Controllable beam reshaping by mixing square-shaped and hexagonal optical vortex lattices

In the present work we show experimentally and by numerical calculations a substantial far-field beam reshaping by mixing square-shaped and hexagonal optical vortex (OV) lattices composed of vortices with alternatively changing topological charges. We show that the small-scale structure of the observed pattern results from the OV lattice with the larger array node spacing, whereas the large-scale structure stems from the OV lattice with the smaller array node spacing. In addition, we demonstrate that it is possible to host an OV, a one-dimensional, or a quasi-two-dimensional singular beam in each of the bright beams of the generated focal patterns. The detailed experimental data at different square-to-hexagonal vortex array node spacings shows that this quantity could be used as a control parameter for generating the desired focused structure. The experimental data are in excellent agreement with the numerical simulations.

Scientific RepoRts | (2019) 9:2128 | https://doi.org/10.1038/s41598-019-38608-5 TC of an isolated OV or even of a large OV array [24][25][26] is an essential part of the physics of the present results. We demonstrate that this is not only valid for a single OV, but also holds for OV lattices (square-shaped and hexagonal) composed of hundreds of OVs on a single background beam. We show that the small-scale structure of the observed pattern results from the OV lattice with larger array node spacing, whereas the large-scale structure comes from the OV lattice with the smaller array node spacing. The detailed experimental data at different square-to-hexagonal vortex array node spacings presented in this paper prove that this quantity can be used to control the generation of the desired focused structure. We confirm the ability to host an OV, a 1-D or a quasi-2-D dark beams in each of the bright beams of the far-field pattern. The numerical data are shown to be in excellent agreement with the experimental results.

experimental setup and Numerical simulations
The experimental setup is shown in Fig. 1. Pump beam from a continuous-wave frequency-doubled Neodymium-doped yttrium orthovanadate (Nd: YVO 4 ) laser is first expanded by the beam expander BE and then illuminates the first reflective spatial light modulator SLM1. This SLM modulates the phase of the input Gaussian beam (and, as a consequence, also its amplitude/intensity) and redirects it to a second spatial light modulator SLM2 of the same type. The singular beam reflected from SLM2 is then focused by a lens L (f = 100 cm) onto a CCD camera chip of 1600 × 1200 pix. (7.2 × 5.4 mm). The SLM2-to-lens distance is 95 cm. A reference beam is split off the laser beam before SLM1 by a beam splitter (BS1). For diagnostic purposes, the object and the reference beams are recombined by a second beam splitter (BS2) to interfere at the CCD camera chip. Intensity distributions of the resulting optical beams and the respective interference patterns are recorded by the same camera by blocking/unblocking the reference laser beam. SLM1 and SLM2 are aligned in parallel with a distance of 49 cm. At the SLMs the angle of incidence of the laser beam with respect to the normal incidence is 4°. The efficiency of the single OV lattice generation is 71% ensuring overall efficiency of nearly 45%. The reason to use two SLMs instead of imaging SLM1 onto SLM2 is to include in the experiment the amplitude modulation resulting from free-space propagation between the modulators and from the second modulator to the lens. According to this choice, in the numerical simulations we considered the evolution of the manipulated light field after each of these elements until the artificial far field is reached. Both the setup and the simulations would simplify considerably when imaging SLM1 onto SLM2 thereafter Fourier-transforming the plane of SLM2. The amplitude and phase modulation could be tailored in first diffraction order of the SLM by applying a suitable weighted blazed grating to the modulator 27 .
Since the propagation of the laser beam in the object arm of the setup is linear (Fig. 1), its evolution was numerically calculated by using the linear paraxial model equation for the slowly-varying optical beam envelope amplitude E Here the transverse part of the Laplace operator is denoted by Δ T , the diffraction length of an individual OV -by L Diff = ka 2 , and the wave number in air is k. In a computational window spanning over 1024 × 1024 points the half width at the 1/e 2 intensity level of the host Gaussian beam was 205 pix. By phase modulation, using SLM1, we generated a single OV and recorded its profile right after SLM1 and at the position of the second SLM. From the data we concluded that the distance between SLM1 and SLM2 corresponds to 1.5L Diff and the distance from SLM2 to the lens -to 3.0L Diff . The presence of a lens in the experimental setup was accounted for by the transmission phase function T(x, y) of a thin lens with a focal length f 2 2 in the respective plane. The beam evolution was then numerically followed to the focal plane (41.0L Diff behind the lens).
