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Abstract
Typically a rigid body can have three degrees of rotational freedom. Among these, there can be two types of out-of-plane rotational modes, called the pitch and the roll. The pitch motion is typically to turn the particle along an axis orthogonal to the axis of symmetry. However, rotation about the axis of symmetry (called the roll motion) has so far not been shown in optical tweezers. It is here that we use a hexagonal shaped particle (NaYF4) which prefers to align side on with the optical tweezers [Rodriguez-Sevilla et al., Nano Letters 16, 8005 (2016) [CrossRef] ]. In this work, we find that the stable configuration of the hexagonal particle changes while using one beam and two beams, so that when one of the tweezers beams is switched on and off, the particle tends to switch between the different configurations. Thus we get a controlled roll motion. This is the first time that controlled partial roll motions have been generated in optical tweezers.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Tracking the rotation of a microscopic particle can serve as a sensitive tool for probing the local environment in both biological [1–3] and other systems [4]. In three-dimensional space, roll motion (i.e., rotation about the cylindrical symmetry axis of the particle) is arguably the subtlest and the most sensitive motion. This generally requires just nanoscale forces to actuate [5]. The other five degrees of freedom are translations in the x, y, z directions, rotation along in-plane direction (also called yaw), and rotation out-of-plane [6] (called pitch) as shown in Figs. 1(a),(b). There has been some work on rotational out of the plane motion [7], where an extended object is usually held in a couple [8,9] or a set of optical tweezer beams [10] and turned, which can be called the pitch rotation too. However, that still leaves out one degree of rotational freedom which cannot be performed with just 2 beams, namely the rotation about the axis of symmetry of the object or the axis of elongation of the object, since the 2 tweezers beams generally hold the ends of the extended object. This rotation can be called the roll rotational motion.
Fig. 1. (a) A sketch of an optical beam with hexagonal particle showing the reference frame representing three translational degrees of freedom and three rotational degrees of freedom with (b) air-line nomenclature. (c) A cartoon of the hexagonal upconverting particle of size 5 $\mu$m enclosed in an imaginary pill box depicting the stable configuration in a single beam trap of wavelength 1064 nm with beam waist of 500 nm at $\theta$ = $0^{\circ }$ (flat edge on top) and (d) in two beam configuration where the beam separation is 4.5 $\mu$m at $\theta$ = $30^{\circ }$ (sharp edge on top). (e) A plot showing normalized torque acting on the particle in single beam (blue) and two beam (red) systems at different configurations ($\theta$ = $0^{\circ }$ - $60^{\circ }$). (d) Depicting six degrees of freedom available for a hexagonal particle in an optical trap.
Usually, the translation and rotational freedoms in optical trap are used for Brownian probe particles to study various systems [11,12]. In addition to the small moment of inertia associated with the roll motion, when compared to translations or yaw rotations, this roll motion generally involves the least amount of free space, thereby making it more likely to be observed in a crowded environment where other motions are constrained. Even for the spherical but optically anisotropic probe particles, the generation and detection techniques of yaw [13–15] and pitch [16] rotations are required to study interfaces [17] and surfaces [18,19]. Moreover, the difference between pitch and roll dynamics from the yaw dynamics is evident in the proximity to surfaces.
The tools to track this roll motion utilize either an optical or a structural asymmetry. In 2005, Yajima and Cross introduced a side arm (structural asymmetry) onto a microtubule to experimentally observe microtubule rotation induced by kinesin molecules [20]. Thereafter, Yajima et al. tracked this microtubule rotation with higher precision by labeling such microtubules with nanocrystals (optical asymmetry) and then tracking such rotation with a three-dimensional microscopy [21]. The Yu group tracked the roll motion of dual fluorescent labeled (optically asymmetric) Janus rods ($\approx 500$ nm in diameter) bound to endosomes to study rotational dynamics of these endosomes during intracellular transport [22] and to see how the distribution of ligands on the Janus particle surface affects the cellular internalization process [23]. In a study of rolling motion in nonliving systems, Biswal group introduced a kink (structural asymmetry) in a chain of DNA-linked colloids (of about $\approx 1~ \mu$m diameter) and studied how the no-slip boundary condition affects the roll diffusivity [24]. While these methods yield measurements of roll motion with high precision, their application is limited by complex fabrication and imaging by video microscopy.
