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Abstract
We introduce a new type of multiple and off-axis optical bottles (OBs) based on the chirped circular Pearcey Gaussian vortex beam. This kind of beam allows the generation of the OBs with a perfect bottle shape through coherent superposition. Also, we show that the number and the position of the OBs can be precisely and flexibly controlled. The experimental results agree well with our numerical simulations, and we observe stable trapping of the mesocarbon microbeads particles by the proposed bottle beam.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Corresponding to the focusing effect mediated by an optical lens, the focusing effect of the abruptly autofocusing (AAF) beams originates from their field structure [1]. The main feature of these beams is that their intensity retains a relatively low index during propagation, and abruptly increases by orders of magnitude right before the focal point [2], without the need for the focusing lens. The circular Airy beam (CAB) proposed in 2010 is the first AAF beam to be found [2]. Within the next decade, the hyper-geometric laser beam [3], the circle Pearcey beam (CPB) [4] and the circular swallowtail beam [5] carrying AAF property were consecutively put forward. Compared with the CAB, the CPB has higher peak intensity and can eliminate the oscillation after the focus [4]. So far, we have known the AAF beams show a great potential application value, and have been successfully implemented in the optical tweezers systems [6–8].
In addition, the optical chirp has been an effective tool in the area of the generation and manipulation of beams in the laser systems. Zhang et al. took the chirp into account on the Airy beam firstly, investing the dynamics of the linearly and the quadratically chirped Airy beam with a parabolic potential [9]. The propagation characteristics of the chirped circular Airy beams were investigated by Zhang in 2017 [10], and other kinds of chirped beams have also been widely studied [11–14]. Meanwhile, as the spatial chirp functions similarly to a lens, and has the capability to focus and defocus the beams, the well-known chirp was also utilized to construct the optical bottle (OB). The OB produced by the second-order chirped ring Pearcey Gaussian vortex beams was proposed by Zhang et al. [15]. Also, Xu and his co-workers have successfully generated the off-axis OB from the second-order chirped symmetric Airy vortex beams [16].
At the meantime, the so-called optical tweezers have been utilized to manipulate and guide the particle opportunely in the past decades [17]. Later, as a new concept, the OB which describes a structure of zero intensity surrounded in all three dimensions by the area of higher intensity was proposed [18]. Compare with the traditional optical tweezers, such structure expands the capture of a single particle to a large number of particles, and is able to transport a large number of particles [19–22]. So far, the OB has been implemented in the trapping of nanoparticles and microparticles in various media [23–26]. Except the interference of the Gaussian beam and the Laguerre Gaussian beam [18], many methods to generate the OB were introduced in these years. For example, the OBs generate from the Moiré technique [27], the Fourier-space [28,29], the laser cavity [30], the optical axicon [31] and the conical refraction [32]. Also, the OB generates from the quadratically chirped symmetric Airy vortex beam [16] and the astigmatic-phase ring Airy Gaussian vortex beam [33] has been put forward. In this paper, we propose a new type of multi-off-axis OBs generated from the chirped circular Pearcey Gaussian vortex beam (CCPGVB). Compared with previous research, this kind of bottle beam has higher adjustable degrees of freedom, the location and the number of the OBs can be flexibly controlled as needed. The structure of this paper is organized as follows: In section 2., we construct the theory model in detail. Then, we analyze the characteristics of the OB in section 3.. Further, in section 4., we generate the CCPGVB experimentally, and realize the confinement of the mesocarbon microbeads (MCMB) particles. Finally, in section 5., we have summarized the main conclusions of our discussions.