In Fig. 2 we show numerical results for the separate creation of a large square-shaped (a1-e1) and a hexagonal lattice (a2-e2) of OVs with alternatively changing TCs. In both cases the array node spacing is 41 pix. In the experiment, separate OV lattice generation means that one of the SLMs is either switched off thus acting as a mirror, or it is programmed with a flat phase distribution. The intensity profile of the background beam illuminating SLM1 is shown in frames (a1, a2). The phase distributions sent to SLM1 are displayed in the second column, where the phase profile of the square-shaped OV array is shown in panel (b1) and the one of the hexagonal OV array in panel (b2). The pairs of white circles in frames (b1) and (b2) denote two neighboring OVs. Their opposite TCs can be recognized by the fact that the phase of one of the OVs changes from black to white (i.e. from 0 to 2π) around the center of the circle in clockwise direction, in the neighboring circle -in counter-clockwise direction. The respective simulated intensity distributions just in front of the focusing lens are presented in frames (c1) and (c2). In both cases, a rotation of the OV lattices is suppressed by the opposite signs of the TCs of neighboring OVs as expected 9,10 . In the plane of the lens the phase profile of the lens (identical frames (d1, d2)) is added, causing the beam to converge into a focus. The resultant intensity profiles of the beams in the artificial far field are shown in the last column of frames in Fig. 2. When a square-shaped OV lattice is encoded on one of the SLMs, the calculated far-field intensity profile (frame (e1)) clearly shows the presence of four peaks situated in the apices of a rhomb. When a hexagonal OV lattice is used on one of the SLMs, the calculated far-field intensity (frame (e2)) is composed by three dominating peaks situated in the apices of an equilateral triangle. This bright peak triangular structure is inscribed in a another less intensive triangular structure of beams. The numerical results obtained for two times more dense lattices (node spacing of 21 pix.) are qualitatively the same. In each frame of Fig. 2 spanning over (−6, 6) arbitrary units 6% of the total computational area is shown.

Results and Discussion
In Fig. 3 we show numerical simulations (a-d) and experimental results (e-g) for the case of mixing square-shaped OV array (lattice constant Δ sq = 41 pix.) with a hexagonal OV lattice with lattice node spacing of Δ hex = 101 pix.
In the particular simulation, the square array is encoded on SLM1 and the hexagonal one is encoded on SLM2.
In frame (a) the OVs constituting one elementary cell of the hexagonal lattice are encircled. In this way we show the large ratio between the node spacings of both lattices and the broadening of each OV of the square lattice propagating from SLM1 to SLM2 as compared to the width of the newly-born OVs from the hexagonal lattice just after SLM2. All OVs continue to propagate in free space (and to broaden due to the diffraction) to the focusing lens L (frame (b)). In frame (c) we show the calculated far-field intensity distribution of the mixed square-shaped and hexagonal OV lattices as well as the respective phase profile (frame (d)). As seen in frame (c), the small-scale structure is the one of the hexagonal OV lattice with the larger node spacing. There are three dominating peaks situated in the apices of an equilateral triangle, which is inscribed in a rotated triangle-like structure of gradually less intense beams. Four such small-scale structures appear arranged in the apices of a rhomb thus forming the large-scale structure determined by the square OV array with the smaller lattice constant. The notations small-scale and large-scale structures are of course arbitrary and depend on the ratio between the lattice constants of the respective lattices. Nevertheless, the effect is a beautiful manifestation of the two known features of the Fourier transformation performed by a thin lens, namely: i. the Similarity theorem stating that "wide" functions in the time (space) domain correspond to "narrow" functions in the (spatial) frequency domain. ii. and the Convolution theorem stating that the Fourier transform of the product of two integrable functions is given by the convolution of their Fourier transforms.

Figure 2.
