of 20μm, but the vortex array is imprinted on a reduced central area of 180 × 180 pixels, which assures that all the vortices are contained in the incident beam. A convergent lens of focal length f = 400mm is placed at a distance f from the mask. After the lens, the field is allowed to propagate in free space up to the plane z = 2f, where it is recorded with a beam profiler (BP). This observation plane was chosen because the size of the carrier beam there, in the absence of vortices, would be the same than the original beam impinging on the SLM. In the figure, we illustrate an example of a phase mask representing an ordered lattice of 5 × 5 vortices of the same helicity (−1), which are marked with red circles, and the area of the impinging beam (2w₀) is delineated by the green dashed circumference. The propagated field in the plane of analysis is also shown.

 

Fig. 1 (a) Experimental setup: Spatial Light Modulator (SLM), convergent lens(L), beam profiler (BP). The insets illustrate the phase mask for an ordered square lattice (left), where the red dots indicate the vortices, and the image of the propagated field obtained with the BP. (b)Different cases of arrays with N = 100 vortices. From left to right: Ordered Lattice (OL) with the Same Helicity (SH); Disordered Lattice (DL) with SH; OL with Random Helicity (RH); DL with RH; OL with Anticorrelated Helicity (AH); DL with AH. From top to bottom: schematics of the arrays, where vortices of opposite helicities are represented by red and blue dots; phase masks; simulations of the field at the observation plane z = 2f; experimental results. The phase levels in the masks vary from 0 (black) to 2π (white). The pitch of the OL is P = 13 pixels of the SLM (260μm) and the scales in the simulations are in mm, being the same for the experiments.

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We wonder whether there is a minimum number of vortices necessary to generate a speckle pattern and what are the conditions on the locations of the vortices in the phase mask and the signs of their topological charges. We will start from a completely ordered square lattice with grid spacing or pitch P, expressed in SLM-pixels, and $\sqrt{N}$ vortices per side, so that the side length of the lattice is given by $L=\left(\sqrt{N}-1\right)P$. In order to guarantee that all the vortices are embedded within the carrier beam, the maximum side length is given by $L_{\mathit{max}}=w_0/\sqrt{2}$, which in turn defines a maximum pitch of $P_{\mathit{max}}=w_{0p}/\left[\sqrt{2}\left(\sqrt{N}-1\right)\right]$, for a given N, where w_0p is the beam radius expressed in pixels. Depending on N, we vary P in the range 4 ≤ P ≤ 30. We notice that, even in this case, the size of the carrier beam is slightly larger than the size of the array in the mask, since only 90% of the energy of the beam is contained in an area of radius w₀. Six different cases will be explored, with the aim of studying the role of order or disorder in both position and/or helicity.

In terms of position, we will consider an ordered lattice (OL) and a disordered lattice (DL). In all cases, the latter is built in the following way: each lattice site in the OL is shifted to the new position (x_n + (0.4P)δ_x, y_n + (0.4P)δ_y), where δ_x and δ_y are independent discrete random variables, which may take the values of −1, 0 or 1 with the same probability. The factor of 0.4P was chosen arbitrarily, being a relatively large fraction of the pitch, while trying to avoid overlapping between neighbor vortices. In this way, there are nine possibilities in total for the new location of the n-th vortex, including its original position, all with equal probability. Regarding the sign of the topological charge, three situations are considered: all the vortices having the same helicity (SH), random helicities (RH), and anticorrelated helicities between pairs of contiguous vortices (AH). Therefore, the combinations we will analyze are: OL-SH; DL-SH; OL-RH; DL-RH; OL-AH; DL-AH. It is worth mentioning that in the RH case, the helicity of each vortex (+1 or −1) is chosen randomly with the same probability, implying that the proportion of vortices of opposite helicities might be unequal, specially when N is low. Figure 1(b) illustrates all these cases, as indicated on the top, for N = 100 vortices. The rows correspond, from top to bottom, to schematics of the vortex arrays, the phase masks imprinted on the SLM, the simulations of the field at the observation plane, and the experimental results. The simulations were performed in two steps using the transfer function approach to solve the Fresnel diffraction integral, one step from the SLM to the lens, taking Eq. (1) as the initial condition, and the second step from the lens to the observation plane [35]. A very good agreement between experiments and simulations can be seen. Notice that, except for the cases OL-SH and OL-AH, all the other masks have a random element, either in position, in the helicity or in both. For all these cases Fig. 1 shows typical examples of a single realization.