2. Theoretical models
Under the paraxial approximation, a laser beam propagates along the $z$-axis in free space can be described by the Schrödinger equation [4,15]:
The complex amplitude of the CCPGVB at an arbitrary plane can be obtained by the angular spectrum formulas: [35]
Subsequently, although it is difficult for us to obtain the analytical solution of the CCPGVB in free space, its complex amplitude at an arbitrary plane can be easily obtained by the inverse Fourier transform of $\tilde E\left ( {{k_x},{k_y}} \right )$ with an additional phase $\exp \left ( {iz\sqrt {k^{2} - k_x^{2} - k_y^{2}} } \right )$, which is the so-called split-step Fourier method [36]. In this paper, we assume: ${w_0} = 3$mm, ${x_0} = 0.15$mm, $\chi = 1$, ${l} = 2$ and $\lambda = 532$nm.
3. Numerical simulations and analyses
The typical simulation results are displayed in Fig. 1. We can see the CCPGVBs propagate along an alterable-incline straight line trajectory and spontaneously form four off-axis foci. Two of which have higher intensity and form a perfect shape off-axis OB. Figure 1(b) shows the transverse intensity distributions of the CCPGVBs, from which we can observe that the initial intensity pattern is different from the CPB [4], which can be attributed to the amplitude modulation effect of the $C\left ( {x,y} \right )$. While propagating, the CCPGVBs move slowly to the right and maintain a central symmetric shape. Meanwhile, lots of studies have demonstrated that a stronger chirp always leads to a shorter focal distance [15,37,38]. Thus, it shows that the auto-focusing effect of the CCPGVB carrying the strong chirp and relatively weak chirp forms the head and the bottom of the OB, respectively, as Figs. 1(b2) and 1(b3) depict. Furthermore, the cascading of the CCPGVBs constructs the body of the OB, creating a 3D area with zero intensity surrounded by regions of higher intensity. Here we emphasize that the appearance of OB is attributed to the overlapping of the CCPGVBs rather than the divergence-and-convergence procession of the beam energy. Beyond that auto-focusing effect from the AAF beam, we discern another kind of focus during the CCPGVB evolution, which is chirp-dominated. After the OB is formed, the diffraction of the beam is inhibited, and the beam finally converges to form other two off-axis foci. By numerical simulation, we learn that the positions of these off-axis foci are just the focal points of the quadratic chirps, verifying our conclusion. In addition, the dislocation between the circular Pearcey Gaussian vortex beam (CPGVB) and the quadratic chirp leading to the redirection of beam’s propagation trajectory, from the $z$-axis to an alterable-incline straight line: $\frac {x}{{{x_1}}} = \frac {y}{{{y_1}}} = \frac {{2{c_1}z}}{{kx_0^{2}}}$. Subsequently, those four focal points generate along the same oblique line trajectory. However, it should be emphasized that only when ${x_n}$, ${y_n}$ and ${c_n}$ meet the conditions mentioned abov can these four focal points generate along the same trajectory and form a perfect OB.
Fig. 1. Evolution of the CCPGVBs and the formation of the OB. (a) Side view of the CCPGVBs numerical evolution; (b1)-(b4) corresponding transverse intensity distributions marked by the dotted line in (a). In (b1)-(b3), the original light spot surrounded by the white dotted line is enlarged and redrawn at the position pointed by the white arrow for better observation.
In this paragraph, we pay attention to the influences of the dislocation factors $\left ( {{x_n},{y_n}} \right )$ on the propagation properties of the CCPGVB. On account of the circular symmetry of the CCPGVB, we investigate the ${x_n}$ as an example. Figure 2 depicts the CCPGVBs with different dislocation factors. We can intuitively find that the CCPGVBs with larger $\left | {{x_1}} \right |$ has greater off-axis degree. As the trajectory equation delineates, the displacement degree linearly varies with the dislocation factor. Interestingly, we glean from this that even if the dislocation factor ${x_1}$ is varied and leading to distinct trajectories, the positions of the OB’s head and bottom on $z$-axis are unaltered. This can be confirmed in Fig. 2(a), where the white dotted lines nearly coincide. It indicates that as the $\left ( {{x_n},{y_n}} \right )$ change, the focal positions will only move transversely. The focusing positions along the optical axis are mainly determined by the quadratic chirp factors as the basic parameters remain constant. In addition, one can observe that the body of the OB ets slim as the value of ${x_1}$ goes up. That makes sense, as the invariance of the positions of the OB’s head and bottom on $z$-axis, the whole bottle structure seems to be stretched, as the beam moves away from the optical axis.