In this manuscript, we show a new way of generating partial roll rotation in a hexagonal shaped particle. These particles upon doping with certain elements (Yb $\&$ Er) opens up a novel class of materials known as upconverting particles [25,26]. Further, when combined with optical tweezers these particles find numerous applications in various disciplines of physics [27,28] and Biology [29]. Hexagonal particles aligns side on with the axis of the linearly polarized optical tweezers beam [30], where it has been shown that a particle initially oriented face on with respect to the tweezers beam automatically aligns side on with the axis of polarization of the beam. This can also be called the pitch rotation. However, once aligned with the polarization direction of the optical tweezers beam, the preferred configurations are different for a single beam and two beams (having same polarization) slightly displaced from each other along the axis of polarization. As a consequence, pulsing the second optical tweezers beam causes changes in the configurations of the hexagonal shaped particle, thereby inducing the partial roll rotation. Since the particle has hexagonal symmetry, it only controllably turns by a maximum of 30 degrees.
2. Theory
A particle trapped in a single beam can experience torque due to it’s birefringence either formed (shape) or intrinsic (crystal anisotropy). This torque defines a specific stable orientation for the trapped particle [31,32] as shown in Fig. 1(c). However, the same particle reorients into a different configuration as depicted in Fig. 1(d) when trapped simultaneously with two trapping beams. In our previous work we have shown this stable configuration of a birefringent particle as a function of trap position in axial direction generating controlled pitch [33]. In addition, there are other techniques to generate pitch rotations in hexagonal sodium yttrium fluoride (NAYF$_4$) particles using convection currents [29] and in birefringent particle by trapping the particle close to a surface and moving the stage [18]. Here, we generate partial roll rotation in a hexagonal particles by switching between the two stable configurations.
In order to realize these stable configurations of our particle we perform a simulation with Lumerical FDTD solver and calculate torques using Maxwell stress tensor (MST). An imaginary box around the particle was used to calculate MST and hence the torque experienced by our hexagonal particle in an electric field given by the Eq. (1).
where, $\sigma _{ij}$ is the Maxwell Stress Tensor [34] calculated on the surface S, $\epsilon _{ijk}$ is usual Levi-Civita symbol and $\textbf {r}$ is the moment arm. We apply one Gaussian beam with linearly polarized light into the system as depicted in Fig. 1(c) and two Gaussian beams into the other system as shown in Fig. 1(d) and calculate corresponding MST. In addition, normalized torques calculated using MST for single beam and two beam systems in different configurations ($\theta$ = $0^{\circ }$ - $60^{\circ }$) of hexagonal particle are shown in Fig. 1(e). The hexagonal particle has a stable configuration at $\theta = 30^{\circ }$ in two-beam system whereas, this configuration becomes unstable in a single-beam system. Thus, by pulsing one of the two trapping lasers the particle performs rocking motion in roll sense. It may be noted here that these hexagonal upconverting particles have been known to align side on while trapped in linearly polarized optical tweezers [30]. The axis of the side-on configuration is such that it automatically aligns with the axis of the polarization. Hence, it was believed to be enough to model the two beam configuration thus.The roll motion of a hexagonal particle induces a change in total cross-polarized intensity of the forward scattered light. Lumerical FDTD simulation was performed to understand the relation between roll angle ($\theta$) and total cross-polarized intensity. Here, we apply a plane polarized light and measure the cross-polarized intensity using the monitor placed above the particle as shown in Fig. 2(a). Further, the variation of cross-polarized intensity is as shown in Fig. 2(b) corresponding to the roll angle ($\theta$). We believe that the projection of the particle that the incoming tweezers beam sees (we only detect one of the beams by using an interference filter) changes the cross-polarized intensity.