There are several aspects that can be pointed out from Fig. 1. First of all, it is clear that in order to obtain a speckled pattern it is a necessary condition that, in average, the number of vortices of opposite helicities is approximately the same, as it occurs in well-developed speckle patterns [1]. However, this is not a sufficient condition, as can be seen from the OL-AH case, where the positional order and anticorrelated helicities preserves an ordered lattice in propagation. On the other hand, in the OL-RH case, the randomness in the sign of the topological charge prevails over the positional order of the original lattice mask, giving rise to a speckled pattern. For comparison, it is worth to notice that in the cases where all the vortices have the same helicity, the pattern extent is visibly larger than that obtained for random or anticorrelated helicities. This can be attributed to the repulsive interaction between equally-charged vortices [25]. Also, as reported previously by Indebetouw, the OL-SH displays the same order of the original lattice, but it has rotated rigidly with respect to the initial condition [25]. The rotation direction depends on the sign of the topological charge.

According to Fig. 1, there are three viable candidates for producing speckle: OL-RH, DL-RH, and DL-AH. However, for the sake of simplicity, in what follows we will focus our attention only on the two latter cases, which represent two different distributions of the topological charge. These disordered patterns will be further analyzed in terms of the main control parameters: N and P.

3. Disordered arrays: the role of the number and density of vortices

Figure 2 shows the influence of the number of vortices on the resulting patterns for disordered lattices with (a) random and (b) anticorrelated helicities. The intensity distributions do not resemble a speckle for a small number of vortices, N = 25, but speckles become apparent in the central regions of the patterns with N ≥ 100. In both cases, RH and AH, the speckle size clearly decreases as N increases. There are also some differences between the patterns for a given N, such as the peripheral intensity distribution. While the RH patterns look wavy in the outer region, the AH patterns exhibit a rather square geometry, likely inherited from the topological charge anticorrelation order. Further discussion on the change of the number of vortices and the statistical properties of the speckles in each case will be postponed until Section 4.

 

Fig. 2 Intensity patterns generated by disordered arrays of vortices with (a) random helicities and (b) anticorrelated helicities, for different number of vortices, as indicated on the top of each figure. Simulations are compared with experimental results for each particular realization.

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It is important to realize that the parameter P varies as N does, due to the imposed condition of filling the carrier beam with the array. Hence, a variation of the vortex density is coupled with the change of N. With the aim of investigating the role of the vortex density alone, Fig. 3 illustrates the results of keeping the number of vortices fixed, N = 169, while the pitch parameter is increased from P = 4 to its maximum possible value of P_max = 10 in this case, in steps of 2 pixels. A small value of P implies a large vortex density, ρ_v = N/A, where the area of the array is $A\approx {\left(\sqrt{N}-1\right)}^2{P}^2$. In both cases, RH and AH (Fig. 3(a) and Fig. 3(b), respectively), it is clear that the speckled appearance improves as P grows (from left to right). There is a strong light depletion at the center surrounded by a halo when the vortex density is large (left), specially for RH. In addition, the extent of the pattern is larger for the lower densities (right). These facts can be understood as follows: as the vortices interact, many pairs with opposite charges are expected to annihilate when their separation distance becomes small enough, as it occurs for small values of P, leading to a decrease in the total number of vortices. Any initial imbalance in the number of opposite-charged vortices in the RH case will contribute to the net topological charge of the array, which might reach the density limit discussed by Roux [30] at some point along the propagation through the focal region. The halo is likely related with the diffraction of that part of the carrier beam that is not considerably affected by the phase mask in the SLM, which becomes larger as the size of the array in the mask becomes smaller. Vortex dynamics during the shrinking of the host beam certainly deserves further investigation and will be discussed elsewhere, but this is out of the scope of the present work. In terms of our original goals, after repeating this analysis for different number of vortices, we can conclude that the optimum pitch for producing speckle always corresponds to the condition of filling the host with the array, $P_{\mathit{max}}=w_{0p}/\left[\sqrt{2}\left(\sqrt{N}-1\right)\right]$, as initially considered.

