## Abstract

Clustered speckle, optical lattices, and their optical vortex array are subjects of interest in optical wave manipulation. In this study, disordered optical lattices and vortex arrays with different unit structures were found in the clustered speckles generated by a circularly-distributed multi-pinhole scattering screen when it was illuminated by coherent light. These structures included hexagonal lattices, kagome lattices, and honeycomb lattices. Moreover, optical lattices with asymmetric units generated by modulation of phases with non-integer multiples of 2π were discussed. Theoretical analysis and numerical calculations demonstrated that optical lattices in clustered speckles in the observation plane were generated by the phase modulations of the random scattering screen. The lattice type depended on the number of 2π multiples of the summed phase difference between the pinholes. Additionally, the conditions for the formation of periodical optical lattices and their vortex arrays were given. Different optical lattices and their vortex arrays appearing simultaneously in the clustered speckle were difficult to generate using the common multi-beam interference system. This phenomenon is of great significance in the study of the orbital angular momentum of photons and other fields.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Speckle is an interesting topic for investigation in optical fields [1], playing an important role in bioimaging [2–4], blood flow microcirculation detection [5–7], and underground reconnaissance [8]. Recent studies suggested that speckles with a network-like structure can be generated on Fraunhofer or Fresnel diffraction planes when the scattering screen was illuminated by coherent light after passing through a ring slit. This structure is considered to be an aggregation of speckle grains, and is known as a clustered speckle [9,10]. Clustered speckle is essentially the transition of an optical field from random disorder to order. This transition makes random manipulation of optical fields possible, and holds promise for application in optical trapping [11], micro-manipulation and interferometry. Furthermore, the optical vortex of a speckle field is the basic phenomenon that has initiated the study of optical vortices [12], and has been investigated extensively due to its correlations with the orbital angular momentum of the optical field [13–15]. The presence of phase vortices in clustered speckle is a new aspect of speckle studies and requires further investigation.

Generating optical lattices with a periodic light intensity structure using multi-beam interference is a major topic in optical field manipulation. Berkhout *et al.* [16] generated different forms of optical lattices, such as kagome and hexagonal lattices, by introducing a fixed phase difference between neighboring pinholes that were circularly distributed using Gauss–Laguerre illumination beams of different orders. By introducing similar phase modulation using multi-beam interference and spatial optical modulator, a diffraction-free kagome optical lattice, superstructure optical lattice, and two-dimensional dual-lattice optical structure were obtained by Boguslawski *et al.* [17], Tsou *et al.* [18], and Kumar and Joseph [19]. Pinholes with a spiral distribution were used to generate a vortex light field and optical lattices with plane wave illumination [20]. Over the past few years, optical lattices have been widely used in quantum physics and condensed matter physics, specifically in cold atomic trapping, orbital superfluids, and Bose phase, due to the unique field force formed by the periodic field intensity of the lattices [21–25]. In this study, an opaque screen with multiple scattering pinholes evenly distributed in a circle was used to generate speckle field. The interference of scattered light waves from different pinholes gave rise to speckle clustering or aggregation, and local optical lattice structures were formed. Large-scale inhomogeneous speckle grains were generated because of the limited scattering area of each pinhole. Due to the random phase modulation of scatterer, clustered speckles contained different types of local optical lattices. The clustered speckles were characterized by different optical lattices, and also demonstrated the properties of phase vortices [26]. Compared with the generation of different lattices using complex optical components and systems in the abovementioned studies, kagome lattices, honeycomb lattices, and hexagonal lattices of random distributions can be easily achieved in the clustered speckles in this study. Experimentally, the phase distributions of the speckle field are extracted by realizing interference between the speckle field and reference wave with the Mach–Zehnder interferometer system. Characteristics of optical lattices in the speckle field and their optical vortices were analyzed through the combination of numerical simulation of the optical field and experimental results. Our results indicated that random transitions of optical lattice types were determined by the phase gradient of inhomogeneous large-scale speckle grains within the observation plane. The radius of the circle where pinholes were distributed corresponded to the density of optical lattices.

