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Abstract
A rigid body can have six degrees of freedom, of which three are with rotational origin. In the nomenclature of the airlines, the in-plane degree of rotational freedom can be called yaw while the first out-of-plane degree of freedom can be called pitch with the second one being called roll. Among these, only the yaw sense has been studied extensively in the optical tweezers literature, while the pitch rotation is starting to be explored. In this paper, we show a way to detect the pitch rotation in a hexagonal-shaped particle using photonic force microscopy using the forward scattered light under crossed polarizers and making it incident on a split photodiode. In this way, the pitch angle can be detected at high resolution and bandwidth. We apply this technique to detect continuous pitch rotation and also exhibit a power spectral density for an anisotropic particle optically trapped in a linearly polarized light and exhibiting Brownian motion.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
High-resolution detection and manipulation of nano/micrometer-sized particles is an essential field of research with applications in single molecule biophysics [1–3],nanophotonics [4–7], and material science [8,9]. Optical [10–12], magnetic [13,14], acoustic [15], electrokinetic [16], and thermophoretic [17] are some of the widely used strategies for confinement and micromanipulation of nano/micro particle. Optical tweezers are robust and highly efficient tools to manipulate such particles. Any optically trapped particle has three translational and three rotational degrees of freedom. One of the rotational degrees of freedom can be called yaw [18–20](in-plane rotation), while the others are pitch and roll [21](out-of-plane rotation) according to the nomenclature of the airline.
The precise control, manipulation and detection of all rotational degrees, especially out-of-plane rotations, have eventual applications in the field of multimode rheology [22], cell-membrane fluctuations [23,24] and interface assessments [25–27] in nanoscale regimes. Different strategies are utilized to rotate particles in pitch sense [24,28,29], but high-resolution detection is only studied for spherical birefringent particles [30,31]. Hexagonal upconverting particles can also be rotated in pitch [32] sense through convection flows using thermo-optical tweezers. Upconverting particles are versatile probes for imaging [33], local heating [34],temperature sensor [35] drug delivery [36,37], and biosensing [38,39]. Exploiting the shape asymmetry of the hexagonal particle, it can also be used in multimode rheology. It may be noted here that the out-of-plane rotational modes behave differently than the in-plane modes particularly in proximity to surfaces for both spherical and non-spherical objects. Recently partial roll rotation [40] of these upconverting particles has been studied with strategies to detect it. However, high-resolution detection of out-of-plane rotational motion of these particles still remains elusive.
In this study, we demonstrate the continuous pitch rotational motion of a hexagonal upconverting particle (UCP) [32] and then detect it using photonic force microscopy. We have also detected the pitch power spectral density (PSD) of trapped upconverting particles under a linearly polarised laser beam of wavelength 1064 nm for the first time, which shall help us to explore the pitch dynamics more precisely.
2. Theory
Hexagonal upconverting particles have form birefringence and prefer to align side-on [41] when trapped using a linearly polarized laser beam. We can detect the pitch rotations by exploiting the form birefringence by ascertaining the anisotropy in the forward scattered light passing through a set of crossed polarizers, quite similar to that reported for spherical birefringent objects [30]. In order to explore the proof con concept, we perform a Lumerical FDTD simulation [42] to realize the relation between pitch angle and cross-polarized intensity of forward scattered pattern. In the Lumerical FDTD simulation, we solve the Maxwell wave equation with appropriate boundary conditions and geometry.
Here, $\epsilon$ and $\mu$ are the permittivity and permeability of the medium respectively, and the $\sigma$ is the conductivity of the medium. E and B are the electric and magnetic fields of light propagating through the system and t is time.
We illuminate the hexagonal particle with plane-polarized light in the simulation and measure the intensity of the orthogonally-polarized light using a monitor placed above the particle. We find the electric field of the scattered light in the polarization orthogonal to the incident light due to interaction with the particle.