 

Fig. 3 Simulations and experimental results for the intensity patterns generated by disordered arrays of N = 169 vortices with (a) random helicities and (b) anticorrelated helicities, for different values of the pitch parameter P expressed in SLM pixels, indicated on the top of each figure.

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4. Vortices and statistical properties in the disordered patterns

In this section we present a quantitative analysis, based on ensemble averages over 50 numerical experiments for each of the following cases: N =25, 100, 225, 400 and 900 vortices, for RH and AH. Unless otherwise stated, the study of the optical field propagated to a distance z = 2f from the lens will be limited in the transverse plane to a circular area of radius r_win = w₀. As mentioned before, this area corresponds to the size that the carrier beam would reach in the absence of the vortex array, according to Gaussian beam propagation through an optical system, and it is the same size than the original beam impinging on the SLM, w₀ = 1688μm. We notice that, by considering only the central region of the patterns, we are disregarding some features that differ from typical speckle patterns. However, we believe that this is a fair comparison, since in the standard ways to produce speckle the scattering medium is usually a heterogeneous material whose size is much larger than the incident beam, whereas our beam is slightly larger than the vortex mask.

First, in order to investigate how is the number of vortices modified on propagation through the focal region, we compare the initial phase masks with the phase of the field obtained in the plane of interest, as shown in the examples of Fig. 4 for N = 100, with RH (a) and AH (b). The vortices are identified as the points where the complex field amplitude vanishes [13,19], which correspond to the intersections of the red and blue contour curves representing the zero crossings of its real and imaginary parts, respectively. Blue and red markers are used to distinguish between vortices of opposite charges. The curvature of the carrier’s wavefront, which is divergent at z = 2f, becomes self-evident from the shape and proximity of the outer contours.

 

Fig. 4 (a)–(b) Examples of the numerically calculated phase (left) and contour plots (right) of the real (red) and imaginary (blue) parts of the complex field amplitude for the initial mask with N = 100, and the propagated field, for RH (a) and AH (b). Vortices of opposite charges are indicated with red and blue markers. The phase levels vary from 0 (blue) to 2π (yellow). (c) Ensemble average of the final number of vortices 〈N_f〉 as a function of the initial number of vortices N, for RH (blue curve) and AH (green curve), for an analyzed area of radius r_win = 1.69 mm. The black line represents vortex number conservation: 〈N_f〉 = N.

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We have estimated the average number of vortices in the propagated field enclosed in the area of interest, 〈N_f〉, as a function of the initial number of vortices, N, as shown in Fig. 4(c). Although we applied several consecutive criteria to identify and discriminate the vortex candidates, starting from the zeros of the complex amplitude and including an analysis of the phase around each candidate, we noticed that there are usually a few undetected vortices and also some false positives, so that the error in our estimation is of the order of 5%. This is not considered in the plots of Fig. 4(c), where the error bars rather represent the standard deviation of the ensemble-average values. Expected but noteworthy, the average number of positive and negative vortices was found to be approximately the same, specially for large values of N. The black line indicates a conservation in the number of vortices, i.e., 〈N_f〉 = N. There is a clear trend in both plots: for the smaller values of N, the final number of vortices increases after the interaction through the focal region and subsequent propagation, meaning that vortex pairs have been created. For larger values of N, the number of vortices in the resulting field decreases, meaning that vortex pairs have been annihilated. The latter result is not surprising, since the reduction of the relative distance between opposite-charged vortices favours the attractive interaction, leading to annihilation [25]. This could also explain the saturation behavior suggested by the shape of the curves, even when the net topological charge tends to zero as N raises. The birth of vortices for smaller values of N should not be surprising either, since in the speckle generation by a diffuser, for instance, all the vortices are in fact created by the interference of the randomly scattered waves. Another aspect worth of emphasizing is that the final number of vortices is always smaller for the AH arrays (green curve) than for the RH arrays (blue curves), in consistence with Figs. 4(a)–4(b), where the vortices look closer together in the propagated field for the RH array.