## 2. Experimental results of speckle field generated by the random surface of a multi-pinhole screen

Six pinholes uniformly distributed along a 3 mm diameter circle were used in the scattering screen, as shown in Fig. 1(a). Pinholes with a radius *r* of 0.07 mm were fabricated on a piece of aluminum foil using femtosecond laser ablation. The angle between two neighboring pinholes to the center was 2π/6. The scattering screen was finalized by attaching the pin-holed aluminum foil to a piece of ground glass with an optical square clamper. The experimental setup is shown in Fig. 1(b). The light source used was a 532 nm laser, which passed through the attenuator A1, and was then expanded and filtered by a spatial filter SF. A convex lens L1 was used to collimate it into a parallel beam. The beam went into the Mach–Zehnder interferometry system through the beam splitter BS1. Then, the reference beam transmitting through BS1 passed through the adjustable neutral attenuator A2 and was reflected by mirror M1. The beam reflected by BS1 was again reflected by mirror M2 and restricted by iris aperture IA. Then the beam normally illuminated the random surface sample of the multi-pinhole scattering screen, and exited as a scattered light wave. The sizes of the clustered speckle grains generated on Fraunhofer plane were adjusted by changing the distances between concave lens L2 and convex lens L3 and their distances to observation plane. Then, the speckle field was interfered with the reference light in the observation plane after passing through the beam splitter BS2. Interferogram formed by interference of the speckles and the reference beam was observed and recorded by an S-CMOS (Scientific Complementary Metal Oxide Semiconductor, pixel size 6.5 μm × 6.5 μm, 2560 × 2160 pixels). For a better interference result, A2 was placed between BS1 and M1 to adjust the reference beam to match the intensities of the speckle fields which were greatly weakened by the small pinhole size of the screen. The intensities of the speckle patterns could be recorded directly by blocking the reference beam in Mach–Zehnder interferometry system. The speckle image recorded by the S-CMOS is shown in Fig. 2(a). It was evident that the speckle pattern had a network intensity distribution in the background made up of large-scale speckle grains. The variations of the background resulted in a whole field random change of lattice and thus clustered speckles were formed. In a local region of interest, however, the shapes of small speckle grains in a given cluster did not vary greatly or even remained constant, with a favorable periodicity by which local optical lattices were formed. Across a larger region, the lattice structures comprised of clustered speckles also changed, and the lattice types were in accordance with those of deterministic lattices generated by the multi-beam interference and multi-pinhole screen interference [16]. The structures of various clustered speckle lattices could be observed in different local speckle patterns which were enlarged by adjusting the positions of L2 and L3. Enlarged speckle patterns taken at different locations are shown in Figs. 2(b)–2(d). Each pattern contained different types of optical lattices. To make it more convenient for observation, a sub-region of interest was selected, further enlarged, and shown as the inset pattern in the upper-right corner of each Figure. The patterns in Figs. 2(b), 2(c), and 2(d) were typical single lattice or lattice unit cell of the hexagonal lattice, kagome lattice, and honeycomb lattice, respectively. Those lattice patterns were also observable in large-scale clustered speckle images, as shown in Fig. 2(a).

From interferograms of interference between the speckle field and reference light, and the phase distribution of three speckle patterns was obtained using Fourier transformation, as shown in Figs. 2(e)–2(g). Interferograms and phase maps corresponding to inset speckle patterns were also presented as insets in the upper left and upper right corners of the figures, respectively. Though the optical vortices in the circled areas were determinable from both inset maps, use of the interferograms to obtain the locations and numbers of vortices is more direct and dependable. A fringe fork in interferogram represents appearance of a vortex with fringes gained or lost determine the charge of the vortex [27]. In the circled area in Fig. 2(e), there existed one vortex of charge one, and in the circled area in Fig. 2(f), there existed two vortices with charge one of the same signs. In Fig. 2(g), the case was more complicated. Intuitively, in the circled area there were six vortices, composing of three pairs of vortices. In each pair, the vortices carried charge one but with opposite signs, with total charge equal zero. Such vortex pair could form easily in speckle field [14,28]. In fact, the center of a perfect honeycomb lattice was a point of intensity zero with zero charge [29], but for the practical non-perfect honeycomb lattice, the disturbance in light field from scattering of random screen surface might cause the vortex pairs to form inevitably.

