## Abstract

We investigate the number of vortices embedded in a carrier beam needed to produce a speckle pattern and the necessary conditions in terms of their initial distribution and topological charges. A spatial light modulator is used to imprint arrays of vortices in a Gaussian beam, which is propagated in free space for a given distance and then focused in order to induce interaction among the vortices in the focal region. The resulting optical field is analyzed after propagation up to a transverse plane where the carrier beam would recover its initial size in the absence of vortices. The role of different control parameters for obtaining ordered and disordered patterns is discussed. Our experimental study is complemented with a thorough numerical analysis, from which the statistical properties of the disordered patterns are characterized, and the conditions for obtaining well-developed speckle are determined. We also discuss the creation and annihilation of vortex pairs, depending on the initial conditions.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 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
2*π*-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 *w*_{0}, whose complex amplitude is
given by

*x*,

_{n}*y*) and

_{n}*β*= ±1 define the coordinates and the helicity of the

_{n}*n*-th vortex, respectively, and $i=\sqrt{-1}$. These kinds of fields have been extensively studied before in both the linear and nonlinear regimes [25,26,28–31] including the simplest case of a single vortex pair [32–34]. For example, Indebetouw analyzed its free-space propagation from the beam waist and determined that there is an attractive interaction between vortices of opposite topological charges, which may lead to the annihilation of the vortex pair, depending on their initial separation distance [25]. Roux studied the limitation of the vortex density at a fixed transverse plane, finding that if the net topological charge in a given area exceeds the circumference of the area divided by the wavelength, light inside that region is depleted in the form of evanescent waves [30]. Our study is also situated in the linear regime, but in contrast to previous work, we imprint the vortices in a wide carrier beam, which is then focused with the purpose of reducing the relative distances among them and force a strong interaction, as we shall explain next.

Our experimental setup is depicted in Fig. 1(a).
A collimated laser beam of wavelength *λ* =
532nm and a measured radius of *w*_{0} =
1688*μ*m, 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 20*μ*m, 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* =
2*f*, 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 (2*w*_{0}) is delineated by the
green dashed circumference. The propagated field in the plane of analysis
is also shown.

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
*w*_{0}. 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}* +
(0.4

*P*)

*δ*,

_{x}*y*+ (0.4

_{n}*P*)

*δ*), where

_{y}*δ*and

_{x}*δ*are independent discrete random variables, which may take the values of −1, 0 or 1 with the same probability. The factor of 0.4

_{y}*P*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.

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.

## 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* = 2*f* from the lens will be limited in the transverse plane to a circular area of radius *r _{win}* =

*w*

_{0}. 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,

*w*

_{0}= 1688

*μm*. 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* = 2*f*, becomes self-evident from the shape and proximity of the outer contours.

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.

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

*I*(

*x*,

*y*) and

*Ī*represent the intensity distribution and its mean value for each realization, and the symbol ★ denotes the standard autocorrelation operation as defined in [36]. The results for

*C*, normalized to its maximum value, are presented in the plots of the right column of Fig. 5 for all the different cases. In a well-developed speckle, the FWHM of this function provides an indicative of the speckle size [1]. Although it might be questionable to associate such quantity to the patterns obtained for

_{I}*N*= 25, the trend of a width reduction for larger

*N*is very clear. This is consistent with the examples of patterns presented in Section 3 (see Fig. 2).

## 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 *w*_{0}. 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.

## Funding

Instituto de Física UNAM (PIIF-9); DGAPA-UNAM (PAPIIT projects IN114517, IN103818, IA104917); and CONACyT (Ciencia Básica 254942 and Laboratorio Nacional 293471).

## Acknowledgments

K.V-S also acknowledges support from Cátedra Marcos Moshinsky 2014, and F.J.P-C thanks to Cátedras CONACYT. Authors acknowledges useful discussions with Prof. Víctor Romero-Rochin, Dr. Rosario Paredes and Dr. Francisco Sevilla.

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