The pioneering works of Ashkin on optical trapping of microscopic particles [1] paved a road for a whole range of practical applications in biology, physical chemistry and condensed matter physics [2–4]. When a particle interacts with a laser beam it is subject to the combined action of several forces. The forces associated with the presence of the laser’s electromagnetic wave include the scattering force or radiation pressure, the gradient force and the photophoretic force [1, 3, 5]. The “non-optical” forces which affect the motion of the particle are gravity, buoyancy and, for instance, the forces of thermophoresis and diffusiophoresis.

Optical tweezing of transparent (i.e., non-absorbing) particles submerged into a transparent medium becomes possible by counterbalancing the scattering and gradient force [1, 2]. For absorbing particles, however, the photophoretic force dominates and can be orders of magnitude larger than the scattering and/or gradient force depending on the optical, physical and shape parameters of the micro-object to be optically manipulated [6]. The photophoretic force which originates from a nonuniform heating of absorbing particles in a fluids and gaseous media [5,7] is typically repulsive and tries to push them away from the regions of maximum light intensity. This precludes stable trapping and reliable manipulation of absorbing particles using light fields with smooth intensity distributions. Depending on the wavelength of light, major classes of micro-objects including aerosols, artificial nano- and micro-structures (e.g., semiconducting nanowires, nanotubes, nanoclusters), protein crystals, and biological cells fall under the category of “absorbing” particles whose optical guiding can be achieved only by controlling the photophoretic force.

The ability to levitate a particle photophoretrically in gaseous medium was first demonstrated by using a superposition of counter-propagating ${\text{TEM}}_{01}^{*}$ and TEM₀₀ laser modes [8]. In this case the ${\text{TEM}}_{01}^{*}$ mode confined the lateral motion of the particle, whereas the TEM₀₀ mode levitated the particle due to radiation pressure. A possibility to trap and move (i.e., not just levitate) light-absorbing micron-sized particles in air by the photophoretic force was demonstrated first by employing two counter-propagating vortex beams [6, 9] and later with a single, slowly diverging vortex beam [10]. Recently, photophoretic trapping and back-and-forth transportation of light-absorbing aerosols have been shown inside an optical vortex bottle beam generated with a spatial light modulator [11].

In this paper we show that stable trapping, manipulation and delivery of light-absorbing particles in air can be achieved with the simplest imaginable optical configuration comprised of a single Gaussian laser beam focused with a conventional lens. Our approach utilizes spherical aberration - an inherent property of any lens with two spherical surfaces. The focal region of such a lens consists of a series of dark optical traps, i.e., regions of low or zero intensity surrounded by regions of high intensity. The proposed approach allows one to achieve a full translational and rotational control over the trapped particle on the macroscale by combining the focusing optics with an optical fiber. The presented results also show that our novel technique of photophoretic trapping differs dramatically from the conventional optical tweezers which are based on counterbalancing the gradient and scattering force to trap micro-objects. In the latter case, any distortions to a propagating wavefront (i.e., aberrations) are highly detrimental as they spread the intensity distribution in the focal region and make it increasingly difficult to transfer the optical momentum to matter [13, 14].

The experimental concept of the trap is schematically shown in Fig.1. The output TEM₀₀ mode from a CW solid-state laser operating at λ = 532 nm is coupled into a single-mode optical fiber with a nominal numerical aperture (NA) of 0.12. The transverse component of the electric field inside the fiber core is well approximated by a Gaussian function [15]. After the fiber the light is collected and collimated with an objective O and then focused in free space with a plano-convex lens L of focal distance f = 25.4 mm. We will show later that the entrance aperture of L has to be significantly under-filled to achieve the desired intensity distribution in its focal region. Specifically, the radius of the beam waist w ₀ at L was 1.1 mm, whereas the radius of the entrance aperture of L inside the optical mount a was 12 mm. The beam waist at L can be adjusted by changing the distance between the fiber end and the front focal plane of O. To maximize positive spherical aberration introduced by L its flat surface is oriented to face the incoming collimated light. Other types of positive lenses and orientations thereof are also possible. For positive spherical aberration the rays close to the optical axis intersect it near the paraxial focus position. As the ray height at the lens increases, the position of the ray intersection with the optical axis moves farther and farther from the paraxial focus towards the lens. In a carefully aligned system the only significant distortion is the one introduced by primary (i.e., third-order) spherical aberration. The aberration function ϕ, which defines the deviation of the distorted wavefront from the ideal convergent spherical wavefront, is well approximated in this particular case (viz., the position factor is equal to −1 and the shape factor is equal to 1) by [16]:

 

Fig. 1 Schematic of the bottle beam trapping experiment. The particles are trapped inside the low intensity regions of the aberrated focus of L, as shown in the inset. The red arrow indicates the red (HeNe) beam.

