1. Introduction

Since its inception in the late 70s, the field of optical trapping and manipulation of micron- and submicron-sized objects with light has experienced intense interest and rapid development [1–3]. Optical tweezers, which utilize the presence of mechanical forces arising from the interaction of light with matter, are now an indispensable tool in various physical, biological and medical applications. The recent decade has seen an enormous progress in the field resulting in the implementation of advanced techniques involving, for instance, spatial light modulators, optical fibers, or complex-shaped beams [4–6]. The optimum conditions for the particle trapping are dictated by the optical properties of both the particles and the surrounding medium, as well as the physical nature of the light-mediated trapping forces. For example, while high light intensity attracts and traps transparent high-index objects in a low-index medium, it actually repels low-index particles in a high-index environment [7]. This is why hollow (or doughnut) beams are used for efficient trapping in the latter case. In general, depending on the particular media and application, the robust trapping and manipulation of micro-objects requires tailoring the light beam intensity structure via phase and/or amplitude modulation.

In 2000, Arlt and Padgett [8] introduced the concept of an optical bottle beam to describe a light field having a distinct low- or zero-intensity region entirely or partially surrounded by a high-intensity region. Such a beam can be used as a three-dimensional (3-D) trap. Following this idea various practical implementations of optical bottles have been proposed. Bottle-like light patterns have been produced using phase and amplitude masks [8], spatial light modulators [4, 9], interferometers [10], axicons [11], uniaxial crystals [12], and focusing systems affected by aberrations [13] – to name just a few common techniques. The suitability of an optical bottle for particle trapping and manipulation has been confirmed in experiments with atoms [10, 14–16] and absorbing airborne particles [9, 13]. The concept of an optical bottle has also been recently extended to plasmonic structures [17].

The problem with an ideal, completely closed optical bottle is that once it is formed the light walls actually prevent particles from entering it. This issue becomes even more pronounced when optical manipulation has to be performed with airborne particles. In this case particles can move fast and it becomes technically difficult to turn on the bottle when a particle crosses the region where the minimum of light intensity is supposed to be. This is especially true when the concentration of particles is low and/or their size is comparable with the characteristic size of the bottle. An elegant solution to this problem is to design a bottle in such a way that it could be partially opened and closed to load and unload particles as required.

The purpose of this work is to demonstrate the feasibility of this novel approach. Specifically, we will show that an optical bottle formed by light conically refracted in a biaxial crystal can be opened and closed in a highly controllable fashion by adjusting the polarization state of the input beam. We will then demonstrate loading and unloading of airborne particles into and from such a reconfigurable optical bottle.

2. Theory

Conical refraction refers to the phenomenon associated with the propagation of a light beam along one of the optical axes of a biaxial crystal [18, 19]. Since its discovery in the 1830s by Hamilton and Lloyd [20], the phenomenon has been studied extensively. It turns out that the light intensity distribution after the crystal strongly depends on such parameters of the incident beam as its divergence, polarization and orientation. By varying these parameters one can observe different light patterns including the famous ring pattern. A close observation of this pattern reveals that it actually consists of two concentric bright rings, separated by the so-called Poggendorff dark ring. For a focused input beam, this splitting is most clearly seen at the focal plane of the lens. When the observation point moves away from this plane, either along or against the beam propagation direction, the ring pattern shrinks until a bright on-axis spot, known as the Raman spot, is formed. It is worth noting that the ring intensity pattern combined with the nontrivial polarization structure [21] of the conically refracted light has already been employed in trapping of micro-objects. However, in that case the beam was used to form traditional tweezers for two-dimensional manipulation of particles [22, 23]. Here, we are interested in the 3-D structure of the conically refracted light.

Consider a focused Gaussian input beam whose Rayleigh length and 1/e² waist are denoted by z_R and w₀, respectively. Correspondingly, the crystal is characterized by its length l and conicity parameter α defined as half the intersection angle of the two ellipsoids of refractive indices. The product of these parameters R₀ = αl gives the radius of the Poggendorff dark ring after the crystal in the geometric optics approximation. In general, the conical refraction intensity pattern is determined by the ratio ρ₀ = R₀/w₀, which implies that for a fixed α, ρ₀ can be controlled by varying either l or w₀ [18, 19, 24, 25].