A hexagonal disk shaped particle of transverse width 2 $\mu$m and thickness 1 $\mu$m was placed inside a square shaped cubic test region of 10 $\mu$m width. A 1 $\mu$m wavelength plane polarized light is made incident on the particle and detection performed at orthogonal polarization angles to the incident beam. We observe that the scattered pattern makes a four-lobe pattern. As the particle rotates in the pitch sense, the difference between the two halves of the four-lobe scattered pattern is proportional to pitch rotation, as shown in Fig. 1.
Fig. 1. (a) Lumerical simulation configuration of hexagonal particle placed under cross-polarization. The detection monitor is orthogonal to the incoming light polarization. (b) Rotation of the particle in pitch sense under cross-polarization. (c) Plot showing the relation between normalized intensity difference and pitch angle. (d) Forward scattered pattern for different pitch angles under cross-polarization.
The previous attempts at detecting the pitch angle at high reolution were for spherical birefringent particles. However, in the present manuscript, hexagonal disk shaped particles have been considered with a geometry completely different from the spherical birefringent ones. It is not obvious that the cross-polarized signal would automatically extend to the present case. Thus, we perform the simulation. We find that the difference-in-halves technique behind crossed polarizers also works for the present situation.
This kind of model in Lumerical is able to correctly predict optical trapping forces by calculating the electric and magnetic components of the fields using a Finite Difference Time Domain (FDTD) solver, which solves the Helmholtz equation for such systems [42]. This simplistic model is enough to compute the effects due to the scattering from the particle. The particle changes polarization as a consequence of interaction with such a shape and orientation of the particle. The effect is clearly indicated by the FDTD simulations.
3. Experimental details
Optical tweezers setup built from an OTKB/M kit (Thorlabs, USA), in an inverted microscope configuration, are employed to perform the experiments as shown in Fig. 2. It consists of a 100x, 1.3 Numerical Aperture (N.A.) Olympus oil-immersion objective and an E plan 10x, 0.25 N.A. air immersion Nikon condenser to constitute for the microscope with a 1064 nm wavelength diode laser (Lasever, China) and a 975nm wavelength diode laser (RGB laser systems), both of which have TEM$_{00}$ mode, for the purpose of generating a hotspot and simultaneous detection. It can be noted that no structured beam has been used for trapping and detection. After being reflected by the dichroic mirror 1 (DM-1), both lasers enter the sample chamber and are tightly focused by the objective. The condenser lens collects forward scattered light and directs it towards the detection units by a reflection from dichroic mirror 2 (DM-2). We collect only 1064 nm wavelength light for forward scattered detection. After passing through the PBS (polarizing beam splitter), the cross-polarised light is spilt by an edge mirror and is made incident on two photodiodes. The difference between those photodiodes’ signal give us information about pitch rotation. The sample chamber is illuminated by a white LED light from the top through the dichroic mirror 2 and collected by a CMOS camera (Thorlabs) via the dichroic mirror 1.
Fig. 2. Schematic diagram of the setup used to detect pitch rotation. The dotted line indicates convection currents induced by the laser-generated hot-spot on the gold substrate, as has been reported in [32]. The flows generated are called Couette flows and formed due to the hot-spot.
The sample chamber consists of gold coated (30nm) coverslip at the bottom (Blue Star, number 1 size, English glass, thickness 170 micrometers), a glass slide (Blue Star), 75 mm length,25 mm width and thickness of 1.1 mm, at the top, and the sample containing 15% Fe-doped UCP (Fe-15 NYF) dispersed in water and 20 µl of which is transferred to the sample chamber and mounted on the microscope.
3.1 Preparation of gold coverslip
To make the gold-coated coverslip, we first cleaned the glass substrate with acetone, isopropyl alcohol, and de-ionized water in an ultrasonic bath for 5 min each and then dried it with nitrogen. Then, 5 nm of chromium was first evaporated at rate of 0.05 nm/sec to improve the adhesivity of the gold, followed by 30 nm gold evaporation at rate of 0.1 nm/sec. A quartz crystal monitor was used to monitor the thickness of the deposition while the vacuum of the evaporator chamber was maintained at 1 nbar pressure for the duration of evaporation.