On the other hand, we calculated the Probability Density Function (PDF) of the intensity for the ensembles in the region of interest. It is worth mentioning that the PDF of the intensity is one of the most important properties used to characterize speckle patterns in literature. A standard speckle pattern obtained with a diffuser, for instance, has an exponential decay PDF. This is known as a well-developed speckle [1]. Therefore, we will compare the PDF’s of the ensembles of patterns obtained for the different vortex arrays with exponentially decaying functions. Figure 5 shows, on the left, examples of the analyzed speckled fields and the PDF plots in the center, for RH (top) and AH (bottom). The insets are semilog plots of the PDF’s, to facilitate the comparison with the best exponential fit for N = 900, shown with the dashed curves. For the sake of clarity in the figures, the best exponential fits for the other values of N are not shown. Yet, we can see that the PDF’s for N = 25 (green curves) are far from an exponential, but as the values of N increase, these curves gradually approximate to it. In the AH case, the exponential fit is very good not only for N = 900 (red) but also for N = 400 (yellow), in contrast with the RH case, indicating that the former is the best choice for producing well-developed speckle.

 

Fig. 5 Left: Numerical statistical analysis. Left: Examples of the analyzed fields (N = 900) in the region of interest. Center: Probability Density Functions of the intensity for N =25 (green), 100 (blue), 225 (purple), 400 (yellow) and 900 (red), and their comparison with a negative exponential function of a well-developed speckle (dashed black curves). Insets: semilog plots of the PDF’s. Right: Normalized autocorrelation of the intensity.

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Finally, we calculated the ensemble average of the autocorrelation function of the intensity, according to the following expression

5. Conclusions

We have demonstrated a new path for the generation of speckle patterns by means of phase masks with an array of N vortices with topological charges ±1, imprinted on a carrier Gaussian beam of waist radius w₀. An important condition is that the vortices were forced to interact strongly by passing through a focal region and the resulting field was characterized at a given transverse plane, where the carrier beam would have recovered its original size in the absence of the vortices. We found that it is indeed possible to generate speckled patterns for N ≥ 100, when the initial arrays have a degree of spatial disorder and either random helicities or anticorrelated helicities with respect to the nearest neighbors. However, well-developed speckle was obtained only in the latter case for N ≥ 400, as unveiled from the analysis of the ensemble-averaged Probability Density Function of the intensity. This result is consistent with previous reports on the sign anticorrelation of vortices in speckle patterns generated by standard means [17,27]. Regarding the mean value of the pitch P, we found that the optimum is such that the array of the phase mask is entirely contained within the host beam, imposed by the condition $P\approx w_0/\left[\sqrt{2}\left(\sqrt{N}-1\right)\right]$.

The number of vortices in the propagated field as a function of the initial number of vortices was also investigated. Interestingly, we found that for a small number of initial vortices, there is a trend to the creation of vortex pairs during the interaction, while for a large number of initial vortices the trend is to the annihilation of vortex pairs due to the closer mean distance between them. This leads to a saturation in the final number of vortices, even when the net topological charge tends to zero. In any case, the number of vortices is not conserved in general.

So far, very few investigations have been dedicated to the interaction of a large number of vortices in the linear regime. Our results show that there are still new relevant aspects about vortex dynamics that can be further explored, whose implications may go well beyond the realm of optics, since vortices appear in many other physical systems, such as superfluids, superconductors, Bose-Einstein condensates, acoustical fields, etc.