Vortex lattice generation with one SLM. Numerical results for array node spacing 41 pix. Intensity of the background beam (a1,a2) illuminating SLM1, square-shaped (b1) and hexagonal (b2) phase distributions (with pairs of OVs encircled to underline their opposite TCs) sent to this modulator and respective intensity distributions (c1,c2) just in front of the focusing lens. The phase used for simulating the action of a focusing lens is shown in frames (d1,d2). The respective intensity profiles of the sub-beams in the artificial far field for square and hexagonal OV lattice are shown in frames (e1,e2), respectively. In each frame we show 6% of the total computational area.
Scientific RepoRts | (2019) 9:2128 | https://doi.org/10.1038/s41598-019-38608-5 Experimentally recorded intensity distributions (frames (e, g)) and interference pattern (frame (f)) of the mixed lattices are shown in Fig. 3 too. For better visualization we highlighted the small-scale structures in frames (e, g) with a dashed triangle. The same ratio of their lattice constants is used in the numerical simulations. As seen in frame (e), the experimental data perfectly match the numerical ones (frame (c)). Moreover frame (g) in Fig. 3 is confirming experimentally, that the small-scale triangular structures of sub-beams can be rotated by 180° by inverting the signs of all TCs of the OVs constituting the hexagonal lattice or by changing the order of the creation of the individual lattices on the SLMs. The detailed inspection of the encircled portions of the numerically obtained phase profile given in frame (d) shows that all bright sub-peaks have flat phases. This is in agreement with the observed parallel interference stripes in the experiment (frame (f)) when the beam in the reference arm of the interferometer (Fig. 1) is slightly inclined with respect to the beam in the object arm. In the graph in Fig. 3 we present horizontal phase cross-sections of the peaks composing the small-scale structures marked with dashed triangle in frame (g). These cross-sections obtained by the four-frame technique for interferogram analysis 28,29 confirm the flat phases of the bright peaks. The small curvature of the phase profiles in the graph in Fig. 3 is probably due to a small offset of the CCD-camera chip from the focal plane of the lens.
In this way we arrived at the idea to also use the Convolution theorem of the Fourier transformation for additional structuring of the generated sub-beams. We remind ourselves that an OV nested symmetrically on its host beam remains an OV in the focus of the lens. In the left three phase distributions in Fig. 4 we present some 10% of the calculated and used phase profiles of square OV lattices with a removed OV (plot (a1)), and with an additionally encoded 1-D (plot (b1)) and quasi-2-D phase dislocation (plot (c1)). The arrows in this left panel of plots are intended to guide the eye to their respective positions. In contrast to the truly 2-D point phase dislocation carried by an OV, the one-dimensional (1-D) phase dislocation is just a π-phase step along a line (Fig. 4, frame (b1)) resulting in an 1-D dark beam (see Figs 1 and 2 in 30 ) while the quasi-2-D dislocations are crossed 1-D phase steps (Fig. 4, frame (c1); see also Fig. 3 in 31 ). In Fig. 4(a2,b2,c2) we present measured far-field images resulting from combinations of the mixed OV lattices and an OV (Fig. 4(a2)), a 1-D spatial phase dislocation (Fig. 4(b2)), and a crossed 1-D (i.e. quasi-2-D) spatial phase dislocation (Fig. 4(c2)). Indeed, we can generate a rhomboidal large-scale structure consisting of sub-beams ordered in triangles with an OV nested in each of the 12 bright sub-peaks (see Fig. 4, row (a2-a4)). To this end, a single OV with opposite TC is added to (equivalently -removed from) the phase corresponding to the large square-shaped OV lattice. Using the same approach, we encoded also 1-D and quasi-2-D spatial phase dislocations in the generated sub-beams in the far-field (Fig. 4, rows (b2-b4) and (c2-c4), respectively). The particular position of the 1-D phase dislocation in the square-shaped lattice shown in frame (b1) of Fig. 4 is symmetrical with respect to the two nearest rows of OVs. Numerical calculation and measurements for other positions including the situation of 1-D dislocation crossing a row of OVs showed the same power density distribution (frame (b4)). The same was observed when the quasi-2-D phase dislocation was shifted with respect to the position marked in frame (c1) in Fig. 4. We attribute