Abstract

Gold nanorods can be optically trapped in aqueous solution and forced to rotate at kilohertz rates by circularly polarized laser light. This enables detailed investigations of local environmental parameters and processes, such as medium viscosity and nanoparticle–molecule reactions. Future applications may include nanoactuation and single-cell analysis. However, the influence of photothermal heating on the nanoparticle dynamics needs to be better understood in order to realize widespread and quantitative use. Here we analyze the hot Brownian motion of a rotating gold nanorod trapped in two dimensions by an optical tweezers using experiments and stochastic simulations. We show that, for typical settings, the effective rotational and translational Brownian temperatures are drastically different, being closer to the nanorod surface temperature and ambient temperature, respectively. Further, we show that translational dynamics can have a non-negligible influence on the rotational fluctuations due to the small size of a nanorod in comparison to the focal spot. These results are crucial for the development of gold nanorods into generic and quantitative optomechanical sensor and actuator elements.

© 2017 Optical Society of America

1. INTRODUCTION

The motion of a Brownian particle in solution is strongly affected by thermal agitation from collisions with the surrounding fluid molecules, resulting in well-defined random dynamics dependent on fluid temperature and viscosity. Optical tweezers offer an ideal platform to study Brownian dynamics because the focused laser field introduces deterministic optical forces and torques [15]. By recording the movement of a trapped Brownian particle, it is possible to probe local environmental properties such as viscosity [6] and temperature [7]. A trapped particle can further work as a handle to analyze and control biomolecules, enabling detailed biophysical measurements [810].

Gold nanocrystals are highly interesting from the point of view of optical manipulation and studies of Brownian dynamics because they are biocompatible and support very strong light–matter interactions due to the well-known surface plasmon resonance phenomenon. Consequently, a number of studies on optically trapped colloidal Au nanocrystals have been reported in the literature [1117]. The large electric dipole polarizability associated with plasmon excitation in fact results in optical forces and torques that are enhanced to such an extent that Au particles as small as 10nm can be trapped [18] and gold nanorods can be propelled to spin at rotation frequencies up to more than 40 kHz by angular momentum transfer from circular polarized light [19]. Together with their superior refractometric sensing and local field-enhancement properties, these features make Au nanocrystals potential generic optomechanical sensor and actuator elements. However, plasmon excitation also leads to strongly enhanced electromagnetic fields within the metal nanoparticle, and this in turn leads to photothermal heating effects that can be orders of magnitude stronger than for dielectric particles of similar size [20]. Plasmonic heating can be an advantage or a detriment, depending on the specific application. It is usually a problem in studies of biological cells and tissues, where a small change in temperature can strongly affect intracellular processes [21], biomolecular transitions [22], and local force measurements [9], but it can be very useful in applications like drug release [23], gene regulation [21], photothermal therapy [24,25], and thermophoretic trapping [26]. Naturally, photothermal effects are of particular concern in applications based on optical manipulation since the light intensities involved are then typically very high. Though challenging, measurements of the medium temperature around optically trapped gold nanocrystals are therefore of both fundamental interest and great importance for the development of various applications.

In this work, we elucidate the Brownian dynamics of gold nanorods that are trapped and rotated at high frequency in a 2D optical trap formed by a circular polarized focused laser beam with a wavelength close to the main plasmon resonance wavelength of the particles. Due to photothermal heating, the trapped nanoparticle has a much higher surface temperature than the average temperature of the surrounding water solution. The resulting temperature gradient then leads to so-called hot Brownian motion [2729]. By simultaneously studying the translations and rotation of a particle, while taking into account the spatially heterogeneous temperature-dependent solvent viscosity [30] and actual particle position within the laser focus, we show that the effective Brownian temperatures, Tr and Tt, characterizing rotational and translational degrees of freedom, respectively, are drastically different. The analysis we present is applicable to various other types of light-absorbing particles and therefore relevant to a wide range of applications based on optical manipulation.

