1. Introduction

Optical microparticle manipulation by a single-beam trap is a well-known tool with numerous applications in the fields of biology, chemistry and physics [1]. It is usually studied at the focal plane of a microscope’s objective where a particle may be trapped or moved along certain trajectory according to the field distribution of the applied beam. Different types of beams have been proposed for this task. For example, optical vortices such as the helical Laguerre-Gaussian (LG) modes yield rotation of a dielectric particle around the beam axis [2–4]. The rotation velocity depends on the orbital angular momentum carried by the beam, apart from the particle’s characteristics (weight, size, form, material, etc.) and the surrounding medium. Certain combinations of the LG modes produce spiral beams [5–7] which maintain the form of their transversal intensity distribution apart from rotation and scaling. Such a type of beam was used for example in order to induce the Archimedes spiral movement of the particle [8]. On the other side, the sum of LG modes with opposite topological charge was found useful for particle sorting [9, 10].

Apart from setups designed for plane-curve particle movements, there are more sophisticated trapping systems based on a spatial light modulator (SLM) which provide an interactive three dimensional (3D) control of the particle position almost in real time [11, 12]. These systems, commonly known as optical tweezers, use the computer generated holograms (CGHs) that allow the fast optical beam modification for particle manipulation. They play a relevant role in many applications ranging from the measurement of biological forces to the construction of micro-machines and micro-components [12], but a high amount of computational resources is required for real time synthesis of the CGHs. Nevertheless pre-designed 3D particle movements which are important for example for micro-robotics, can be induced by a single-beam trap. Notice that such an optical setup might use an analogue hologram for beam generation and therefore it is more compact and cheap than using a single SLM [5].

In this work, we study micro-particle’s 3D movements induced by a single-beam trap given as a collinear superposition of the LG modes. The LG modes present a well-defined conical structure in the minimum beam-waist region. Correspondingly, the strongly focused beams over the sample constructed from the LG modes allow to form optical funnels or pods of different configurations depending on the mode indices, weights, waist, etc. In next section we describe the beams chosen for this study. A conventional holographic optical tweezers setup, considered in section 3, is used for the observation of particle-light interaction. Since the quality of the beam and the visualization of particle movement are essential for the correct interpretation of the results we also shortly discuss the methods used for CGH encoding and particle tracking. The movements of dielectric particles suspended in water induced by the optical beams are analyzed in section 4. The paper ends with concluding remarks and perspectives for future research.

2. Beam design for the optical trap

As basic elements for the optical beam-trap design we use linearly polarized monochromatic helical LG modes given by [13]

It has been reported in numerous publications that the beams carrying OAM produce particle rotation at the trapping plane, see for example [2–4]. Nevertheless, to the best of our knowledge, the obvious 3D spiral like trajectory of the particle in the neighbourhood of the focal plane has not been observed yet. Thus, one of the beam trap used for this study is a single helical LG mode

We underline that two beams may have the same OAM but their interaction with a particle may differ due to the different intensity and OAM density distributions, which play an important role in the optical trap performance. The OAM density distribution of a beam constructed as a linear superpositions of the LG modes ${\sum }_{p,l}{a}_{p,l}{\text{LG}}_{p,l}^{\pm }\left(\mathbf{r};0\right)$, where a_p,l are constants, is given by the expression [15]

To study the dependence of particle trajectory from the OAM density distribution we use the following superposition of the LG modes

The next beam of interest is a linear combination of the LG modes with the same intensity distributions and opposite sign of OAM values,

The intensity, phase and OAM density distributions of these three beams are shown in Fig. 1. We observe that the maxima of the OAM densities almost coincide with the maxima of the intensity distribution for ℱ₁ and ℱ₂. The nonuniform distribution of the OAM over the rings for ℱ₂ yields to the larger OAM density in some regions and smaller for others than for ℱ₁. It is expected that particle plane rotation induced by ℱ₂ will be limited by the region with large OAM density and intensity distribution.

 

Fig. 1 Intensity, phase and OAM density distributions of the beams ℱ₁ (a), ℱ₂ (b), ℱ₃ (c), displayed in the first, second and third row respectively, at the minimum beam waist plane z = 0.

