We propose an approach for creating optical vortex array (OVA) arranged along arbitrary curvilinear path, based on the coaxial interference of two width-controllable component curves calculated by modified holographic beam shaping technique. The two component curve beams have different radial dimensions as well as phase gradients along each beam such that the number of phase singularity in the curvilinear arranged optical vortex array (CA-OVA) is freely tunable on demand. Hybrid CA-OVA that comprises of multiple OVA structures along different respective curves is also discussed and demonstrated. Furthermore, we study the conversion of CA-OVA into vector mode that comprises of polarization vortex array with varied polarization state distribution. Both simulation and experimental results prove the performance of the proposed method of generating a complex structured vortex array, which is of significance for potential applications including multiple trapping of micro-sized particles.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Optical vortex beams (OVB) carrying orbital angular momentum (OAM) are arousing great interest owing to their wide applications such as optical tweezers [1, 2], particle manipulation [3–6], optical communication [7–9], quantum information communication [10–12], and optical imaging [13, 14]. The so-called vortex beam is a continuous spiral phase beam whose wavefront phase distribution is whirlpool like. The phase singularity of the beam gives rise to a dark hole at the center region. The vortex beam can be simply regarded as a linear superposition of Laguerre-Gaussian (LG) modes . LG modes are characterized by a radial index p and an azimuthal index l, where a superposition of a few “p” modes describes the far field of an optical vortex. On a different note, a superposition of different “l” modes describes the interference in the azimuthal direction, quantifying the OAM carried by the beam [16–19]. A single optical vortex can be experimentally obtained by generation of laser beam modes with helical wave fronts [20, 21]. Similarly, V. Kotlyar et al. discussed paraxial diffraction at a spiral phase plate of a limited light field and described the formation of a vortex beam with an intensity distribution in the form of a double ring of arbitrary radius . Y. Izdebskaya et al. considered the diffraction of the Gaussian beam in a continuous positioning optical wedge system, and realized that the system was able to form higher-order optical vortices. The topological charge of the vortex is equal to the number of wedges in the stack .
To date, optical vortex array (OVA) that contains multiple vortices have been widely studied as it can provide more flexible and potential applications. Y. Lin et al. proposed a method of creating OVA by the conversion of a standing-wave LG mode. They firstly verified that the crisscrossed Hermite-Gaussian modes can be transformed into a flower-like LG mode theoretically . Some studies involving the generation of special OVA were proposed based on a type of diffractive optical element called Dammann grating. For instance, a series of 3D dipole OVA in the focal region of a focusing objective was generated by using a spiral Dammann zone plate, which enables good uniformity and high efficiency of OVA . Similarly, the generation and detection of broadband multi-channel OVA was achieved by the combination of Dammann vortex grating and meta-reflectarray . The OVA is also able to be produced via overlapping two special optical beams. Based on Michelson and Mach–Zehnder interferometers, S. Vyas et al. modified it to produce three beams of equal amplitudes that can result in optical vortices embedded with high contrast fringes. These interferometers are different from the conventional interferometers where they are capable of producing multiple vortices . P. Vaity et al. observed that OVA can be formed by superposing two Gaussian beams in a Mach–Zehnder interferometer . Huang et al. generated composite vortex beams via coaxial superposition of two, three and four LG beams respectively . Furthermore, some special optical fields can be produced by the overlap of two Bessel beams [30, 31]. The OVA generated by two overlapped optical beams can be well applied in microparticle manipulation. However, the radius of the bright ring in OVA obtained by traditional methods is proportional to the topological charges. As a result, it is inconvenient to overlap two or more constituent optical vortex beams for further creating complicated optical fields.