2. EXPERIMENTS

The experimental setup [Fig. 1(a)] used here is similar to that described in Ref. [19], but with the additional feature of being able to probe real-time nanoparticle lateral displacement (x,y) in the optical trap using a quadrant photodiode (QPD). We prepared gold nanorods with an average size of (84±5)nm×(164±10)nm in aqueous solution according to Ref. [19] [Fig. 1(a) inset]. The gold nanorod sample has a plasmon resonance centered at 760 nm in solution and the trapping laser wavelength is 830 nm. A combination of a half-wave and quarter-wave plate was employed to create an almost perfect circular polarization in the laser trap [31]. The comparatively large scattering cross-section of the gold nanorods boosts transfer of both angular and linear momentum. The latter results in strong radiation pressure, which makes it challenging to achieve stable optical trapping in 3D. As a result, the gold nanorods are trapped in 2D against a glass slide and rotated about their short axis in a plane normal to the optical axis [19]. The back-scattered laser light from the trapped particle is split into two beams. One of the beams is recorded by the QPD (First Sensor), located at the objective back focal plane [32,33], and the other beam is passed through a linear polarizer and collected by a fiber-coupled avalanche photodiode (APD) connected to a hardware autocorrelator (ALV-5000). This setup can be used to measure both translational and rotational Brownian motion dynamics of a trapped nanoparticle with high temporal and spatial resolution [32,34].

 figure: Fig. 1.

Fig. 1. Measurement of the rotation dynamics of a 2D trapped gold nanorod. (a) Schematic of the optical tweezers setup. (Inset) Scanning electron microscopy image of the used gold nanorods. (b) Measured QPD voltage dependence on immobilized nanorod displacement from the laser focus at different laser powers. The dashed line shows a fit to the central linear range of the data measured at 11.7 mW. (c) and (d) QPD-measured trajectory of a rotating nanorod in the optical trap (c) and distribution of the rod position (d). The laser power was set at 6.4 mW. (e) PSD of the measured nanorod x displacement. The red smooth curve is the fitting at ffavg to obtain the corner frequency.

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The QPD gives three output voltage signals, Vtotal, Vx, and Vy, indicating both total back-scattered laser intensity reaching the detector (Vtotal) and the displacement of a nanoparticle in the optical trap with respect to the laser focus (Vx,Vy) [18]. Vx and Vy can be converted to particle position in order to track translational Brownian motion [18]. The conversion was based on calibration measurements using individual immobilized nanorods (Section 1 of Supplement 1) [32,33] that were scanned through the laser focus using an XY piezoelectric scanning stage [Fig. 1(b)]. Vx, and Vy are linearly proportional to the particle displacements, Vx=bx, Vy=by, when the nanorod is within 250nm from the laser focus and varies with laser power. Unfortunately, for a fixed laser power, b also varies between different nanorods because of their varying resonant scattering properties, which are highly sensitive to the absolute size and shape. We found that the measured b for different nanorods displayed a variance less than 15% for fixed laser power. Figures 1(c) and 1(d) show the trajectory and corresponding displacement distribution, respectively, of a rotating gold nanorod obtained using the average b value from measurements on several immobilized nanorods. The particle displacement follows a Gaussian distribution centered at the laser focus, as expected. According to an harmonic approximation of the optical trapping potential, the standard deviation of the particle lateral displacement σ is expected to follow σ=(kBTt/k)1/2, where k is the trap stiffness (force constant). We also calculated the power spectral density (PSD) of the measured nanorod displacement [Fig. 1(e)]. The PSD can be used to obtain the corner frequency fc, which is determined by the trap stiffness k and the nanorod translational friction coefficient γt(Tt) through fc=k/(2πγt), via a fit to PSD(f)=C/(f2+fc2) [35]. Since the conversion factor b is now included in the fitting parameter C, we can estimate fc directly from the PSD of Vx or Vy without converting to absolute displacement. The intensity profile in the laser focus can also be obtained from the calibration experiment via a fit of the measured Vtotal(x,y) to a Gaussian function. This measurement is quite robust since it is independent of the absolute laser power and properties of the immobilized nanoparticle. We obtained a laser beam waist size w0541nm for the present experiment. The beam waist and corner frequency are the main experimental parameters used for estimating the translational Brownian temperature, as discussed further below.

Since the dichroic mirror in the setup has different reflectivity to s- and p-polarized light, the PSD data also reveals the rotational motion of a trapped anisotropic particle. For an object with two-fold symmetry, like a nanorod, the PSD will exhibit a peak centered at fQPD=2favg, where favg is the average rotation frequency in the trap. Although the rotation peak is clearly visible in the trace displayed in Fig. 1(e), it is much more obvious in the PSD of Vtotal, shown in Fig. 2(a). A peak fit to a Lorentzian function can be used to determine the peak width Γ, which describes the fluctuation in rotation frequency. Note that the appearance of a rotational peak in the PSD of Vx or Vy implies that the fit to obtain the corner frequency has to be performed for ffQPD.

 figure: Fig. 2.