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All three beams form a funnel-like structure in the neighbourhood of the focal plane (z = 0) as it can be appreciated from Fig. 2. For the numerical simulations we used the parameters corresponding to the experimental setup: a 100× microscope objective lens (MO) with 1.32 NA, wavelength λ = 532 nm and 0.25× relay optics (telescope). Then a minimum beam waist value of w₀ = 3 μm is reached at the focal plane of the MO. As observed in Fig. 2, while the ℱ₁ and ℱ₃ beams remain invariant (except scaling), the ℱ₂ one presents an additional clockwise rotation yielding a double helix intensity structure. We also note, that since the diameter of the particle used in the experiments for the analysis of 3D movement is 10 μm, then the average intensity distribution over the particle cross-section is almost the same for the beams ℱ₁ and ℱ₃. Therefore the main difference in their interaction with the particle is the presence of the OAM in the case of ℱ₁ beam.

 

Fig. 2 Three-dimensional representation for the propagation of beams ℱ₁ (a), ℱ₂ (b), ℱ₃ (c). Minimum beam waist w₀ = 3 μm is obtained at focal plane (z = 0) of the 100× microscope’s objective lens.

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It is also possible to design other attractive trapping beams by combining two or more optical funnels. Along the last decade, different techniques were proposed to generate beams with cylindrical hollow axial structure (so-called optical bottles) [16–18]. Here we consider the collinear superposition of two optical funnels with a controlled focusing for each one then allowing generation a beam with pod-like structure. Indeed the position of the minimum beam waist value can be displaced a distance d away from the focal plane by modulating the input beam ℱ_n (r) with a spherical phase distribution

 

Fig. 3 Three-dimensional representation for the propagation of beam 𝒫_1,1 (a) and 𝒫_2,2 (b). To help visualization a 3D cut-out view of such beams is displayed in the second row.

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The cross sections of the intensity distribution for all beams studied in this work at the distances z = 0 (first row) and z = −11 μm (second row) are shown in Fig. 4.

 

Fig. 4 Transversal intensity distributions for propagation distance z = 0 (first row) and z = −11 μm (second row) for the beams: ℱ₁ (a), ℱ₂ (b), ℱ₃ (c), 𝒫_1,1 (d) and 𝒫_2,2 (e).

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3. Experimental implementation

To generate these trapping beams we used a conventional holographic optical tweezers setup (see Fig. 5), which consists from a hand-made inverted light microscope with Zeiss 100× objective (1.32 NA, oil immersion, Plan-Apochromat), a reflective LCoS-SLM (Holoeye LCR-2500, twisted nematic, 8-bit grayscale signal, pixel size δ = 19 μm) operating in phase-only modulation, and Nd:YAG laser with wavelength λ = 532 nm and 400 mW. Theory and experimental implementation of such systems are studied in [19] and [20], to name a few. The CGH is addressed onto a phase-only SLM that modulates a collimated laser beam. The resulting beam, B1, is focused over the sample by using a Keplerian telescope and the microscope objective, as sketched in Fig. 5.

 

Fig. 5 Schematic representation of the holographic optical trapping system. SLM displays the hologram that is imaged into the back focal plane of the microscope objective (MO) by using two relay lenses, L₁ and L₂, working as 0.25× telescope. The sample is illuminated by an additional laser beam B2 in order to acquire its image as in-line hologram, which is recorded in real time by a CCD camera.

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Particle tracking is achieved by using an additional collimated laser beam B2 (in our case λ = 532 nm and 10 mW) that replaces the conventional incandescent illuminator, see Fig. 5. Its irradiance is comparable to that of conventional microscope illumination and therefore it does not exert measurable forces on the illuminated particles (suspended in the sample). Light scattered by the sample is recorded by a CCD camera (pixel size of 4.6 μm and 30 frames per second) in real time as an in-line hologram, which can be numerically analyzed in order to obtain the 3D particle position. In this work we use Rayleigh-Sommerfeld (RS) back-propagation for the in-line hologram reconstruction because it permits faster particle tracking. This technique was studied in detail for holographic video microscopy in [21] and was found successful for micro-sized colloidal spheres of polystyrene and silica tracking. The in-line hologram results from interference between the light scattered by the particle and the unscattered portion of the illumination laser beam B2 at the focal plane of the objective. In our case this hologram is 100× magnified by the MO lens and recorded as a raw-video file by using a CCD camera (exposure time was 1 ms). The hologram reconstruction is achieved by computing the RS diffraction integral which makes possible to render a volumetric reconstruction of the light scattered by the particle. The particle’s position is found by locating maxima in such a volumetric reconstruction [21].

Since the SLM operates in phase-only modulation, the beam complex amplitude distribution has to be encoded into the phase hologram. In the last decades several types of CGHs have been proposed to overcome this problem, see for example [22–24]. To shape both amplitude and phase of the trapping beam we consider the method described in [24], that yields an accurate beam synthesis [25] ensuring its stability under propagation.