this result to the known feature of the Shift theorem of the Fourier transformation: Shift in the spatial domain results in a linear phase term in the spatial frequency domain. In column (a3-c3) of Fig. 4, we present the respective experimentally recorded interference patterns. In frame (a3) upward interference lines splittings can be recognized confirming that all 12 OVs nested on the bright sub-beams have the same topological charges. In Fig. 4(b3,c3) the interference lines are shifted along a line (b3) or along two crossed lines (c3) by half a period, thus indicating 1-D and quasi-2-D phase shifts of π and, hence, confirming the presence of spatial phase singularities. We confirmed that the results remain similar after changing the OV lattice node spacings while keeping the square-to-hexagonal lattice node spacing ratio Δ = Δ sq /Δ hex less than unity. For the data shown in Fig. 4 Δ = 0.41. Qualitatively similar results were recorded for four additional values of Δ ranging from 0.44 to 0.54. The TCs of the OVs placed in the bright focal peaks in row (a1-a4) in Fig. 4 can be reversed e.g. by erasing an OV with an opposite TC (in our experiments -from the square-shaped vortex arrays). In frames (a4, b4, c4) in Fig. 4 we show the respective computational results obtained for Δ = 0.41. As seen, the simulated additional pattern structuring by adding OVs or 1-D dark stripes is in excellent agreement with the experiment. The numerical data in the last case of structuring the peaks with quasi-2-D dark beams also agree well with the experiment, however in both cases the beam's fine sub-structures appear closely spaced and partially overlap. This can be improved by optimizing Δ = Δ sq /Δ hex .
Following the style of presentation used in Fig. 3, we show numerical simulations (a-d) and experimental results (e-g) for the case of mixing a square-shaped OV array with a lattice constant of Δ sq = 151 pix. with a hexagonal OV lattice with a lattice node spacing of Δ hex = 61 pix. in Fig. 5. The well-pronounced difference between these two figures (compare, e.g., frames (c) and (e) of Figs 3 and 5) is due to the fact that Δ = Δ sq /Δ hex = 0.41 for Fig. 3 but Δ = 2.48 for Fig. 5. In Fig. 5(a) the OVs constituting one elementary cell of the square lattice are again encircled in order to emphasize the large ratio between the node spacings of both lattices and the broadening of each of the OVs of the hexagonal lattice on propagation from SLM1 to SLM2 as compared to the width of the newly-born OVs of the square-shaped lattice just after SLM2. All OVs continue to propagate in free space and broaden due to diffraction until they reach the focusing lens L (frame (b)). In frame (c) we show the calculated far-field intensity distribution of the mixed square-shaped and hexagonal OV lattices for Δ = 2.48 and the respective phase profile (frame (d)). In contrast to the results for Δ = 0.41 shown in Fig. 3, the small-scale structure in frame (c) of Fig. 5 is formed from the square OV lattice with a larger node spacing (151 pix.) -peaks arranged in the corners of three rhombs, whereby the rhombs are arranged in the apices of an equilateral triangle (Fig. 5,  frame (e), dashed rhomb). The triangular large-scale structure is inscribed in a rotated triangle-like structure of much less intense beams. Once again here we see clear manifestation of both Similarity and Convolution theorems for the Fourier transformation. Experimentally recorded intensity distribution (frame (e)) and interference pattern (frame (f)) of the mixed lattices for the same ratio (Δ = 2.48) between their lattice constants are shown in Fig. 5 too. As seen in frame (e), the experimental data perfectly match the numerical ones (frame (c)). It was confirmed experimentally (frame (g)) that the large-scale triangular structure of rhomboidal beam sub-structures can be rotated at 180° by inverting the signs of all TCs of the OVs of the hexagonal lattice or by changing the order of the creation of the individual lattices on the SLMs. A close inspection of the encircled portions of the numerically obtained phase profile given in frame (d) shows that all bright sub-peaks have flat phases, which is unfortunately not obvious without image magnification. This is, however, in agreement with the observed parallel interference stripes in the experiment (frame (f) in Fig. 5) and with the phase cross-sections of the peaks marked in frame (g). The data are obtained by the four-frame technique for interferogram analysis (last graph in Fig. 5).