Fig. 2. Propagation of the CCPGVBs with different dislocation factors. (a1)-(a3) Side views of the CCPGVBs numerical evolution, the white dotted lines are used for marking the focusing positions of the CCPGVBs in the $z$-direction; (b1)-(b3) transverse intensity distributions at initial, which are normalized by the maximum intensity showed at (b1); (c1)-(c3) normalized light intensity diagrams. All other parameters are the same as those in Fig. 1.
Moreover, as Figs. 2(b1) and 2(b3) depict, we point out that the OB generated from the CCPGVBs usually has an inhomogeneous intensity distribution, this may be a defect for confining the particles. So as to create an OB with uniform energy distribution, the dislocation factors can be adjusted, and we observe a good result as we set $\sqrt {x_1^{2} + y_1^{2}} = 0.5$mm. Meanwhile, we can learn from Fig. 2(c) that the CCPGVBs seem to have higher normalized peak intensity, as the value of $\left | {{x_1}} \right |$ rises. What we need to outline here is that the dislocation factors do not actually affect the focusing ability of the CCPGVB. The reason we observe this index growth is that the rise of the dislocation factors resulting in the decrease of $\left | {C\left ( {x,y} \right )} \right |$, which ensues the decline of the maximum intensity at the initial plane, and the intensity diagrams in Fig. 2(c) are normalized by the maximum intensity of the initial plane.
The quadratic chirp factor also shows tremendous influences on the properties of the CCPGVB. Figure 3 demonstrates the CCPGVBs with different quadratic chirp ${c_2}$. For a higher value of ${c_2}$, we can see that the OB shrinks. Specifically speaking, when the ${c_2}$ increases, the length and the width of the OB reduce, leading to a bottle with a smaller volume. This allows us to accommodate the particle with different bottle sizes as we implement the bottle beams in the optical tweezers systems. Meanwhile, we are interesting in the effect of the quadratic chirp on the focusing distance since it is essential for controlling the spatial location of the OB. It has been concluded in previous studies that the focusing distance of the CPGVB is irrelevant to the topological charge and has a fixed value: ${z_e} = 2kx_0^{2}$ [4]. Likewise, for an arbitrary quadratic chirp $c$, the focal length of the chirped circular Pearcey beam (CCPB) can be calculated from the Fresnel diffraction integral: $L = \frac {{2kx_0^{2}}}{{1 + 4c}}$ [38,39], which gives a good prediction about the focal distance of the CCPGVB, showed in Fig. 3(c). Further, the spatial locations of the foci can be derived from the trajectory equation analytically: $\left ( {\frac {{4{c_1}{x_1}}}{{1 + 4{c_n}}}\textrm {{, }}\frac {{4{c_1}{y_1}}}{{1 + 4{c_n}}},\frac {{2kx_0^{2}}}{{1 + 4{c_n}}}} \right )$. This gives us the possibility to accurately control the location of each OB, even in the case of multiple OBs. In conclusion, the quadratic chirp factors exhibit a great contribution to the high tunability of the CCPGVB.
Fig. 3. Propagation of the CCPGVBs with different quadratic chirp factors. (a1)-(a3) Side views of the CCPGVBs numerical evolution, the white dotted line is used for marking the position of the second focus of the CCPGVBs in the $z$-direction; (b) normalized peak intensity curves with different ${c_2}$; (c) the solid red line indicates the focal length of the CCPB as the function of ${c_2}$, and the blue dots represent the focusing positions of the CCPGVB with different ${c_2}$. All other parameters are the same as those in Fig. 1.