Let us now describe the beam generation approach. It is well-known [22] that an arbitrary scalar complex field described by s(x, y) = a(x, y) exp[iϕ(x, y)], where a(x, y) and ϕ(x, y) are the amplitude and phase distribution respectively, can be encoded as a phase CGH with a transmittance function H(x, y) = exp[iψ(a, ϕ)]. To obtain an appropriate dependence of ψ(a, ϕ), function H(x, y) has to be represented as a Fourier expansion in the domain of ϕ(x, y). As demonstrated in [24], the encoded signal s(x, y) is recovered from first-order term of such Fourier expansion by using the following hologram phase modulation:

Wavefront distortion φ_ab arising from the SLM non-flatness, mainly astigmatism and spherical aberration, was calibrated following the suggestions reported in [20]. The SLM also allows compensating the wavefront aberrations corresponding to the optical setup, see Refs. [27, 28]. To compensate such aberrations and therefore to optimize the optical trap performance, the hologram phase modulation Eq. (8) is evaluated as ψ(a,ϕ − φ_ab + φ_c) mod 2π. Notice that aberrations such as coma and astigmatism mainly degrade the lateral trapping efficiency while the spherical aberration mostly degrades the axial trapping performance [27]. A further wavefront correction can be achieved as proposed in [28].

The corrected beam B1 is focused over the sample, which is enclosed into a flow cell made by attaching a glass coverslip (thickness 0.17 mm) to a standard microscope slide with double-sided tape, see Fig. 5. The coverslip partially reflects the trapping light back into the objective lens which images this beam through a 90% dichroic mirror (at wavelength λ = 532 nm) to the CCD camera, see Fig. 5. Then it is possible to obtain a volumetric reconstruction of the trapping light by translating the coverslip with respect to the objective lens. This approach has been previously demonstrated in [29] by using a mirror instead of a coverslip. In this work we use the coverslip for this task because it allows in-situ testing of the beam degradation caused by its lack of flatness. Figure 6 shows the intensity distributions of the trapping beams corresponding to the optical funnel ℱ₁ (a), ℱ₂ (b) and ℱ₃ (c) and the optical pods 𝒫_1,1 (d) and 𝒫_2,2 (e), measured by translating the coverslip (see also Media 1). These experimental results are in good agreement with the numerical simulations displayed in Fig. 4. Nevertheless there exists a wavefront distortion due to the residual aberrations of the optical setup, which may be minimized by performing further wavefront correction as reported in [28]. In the next section we study the action of these beams over polystyrene spheres suspended in distilled water.

 

Fig. 6 Experimental results ( Media 1): Transversal sections at z = 0 (first row) and z = −11 μm (second row) for the beams: ℱ₁ (a), ℱ₂ (b), ℱ₃ (c), 𝒫_1,1 (d) and 𝒫_2,2 (e).

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4. Microparticle manipulation: experimental results

We start from the analysis of particle movements (2 μm diameter polystyrene sphere, Spherotech SPV-20-5 Lot W01) at the focal plane of the MO, which in this case is near to the upper covering glass in order to compensate the axial scattering force. Figure 7 shows the motion of the trapped particle along the multiringed beam profile following the exerted torques (see also Media 2 and Media 3). As expected, the particle is confined in the high-intensity circular fringes and rotates around the beam ℱ₁ axis whereas for the case of beam ℱ₂ it only rotates in the angular range limited by the nonzero OAM density. The trapping with the beam ℱ₃ does not produce any rotation due to zero OAM density.

 

Fig. 7 In-line holograms showing the trapped particle (2 μm diameter) for the beam ℱ₁ Media 2 and ℱ₂ Media 3, first and second row respectively.

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Let us now consider the 3D motion of 10 μm diameter polystyrene spheres (Duke Scientific Lot 9949) suspended in water, which is induced by the proposed trapping beams. The flow-cell thickness is about 120 μm and the particle motion starts from the lower coverslip surface. Figure 8(a)–8(c) displays the particle’s tracking for the beams ℱ₁, ℱ₂ and ℱ₃, respectively. Although both beams ℱ₁ and ℱ₂ have the same OAM (J_z = 18), they exert different forces and torques on the particle according to the geometry of their 3D intensity and OAM density distributions. This yields different types of helix-like trajectories, see Fig. 8(a) Media 4 and Fig. 8(b) Media 5. In contrast, the beam ℱ₃ does not transfer OAM and therefore the particle just goes up to the minimum beam waist plane (z = 0), see Fig. 8(c). Notice that these trapping beams are displayed for the trap-region of interest, given between the lower coverslip surface at z = −30 μm and the focal plane z = 0. Several video snapshots corresponding to the in-line holograms obtained for the case ℱ₁, ℱ₂ and ℱ₃ are also showed in Fig. 8(d)–8(f). We underline that clockwise rotation direction is obtained for the complex conjugate beams ${ℱ}_1^{*}$ and ${ℱ}_2^{*}$.