In contrast to Fig. 4, in which the large-scale structure is a rhomb, in Fig. 6 we demonstrate a triangular large-scale structure. In Fig. 6 we show results confirming the possibility for additional structuring the far field intensity profiles of the mixed square-shaped and hexagonal OV lattices. This was done by adding an OV with an opposite TC (a1-a3), by additionally changing the order of the generation of the respective OV lattices on the SLMs (b1-b3), and by adding a 1-D dark beam (c1-c3). The first column in Fig. 6 shows measured far-field intensity beam profiles. In the second column, using the same ordering, we show the respective experimentally recorded interference patterns. In frames (a2) and (b2), downward and upward fork-like splittings of interference lines can be recognized confirming that: i) all 12 OVs nested on the bright sub-beams have the same topological charges and ii) all topological charges reverse in sign when the ordering of the phases projected on the SLMs is reversed or by deleting an OV with an opposite TC from the input phase structure (Fig. 4(a1)). The data prove also that when the ordering of the phases projected on the SLMs are reversed, the orientation of the large-scale triangular structure consisting of rhomboidal beam ensembles carrying OVs changes its orientation from (arbitrary) left orientation (frames (a1-a3)) to right orientation (frames (b1-b3)). In frame (c2) the interference lines are shifted along a line by a half a period, which is a clear indication for a 1-D phase shift of π confirming the presence of spatial phase dislocation. The experimental results presented in Fig. 6 refer to Δ = 2.48 and remained qualitatively similar to these for square-to-hexagonal lattice node spacing ratio Δ = 3.73. The data shown in Figs 4 and 6 are directly comparable in spatial extent. The numerical data shown in the last column of frames in Fig. 6 refer again to Δ = 2.48, however, different from the case shown in Fig. 5, here Δ sq = 101 pix. and Δ hex = 41 pix.
In Fig. 7 we show different experimentally recorded far-field bright beam intensity distributions obtained by varying the node spacing Δ sq of the square-shaped (row (a)) or Δ hex of the hexagonal OV lattice (row (b)). These data are selected from a larger set of measurements demonstrating the change in the symmetry and in the size of the far-field beam structures by changing the vortex-to-vortex node spacing of one of the lattices, keeping the respective spacing for the other lattice unchanged. In case (a), the hexagonal lattice node spacing is Δ hex = 41 pix. One can clearly see that for Δ sq = 21 pix. (row (a), left frame) the far-field beam profile has the small-scale structure resembling an equilateral triangle with three dominating peaks situated in its apices. In other words, the small-scale structure is the one of the hexagonal OV lattice, which has a two-fold larger node spacing, i.e. Δ hex = 41 pix. The large-scale focal structure resembles a rhomb with the aforementioned triangular small-scale structures in its apices. Thus the large-scale structure is the one formed by the square OV lattice with the smaller node spacing Δ sq = 21 pix. In the other limiting case shown in the most right frame of Fig. 7(a), Δ hex = 41 pix. is kept unchanged and Δ sq is increased from 21 pix. to 151 pix. The change in the symmetry of the structure is impressive. For Δ sq > 80 pix. the small-scale structure becomes rhomboidal and the peaks arranged in the apices of the rhombs are ordered in a triangle-like large-scale structure. In other words, in this limiting case the small-scale structure is this of the square lattice with the larger Δ sq = 151 pix., whereas the large-scale structure is formed by the hexagonal lattice with the smaller Δ hex = 41 pix. The transition from one symmetry of the distribution of the bright peaks to another symmetry seems to happen (Fig. 7, row (a)) approximately when Δ hex = 41 pix. and Δ sq = 61 pix. At equal node spacings the large-scale structure can be still recognized to be this of the square lattice. For Δ sq = 81 pix. it is already changed to the form coming from the hexagonal lattice. The further increase of the node spacing of the square lattice shows well pronounced shrinking of the large-scale triangular structure. The frames shown in row (b) in Fig. 7 provide a nice visualization of the mentioned Similarity theorem for the Figure 6. Additional structuring of the far field intensity profiles of mixed square-shaped and hexagonal OV lattices by adding an OV with an opposite TC (a1-a3) (effectively -by removing it), by changing the order of the generation of the respective OV lattices on the SLMs (b1-b3), and by adding a 1-D dark beam (c1-c3).