3.2 Experimental method
We use two lasers for the experiment having wavelengths 975 nm and 1064 nm. Here, the 975 nm wavelength laser is employed to create the hotspot on the gold-coated coverslip, and the laser of 1064 nm wavelength is used to detect the dynamics of the particle. The sample is held at the stage in such a way that the gold surface is at the bottom as shown in Fig. 2. The 975 nm laser with a power of 10 mW is tightly focused using an objective onto the gold surface to create a hotspot due to the localised plasmonic heating [43]. This hotspot induces thermoplasmonic convection currents in water. The hexagonal UCP dispersed in water is captured by the convection flows towards the hotspot and rotates in the pitch sense. Then we focus the second laser of 1064 nm wavelength with a low power of 1 mW, at the confined position of the rotating particle and collect it using a pair of photodiodes along the laser’s propagation direction.
3.3 Preparation of upconverting particle
Hexagonal upconverting particles of NaYF$_4$: Er$^{3+}$, Yb$^{3+}$ are used in this experiment.The diagonal length of these particles is around 5µm. The conventional hydrothermal method with modifications [44] was followed to synthesize NaYF$_4$: Er$^{3+}$, Yb$^{3+}$ microparticles. A quantity of 1.2548 g of Yttrium nitrate (Y(NO$_3$)$_3$) and 1.2321g of Sodium citrate (Na$_3$C$_6$H$_5$O$_7$) are mixed with 14 ml of water and vigorously magnetically stirred for 15 min. Then, 0.3773g of Yb(NO$_3$) and 0.0373 g of Er(NO$_3$) in 21 ml of H$_2$O ( taken from miliQ) are added to the earlier made solution, which then turned into a milky white solution. Subsequently, 67 ml aqueous solution of 1.411 g NaF is added to the milky solution and stirred for one hour to get a transparent solution. the transparent solution is then transferred to a tightly sealed 200 ml Teflon-lined autoclave and heated in a muffle furnace for 12 hours at 200 degrees C. After the reaction, white-coloured powders are formed, rinsed with ethanol/water and subsequently dried for a duration of 12 hours at 100 degrees C. Then, FE-SEM images are captured to verify the hexagonal shape of the particles with average diagonal length of 5 $\mu m$. The images of such particles are shown in Fig. 3.
Fig. 3. FE-SEM image of hexagonal upconverting NaYF$_4$: Er$^{3+}$, Yb$^{3+}$ particle. The average diagonal length is 5 µm and thickness is 1 µm.
4. Results and discussions
We rotate a hexagonal-shaped upconverting particle by hotspot-induced convection currents [32] in pitch sense by generating a hotspot on a gold coated surface and then inducing convection currents. We detect the same rotation using our experimental setup shown in Fig. 2, relying upon the asymmetry in the scatter pattern behind crossed polarizers. The corresponding results of the experiment have been shown in Fig. 4. We can find a periodic modulation of the pitch signal when the particle rotates, the amplitude of which is the degree of rotation of the particle. In addition to video microscopy, we can obtain the rotation frequency from the time series of pitch motion, and both results are matching well. We also show the displacement along the x-direction in the same figure and find that this detection mode does not resolve the pitch motion. The theoretically expected signal for the corresponding experimental time series shown in Fig. 4(b) is given in Fig. 1(c).
Fig. 4. (a) Pitch rotation of UCP in the presence of hotspot-induced convection current. (b) Time series for pitch motion and the transverse x motion. (c) Power spectral density of pitch motion and transverse x motion.
We then optically trap a hexagonal-shaped UCP in a linearly polarized optical trap. The particle aligned side-on as expected from the reports in the literature [41]. We then use the pitch detection setup to ascertain the pitch Brownian motion. The result has been shown in Fig. 5.
Fig. 5. Pitch power spectral density (PSD) of a hexagonal UCP trapped by a laser of wavelength 1064 nm. It is fitted to the Lorentzian function for calibration purposes.