In the remainder of this section, we concentrate on the multi-OB effect of the CCPGVBs. To generate the multi-OB, we cascade more CCPGVBs. Figure 4 shows the intensity volumes of the bottle beam with different N. Although the modulation term $C\left ( {x,y} \right )$ is generally asymmetric, its amplitude $\left | {C\left ( {x,y} \right )} \right |$ has a centrosymmetric form, leading to the variant radial variation patterns at initial. It is observed that many side lobes accompany the generation of the OBs. For a determined $N$, $N-1$ OBs are created, each CCPGVB cooperates well with others, forming a series of olive-like OBs located on the same oblique line, with a homogeneous intensity distribution. Also, here we plot the three-dimensional isosurfaces of the CCPGVBs in Fig. 5 in order to better observe the bottle structure. It is clearly shown that the CCPGVBs form almost completely closed three-dimensional zero intensity dark cores, surrounded by the area of higher intensity. This demonstrates the existence of the high potential barrier around the OBs in the case of different $N$ values. In addition, we know the effect of the modulation term is equivalent to the superposition of multiple CCPGVBs. Consequently, this term will equally divide the energy of the CPB into several parts. When the CCPGVBs propagate, only a part of the beam’s energy converges in the corresponding planes, resulting in the decline of the focusing ability.
Fig. 4. Numerical simulations of the bottle beam intensity volume with different values of $N$, which cut from the y=0 plane. The parameters are as follows: ${c_1} = 0.35$, ${x_1} = 0.5$mm, ${y_1} = 0$mm, ${c_2} = 0.25$, ${c_3} = 0.18$, ${c_4} = 0.13$. All other parameters are the same as those in Fig. 1.
Fig. 5. Numerical simulations of the three-dimensional isosurfaces of the CCPGVBs with different values of $N$. All parameters are the same as those in Fig. 4.
4. Experimental generation of the CCPGVBs and the trapping demonstration
So as to generate the CCPGVBs, a Gaussian beam is generated by the solid-state laser ($\lambda = 532$nm) and expanded by a beam expander BE first. Then, it passes through a half-wave plate HWP and a Gran prism GP to control the polarization direction and the power of the beam. The spatial light modulator SLM (Santec SLM-200) is previously loaded with the amplitude and the phase information of the targeting beam [40]. After then, the expanded beam will be reflected by the SLM and enter the 4f filtering system (lenses ${L_1}$ and ${L_2}$, with a focal length of $300$mm), where the Fourier filter FF will select the positive first-order fringes of the incident beam. Before entering the particle capture system, the modulated beam would be rescaled by a telescope (lenses ${L_3}$ and ${L_4}$, with focal lengths of $125$mm and $25.4$mm) for accommodating the particle, and OB will form inside the glass cuvette C finally. In order to illuminate the trapping plane, the illumination light I is coupled into the glass cuvette by a shortpass dichroic mirror DM with a cut-off wavelength of $650$nm. At the imaging system, the trapping plane is imaged by a charge-coupled device CCD (MindVision MV-GE500M-T) by means of a micro-objective MO (10$\times$, 0.25 NA) and a lens ${L_5}$ with a focal length of $70$mm. A dichroic longpass filter F with a cut-on wavelength of $585$nm is used to block the laser beam. Also, a camera is placed to shoot the scattered light from the trapped particles.
As the CCPGVBs are restructured at the back focal plane of the ${L_2}$, the details of the beam structure can be recorded by a beam quality analyzer (not drawn in Fig. 6). Figure 7 shows two sets of the numerical and experimental intensity distributions. In the first group, it is clear that the focusing points shown in Figs. 7(b2), 7(b4), and 7(b6) form the "closed-neck" structures, while the ring intensity distributions at Figs. 7(b3) and 7(b5) construct the bodies of the OBs. Such structures can also be observed in the second group. These represent the appearance of the bottle structure. Moreover, in two sets of experiments, we can find that the positions of the bottles are consistent with the simulations in transverse and longitudinal directions, which experimentally demonstrate the precise and flexible control of the bottle.