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The intensity distribution in the focal region of a lens illuminated with a Gaussian beam may be found by evaluating the scalar Debye integral, which is based on the Huygens-Fresnel principle and uses the boundary conditions approximated by the Kirchhoff diffraction theory [17]. The spatial variation of the intensity I is then:

As it was mentioned earlier, optical trapping of absorbing particles floating in a transparent medium is expected to occur inside regions of low or null light intensity due to the thermal nature of the photophoretic force [6]. The insets in Fig.2(a) present the theoretical and experimentally measured focal intensity distribution in a plane containing the optical axis, whereas the graph compares the corresponding axial distributions. Both the simulations and experiments were performed at w ₀ = 1.1 mm. In our simulations we used (2) with the aberration function given by (1). In the paraxial approximation the 3D intensity distribution near focus is cylindrically symmetric. The coordinate z represents the distance measured axially from the Gaussian focus, i.e., the point where the paraxial rays intercept the optical axis in the absence of spherical aberration. Fig.2(a) clearly demonstrates that the aberrated focal area consists of discrete dark regions of low intensity surrounded by local intensity maxima. In 3D the focus is essentially a structured array of dark traps or “bottle beams” [18]. We showed earlier that field structures consisting of multiple, randomly distributed speckles generated from a coherent laser beam are suitable for a photophoretic trapping of carbon particles [19].

 

Fig. 2 Analysis of the aberrated focus.(a) Measured (circles) axial light intensity distribution in the aberrated focus used in the particle trapping experiment shown in Fig.1. The solid line represents theoretically calculated intensity. The experimental plot was obtained by scanning the imaging optics (not shown in Fig. 1) with a 2 μm step along the beam propagation direction z, i.e., from left to right. The Gaussian focus is at z = 0. The insets represent respectively the simulated and experimentally obtained intensity distributions in the xz-plane inside the aberrated focal region of lens L. In both cases w ₀ = 1.1 mm. (b) Normalized axial light intensity distributions calculated for different beam width w ₀. The normalization is performed with respect to the peak intensity of the unaberrated focus (i.e., ϕ = 0) for w ₀ = 1.1 mm. The inset zooms into the intensity distributions for different values of w ₀ inside the first minimum.

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To optimize photophoretic trapping the intensity on the “light walls” should be as high as possible, whereas the intensity inside the dark regions should be as low as possible - ideally zero. The graphs in Fig.2(b) show that for given f and ϕ these conditions can be achieved only with certain ratios of the beam waist and entrance aperture diameter w ₀/a. As the ratio is reduced, the intensity variations between neighboring minima and maxima become weaker and weaker which, in the limit, makes it impossible to trap particles in a smooth Gaussian beam. By increasing this ratio significant intensity variations between the minima and maxima can be achieved, but in this case the intensity is non-zero in the minima located on the optical axis. The non-zero intensity inside such traps leads to undesirable heating of the particles, which also affects the stability of the trap due to the air/gas currents caused by convection. Moreover, the presence of a particle inside a non-zero intensity region affects the intensity distribution behind it and hence makes trapping of subsequent particles less controllable. There exists, however, a range of the w ₀/a - ratios when a series of real dark traps can be produced along the optical axis. By tuning this parameter one can reduce the light intensity almost to zero inside one or several on-axis traps. In fact, Fig.2(a,b) show the situation when the first intensity minimum is optimized in this manner.

Figs. 3(a,b) present the 3D structure of a spherical aberration-induced dark trap located between the first and second on-axis intensity maxima (i.e., the first intensity minimum), whereas Figs.3(c,d) show a “secondary” trap located upstream along the beam, i.e., closer to the focusing lens. Either of the traps consists of a cylindrical light wall which is closed with “light plugs” from both ends. The bottle beam shrinks both in the lateral and axial direction as one moves to the secondary intensity minima (see the captions to Fig.3). The traps demonstrated in Fig.3(a,c) were generated at w ₀ = 1.1 mm and 1.4 mm, respectively, in order to achieve zero intensities at the corresponding minima. The shown trapped micro-objects are agglomerates of carbon nanoparticles with complex internal structure which were produced by high-repetition-rate laser ablation of graphite targets. By using the same focused beam we also trapped solid graphite particles and agglomerates of carbon nanotubes. In all experiments the trap was loaded by first floating the particles in air and then turning on the trapping beam.

