1. Introduction

Optical tweezers have proven themselves as a powerful tool in diverse research fields, such as biology [1,2] and colloidal physics [3,4]. Research works related to optical tweezers commonly exploit three-dimensional (3D) trapping of particles using the gradient force in tightly focused laser beams. The intensity-gradient optical tweezers provide high precision on spatial control of particles and the possibility of measurement of small force and displacement. In contrast, the scattering force drives the particle away from the trap center, thus guiding the particle along the beam propagation path [5]. Based on the scattering force, optical guiding of microparticles was first demonstrated in 1970 by Ashkin [6], who realized optical acceleration of suspended microparticles using a loosely focused Gaussian beam. However, because of the limited Rayleigh length, the Gaussian beam is not an ideal tool for this goal. Compared with Gaussian beams, nondiffracting beams like Bessel beams have two appealing features: propagation-invariance and self-healing ability, making them more advantageous for particle guiding over long distances [7,8]. Many efforts have been made to shape novel nondiffracting beams with desired trajectories for practical purposes [9–17]. By controlling the trajectory of nondiffracting beams, particles can be flexibly delivered to the desirable destination over long distance using these beams. These feature makes nondiffracting beams a potential tool for optical guiding of microparticles in application of biomedicine [18] and aerosol science [19,20], etc., where transport and sorting of small samples are highly desirable.

In spite of the wide applications of nondiffracting beams in optical guiding, few studies have delivered a direct visualization of the whole dynamic process under tight focusing condition [11,12,14]. Some works have been reported on direct observation of low numerical aperture (NA) optical trapping and guiding in the axial plane (containing the z axis) [21–23]. In this letter, we report a technique that provides a direct visualization of the guiding performance of nondiffracting beams in the axial plane in the tight focusing condition. By investigating the particle dynamics, we demonstrate a parabolic path optical guiding of microparticles over an axial distance of more than 100 µm and maximal lateral deviation of 20 µm. To realize the flexible control of particle direction, we introduce a kind of nondiffracting beam, termed the tilted zeroth-order quasi-Bessel beam, to optical trapping. The tilted quasi-Bessel beams are commonly used for scanning two-photon fluorescence stereo microscopy [24–26]. When applying this kind of beams to optical trapping, we achieve the tilted-path optical guiding of 4 µm polystyrene microbeads with a maximum tilt angle of up to 18.8° with respect to the optical axis over a distance of 300 µm, using an objective lens with an NA of 0.95. The results presented in this work demonstrate the first tilted-path optical guiding of microparticles. As the tilt angle of such beams can be readily controlled, the tilted quasi-Bessel beam provide more possibilities to optical guiding and sorting of microparticles with a much more flexible range of applications. We believe that the presented measuring system and measuring methodology are of significance to the study of optical trapping with nondiffracting beams and will lead to broader applications of novel nondiffracting beams.

2. Materials and methods

To visualize the particle transport by nondiffracting beams, we developed an experimental setup for simultaneous axial-plane trapping and imaging. The setup consist of a home-built holographic optical trapping module and two fluorescence imaging modules (Fig.  1(a)). The linearly polarized CW laser (FB-1064, RGBLase LLC Inc., USA, λ = 1064 nm) for optical trapping has a maximal power of 2 W. The output Gaussian beam is expanded and collimated by a telescope consisting of Lenses 1 and 2 to one with diameter of 10 mm before it is incident onto the spatial light modulator (SLM). For high efficient optical trapping and manipulation, a pure-phase SLM (Pluto-NIR-II, HOLOEYE Photonics AG Inc., Germany) with a resolution of 1920×1080 pixels and pixel pitch of 8 µm was used for beam shaping. A specially designed triangle reflector was employed to reflect the collimated input beam onto the spatial light modulator, and then to guide the retro-reflected modulated beam to the relay system consisting of Lenses 3 and 4. The modulated beam was then directed to the trapping objective (Objective1, 40×, NA0.95, CFI Plan Apo, Nikon Inc., Japan) by Dichroic1 (FF700-SDi01-25×36, Semrock Inc., USA) and focused into the sample chamber, where the beam power measured approximately 30 mW. To obtain a high signal-to-noise ratio image simultaneously in the lateral and axial plane, we employed two fluorescent imaging modules in the setup using excitation LED light with peak wavelength of 470 nm and power of 2W. Two cameras (Point Grey GS3-U3-41C6M-C, 2048×2048, 5.5µm, 90fps@fullresolution, FLIR System Inc., USA) with two band pass filters (BrightLine full-multiband filter set, LED-DA/FI/TR/Cy5-A-000, SemRock Inc., USA) were implemented to simultaneously render the fluorescent images in the lateral and axial planes. Two dichroic mirrors (Dichroic2 and Dichroic3, FF500-Di01-25×36, Semrock Inc., USA) are used for splitting the LED illumination light and fluorescence signal. Here, the lateral plane is defined as the one parallel to the focal plane of trapping objective, i.e. the X-Y plane, whereas the axial plane is perpendicular to the focal plane, i.e. the X-Z or Y-Z plane. Objective1 served for optical guiding/trapping and lateral-plane imaging, while Objective2 (20×, NA0.45, CFI Plan Apo, Nikon Inc., Japan) was used to collect the epi-fluorescence signal for axial-plane imaging.