Fig. 6. Schematic diagram of the experimental setup for the generation of the CCPGVBs and the realization of trapping MCMB particles.
Fig. 7. (a1)-(a6) Snapshots of the numerical simulation results with variant propagation distances, where the parameters are the same as those in Fig. 4(a2); (b1)-(b6) experimental results corresponding to (a1)-(a6); (c1)-(c6) snapshots of the numerical simulation results with variant propagation distances, the parameters here are $N = 3$, ${c_1} = 0.28$, ${x_1} = 0.6mm$, ${y_1} = 0.2mm$, ${c_2} = 0.21$ and ${c_3} = 0.16$; (d1)-(d6) experimental results corresponding to (c1)-(c6). For better visualization, (a1), (b1), (c1) and (d1) have different scale bars with others.
The photophoretic trap of the light-absorbing particles is known as one of the applications of the OB. In gaseous media, if a light-absorbing microparticle is illuminated unevenly, resulting in the nonuniform temperature distribution, it radiates heat to the environment near the illuminated part and increases the kinetic energy of the gas molecules. Subsequently, these gas molecules exchange the linear momentum with the particle, pushing it away from the light source [41]. In our experiment, we use the MCMB as the sample, with the median particle diameter $D50 = 23\mu$m. These particles were preloaded in the glass cuvette for preventing the ambient perturbations during the trapping procedure [32]. In order to load the OB, we shake the glass cuvette to fill it with the floating MCMB.
The OB we implement here is about $2$mm in length and $25\mu$m in diameter, and the beam power is approximate $25$mW. As Fig. 8(a) describes, the CCPGVBs successfully confine a MCMB particle suspending in air, which can also be accomplished in two OBs cases. Figure 8(b) shows the motion stages of a MCMB particle in the case of a single OB. Although the trapping is robust and can sustain more than an hour, the MCMB particle is generally oscillates with a high frequency inside the OB. Since the OB’s length is much larger than its width, the trapped particle also oscillates longitudinally and out of focus at $t = 0.67$s, as shown in Fig. 8(b6). For a larger particle, such as a $10\mu$m MCMB particle showed in Fig. 8(c), the trapping is rather stable and nearly eliminates the oscillation.
Fig. 8. (a1)-(a2) Scattered light from the trapped particle taken from the side of the glass cuvette, and the positions of the particles are marked by the white dotted circle; (b1)-(b7) microscope images of a $6\mu$m MCMB particle trapped by the same CCPGVBs in (a1), and the position of the particle is marked by the white dotted circle; (c) microscope image of a $10\mu$m MCMB particle trapped by the same CCPGVBs in (a1). The beam parameters in (a1) and (a2) are the same as those in Figs. 4(a1) and 4(a2) respectively, except ${c_2} = 0.29$ and ${c_3} = 0.26$.
5. Conclusions
In summary, we propose numerically and experimentally a new kind of OB developed from the superposition of the CCPGVBs. Results confirm that the CCPGVB follows an alterable-inline straight line while propagating, and the cascade of the beams allows to generate the multi-off-axis OBs. The displacement factors are verified that they only move the focusing positions transversely, and are capable of adjusting the energy distribution around the bottle. Also, the OB’s volume can be regulated by the quadratic chirp factor. Moreover, the location of the beam focus is derived, which enables the possibility to precisely control the locations of each OB and gives us a high degree of control freedom. Experimental results indicate the CCPGVBs are able to constrain the particles, and this kind of OB, may expand the possibility of the optical tweezers systems.
Funding
National Natural Science Foundation of China (11775083, 12174122); Science and Technology Program of Guangzhou (2019050001); The Extracurricular Scientific Program of School of Information and Optoelectronic Science and Engineering, South China Normal University (21GDKB02).
Disclosures
We declare that we have no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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