 

Fig. 8 3D particle-tracking reconstruction for the beams: ℱ₁ (a) Media 4, ℱ₂ (b) Media 5, ℱ₃ (c). Particle’s trajectory is also projected into x − y, x − z and y − z planes (grey lines). It is experimentally measured from in-line holograms stored as video as the ones showed in (d) Media 6, (e) Media 7, and (f), correspondingly. The trajectories start from the lower coverslip surface at t = 0 s (z = −30 μm) and ends at the focal plane z = 0 μm.

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Since the intensity distribution of the beams ℱ₁ and ℱ₃ posses angular symmetry the particle may be trapped in any point of the plane where the light force is sufficient, while for the beam ℱ₂ there are two well-defined regions around the intensity maxima where it may happens. Moreover, due to the relatively big particle size, two particles can be simultaneously trapped by beam ℱ₂ following different helical channels as it is observed in Fig. 8(b) and 8(e). This may be useful for the organization of different sorting particle flows in the same cell for the observation of their interaction in the focal plane.

While the direction of the particle rotation trapped by the considered optical funnels is fixed, it may varies with the distance for the pod-like structures. As an example, we consider the 3D particle movement induced by the beam 𝒫_1,1. Figure 9(a) together with Media 8 show the particle trajectory. For better visualization a cut-out view of the beam and the particle’s trajectory are represented in Fig. 9(b). The video snapshots, several of them are presented in Fig. 9(c), corresponding to the in-line holograms used for the 3D particle-tracking reconstruction are found in Media 9. In this case the particle describes a spiral trajectory from the starting point A [z = −30 μm, see Fig. 9(b)] with clockwise rotation (due to the ${ℱ}_1^{*}$ beam), until it reaches the distance z = −25 μm corresponding to the point B. At this point the particle is simultaneously trapped by both optical funnels ${ℱ}_1^{*}$ and ℱ₁, which exert opposite torques over the particle according with their OAM value (J_z = ∓18, respectively). Therefore the particle goes straight up until the point C, in which its interaction with the ℱ₁ becomes stronger. From the point C to D the particle, as expected, follows a new spiral trajectory with anticlockwise rotation. The relatively short elevation (with respect to the funnel trapping) of the particle during spiral movement (A–B and C–D tracks) is explained by the fact that only one half of the beam energy is participated in light-particle interaction in these regions. We underline that the movements of such particles, induced by the designed beams, have been systematically reproduced. For smaller polystyrene sphere the longer spiral trajectories are expected due to the lager longitudinal regions where the particle is trapped by only one of the funnels ( ${ℱ}_1^{*}$ or ℱ₁) composed the pod.

 

Fig. 9 3D particle-tracking reconstruction for the trapping beam 𝒫_1,1 (a) Media 8 and (b). Particle’s trajectory is also projected into x − y, x − z and y − z planes (grey lines). The in-line holograms (as the ones shown in (c)) used for the reconstruction are stored as video, see Media 9.

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This pod beam allows trapping of multiple particles, which simultaneously rotate in clockwise and anticlockwise directions at different distances z. It could be useful for twisting of rod-like particles or filaments which ends are trapped in the regions with opposite OAM sign.

5. Conclusions

The design of funnel and pod beams suitable for volumetric optical trapping and the method for their correct generation have been discussed. The interaction of these beams with dielectric colloidal particles was experimentally observed. It has been demonstrated that the trapped particles describe a 3D spiral trajectory governed by the forces and torques exerted by the beam carrying OAM. In contrast to a single helical-LG funnel the double-helix funnel beam provides two well-defined 3D channels for the particle movement. The capacity of the proposed bifocal pod beam to exert opposite particle rotations, at different planes during its propagation, is attractive for various micro-object manipulation techniques including all-optical twisting. Moreover the trapping of a rod-like particle by a similar pod beam with the same OAM sign in both focal regions could be used for optically driven stirring or pumps. These facts result useful for volumetric trapping and provide an additional level of control over the trapped particles. The manipulation of particles with different geometries, size and material by these and other funnel and pod beams will be the subject of further research.

While the described setup uses the SLM for the single-beam trap generation, the application of analogue hologram for this task will significantly increase the energetic layout of such optical traps providing setup compactness and reducing its cost.