Our results showed that the clustered speckles generated by the multi-pinhole scattering screen included different types of lattices with variable unit cells and vortex arrays. These results differ from published studies, in which only one type of optical lattice or lattice array could be generated using a specific method or optical system, and it was difficult to obtain such abundant optical lattice arrays within one optical field [16]. To investigate this experimental phenomenon and the mechanism of formation, theoretical analysis and simulation of the speckle field generated by the multi-pinhole scattering screen were conducted in the following.

## 3. Numerical calculation of clustered speckle fields generated by a multi-pinhole scattering screen

The light waves scattered from the rough surface could be considered as being randomly modulated by the scatterers on the surface, and the wave field at the observation plane was the coherent superposition of numerous scattered wavelets with independent phases. As shown in Fig. 3, a parallel laser beam of wavelength *λ* and unity amplitude perpendicularly illuminated a circular-aperture followed by random scattering screen located at *Ox*_{0}*y*_{0}, and the complex amplitude of scattered wave field formed at point *Q*(*x*,*y*) on Fraunhofer plane *xyz*_{0} is

*h*(

*x*

_{0,}

*y*

_{0}) is the height function of a random surface,

*P*(

*x*

_{0},

*y*

_{0}) is the aperture function of the scattering area,

*z*is the distance between the observation surface and the random surface, and

*n*is the refractive index of transparent screen. Ground glass was made with transparent dielectric glass plate, its absorption could be neglected. The random surface height distribution might cause small variation of local incident angle of light, and could result in minor variations of transmittance and polarization orientation when light emerged from ground glass. Under the small slope approximation in the theory of elastic scattering of light [30], and paraxial approximation that was satisfied in our experiment, the variations were so small that they had little influence on the properties of the speckles, and pure phase modeling for ground glass was safe.

The complex amplitude *F*(*x*,*y*) at *Q*(*x*,*y*) on the observation plane can also be written in the general form *F*(*x*,*y*) = *A*(*x*,*y*)exp[*iφ*(*x*,*y*)], and it can also be denoted as *F*(*x*,*y*) = *f _{re}* (

*x*,

*y*) +

*if*(

_{im}*x*,

*y*), where

*f*and

_{re}*f*are the real and imaginary parts, respectively. The amplitude is

_{im}*A*(

*x*,

*y*) = {[

*f*(

_{re}*x*,

*y*)]

^{2}+ [

*f*(

_{im}*x*,

*y*)]

^{2}}

^{1/2}and the phase is

*φ*(

*x*,

*y*) = arctan(

*f*/

_{im}*f*). The aperture function

_{re}*P*(

*x*

_{0},

*y*

_{0}) took the value of one if it fell within the range of pinholes and zero if it was outside the ranges pinholes. Then, the aperture function of the

*l*

^{th}pinhole in a six-pinhole scattering screen could be written as

*x*

_{0}

*,*

_{l}*y*

_{0}

*) is the center and*

_{l}*r*is the radius of the aperture. The six-pinhole aperture function was $P\text{(}{x}_{0},{y}_{0}\text{)}={\displaystyle \sum _{l=1}^{6}{P}_{l}\text{(}{x}_{0},{y}_{0}\text{)}}$. In the numerical calculation of the speckle field, the surface height function

*h*(

*x*

_{0},

*y*

_{0}) was replaced by a set of random numbers, with the factor 2π(

*n*−1)

*h*(

*x*

_{0},

*y*

_{0}) uniformly distributed in [0, 2π]. The radius of each pinhole was 70 μm, the radius of the circle on which the six pinholes were evenly distributed was 1500 μm, and

*z*= 45 cm, and

*λ*= 0.532 μm. All other parameters were set to those used in the experiment. Based on Eq. (1), the real part

*f*and imaginary part

_{re}*f*of the complex amplitude of the speckle field were calculated, and the intensity

_{im}*I*(

*x*,

*y*) and the phase

*φ*(

*x*,

*y*) were obtained.