In the paraxial regime the light intensity distribution after a biaxial crystal can be analytically described as [24, 25]

$\begin{array}{l}{B}_C(\rho ,Z)={\displaystyle {\int }_0^{\infty }\eta a(\eta )}{e}^{-i\frac{Z}{4}{\eta }^2}\cos \left(\eta {\rho }_0\right)J_0\left(\eta \rho \right)d\eta ,\\ B_S(\rho ,Z)={\displaystyle {\int }_0^{\infty }\eta a(\eta )}{e}^{-i\frac{Z}{4}{\eta }^2}\sin \left(\eta {\rho }_0\right)J_1\left(\eta \rho \right)d\eta ,a(\eta )={\displaystyle {\int }_0^{\infty }\rho E_{in}(\rho )}{J}_0\left(\eta \rho \right)d\rho .\end{array}$ Here E_in(ρ) denotes the amplitude of the input beam, $\eta =\left|{\overrightarrow{k}}_{\perp }\right|w_0$ is proportional to the absolute value of the transverse components of the wave vector projected onto the entrance surface of the crystal, and J_m denotes a Bessel function of the first kind of order m. From Eqs. (1) and (2), the on Z-axis intensity distribution is defined solely by the function B_C (ρ = 0, Z) since J₁(0) = 0. According to Eq. (1), for a circularly polarized input beam the light intensity distribution after the crystal has a cylindrically symmetric structure. One can also show that if ρ₀ >> 1, a well defined optical bottle between the two Raman spots is formed [18, 19, 24, 25]. This situation is illustrated in Fig. 1(a) where we show the 3-D light intensity distribution after the crystal for ρ₀ = 10. In this Figure one can clearly observe the existence of a dark region surrounded by high light intensity. The radius of the bottle at its widest point, which is located at Z = 0, with a high degree of accuracy can be approximated by R₀ and is thus determined solely by the crystal. On the other hand, the distance between the Raman spots (i.e., the length of the bottle) depends on both the beam and the crystal and is given by

 

Fig. 1 Calculated light intensity distributions of a light beam conically refracted in a biaxial crystal (ρ₀ = 10). (a) a circularly polarized input beam (Eq. (1) leads to a fully closed optical bottle (Media 1); (b) a linearly polarized input beam (Eq. (2) leads to an open-top optical bottle (Media 2). The 2-D images depict the principal cross sections of the light intensity distribution. Bright (dark) corresponds to high (low) light intensity.

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A more complex intensity pattern can be realized with a linearly polarized input beam. From Eq. (2) one finds that linear input polarization causes the loss of perfect cylindrical symmetry. Instead of the rings, the transverse light intensity distribution now takes the form of crescents, which are produced within a certain interval along z. In 3-D, this implies the formation of a gap in the side wall of the otherwise perfect optical bottle. This situation is depicted in Fig. 1(b). More importantly, the angular position of the gap can be varied by rotating the azimuth of linear polarization. The appearance of a hole in an optical bottle and the ability to vary its position and size provide a unique opportunity to control the trapping conditions. Making a hole in the light wall would allow one to quickly load particles into the trap after which it can be closed by switching over to circular input polarization. Reverting to linear polarization with a different azimuth would enable opening the trap at any desired angular location to unload the particles.

3. Experiment

The experimental setup is shown schematically in Fig. 2. The light beam from a cw laser (λ = 532 nm, power 100 mW) passes consecutively through a half- and quarter-wave plate and after focusing with a lens propagates along the optical axis of a monoclinic KTP crystal (α = 0.0177 rad) cut perpendicular to one of its optical axes. The crystal length is l = 11.3 mm inthe beam propagation direction (i.e., z-axis). The light intensity distribution inside the optical bottle, which is formed in the focal region of the focusing lens, is mapped by translating a CCD matrix along the beam propagation direction in 500 µm steps and recording the corresponding transverse light intensity patterns. A sequence of intensity slices is then used to reconstruct the full 3-D structure of the optical bottle. In order to allow observation of particles that are trapped inside the bottle, an objective is used to form their images on the CCD matrix (not shown in Fig. 2).

 

Fig. 2 Experimental setup. λ/2 and λ/4 respectively denote the half- and quarter-wave plate used to reconfigure the bottle.