In 2013, an advanced concept of “perfect optical vortex” (POV) was introduced, where the ring radius is independent of topological charges . Afterwards, the generation of the POV was explored based on the width-pulse approximation of Bessel function , digital micromirror device  and Fourier transformation of a Bessel beam . Further, J. Yu et al. generated multiple POVs arranged into a square lattice regularly by using a type of two-dimensional (2D) encoding continuous-phase gratings . POV array with controllable diffraction order and topological charge was demonstrated by a holographic grating under special design . Then, a method of creating 3D POV array with high quality and uniform intensity  was likewise investigated in detail. M. K. Karahroudi et al. also reported the generation of several various POVs with arbitrary charges by use of the curved fork grating illuminated by Bessel-Gaussian beam . Recently, a novel OVA named circular optical vortex array, was generated by superposing two concentric POVs with different topological charges . The number of optical vortex as well as the radius of the circular array trajectory can also be easily regulated by this new structure of OVA, which paves the way for new applications such as optical tweezers and micro-particle manipulation under circumstances of complex structure optical field. But this type of OVA is merely suitable for circular alignment. Besides, due to the non-diffraction feature, this kind of multiple vortex array based on POV technique is unable to produce high intensity gradient upon axial propagation in focal region, limiting its further utilization.
In this paper, we propose a method to generate a more general and fancy OVA where all of the optical vortices are arranged along arbitrary path of curves. The scheme is based on the coaxial superposition of two 2D curve laser beams with independently controllable radial dimensions and phase variation (topological charges) along each curve. We exploit and modify the holographic beam shaping technique  by introducing an extra modulation of the radial width dimension of the 2D light curves in the calculation of computer generated hologram (CGH). We also perform an exploratory investigation for further converting the CA-OVA into vector mode, where each hollow annulus along the curve path exhibit linear polarization state variation around its singularity. Our method can produce distinctive curvilinear permutation OVA which might have potentials in aspects of complicated and multi-target optical manipulation and imaging.
2. Principle of the technique
2.1 Shaping of width-controllable 2D beams along curves
The proposed technique is based on the theory reported in , which allows designing scalar beams whose intensity distribution follows a prescribed 2D curve in the focal plane of a Fourier lens. Briefly, the complex field amplitude that acts as a complex CGH at the incident plane of a focusing (Fourier transform) system is written specifically asEq. (1) is determined respectively by
In Eq. (3), σ is a free parameter that allows varying the phase gradient along the curve if needed and is also independent from the size of the beam, resulting in the ability of shaping such kind of “perfect helical beams” in the focal plane. The parameter φ0 is he initial phase factor, playing the role of dominating the starting point of the phase gradient along the curve. Actually, the obtained focal beam shaped by Fourier transform of the incident field H(x, y) is composed of numerous consecutive tiny spots along the curve path. These extremely focused spots yield both of high transversal and axial intensity gradient in the focal plane , looks like a “needle-tip” slipping through the curve.
In our work, we extend the design of the above 2D light curve into a more general shape of width-controllable light curve to enable the generation of OVA arranged along prescribed trajectory, by overlapping two component curves with differently allocated phase gradient and curve’s widths. First of all, in order to generate each single component curve with the desired width, we introduce an increment Δd in the complex CGH calculation procedure. As an illustration, we consider the generating of a 2D ring curve of x0(t) = R·cost, y0(t) = R·sint with different widths (or namely “thickness”) and demonstrate the performance of our scheme followed by simulation. We first set an initial radius of the ring by R0. Afterwards we define a demand-dependent integer n to control the desired width of the curve. The complex CGH that shapes the ring curve with a certain width of n is simply calculated by accumulating the incident fields from all the curves of different radius ranging from R0 to Rn = R0 + Δd·(n-1) as followsEq. (1) responsible for shaping the i-th curve with radius Ri. As a result, the shaping of an optical ring curve with desired width along its path is achieved by changing the parameter of n. This procedure can be interpreted as to “thickening” a curve through overlying multiple layers of single curve of different radial dimensions. Ai is a weight factor responsible for regulating the amplitude distribution of the curve. Figure 1 shows the simulated results of the shaped rings by Fourier transform of complex CGH E(x, y), which is calculated under the same initial radius R0 = 0.15mm but different width parameters of n = 1, 4 and 8. The vector coefficient A = [A1, A2,…,An] obeys the one-dimensional Gaussian profile by our setting. It is obvious that the slender ring curve generated when n = 1 is notably thickened after increasing the width parameter of n.