A typical speckle intensity pattern is presented in Fig. 4(a). The display region was 5 mm × 5 mm. This intensity pattern also indicated that clustered speckles were distributed in lattices, in agreement with the experimental results. Enlarged patterns of the three typical random lattices are presented in Figs. 4(b)–4(d), in which three unit cell lattices are marked with yellow circles. Phase distribution of enlarged speckle images are shown in Figs. 4(e)–4(g). These images suggested consistency between the calculated clustered speckle lattices and experimental results. Thus, additional results of clustered speckles could be analyzed by means of numerical calculation.

## 4. Theoretical analysis of a scattering multi-pinhole aperture-generated clustered speckle field

In experimental and calculated speckle patterns, the large-scale speckle grains were the slowly-varying background and were formed by fields scattered from the random surface in pinholes of limited size, while the fast-varying small-sized local optical lattices were originated from interference of the fields related to the central positions of pinholes. This fact reflected the main contributions of the two factors to the formation of the clustered speckle, though more seriously, their contributions were actually correlated. To justify this mechanism, further analyses were needed for clustered speckles. Beginning with speckle field generated by a single pinhole, from Eq. (1), the speckle field ${U}_{fl}\text{(}{x}_{f},{y}_{f}\text{)}$ generated by the *l ^{th}* pinhole is Fourier transformation of product aperture function ${P}_{l}\text{(}{x}_{0},{y}_{0}\text{)}=\text{circ(}\sqrt{{x}_{0}^{2}+{y}_{0}^{2}}\text{/}r\text{)}$ $\otimes \delta \text{(}{x}_{0}-{x}_{0l},{y}_{0}-{y}_{0l}\text{)}$and phase factor $\phi \text{(}{x}_{0},{y}_{0}\text{)}=-2\text{(}n-1\text{)}\pi h\text{(}{x}_{0},{y}_{0}\text{)}/\lambda $. Here $\otimes $ represents convolution operation. With the center of the pinhole at $\left({x}_{0l},{y}_{0l}\right)$,${U}_{fl}\text{(}{x}_{f},{y}_{f}\text{)}$ can be written as

In the above equation, the subscript *A _{l}* means that the integral of Fourier transformation is only limited to the integral of the

*l*pinhole, and subscript

^{th}*C*represents the Fourier transformation after translating phase term $\phi \text{(}{x}_{0},{y}_{0}\text{)}$ to the center (

*x*

_{0}= 0,

*y*

_{0}= 0) while maintaining the corresponding height distribution inside the

*l*pinhole, which is actually a result of translation theorem of Fourier transform. For ease of description, we first considered the condition of infinitely small pinhole,

^{th}*i.e.*$r\to 0$. Then, ${U}_{fl}\text{(}{x}_{f},{y}_{f}\text{)}$ could be written as

*r*decreased to the size of a scatterer. At this extreme (

*r*equivalent to the size of the scatterer), the light wave generated by the pinhole on the Fraunhofer plane is regarded as planar wave modulated by the random phase ${\phi}_{l0}$. Similarly, in Eq. (3), the spectrum of the speckle field ${U}_{fl}\text{(}{x}_{f},{y}_{f}\text{)}$ was determined by the high-frequency plane wave factor $\text{exp[}i2\pi \text{(}{x}_{0l}{x}_{f}+{y}_{0l}{y}_{f}\text{)]}$ and the light wave ${E}_{l}\text{(}{x}_{f},{y}_{f}\text{)}$ filtered by the circle function of the pinhole of radius

*r*. Intuitively, the grain sizes of the speckle field ${U}_{fl}\text{(}{x}_{f},{y}_{f}\text{)}$ given by Eq. (3) were enlarged by the smoothing convolution of${E}_{l}\text{(}{x}_{f},{y}_{f}\text{)}$with the point spread function ${J}_{0}\text{(}2\pi \rho r/\lambda f\text{)/(}\pi \rho r/\lambda f\text{)}$, which had an extended range. Compared with the phase of high-frequency plane wave factors, the intensity and phase of this speckle field varied slowly as $\text{(}{x}_{f},{y}_{f}\text{)}$ changed within a large speckle grain of low spatial spectrum. Therefore, the optical field generated by a single pinhole could be considered a low-frequency speckle field modulated by high-frequency plane wave factors. Thus, ${U}_{fl}\text{(}{x}_{f},{y}_{f}\text{)}$ is