 

Fig. 1. Optical guiding of microparticles using nondiffracting beams. (a) Schematic of the experimental setup for simultaneous axial-plane trapping and imaging. (b) Principle of the axial-plane imaging using a right-angle microreflector. Red arrow: the nondiffracting beam for optical guiding of microparticles (λ = 1064 nm); blue arrow: the excitation light; green arrow: the emitted fluorescence in the direction perpendicular to the axial plane. (c) The 3D intensity profile of an Airy beam. (d) The 3D intensity profile of a quasi-Bessel beam. (Font size)

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Figure  1(b) depicts the principle of the axial XZ-plane imaging using a right-angle microreflector. The sample solution containing polystyrene fluorescence microbeads 488/515 nm, diameter 4µm, BaseLine Inc., China) is injected into a square microtube made of Borosilicate glass with a square inner diameter of 800 µm and wall thickness of 160 µm (Vitro Tube #8280-50, VitroCom Inc., USA), and illuminated with the SLM-shaped guiding laser beam propagating along the Z-axis. The right-angle microreflector with a size of 1×1×1 mm³ coated with a silver film on the hypotenuse face then enabled recording images of the guided particles in the axial XZ-plane using Camera2. Our system can be employed for investigating the guiding and trapping performance of the nondiffracting beams. Figures  1(c) and 1(d) present the 3D intensity profiles of two nondiffracting beams that are widely studied, i.e., an Airy beam and a quasi-Bessel beam. When used to manipulate particles, these beams can hold the particles laterally in the main intensity lobe by the gradient forces. Furthermore, owing to the scattering force that plays a major role in nondiffracting beams along the propagation direction, the particle will be guided along the beam propagation trajectory. Therefore, the particle is expected to be transported along a parabolic path by the Airy beam, and along a straight path by the quasi-Bessel beam. The axial-plane imaging then provides an efficient way for investigating the trapping performance of these beams.

3. Results

Because of the limited field of view in high-NA objective optics, it is still a challenging issue to investigate the trapping performance of nondiffracting beams under tight focusing conditions. To analyze the particle behavior in nondiffracting beams, multiple lateral-plane images are commonly acquired at different axial positions by using a translation stage [10,11,27]. This method is confronted with the challenge of tracking the particles in real time and quantitatively analyzing particle dynamics, as the mechanical translation is slow. Below, we demonstrate how the presented axial-plane imaging technique can overcome this difficulty in investigating the optical trapping performance of Airy beams and tilted Bessel beams.

3.1 Curved-path guiding using the Airy beams.

The Airy beams are playing important roles in optical trapping as they permit transporting particles along a curved trajectory and bypassing obstacles [10,11]. An Airy beam can be generated from a Gaussian beam incident on an SLM by loading a cubic phase profile onto the SLM, written as [28]

As the propagation properties of Airy beam have been widely studied, here we directly investigate its trapping performance. Although guiding particles using an Airy beam has been demonstrated by Baumgartl et al. [10], the observation was made in an indirect manner, and only the z > 0 part of the parabolic trajectory was used. Here, the axial position of z = 0 represents the focal plane of the objective lens and the middle point of the Airy beam intensity profile before it is axially shifted. To make full use of the trajectory, we shifted the beam by imprinting a lens phase exp(-ik_zz), with k_z and z denoting the longitudinal wavenumber and coordinate, to the cubic phase such that its curve trajectory lies in the z > 0 region. With the help of the axial-plane imaging, we obtain a complete parabolic guiding trajectory, which is directly visualized in the axial XZ plane (Fig.  2(a), also see Visualization 1). The cyclic curves show the guiding paths of the trapped beads, suggesting that the curved-path guiding of microparticles has been realized. Note that the brightness of the moving particle image changes along its trajectory, which might be caused by the inhomogeneous illumination of excitation light.