First column of frames -experimental intensity profiles, second column -recorded interferograms. The lattice node spacings as encoded on the SLMs are Δ sq = 151 pix. and Δ hex = 61 pix. for the square-shaped and for the hexagonal OV lattices, respectively (Δ = 2.48). For the numerical data (last column showing some 10% of the total computational area) again Δ = 2.48, however Δ sq = 101 pix. and Δ hex = 41 pix.

Figure 7.
Experimentally-recorded far-field beam reshaping by varying the node spacing of the square-shaped (row (a)) or of the hexagonal OV lattice (row (b)). In case (a) the hexagonal lattice node spacing is Δ hex = 41 pix. and Δ = Δ sq /Δ hex varies between 0.5 and 3.7. In case (b) Δ sq = 21 pix. and Δ varies between 0.5 and 0.17.
Scientific RepoRts | (2019) 9:2128 | https://doi.org/10.1038/s41598-019-38608-5 Fourier transformation: For Δ hex > Δ sq , the increase of the node spacing of the hexagonal lattice Δ hex from 41 pix. to 121 pix. results in a shrinking of the focal small-scale structures. It is worth noting that the "centers of mass" of the four small-scale triangle-like structures do not change their mutual positions because Δ sq remains the same. Figure 8 is intended to provide more insight into the dynamics shown in Fig. 7. The results in Figs 7 and 8 clearly indicate that Δ = Δ sq /Δ hex could serve as a control parameter for generating the desired focused structure.

Conclusion
In this work we show significant far-field beam reshaping by mixing square-shaped and hexagonal optical vortex lattices. Each of the singular lattices used here is composed of vortices with alternating TCs. We showed that the small-scale structure of the observed pattern is a result of the OV lattice of larger array node spacing and, conversely, that the OV lattice with smaller array node spacing determines the large-scale structure. The orientation of the mixed far-field structures is proven to rotate by 180° when all TCs are inverted. We succeeded to additionally host singly-charged OV, 1-D or quasi-2-D singular beams in each of the bright peaks of the mixed far-field patterns by erasing the TC of an OV of one of the arrays or by adding to it the respective singular phase profile of the aforementioned 1-D and quasi-2-D singular beams. We attribute the somewhat "noisy" intensity dots in the experimentally generated focal bright structures to the different sizes of the OV cores of the mixing lattices and to the different reshaping of the neighboring part of the background beams due to the diffraction. This could be improved by imaging the first SLM onto the second modulator and would even enable the use one SLM only. The detailed experimental data for the evolution of the far-field patterns at different square-to-hexagonal vortex array node spacings Δ = Δ sq /Δ hex shows that this quantity could serve as a control parameter for generating the desired focused structure. It will be intriguing to extend the presented approach to other shapes of lattices, e.g. rhomboidal OV lattices, periodic kagome lattices, and to quasi-periodic patterns (Penrose, decagonal; see the systematic description in 12 ). These results, in addition to the previously published ones 11 , may appear particularly interesting, as a new degree of freedom, for modifications in stimulated emission depletion (STED) microscopy, for extending the possibilities of generating singular higher-order vector fields 15 , for controllable writing of parallel optically-induced waveguide structures e.g. in (photorefractive) nonlinear media (see, e.g. 14 ,) and may appear applicable for orbital angular momentum multiplexing of information 16 for data transfer using complex optical fields [17][18][19][20][21] .

Data Availability
The datasets generated during and analyzed during the current study are available from the corresponding author on reasonable request.

Figure 8.
Experimentally-recorded far-field bright beam patterns for different square-to-hexagonal OV lattice node spacings Δ = Δ sq /Δ hex encoded on the SLMs. The top row of frames corresponds to Fig. 7(a), the left column of frames -to Fig. 7(b).