Next, the speckle field generated by the multi-pinhole scattering screen was considered. Taking the uniformly distributed six-pinhole screen as an example, the sum of light waves generated by six pinholes is

In the above equation, superscript $\ast $ denotes the complex conjugate. ${U}_{fl}^{\text{(}s\text{)}}\text{(}{x}_{f},{y}_{f}\text{)}{U}_{fm}^{\text{(}s\text{)*}}\text{(}{x}_{f},{y}_{f}\text{)}$ represents the interference of low-frequency speckle fields produced by any two pinholes, and from statistical point of view, each of the low-frequency speckle field is the same as the optical field generated by a single pinhole. When only the interference of low-frequency fields was considered, the large grain speckles are formed, which are the speckled background in Fig. 2(a) and Fig. 4(a). The term in the summation symbol of Eq. (9) can be denoted in the form of fringe:

*q*π, the lattices with different shapes could be obtained for different

*q*, as shown in Figs. 2(b)–2(g) and Figs. 4(b)–4(g). This indicated that the speckle field at points in localized areas in the observation plane underwent different phase modulations caused by height variations of the random surface. To investigate the patterns and phase vortices of lattices in clustered speckles at different phase differences$\Delta {\phi}_{lm}$ for two neighboring pinholes, the patterns and phases of the optical lattices with deterministic phase modulation were calculated. The height function $h\text{(}{x}_{0},{y}_{0}\text{)}$ and phase $\phi \text{(}{x}_{0},{y}_{0}\text{)}$ within a pinhole were set as constants, the phase difference of any two neighboring pinholes was the same, and the total phase difference was 2

*q*π. Calculation results are shown in Fig. 5. Phase modulations $\phi \text{(}{x}_{0},{y}_{0}\text{)}$ of all the pinholes are presented in Figs. 5(a)–5(f). Along the circumference, the total phase modulation values of the pinholes were 1π, 2π, 3.5π, 4π, 5.5π, and 6π in Figs. 5(a)–5(f), respectively, corresponding to

*q*0.5, 1, 1.75, 2, 2.75, and 3, respectively. The intensity patterns of the deterministic lattices generated by the interference between light waves of six pinholes are shown in Figs. 5(g)–5(l), with the corresponding phase maps presented in Figs. 5(m)–5(r). These results suggest that when

*q*is an integer, the lattice patterns are similar to the results shown in Fig. 2 and Fig. 4. It should be noted that it would be more difficult to implement optical lattices and vortices using multi-beam interference when

*q*was not an integer, as shown in rows 1, 3, and 5 in Fig. 5, and few researchers have investigated this before. As shown in the figure, the patterns of the lattice unit cells are marked by circles. Those cells were not simultaneously axially or rotationally symmetric, which differed from the patterns observed when

*q*was an integer. Compared with the case when

*q*was an integer, the distribution of phase vortices was complex, but there were no phase vortices with a topological charge of the non-integer

*q*or high-order vortices with topological charges larger than 1. Identical or similar lattice patterns were identified upon careful observation of large-scale clustered speckle patterns in Fig. 2(a) and Fig. 4(a), which indicated that there were lattices generated by phase modulation where

*q*was not an integer in the clustered speckles by the multi-pinhole scattering screen.

It was interesting to note that the non-integer *q* for phase difference modulation could not generate the optical vortex of non-integer charge. This might be originated from the fact that non-integer charge carrying optical vortices could not propagate and that they were known to scatter into optical vortex with its charge equal to nearest integer value [31]. In addition, the value of integer *q* was not always related to the topological charge of optical vortex. This could be demonstrated in the case of *q* = 3 in Fig. 5(r), where the phase profile of honeycomb lattice is binary valued, and they were intensity zeroes with zero charge. Moreover, by use of gradient basis structures [29], honeycomb lattice had been generated as the superposition of linear combination sets of symmetrical 6-beams and 3-beams carrying zero starting phases, i.e., with *q* = 0, and this further demonstrated the zero charge of lattice.