 

Fig. 2. Particle guiding results using an Airy beam with P = 6 pixels. (a) Time-lapse images of optical guiding of a single 4 µm polystyrene microbead (see Visualization 1). The cyclic curves show the guiding paths of the trapped beads. (b) Particle guiding trajectory in the XZ plane. The blue solid line represents a parabolic curve fitted to the experimental data. (c) The magnitude of particle velocity as a function of the axial position.

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More importantly, the axial-plane imaging technique provides more possibilities to study particle dynamics. To quantify the particle’s motion, we measured the position and velocity of the microbead during its transport with imaging rate of 400 fps (Figs.  2(b) and 2(c)). The velocity was calculated from the measured position data by dividing the displacement at various times with time interval. Quantification of the position and velocity reveals several insights. First, we attain a long transport distance up to 90 µm in the axial Z-direction for a high NA of up to 0.95. Second, the fitted parabolic curve in the middle region of Fig.  2(b) coincides with the path of an Airy beam predicted by theory. Finally, the results given in Fig.  2(c) show that the guided bead moves in the middle region of the path at a velocity four times greater than that at either end. In the central part of the region of existence of an Airy beam, the intensity gradient along the beam propagation direction is quite small and the scattering force dominates. But under such a tight focusing condition (NA = 0.95), the on-axis intensity of an Airy beam varies at the edges of the beam trajectory, which is caused by the destruction effect in tight focusing [29]. This leads both to the generation of axial gradient forces and to the modulation of the amplitude of scattering forces, which is directly proportional to the on-axis beam power. Since the studied system is overdamped, the observed particle velocity is directly proportional to the local magnitude of the net optical force. Consequently, the particle will move at a quite large velocity in the middle region where the scattering force is much larger than the gradient force, while it moves much more slowly with variant velocity at the edges because of the large axial gradient force in these regions.

3.2 Generation of the tilted zero-order quasi-Bessel beams.

In theory, the angular spectrum of a zeroth-order Bessel beam is a perfect ring with infinitely small ring width in the Fourier plane [8]. Therefore, the inverse spatial Fourier transform of a ring-like input field present at the entrance pupil plane of the focusing objective lens results in a Bessel beam in the focal region of the objective (Fig.  3(a)). However, due to the finiteness of the ring width, the generated beam is quasi-Bessel in practice (blue line, Figs.  3(a) and 3(c)), which propagates invariant over a limited distance, while an ideal Bessel beam stays invariant in all space during the propagation. A tilted quasi-Bessel beam (yellow line, Fig.  3(c)) can be obtained by laterally shifting the ring illumination pattern in the entrance pupil plane by a displacement of Δu. To achieve such illumination, the conventional method uses an apodization mask with an annular slit to tailor the input field at the entrance pupil plane. As a result, the apodization method inherently suffers from the low utilization of the incident power.

 

Fig. 3. Principle of generating a tilted zeroth-order quasi-Bessel beam with two methods. (a) The apodization method. The red and green light paths denote the illumination light path when the center of the ring illumination pattern is located on and shifted away from the optical axis, respectively. (b) The method of using an off-axis axicon hologram. (c, d) The enlarged view of the regions marked by the dashed squares in parts (a) and (b).

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To address this issue, an alternative method uses an off-axis an off-axis axicon phase hologram (Fig.  3(b)) with the phase profile ϕ(u,v) expressed as

In this letter, we adopt the decentered axicon method for generation of a tilted Bessel beam by use of an SLM. We carried out numerical simulations to investigate the propagation properties of such a tilted quasi-Bessel beam using the Richards-Wolf diffraction theory [34]. The focusing numerical aperture (NA) is set to be 0.95 and the light wavelength is 1064nm. Through this letter, the phase period ρ₀ is set to be 4 pixels and the origin of the coordinate system (u, v) is located at the center of the SLM chip. As an example, Fig.  4(a) shows an off-axis axicon phase hologram with a resolution of 1080×1080 pixels and the center position at (u_c, v_c)=(−150, 0) pixels. Note that the ring-like optical field in the focal plane of the objective lens exhibits non-uniform intensity distribution along the azimuthal direction (Fig.  4(b)). In this case, the intensity of the ring at x < 0 is obviously stronger than that at x > 0. Propagating from the focal plane over a distance of ∼ 20µm, the ring-shaped field evolves into a tilted quasi-Bessel beam (Fig.  4(c)). The simulated 3D intensity profile clearly presents a quasi-Bessel beam propagating along a tilted straight path in the XZ plane (Fig.  4(d)). The slices P1∼P3 shown in Fig.  4(d) depict the intensity distributions in some transverse planes during the propagation, showing transverse intensity patterns with concentric rings.