This fits well with the Lorentzian expected for pitch motion in the overdamped limit [45]. The PSD arising from the pitch motion of the hexagonal UCP can be expressed as
The calibration factor $\beta$ and the rotational trap stiffness $\kappa$ are given as
Here $k_B$ is the Boltzmann constant, $T$ is the ambient room temperature, $\gamma$ is the rotational drag coefficient of the hexagonal particle. Corner frequency $f_c$ can be obtained from the Pitch PSD fitting parameters. The drag coefficient is assumed for an oblate spheroid, given as [46]
for a particle of 5$\mu m$ major axis and 1$\mu m$ of minor axis the drag coefficient becomes $\gamma$= 1/3$\gamma _0$, where $\gamma _0$ is the rotational drag coefficient of a spherical particle of diameter 5$\mu m$. In this figure, we show a 0.1 degree motion over 1 Hz detection, while the bandwidth clearly extends to 200 Hz. This detection amplitude and the bandwidth can be further improved with better and more sensitive detectors.5. Applications: pitch rotation of trapped active particle
Upconverting particles (UCP) absorb low frequency photons and emit higher energy photons relying upon multiphoton processes. NaYF$_4$: Er$^{3+}$, Yb$^{3+}$ upconverting particles absorb 975 nm radiation (pump wavelength) and emit in visible wavelength [47] . When we trap a single UCP using a linearly polarised 975 nm wavelength laser, the particle aligns in a side-on configuration [41]. In our experimental setup, the laser is incident on the bottom side of the particle and mainly re-emitted in the backscatter direction. Some light is also emitted in the forward scatter direction, but the intensity is very low [44]. The efficiency of this multiphoton process is about 2 percent, and most of the absorption is converted into heating by photon-to-phonon conversion [48]. Since most of the absorption happens on the bottom side, this side becomes hotter than the top. As a result, we observe a temperature gradient along the laser propagation direction and self-propels in the same direction [44].
This absorption-induced self-propulsion can be considered an active self-thermophoretic motion inside an optical trap. These particles tend to move from a hotter region to a colder region, which is evident in the axial time series, indicated in Fig. 6(a) and (c). As reported in the literature, we observe the asymmetry in the axial position histogram shown in Fig. 6. Simultaneously, we detect the pitch rotation angle of the same particle and observe the pitch angle time series and histogram, shown in Fig. 6(b) and (d). The histogram of the pitch angle rotation is completely symmetric and can be fitted well to a Gaussian. This facet of the process where the pitch angle distribution remains Gaussian is not obvious. The tail region of the Gaussian can become non-Gaussian. The actual calibration of the of the pitch time series was performed using a method outlined in the previous section.
Fig. 6. This figure indicates a typical application of the pitch angle determination technique for hexagonal disk-shaped particles. (a) The z position time series of the particle when optically trapped at the absorption resonance at 975 nm. (b) The simultaneous corresponding pitch angle time series of the particle. (c) Histogram of the z time series and (d) histogram of the pitch angles. It is noticable that there is a significant skewing in the z position while the pitch angle distribution remains a very good Gaussian.
6. Conclusions
Thus, to conclude, we show a new way of detecting pitch rotation in an asymmetric hexagonal-shaped particle by taking scattered light passing through crossed polarizers and then illuminating it onto a split photo-detector, similar to what has been performed for a spherical birefringent particle. We show the application of this technique to detect continuous pitch rotation close to a gold coated surface and the pitch power spectrum for a particle optically trapped in a linear polarized beam. This can be useful to detect pitch rotation at a high resolution with a high bandwidth using photonic microscopy. We also show that the distribution of pitch angles for a hexagonal disk shaped upconverting particle optically trapped on absorption resonance is a Gaussian, while the axial position distribution differs from a Gaussian signficantly.
Funding
Department of Biotechnology, Ministry of Science and Technology, India (IA/I/20/1/504900).
Acknowledgments
We thank the Indian Institute of Technology, Madras, India for their seed and initiation grants to BR. This work was also supported in parts by the DBT/Wellcome Trust India Alliance Fellowship IA/I/20/1/504900 awarded to BR.
Disclosures
There are no conflicts of interest to declare.
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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