Fig. 2. (a) Schematic of the hexagonal particle placed between an incoming plane polarized wave and monitor in orthogonal direction showing our detection mechanism used in our simulation. (b) A plot depicting the cross-polarized intensity corresponding to roll angle ($\theta$) calculated from FDTD solver in Lumerical software.
3. Materials and methods
The Hexagonal particles (NaYF$_4$) used in the experiment were synthesized by the conventional hydrothermal method [35] with required modifications. A quantity of $1.5813$ g of Yttrium nitrate $(\textrm{Y}(\textrm{N}0_3)_3)$ and $1.2321$ g of sodium citrate were dissolved in $35$ ml water by stirring it vigorously for about $10$ minutes. Subsequently, an aqueous solution of $67$ ml with $1.411$ g of NaF (Sodium fluoride) was added to the above mentioned solution. The resultant solution was heated at $200^{\circ }$C for $12$ hours in a teflon lined stainless steel vessel ($200$ ml). Then, the solution obtained was cooled to room temperature and washed with water and ethanol for 4-5 times. Finally, by calcinating at $100^{\circ }$ C for $12$ hours we get a white powder which can be stored at room temperature. This powder was then added to de-ionised water to prepare the hexogonal particle suspension and used in our experiments. Further, the morphology of the particles prepared were confirmed by scanning electron microscope (SEM) images shown in Figs. 3(a),(b). Additionally, these hexagonal particles can be doped with ytterbium (Yb) and Erbium (Er) which emits visible light upon excitation with $975$ nm laser generally known as upconverting particles (UCPs). The protocol followed to synthesize these UCPs was as described in Ref. [29].
Fig. 3. Scanning Electron Microscopy (SEM) images depicting the morphology of (a) a cluster of hexagonal particles with random orientations and (b) a single hexagonal particle. (c) Schematic of the two laser beams optical tweezers setup showing our detection, imaging systems and an inset depicting degrees of freedom available for a hexagonal particle in an optical trap.
The schematic diagram of the experimental setup is shown in Fig. 3(c). Two similar polarized lasers of wavelengths 1064 nm (diode laser from Lasever) and 975 nm (diode laser from Thorlabs) with maximum powers of 1.5 W and 0.3 W respectively are combined using a Beam Splitter (BS) and reflected to the objective by a dichroic mirror. An Olympus 100X infinity corrected oil-immersion objective with Numerical Aperture (NA) of 1.3 is used to tightly focus the laser into a sample chamber and image the sample plane with CMOS camera inverted microscope configuration as shown in Fig. 3(c). The sample chamber comprises of a thin layer of $20~\mu$l solution of hexagonal particles dispersed in water which is placed between the glass slide (Blue star, 75mm x 25mm x 1.1mm) and a coverslip (Blue star, number 1 size, english glass). This chamber is illuminated by visible LED and is combined with a dichroic mirror which also allows the Nikon E-Plan 10x/0.25 NA condenser lens to collect the forward scattered light for our detection unit. The forward scattered light is split by a PBS and light in one arm is directed towards quadrant photodiode (QPD from Thorlabs Inc.) to detect the laser pulsing. Further, the light in other arm is cross-polarized by turning the half-waveplate (1064 nm) and filtered by a band-pass filter (1064 $\pm$ 10 nm) to detect rotational motion of the trapped particle.
The 1064 nm laser was continuously illuminated onto the particle while the 975 nm laser was pulsed periodically. A function generator was used to generate laser pulses at $5$ Hz frequency which was put onto the current controller of the 975 nm diode laser. This switched the configuration of the system from one beam to two beams periodically.