 

Fig. 4. Simulated propagation of a tilted zeroth-order quasi-Bessel beams. (a) The off-axis axicon phase hologram for generating a tilted zeroth-order quasi-Bessel beam. The axicon center is at (u_c, v_c) and the red dot indicates the origin of the coordinate system (u,v). (b) The intensity profile of the propagating beam in the focal plane of the objective lens. (c) The intensity profile in the axial plane…” as “The intensity profile of the propagating beam in the axial XZ plane. (d) The 3D intensity profile of the propagating beam. The slices P1-P3 show the transverse intensity distribution at various axial locations. (e) The tilt angle θ of the propagating beam as a function of the off-axis displacement u_c in the entrance pupil plane.

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The most attractive property of the tilted quasi-Bessel beam is that the tilt angle θ with respect to the z axis can be flexibly controlled by simply altering the position of the axicon center (u_c, v_c). For simplicity, we only vary the parameter u_c while keeping v_c=0. The simulation results show that the tilt angle θ increases linearly with the parameter u_c (Fig.  4(e)). For u_c=0, 30, 60, 90, 120, 150 and 180 pixels, the tilt angle θ takes the value approximately 0, 3.1, 6.3, 9.5, 12.7, 16.1 and 18.8°, respectively. A larger tilt angle can be obtained by setting a larger off-axis displacement |u_c|. However, the problem of intensity degradation arises with further increasing the tilt angle, leading to a poor trapping performance in experiment. To avoid this issue, the displacement u_c is limited to 0 ∼ 180 pixels for NA = 0.95, the validity of which can be seen from the results of optical guiding of microparticles presented below. Another noteworthy thing is the axial intensity uniformness of the beam during the propagation. At the beginning of the beam path, the intensity is high (z = 25∼50 µm, Fig.  4(c)), thereafter decreases slowly at z = 50∼100 µm, and then becomes a plateau distribution. In addition, note that the propagation direction of the tilted quasi-Bessel beam can be varied within a solid angle Ω=2π[1-cosθ_max] with θ_max denoting the maximum propagation angle of the tilted quasi-Bessel beam with respect to the optical axis. This is of significant importance for optical guiding of microparticles at various directions.

3.3 Tilted-path guiding using the tilted quasi-Bessel beams.

In optical manipulation with the quasi-Bessel beam, the particles will be held transversely in the main intensity lobe by the gradient force, and be delivered along the propagation path by the scattering force. Using the tilted beams mentioned above, we demonstrate the optical guiding of microparticles along a tilted straight path. As an example, we first present the guiding results obtained with a tilted quasi-Bessel beam with u_c=150 pixels in Fig.  5. The figure shows an overlapped image of the guided particle during the guiding process at t = 0.0∼4.6 seconds. As shown in the figure, the particles initially wandering nearby the ring-like focal field are attracted towards and then guided by the quasi-Bessel beam. Clearly, the time-dependent positions of the guided particle lie on a tilted moving path with a tilt angle of about 15.9°. Besides, it is seen that the whole transport process falls roughly into three stages: initial acceleration over a distance of ∼40 µm (t = 0.0∼0.4 sec.), followed by a slight deceleration with an elapsed distance of ∼60µm (t = 0.4∼1.0 sec.), and eventually an approximately uniform speed (t = 1.0∼4.6 sec.). Since the scattering force proportional to the intensity of light plays a major role in the tilted quasi-Bessel beams, the trapped particle tends to move fast in the high-intensity region, and moves slowly and smoothly when the intensity is weak. This has been verified by the result given in Fig.  5, which is consistent with the intensity profile of a tilted Bessel beam. But as non-uniform axial profile of the beam intensity will also generate the axial gradient forces. These forces in turn will influence the behavior of the manipulated particles, together with the scattering force. Therefore, the velocity fluctuates during the transport of particles.