 

Fig. 3 Agglomerates of carbon nanoparticles trapped inside the aberrated focus. The trapping power is ∼ 5 mW. (a) and (b) correspond to the opposite side views of the 3D intensity distribution near the 1^st axial intensity minimum (see Fig. 2). (c) and (d) are two side views of a “secondary” axial trap which corresponds to the 7^th axial intensity minimum. The corresponding axial intensity maxima (i.e., the “light plugs”) confine the axial motion of the micro-objects. The radius of the 1^st bright ring (i.e., the radius of the ’bottle beam’) is 5.2 μm in (a) and 3.1 μm in (c). The apparent size of the trapped micro-objects along z is enlarged by the point spread function of the imaging optics. The beam propagation direction is denoted by arrows.

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In another set of experiments a microscope objective with a variable amount of positive spherical aberration was used to study particle trapping in a much tighter focusing geometry and also to investigate whether both positive and negative spherical aberrations are suitable for this purpose. For negative spherical aberration the marginal rays intersect the optical axis behind the paraxial focus position. Positive and negative spherical aberrations of the same magnitude produce identical intensity distributions except that they are mirror images of each other with respect to the Gaussian focal plane. We introduced negative spherical aberration by placing a 1 mm-thick plane-parallel plate of fused silica in front of the objective [16, 21]. The induced negative spherical aberration increases linearly with the plate thickness and by undercorrecting or overcorrecting it by the same amount one can easily flip the focal intensity distribution. We have confirmed this by trapping and moving micron-sized agglomerates of carbon nanoparticles using spherical aberration of both signs. Micromanipulation based on negative spherical aberration allows one to position micro-objects on various substrates while keeping the region of the highest intensity away from the surface.

Fig. 4 together with a series of videos demonstrates the capability of our technique to robustly trap and manipulate light absorbing micro-objects in air. The photophoretic force dwarfs the forces of radiation pressure and gravity and hence allows one to move the trapped particles along any trajectory in 3D using any orientation of the trapping beam with respect to gravity. The latter capability should be assessed in the context of the earlier reported experiments on optical levitation where the trapping beam can be directed only in one direction - against gravity. With a trapping power of only 0.5 – 30 mW the aberration-based trap was used in open air as it could easily withstand air currents estimated at 10 – 50 cm/s depending on the focusing geometry and the type and size of the particles. For a fixed trapping power a tighter focusing provides a more stable trap because in this case the intensity minima are surrounded by regions with higher light intensity. The size of the trap can be varied by tuning the effective NA of the focusing optics. Generally, the remarkable simplicity of the proposed approach allows it to be employed in a wide range of experimental geometries. If the light beam is delivered through an optical fiber, the 3D manipulation is restricted only by the length of the fiber and the maximum power which can be handled by the fiber. The spectral range of the trapping light is limited solely by light absorption in the fiber core and the lens (i.e., λ ∼ 0.3 – 3μm for glass). Polychromatic light in combination with achromatized focusing optics can be used for trapping as the focal intensity pattern itself depends only weakly on the spectral bandwidth. Experiments inside chambers filled with different gases within a broad pressure range including trapping and manipulation through optical windows are also possible. In fact, we trapped and moved the absorbing particles inside a glass container with 1.2 mm-thick walls using the same planoconvex focusing lens. The focal intensity which was affected by negative spherical aberration introduced by the container’s walls was optimized by adjusting the diameter of beam waist.

 

Fig. 4 3D photophoretic macro-manipulation of absorbing particles in air. (a) (Media 1): a solid graphite particle (encircled) with an estimated characteristic size of 5 μm trapped inside the aberrated focus of the lens L. The trapping power is 25 mW. In (Media 1) manipulation is performed inside a glass container (wall thickness 1.2 mm). (b) (Media 2): an agglomerate of carbon nanoparticles inside the focus of a microscope objective (Olympus LUCPlanFLN 60x, NA = 0.70, 0.1 – 1.3 mm glass thickness correction). The radius of the beam waist is 0.8 mm, the radius of the entrance aperture of the objective is 2.5 mm. The estimated radius of the bottle for these parameters is 0.7 μm (in our calculations we used the vectorial Debye integral discussed in [20]). The trapping power is 1 mW in (b) and 5 mW in Movie 2. (c), (d) (Media 3): controlled insertion and manipulation of a 5 μm graphite particle inside a capillary with an inner diameter of 0.8 mm. The trapping power is 30 mW. In (a–d) gravity is directed from top to bottom.

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While we were considering here trapping in gaseous media our technique should be, in principle, applicable to fluids. In fact, photophoretic agglomeration of colloidal particles in water has been recently reported [22]. However, as the nature of the photophoretic forces in liquid is more complex [23] than that in gases the issue of efficiency of particle trapping and manipulation in liquids requires further investigation.