Fig. 2. Experimentally measured rotation dynamics and estimated Brownian temperatures of a rotating gold nanorod under varying laser powers. (a) PSD of the backscattered laser light intensity collected by the QPD. (b) Autocorrelation plots of scattering intensity after an analyzer collected by the APD. (c) Laser power-dependent rotation frequencies of nanorods measured by APD and QPD. Error bars reveal the standard deviations of the autocorrelation fitting (red) and of a Lorentzian distribution fit to PSD peak (green), respectively. (d) Power spectrum peak widths of signals shown in (a) and ACF decay rates of signals shown in (b). (e) Estimated Tt and Tr from the experimental corner frequency fc and ACF decay time τ0, respectively. Error bars represent 95% confidence intervals. The red solid line indicates the calculated surface temperature of the nanorod while the green solid line indicates the calculated temperature in solution 110 nm away from the nanorod surface. (f) Calculated temperature profile around the gold nanorod when heated by the circularly polarized trapping laser. The yellow and white dashed circles indicate the sizes of the laser beam waist and the movement range of the nanorod center of mass, respectively, with the latter representing two standard deviations of the nanorod displacement.

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The rotational motion of the nanorod can also be independently analyzed through the APD/autocorrelator setup. The intensity of the total back-scattered light emanating from the nanorod, after passing through a linear polarizer, is IscaP=σsca,CPI(x,y)cos2φ. Here, σsca,CP is the nanorod scattering cross-section for circular polarization at the laser wavelength, I(x,y) is the laser intensity at the particle position, and φ is the rotation angle of the nanorod relative to the polarizer [19,36]. The autocorrelation function (ACF) of IscaP, which is directly obtained from the hardware autocorrelator [Fig. 2(b)], can be fitted by C(τ)=I02+0.5I12exp(τ/τ0)cos(4πfavgτ), where I0 is the average intensity, I1 is the amplitude of the intensity fluctuation, and τ0 is the autocorrelation decay time [19]. The rotational fluctuation is determined by the particle rotational friction coefficient γr and the effective rotational Brownian temperature through τ0=γr/(4kBTr), and it relates to the PSD rotational peak width as Γ=(πτ0)1.

We measured the PSD and ACF of a trapped rotating nanorod as a function of applied laser power [Figs. 2(a) and 2(b)]. As indicated previously, the average rotation frequency, favg, can be obtained either by analyzing the PSD obtained from the QPD data or by analyzing the ACF of the APD signal. Figure 2(c) demonstrates that the two measurements are in excellent agreement. Moreover, the rotation fluctuations estimated from the ACF decay rate and the PSD peak width also match well with each other [Fig. 2(d)]. The agreements is in principle not surprising since both measurements record the scattering intensity of the rotating nanorod after a polarizer, and the PSD of a signal essentially is the Fourier transform of the ACF of the same intensity–time series.

We are now able to analyze the four parameters σ, fc, favg, and τ0 (or Γ):

{σ=(kBTt/k)1/2fc=k/(2πγt(Tt))favg=Mopt/(2πγr(Tr))τ0=γr(Tr)/(4kBTr),
which are all directly or indirectly dependent on temperature [19,35,36].

The friction coefficient temperature dependence can be found in Section 2 of Supplement 1. To estimate Tt and Tr, we proceed as follows: We first note that, for given laser properties, the optical driving torque Mopt and the trap stiffness k are only determined by the nanorod resonance properties, which are strongly dependent on the rod size and shape. The two friction coefficients, γt and γr, are also strongly size-dependent. However, the exact size of the trapped nanorod is unknown in the experiments. We first solved the equations of favg and τ0 to obtain Tr and the nanorod length from the measured (favg,τ0) at a specific laser power P=6.4mW by assuming that the nanorod diameter is fixed at 84 nm. With the derived particle size, we next calculated Tt and Tr versus laser power from the measured (fc,τ0). Figure 2(e) shows that Tt and Tr both increase with laser power. It is also clear that we always have Tt<Tr. This is expected for a particle performing hot Brownian motion because the fluid velocity field affecting rotational motion is more localized near the hot particle surface than the velocity field affecting translation [28].

The Tt and Tr values we extract from the preceding experiment are crucial to understanding the non-equilibrium dynamics of a trapped nanoparticle. However, it is not obvious how they relate to the exact temperature distribution around the particle, although we can assume that Tt is basically set by the distribution of temperature within the volume of water that has to move away from the particle as it moves forward through the liquid [27], while Tr is mostly dependent on the properties of the solution very close to the particle surface. To investigate this, we calculated the temperature profile around a trapped nanorod, assuming the same particle and trapping parameters as before, using finite element simulations (FEM, COMSOL Multiphysics) as shown in Fig. 2(f). Figure 2(e) shows that the calculated temperatures on and 110 nm away from the nanorod surface are very close to the measured Tr and Tt at the same laser power. A slight deviation is not surprising given the uncertainty in particle properties mentioned previously. In any case, the fast temperature drop away from the nanorod surface seen in Fig. 2(f) qualitatively explains why Tt<Tr.