 

Fig. 5. The overlapped axial-plane image of optical guiding of a 4-µm polystyrene microbead at different times using a tilted Bessel beam with u_c=150 pixels.

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As indicated in Fig.  4(e), optical guiding of microparticles in different directions can be readily achieved by alerting the off-axis displacement of the axicon phase hologram in the entrance pupil plane. To investigate this, we applied the tilted quasi-Bessel beams with u_c=0, 30, 60, 90, 120, 150 and 180 pixels to optical guiding of microparticles. The particles’ positions during their transport obtained from the time-lapse videos of the guiding process (see Visualization 2) show that the tilt angles for u_c=0, 30, 60, 90, 120, 150 and 180 pixels are 0, 3.9, 6.9, 9.8, 12.9, 15.9 and 18.8°, respectively (Fig.  6(a)), agreeing well with the numerical simulations (Fig.  4(e)). Furthermore, the particle dynamics are quantitatively assessed via the velocity curves of the guided particles plotted against their moving distance for various u_c in Figs.  6(b)–6(h). A qualitative analysis shows that the particles move in three phases for all tilt angles: moving at relatedly high speed (50∼100 µm); slowing down to a moderate speed (100∼150 µm) and moving at steady-state rate (>150 µm), corresponding to the intensity distribution of the beam. In addition, the maximal velocity attained by a particle is dependent on the tilt angle. The results indicate that the maximal velocity decreases with increasing the tilt angle. The maximal velocities for u_c=0, 30, 60, 90, 120, 150 and 180 pixels are 691.2, 442.8, 454.7, 432.4, 387.3, 340.5, and 315.6 µm/s, respectively. However, the velocity at the steady state for all cases, which is about 30 µm/s, shows no obvious difference between various tilt angles regardless of measurement error.

 

Fig. 6. The dynamics of motion of a single guided 4-µm polystyrene microbeads in the Bessel beams with different tilt angles (see Visualization 2). (a) The particle trajectories. (b-h) The particle velocities dependent on the displacement along the particle trajectory for u_c=0, 30, 60, 90, 120, 150 and 180 pixels, respectively. The distance is measured along the actual particle path.

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4. Conclusion

In conclusion, we have reported a technique for investigating the optical trapping performance of the nondiffracting beams under the tight focusing condition by directly visualizing the particle’s motion during the guiding process. To this end, we developed a simultaneous axial-plane trapping and imaging technique based on a right-angle microreflector to achieve direct imaging in the axial plane. By doing so, the axial-plane information was recorded directly by the camera. With the axial-plane imaging technique we showed a parabolic path guiding of microparticles using Airy beams with a guiding distance of more than 100 µm and a maximal lateral deviation of 20 µm for NA = 0.95. To realize the tilted path particle guiding, we presented a method of using an off-axis axicon phase hologram to create the tilted zeroth-order quasi-Bessel beams. Numerical simulation results showed that the propagation direction of such beams can be flexibly controlled by simply altering the axicon phase hologram. Using this kind of beams, we revealed the tilted-path optical guiding of microparticles with a maximum tilt angle θ_max with respect to the optical axis up to 18.8° for NA = 0.95. Note that the propagation direction of the tilted quasi-Bessel beam can be varied within a solid angle of Ω=2π(1-cosθ_max). Particle dynamics were analyzed by recording the particle’s time-dependent positions and velocity. While observing the microbeads along the tilted beam, we noticed that the maximal guiding velocity decreased with increasing traveled distance and tilt angle whereas the steady-state guiding velocity was largely independent of the beam tilt angle.

In our work, we employed an objective lens of NA = 0.95 for investigating the guiding properties of with nondiffracting beams with the axial-plane imaging. In practice, high NA can be realized in our setup by using oil or water immersion objective lenses. Importantly, the axial-plane imaging technique provides the ability to investigate the particle dynamics via quantitative measurement of the particle’s trajectory and velocity, and further to investigate light-matter interactions with different angle from the conventional lateral imaging methods. This feature makes our scheme a promising technique for characterizing optical trapping and guiding of microparticles by using nondiffracting beams. Providing direct visualization of the guided particles, our technique also pave the way to investigate the effect of controllable longitudinal intensity profiles of novel beams under the tight focusing condition, for example arbitrarily controlled intensity profile along the propagation trajectories of Airy and Bessel beams [23,35–43]. These novel beams have found increasing applications in microscopy, communications, etc. [44–47]. The presented measuring system and method may help enlarge their range of applications besides optical trapping by investigating their properties.