2.2 Curvilinear arranged optical vortex array (CA-OVA)
By using the width-controllable light curves shaping technique, we report the generation of a tunable optical vortex array named as “Curvilinear Arranged Optical Vortex Array (CA-OVA)” by superposing two concentric curves. The total complex CGH at the incident plane of the focusing system aiming to generate the two concentric curves in the focal plane simultaneously is calculated byEq. (4) under different initial curve radii R01, R02 and different initial phase φ01, φ02. As an example, assuming that there are two concentric ring curves and each of which is generated by the aforementioned method. We consider the two width-controllable ring curves having the same width of n = 8 but different initial radii of R01 = 0.15mm and R02 = 0.19mm as shown in Fig. 2(a) and 2(b). The values of R01 and R02 as well as the width are selected to guarantee the optimized overlapping ratio (approximate 36%) between the adjacent inner and outer rings . The phase distribution of each ring is also well defined along the curve under different topological charges of l1 = 4 and l2 = −4, as illustrated in Fig. 2(d) and 2(e). The pre-specified initial phases for both ring curves are φ01 = φ02 = 0. Furthermore, we define and derive the radius of each ring’s center line marked in Fig. 2(a) and 2(b), represented by Rc1 = R01 + Δd·(n-1)/2 and Rc2 = R02 + Δd·(n-1)/2.
According to the theoretical analysis depicted in , when the two ring curves are superposed with each other, interference will occur where the two rings overlap while the intensity profile remains at the non-overlapped regions, producing annularly distributed intensity vanishes (dark dots) within the overlapped region (see Fig. 2(c)). The center radius of the interference pattern comprising dark dots is deduced by Rc = (Rc1 + Rc2)/2 = (R01 + R02 + Δd·(n-1))/2. The number of dark dots along the ring is calculated by N = |l2-l1|. Counting the dark dots in Fig. 2(c), it can be observed that there are eight dark cores, which is consistent with the relation N = |−4−4| = 8. Meanwhile, the existence of the local optical vortices on the position of each dark dot is also confirmed by the phase pattern of the superposition of the two ring curves as shown in Fig. 2(f). Therefore, the position of each dark dot along the ring zone in Fig. 2(c) is a phase singular dot. The phase increases from 0 to 2π surrounding each singular dot defining a negative vortex of topological charge (l = −1). Note that the change of the helical phase patterns in Fig. 2(d) is contrary to Fig. 2(e) due to the opposite topological charges, which is pivotal to guarantee the optical vortex. For the sake of clarity, the existence of optical vortices on dark dots can also be verified by means of interference with a spherical wave , which will be given by both of simulation and experiment in the next section.
For further creating arbitrary CA-OVA, the two concentric curves are configured in the focal plane by using more general curve expressions. We demonstrate the generation of different types of CA-OVA by shaping different width-controllable curves. In Table 1 we list the parametric expressions of each curve (x0(t), y0(t)) in our simulation, where t∈[0, 2π] for all cases, used for the generation of CA-OVA. It should be noted that, the positive integer c in the expression of the star-like curve represents the number of horns of the star.
Figure 3 shows the simulated results of three types of CA-OVA, i.e., OVA along square curve, four-horn star-like curve (c = 4) and five-horn star-like curve (c = 5). The intensity and phase patterns of the inner, outer and overlapped curves for each structure are clearly displayed, and the specific parameters for generating each CA-OVA are described in detail in the figure caption. The number of dark dots (optical vortices) is N = 8 in the CA-OVA along square and four-horn star-like paths while N = 10 in the CA-OVA along five-horn star-like curve, which proving the capacity of controlling the numbers of optical vortex in the CA-OVA by modifying the phase gradient of the two component curves. As a consequence, by combining the holographic shaping technique of arbitrary width-controllable 2D curves using Eqs. (1)-(4), the generation of optical vortex array (dark dots) arranged in arbitrary curvilinear permutation after the interference of two width-determined curves can be actualized as long as the prior knowledge of the parametric expression of a curve.