4. Results and discussions
The time series for the forward scattered 1064 nm, placed under crossed polarizers has been shown in Fig. 4. The gradual change in the intensity of the scattered light is visible. At one edge of the time series, the 975 nm laser light is suddenly turned off then the particle gradually starts to turn towards the single beam configuration. However, when the 975 nm beam is turned on the particle goes back to the original two-beam configuration which is shown as the sharp raising edge (blue arrow) in the time series of Fig. 4. A raise in total intensity of the time series is due to a small leakage of $975$ nm laser into the detection system. The two trapping lasers are separated by $4~\mu$m with one of the lasers pulsing at $5$ Hz and pulse width of $100~$ms. Also, the hexagonal particle stabilizes faster in two beam configuration due to higher trap stiffness. In order to confirm the roll rotational motion, we perform the same experiment under the same conditions while detecting with a video camera.
Fig. 4. The figure shows a pulse sequence (on and off) of $975$ nm laser (red) and the time series of total cross-polarized intensity which is correlated to roll angle of the hexagonal particle. Also, a gradual change (black arrow) in the total intensity when laser is off and a sharp raise (blue arrow) followed by constant intensity region (two-beam stable conformation) when laser is on is shown.
We can see a fringe moving across the side of the particle while the configuration is changed from one beam to two beams, in Fig. 5 (also see Visualization 1). Under these settings a roll rotation of about 20$^{\circ }$ was generated as the particle experiences a new pulse before it relaxes into single beam configuration in about 100 msec. It may be noted here that although maximum permissible rotation is 30$^{\circ }$ for this shape, the particle only gets time to turn by 20$^{\circ }$. The time scale confirms that we are measuring the same effect with both the techniques. The angle turned by the particle from distance moved by the fringe was calculated using Eq. (2).
where, d is the distance moved by the fringe in $\mu$m, r is the distance from center to edge of the hexagonal particle in $\mu$m and $\theta$ is the roll angle. In addition, a hexagonal shaped UCPs comprising of NaYF$_4$ and doped with Yb and Er, of size $3~\mu$m was also rotated in roll sense employing the same technique (Fig. 6, see Visualization 2 and Visualization 3). This particle emits light in visible range when excited with $975$ nm laser which appears in video microscopy. However, a small particle attached to the trapped UCP showed rotation in roll sense from one configuration (6(a)) to other configuration (6(b)) when the laser was pulsed at a rate of 2.5 Hz.Fig. 5. Snapshots of the hexagonal particle trapped side-on in the optical tweezers beam depicting the fringe enclosed in the white box (a) immediately after $975$ nm laser was turned off (t = 0s) and (b) shortly before the laser was turned on (t = 0.1s). (c) Roll angle ($\theta$) in degrees calculated from Eq. (2) plotted against time in seconds showing the gradual change of $20^{\circ }$ in $0.1$s.
Fig. 6. Snapshots of the upconverting hexagonal particle (NaYF$_4$:Yb, Er) with a tiny speck attached to it (white arrow), depicting two orientations, one (a) immediately after $975$ nm laser pulsing at 2.5 Hz was turned off (t = 0.00 s) and the other (b) shortly before the laser was turned on (t = 0.18 s)
Due to the hexagonal symmetry of the particle, the roll rotation is only possible to the maximum extent of 30 $^{\circ }$, after which the configuration becomes similar. Thus the angle of 30$^{\circ }$ is limited by the shape of the particle. It can be increased by using particles which have less sides and yet symmetric, such as squares, triangles or pentagons. We have not performed the FDTD simulations for these but believe that the rotation angles might be higher in these shapes.
5. Conclusion
Thus, in conclusion, we show a new way of generating roll motion in a hexagonal NaYF$_4$ particle optically confined in the side-on configuration. The particle prefers different configurations with one and two optical tweezers beams and hence switching one of the trapping beams on and off periodically causes the particle to turn in the roll sense controllably to an extent of 20 degrees. It could be detected by video microscopy upon observing the fringes on the particle move and also by detecting the scattered light under crossed polarizers. Thus, generation of partial roll motion and high resolution detection is made possible. One can use it for performing rheological measurements of the local environment of the particle.
Funding
Department of Biotechnology, Ministry of Science and Technology, India (IA/I/20/1/504900).
Disclosures
The authors declare no conflicts of interest.
Data availability
The data for the manuscript are available upon reasonable request to authors.
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