Lattices could also be obtained using the *q*-order Gauss–Laguerre light beam to illuminate the six-pinhole screen [16] or using the parallel light to illuminate the screen along a spiral [20] when *q* was an integer. When *q* was not an integer, however, the former method required non-integer-order Gauss–Laguerre beam for the illumination, which was difficult to realize. For the latter method, the phase differences from the centers of the six pinholes along the spiral to a point near the origin on the observation plane must be non-integer multiples of 2π, which was possible, but there has been no research generating lattices with such schemes. Since the screen contained abundant phase components and was highly capable in phase modulation, a variety of optical lattices and vortex arrays were included in clustered speckle fields generated by the scattering screen.

## 5. Discussion

Similar to the common speckle system, the characteristics of clustered speckles were subject to the size of the scattering aperture and position of the observation plane. Variations in clustered speckle with pinhole size and diameter of the circle on which the pinholes are arranged were calculated. Clustered speckle patterns with *R* = 2000 μm and *r* = 100 μm, *R* = 2000 μm and *r* = 200 μm, *R* = 1000 μm and *r* = 100 μm, *R* = 1000 μm and *r* = 200 μm are shown in Figs. 6(a)–6(d), respectively. The evolution of the speckle lattice unit cell and large-grain background with the sizes of pinhole and the circle is intuitively demonstrated in these figures. Background grains primarily depended on the pinhole size; the smaller the pinhole, the larger the background grains were. The size of lattice unit cell was determined by the radius of circle. A larger circle corresponded to a smaller unit cell lattice. The speckle pattern in the case of the largest circle radius and smallest pinhole radius is shown in Fig. 6(a), where the ratio between the circle radius and pinhole radius was at the maximum of 20, and with which the lattice generated showed the best periodicity and regularity. The speckle pattern in the case of the smallest circle radius and largest pinhole radius is shown in Fig. 6(d), where the ratio between the circle radius and pinhole radius was at the minimum of 5, and with which clustered speckles were irregular. Unit cell lattices could still be found, but the periodic distribution of lattices in clustered speckle was diminished, which was consistent with previous studies on clustered speckles [32]. According to the results in this study, when $R/r<5$, it is difficult to obtain optical lattices and optical vortex arrays with a good periodicity in clustered speckles.

## 6. Conclusion

In this study, optical lattices and vortex arrays generated by multi-pinhole scattering screens and their properties were investigated experimentally and theoretically. Using a six-pinhole scattering screen as an example, the modulation of the multi-pinhole scattering screen on clustered speckles was explored, and clustered speckles with different unit cell optical lattices were obtained. These unit cell lattices include typical hexagonal lattices, kagome lattices, and honeycomb lattices. This study is the first to investigate the lattices generated by phase modulation with non-integer multiples of 2π. The phenomena observed in the experiment were further verified by simulation calculation, and the underlying mechanism was analyzed. Specifically, multiple optical lattices are generated due to the abundant phase components of the scattering screen and their ability for phase modulation. Phase near an observation point varies due to the influence of pinhole position, scattering unit cells within individual pinholes, and interference between light waves from all the pinholes. If the sum of the phase differences of light waves between pinholes is a multiple of 2π at the area near an observation point, the clustered speckles of corresponding types of optical lattices will form. Further analysis indicates that density of lattices in clustered speckles are related to *r* and circle radius (*R*); for optical lattices and vortex arrays with good periodicity and regularity to be formed, the condition of *R/r ≥* 5 must be satisfied. The findings of this study are of great significance for optical lattice design, orbital angular momentum of photons, and speckle measurement, as well as their application.

## Funding

National Natural Science Foundation of China (NSFC) (11574185, 11604183).

## Acknowledgments

We thank LetPub (www.letpub.com) for its linguistic assistance during the preparation of this manuscript.

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