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The results are presented in Fig. 3 and the accompanying movies. We start with a circularly polarized input beam as depicted in Fig. 3(a). It is evident that the conically refracted light does form an optical bottle with a dark central region entirely surrounded by light. The transverse size and the length of the bottle can be adjusted by varying the focusing geometry. In the next step, the polarization of the input beam is changed to linear. The resulting 3-D light intensity distribution is shown in Fig. 3(b) and the accompanying movie. The structure is no longer cylindrically symmetric, with the top wall of the bottle featuring an opening, in agreement with the theoretical prediction (see Fig. 1(b)).

 

Fig. 3 Experimentally recorded 3-D structure of an optical bottle beam formed by conical refraction of light in a KTP crystal (ρ₀ = 18). (a) The bottle is fully closed for a circularly polarized input beam (Media 3) and (b) opened for a linearly polarized input beam (Media 4). The hole in the top wall of the bottle is clearly visible. The 2-D images depict the principal cross sections of the light intensity distribution. The size of the 2-D images is 970 µm along the x- and y-directions, and 26000 µm along the z-direction. Bright (dark) corresponds to high (low) light intensity.

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We used the optical bottle depicted in Fig. 3 to demonstrate trapping and manipulation of light absorbing particles in air. Such particles can be efficiently confined by employing the photophoretic force [26, 27]. In this case the illumination of particles leads to their heating and nonuniform temperature distribution. Interaction with the surrounding air results in the appearance of the photophoretic force, which tends to repel particles from high intensity regions. We have recently demonstrated efficient photophoretic trapping and transport of micron-sized particles over tens of centimeters [28]. In our experiments with the aforementioned optical bottle we used glass shells covered with a thin layer of carbon (~200 nm) in order to make them light-absorbing. The external diameter of the shells varied from 10 μm to 100 μm, whereas the wall thickness was ~400 nm based on SEM-assisted measurements performed on several crushed shells. To prevent accidental air flows from affecting the trapping process, the optical bottle was formed inside a transparent cylindricalglass cell placed immediately behind the biaxial crystal. To load the bottle, the cell was rotated and some shells eventually detached from its walls, fell and got trapped inside the intensity minimum at the beam axis.

We found that while a particle could be trapped using either a fully closed (i.e., circular input polarization) or opened (i.e., linear input polarization) bottle, the loading process was tens of times faster in the latter case. As the internal diameter of the bottle was rather large (~400 µm) the bottle could accommodate a great variety of shells. In Fig. 4 we show three examples of shells with different sizes confined in the trap. The trapping was generally very robust, with the particles stably resting at the lower light wall due to gravity. However, sometimes we observed the trapped particles to oscillate inside the trap with the oscillation frequency increasing with the optical power. Such a dynamics was observed in the case of complex objects, such the two fused shells in Fig. 4(d).

 

Fig. 4 Experimental images of hollow glass shells of different size trapped inside the bottle beam (ρ₀ = 18). In (a-c), the shell’s diameter is respectively 40 µm, 85 µm and 95 µm. (d) Simultaneous trapping of two 40 µm glass shells.

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The ability to open or close the bottle at will by varying the input beam polarization gives a unique opportunity not only to easily trap micro-objects but also to release them. This functionality is demonstrated in Fig. 5. The sequence of image represents different stages of reducing the light intensity of the bottle’s lower wall. It is clearly seen that the initially stably trapped shell drops out of the trap under gravity when a hole in the lower wall is created.

 

Fig. 5 An experimentally recorded sequence of images illustrating unloading of the trapped 40 µm glass shell from the bottle (ρ₀ = 18). The sequence of graphs (i-iv) represents different stages of opening the bottle.

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4. Conclusion

In summary, we have employed the phenomenon of conical refraction of light to create an optical bottle for photophoretic trapping and manipulation of airborne particles. By changing the input beam polarization from circular to linear the light wall can be opened up to let the particles under study enter the otherwise almost impermeable trap. If necessary, the trapped particles can later be released by rotating the plane of the linearly polarized input beam and thus adjusting the angular position of the exit opening in the trap. The ability to perform such manipulations in ambient air has been demonstrated with relatively large and heavy light-absorbing glass shells.