In interference theory, the positions of the fringes will move by altering the initial phase difference. As a result, the rotating of the CA-OVA can be easily realized by introducing an initial phase shift between the two generated curve components. The initial phases are all zero in the above-discussed situations in Fig. 2 and Fig. 3. In general, the relative phase shift between the two curves is defined as Δφ0 = φ02-φ01. Figure 4 shows the simulation results of rotating optical vortex (hollows) in the generated three types of CA-OVA under different relative initial phases shifts of Δφ0 = 0, π/2, π, 3π/2 and 2π, respectively. For the sake of simplicity, we set φ01 = 0 and change the value of φ02 from 0 to 2π in steps of π/2 in the CGH calculation. A blue dashed line is marked in each figure as a reference to reveal the sites variation of the hollows. The rotation angle of CA-OVA is calculated by Δφ0/|l2-l1| = Δφ0/N. The sign of Δφ0 determines the direction of the OVA rotation. In this case, since the sign of Δφ0 is positive, all of the CA-OVA rotate clockwise as shown in Fig. 4.
2.3 Orbital flow density of the CA-OVA
As well-known, a vortex field possesses orbital angular momentum, which can be elegantly depicted by the internal energy flow in the field . Let us now analyze the orbital angular momentum in the generated CA-OVA by calculating the momentum density of the beam. The momentum density of a monochromatic light field follows from the cycle-average Poynting vector is given as Eq. (6) can be separated into spin and orbital parts as where the spin constituent and the orbital constituent of the total energy flow are represented by
Figure 5 illustrates the calculated transverse orbital flow density for four types of CA-OVA structures corresponding to Fig. 2(c), Fig. 3(e1), Fig. 3(e2) and Fig. 3(e3). The blue arrow heads inset indicate the direction of each flow. It is observed that the generated CA-OVA carry rotational energy flow around each hollow site, owing to the spatial inhomogeneous intensity distribution and local phase vortex at each singularity. In particular, the energy flux around each vortex induced from the orbital flow density may transfer from the electromagnetic radiation to the particles, resulting in the radiation force exerted on the particles, which enables it an ideal tool for studying optical trapping in more diverse geometry circumstances.
2.4 Curvilinear arranged polarization vortex array (CA-PVA)
The generated scalar CA-OVA in the above section is able to convert into the vector mode since the azimuthally phase vortex can be regulated by varying the topological charge of the two width-controllable curves. We use the general idea of vector beam generation scheme [44, 45], upon which the conversion of curvilinear arranged scalar “phase vortex array” into vector “polarization vortex array (PVA)” is realized by spatially overlapping two orthogonally polarized CA-OVA components, each CA-OVA has the same intensity patterns (i.e. the same number and position of hollows) but different phase vortex at each hollow. In this way, the variation of the polarization state around each hollow along the curved beam is introduced due to the relative amplitude and phase (i.e., phase difference) of the two generated base CA-OVA structures. We use the left and right circularly polarized CA-OVA as the two orthogonal polarized base components in our work although the linear polarization of x- and y- direction is also an alternative choice.
The theoretical demonstration of generating the curvilinear arranged PVA (CA-PVA) is shown in Fig. 6. The CGH to generate each orthogonal circularly polarized base of CA-OVA is denoted by Etotal-L(x, y) and Etotal-R(x, y) which are calculated by using Eq. (5). In the calculation process of Etotal-L(x, y), the topological charges of the inner and outer width-controllable curve are given by l1 = 4 and l2 = −4 in the ring curve example. In contrast, the topological charges are exchanged to l1 = −4 and l2 = 4 in the calculation of Etotal-R(x, y). Besides this, all the other parameters in the calculation of Etotal-L(x, y) and Etotal-R(x, y) are the same. As a result, the generated left and right circularly polarized CA-OVA have opposite helical phase distribution at each hollow annulus as shown in Fig. 6(a). The angle of linear polarization distribution in the resulted PVA can be easily deduced according to the phase difference between the two orthogonal CA-OVA components as plotted in Fig. 6(b). It is observed that there is cylindrical-symmetry linear polarization distribution variation (polarization vortex, marked out by dotted lines) around each hollow cross section, yielding a curvilinear arranged polarization singularity array that is transformed from preceding phase singularity array. The phase distribution of the PVA beam is uniform as shown in Fig. 6(c), indicating the vanish of phase gradient. The simulated results about the Stokes parameters (S1, S2, S3) of the optical field of PVA are also shown in Figs. 6(d)-6(f). It should be pointed out that each polarization vortex hollow is actually corresponded to the so-called high-order Poincare sphere beam . Therefore the generated vector mode can be considered as a type of curvilinear arranged high-order Poincare beam array.
3. Experiment of CA-OVA
The experimental arrangement for creating the scalar CA-OVA is sketched in Fig. 7. The optical setup for generating the CA-OVA is composed of a Liquid Crystal spatial light modulator (SLM), a 4-f filtering system, and a Fourier transform (focusing) lens. A solid-state laser with a wavelength of 532 nm is collimated to plane wave illumination by passing through a beam expander composed of two convex lenses. The SLM (Holoeye Leto, 6.4μm pixel pitch, 1920 × 1080 resolution) is utilized to address a phase-only CGH which is encoded from the complex CGH Etotal (calculated from Eq. (5)) by using double phase method . The beam modulated by SLM is then projected to the back-aperture of the Fourier transform lens (L3, f = 100mm) through a 4f optical filtering configuration. The complex amplitude Etotal, namely the incident beam of the Fourier lens, is focused to generate the designed CA-OVA at the focal region. A charge coupled device (CCD) camera is placed at the Fourier plane of the focusing lens to record the intensity patterns. The optical diagram also includes an extra composition of interference between the generated CA-OVA and a spherical reference beam, indicated by red-dotted lines in Fig. 7. A lens (L4, f = 200mm) in the reference optical path is employed to produce a spherical wavefront in order to confirm the existence of the optical vortex via coaxially interfering with the CA-OVA.
In order to verify the existence of CA-OVA, we perform the interference between the CA-OVA along ring trajectory and a spherical wave. The results of simulation and experimentation are illustrated in Fig. 8. Figure 8(a) is the simulated interference fringe between CA-OVA and a spherical wave where the number of vortex is N = 8. The rest parameters are the same as that in Fig. 2. There is fork intensity profile on the position of each dark dot due to the vortex phase property in the interference. The number of forks is equal to the number of dark dots that obeys the formula of N = |l2-l1| = |-4-4| = 8. The corresponding experimental result shown in Fig. 8(b) is consistent with the theoretical one. Accordingly, Figs. 8(c) and 8(d) are the simulated and experimental results in which the number of vortex is changed to N = 10 by modifying the phase gradient into l1 = 5 and l2 = −5 in the calculation of two component curves. The results also exhibit a high degree of matching and demonstrate the existence of CA-OVA along curves.
The intensity distributions of different CA-OVA generated by the proposed method are experimentally measured at the focal plane by the CCD. The first row in Fig. 9 illustrates the generated CA-OVA along ring, square and four-horn star-like curves under the same number of vortex as N = 8, whereas the second row corresponds to the shaping of CA-OVA along three curve structures (ring, square and five-horn star) where the dark dots number is N = 10. Besides, the experimental results of Figs. 9(a)-9(c) and 9(f) are also in good agreement with those simulated ones as previously shown in Fig. 2 and Fig. 3. It should be emphasized that similar concept of creating so-called “Optical Ferris wheel” in Fig. 9(a) and 9(d) has been demonstrated in . Nevertheless here we show much better agreement between experiment and theory data. In particular, we also extend the concept to a square lattice as well as star-like lattice where we use the superposition of perfect helical beams with different radius in all cases. Figure 10 also complements the experimental results of rotating the generated CA-OVA around the corresponded beam circumferences. The parameters are all consistent with the simulations in Fig. 4. By adjusting suitable initial phase shift, it is capable to accurately tune the position of each vortex in the CA-OVA. This property is useful to trap as well as accelerate the micro particles in optical manipulation and other applications.
Next, we demonstrate that the proposed method is able to simultaneously shaping hybrid CA-OVA that comprises of multiple OVA structures along different respective curves. Each sub-CGH responsible for shaping each individual CA-OVA is firstly calculated by using Eq. (5), then all the sub-CGHs are multiplexed into a final hybrid CGH after multiplying a shifting phase factor assigned to each sub-CGH expressed in the formEq. (5) to shaping the i-th CA-OVA along a specific curvilinear structure. k = 2π/λ is the wave number and (ui, vi) is the frequency coordinate aiming to independently control the transversal position of each generated CA-OVA in the focal plane. Based on Eq. (9), we can calculate a CGH Ehybrid(x, y) for simultaneously generating more than one CA-OVA structure, giving rise to a hybrid CA-OVA distribution in focusing region. Figure 11 provides an evidence of generating hybrid CA-OVA in two different types of layouts. Figures 11(a) and 11(b) are the simulated and experimental intensity distribution of hybrid CA-OVA that contains four CA-OVA structures in a “diamond” layout, whereas in Figs. 11(c) and 11(d) the four CA-OVA structures are shifted to the vertex angles of a square layout. The number of vortex in each individual CA-OVA structure along ring, square and four-horn star curve is 8 while in the five-star curvilinear arranged CA-OVA is 10.
4. Experiment of CA-PVA
Finally, we carry out the optical experiment of generating CA-PVA as depicted in Section 2.4. In this case, we use our proposed vector optical field generation system [44, 45] to create the desired structures of CA-PVA. The complex CGHs of Etotal-L(x, y) and Etotal-R(x, y) are encoded into a phase-only holographic grating using double-phase technique [45, 47]. The experimental arrangement is shown in Fig. 12. Compared with the system of generating scalar type CA-OVA in Fig. 7, two quarter-wave plates (QWPs) are placed at the filtering plane to convert the separated beam into mutually orthogonal left and right circular polarization components, which serve as a pair of base beams for the subsequent vectorial superposition. A Ronchi grating is also appended at the output plane of the 4f configuration to enable the re-collinear propagation of each beam. We place a polarization analyzer (linear polarizer P2) in front of the CCD to record the intensity variation induced by specific polarization distribution of the generated vector structure.
The four pictures in the first row of Fig. 13 plot the concrete polarization distribution of four generated CA-PVA structures. The results presented from the second to fifth rows are the theoretical and experimental intensity distributions at CCD plane when an analyzer is inserted by rotating its angle to 0°, 45°, 90° and 135° respectively, where the odd columns show the theoretical results while the corresponded optical experimental records are given in even columns. It is clear that the experimental data agree well with the simulated ones and the pattern extinction phenomenon appeared in each situation also validates the match in polarization state variation of our designed CA-PVA. The generated CA-PVA structures exhibit polarization gradient around central singularity of each hollow as well as lattice arrangement along arbitrary curve trajectory, which may provide more complicated vector optical field in latent applications such as micro-particle manipulation and light-matter interactions.
Conventionally, the generation of versatile optical vortex lattice is achieved by individually modulating each optical vortex site in the lattice. The site of each vortex is governed by imposing independent phase shifting factor, through which the position coordinates of each vortex should be the prior parameters. As a result, the requirement of separate operation for every vortex increases the complexity as well as producing inconvenience for OVA generation, especially when the number of desired vortex arises. However, in the proposed method we can modulate, only in one step, all of the vortex sites in an arbitrary curvilinear lattice arrangement by simple superposition from two pairs of width-controllable curve beams, without laborious individual parameters allocation for each vortex. Therefore, the proposed method gives rise to an optimized and high efficient way of generating and modulating a variety of OVA beams.
In summary, we present a method of generating a distinctive optical vortex array along arbitrary curvilinear arrangement. The CA-OVA is obtained by coaxially superimposition of two width-controllable curves with different initial radius and topological charges. The existence of OVA long curvilinear path is proved by interference with a spherical wave. The number of optical vortex in CA-OVA is determined by the topological charges of the inner and outer curves, respectively. Both of the numerical and optical results verify the performance of the proposed method for shaping different types of CA-OVA structures in the focal plane as well as simultaneously generating multiple CA-OVA structures in different hybrid layouts. We also prove the ability of converting the curvilinear arranged scalar phase vortex array into vector polarization vortex array where the helical phase distribution in each vortex annulus is transformed to polarization gradient by using the vector beam generation technique. We expect that the proposed technique can be employed in optical trapping and manipulation in which the driving of multiple micro-particles in complex field is necessary.
National key R&D program of China (2017YFA0303700)); National Natural Science Foundation of China (NSFC) (61605080, 61775097, 91750202, 11474156).
We particularly thank Yijun Qi in Southeast University for giving helpful discussions with this work. We also would like to thank the two anonymous reviewers for insightful comments.
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