3. SIMULATION OF HOT BROWNIAN MOTION

In the analysis of the preceding experimental results, we have implicitly assumed that the translational and rotational Brownian dynamics describes independent degrees of freedom. However, since the nanorod is significantly smaller than the laser spot, its translational motion will lead to a spatially varying optical torque and, hence, a fluctuation in the nanorod rotation frequency irrespective of the rotational Brownian motion. The nanorod Tt thus also contributes in the measured Tr. To investigate this effect in more detail, we turn to numerical simulations.

The (angular) velocities of hot Brownian nanoparticles are Maxwell–Boltzmann distributed according to nonuniversal effective temperatures [29]. This allows for an effective equilibrium description in terms of Langevin equations. Since the optical heating affects the translational and rotational movements differently due to the intrinsic nonequilibrium nature of the phenomenon, the translational and rotational effective Brownian temperatures have different values, as also observed in our experiments. For simplicity we still denote the two effective Brownian temperatures by Tt and Tr. We should always have T0TtTrTs, where T0 is the ambient temperature and Ts is the nanorod surface temperature [28]. Given that the rod is trapped in 2D and rotating continuously, we assume that its in-plane translational motion is isotropic. The nanorod translational and rotational motions can then be modeled by the following Langevin equations [19,36,37]:

mx¨(t)=γtx˙(t)+kxx(t)+(2kBTtγt)1/2Wx(t),my¨(t)=γty˙(t)+kyy(t)+(2kBTtγt)1/2Wy(t),Jφ¨(t)=γrφ˙(t)+Mopt+(2kBTrγr)1/2Wφ(t).
Here, x and y are the in-plane particle positions, φ is the rotation angle, m and J are the mass and moment of inertia, γt and γr are the translational and rotational friction coefficients [38] as before, and kx=ky=k are the trap stiffnesses along the two lateral directions. (2kBTtγt)1/2Wx,y(t) and (2kBTrγr)1/2Wφ(t) are the fluctuating translational force and thermal stochastic torque, respectively, due to random impulses from the fluid molecules [39,40].

Equation (2) can be solved numerically through a Brownian dynamics simulation under the non-inertial approximation (m and J set to zero). The continuous-time solution is then approximated by discrete-time sequences xi, yi, φi at ti=iΔt, where Δt is assumed to be much larger than the friction relaxation times [37]:

xi=xi1(kxi1/γt)Δt+(2DtΔt)1/2wi,x,yi=yi1(kyi1/γt)Δt+(2DtΔt)1/2wi,y,φi=φi1Mopt(xi1,yi1)γr·Δt+(2DrΔt)12wi,φ.
Here, wi,x, wi,y, and wi,φ are Gaussian random numbers with zero means and unit variances. Dt and Dr are the diffusion coefficients for translational and rotational motions. The discrete time interval Δt was selected to be much smaller than the time scale on which the restoring force acts, ϕt=γt/k [37]. The nanorod was modeled as a prolate ellipsoid for calculating the friction coefficients. All laser trapping parameters, including the laser beam intensity profile, trapping stiffness, and optical torque acting on the nanorod, were selected to mimic actual experimental conditions. Details of the parameter calculations are given in Section 2 of Supplement 1.

A. Translational Fluctuations

We first studied how the temperatures affect the translational motion of the nanorod. According to Eq. (3), only Tt determines the nanoparticle translational motion. We calculated the nanorod trajectory in the optical trap and verified that the rod follows a confined random walk in a potential of U=kr2/2 [Fig. 3(a)]. The particle position exhibits a Gaussian distribution with a standard deviation σx=σy=σ=80.6nm [Fig. 3(b)] at Tt=293.15K. In Fig. 3(c), we also plot the PSD of the particle x-displacement, which can be fitted by P(f)=C/(f2+fc2) [42] to obtain a corner frequency fc=91.8±8.2Hz. The values of both σ and fc agree well with those predicted from Eq. (1): σ=81.6nm and fc=93.5Hz. We further varied Tt and acquired the corresponding σx and fc by fitting the calculated probability distribution histogram and the PSD of x. Both σx and fc increase with Tt for constant laser power [Figs. 3(d) and 3(e)].

 figure: Fig. 3.

Fig. 3. Simulation of the translational movement of a rotating Brownian gold nanorod in a 2D optical trap. The incident laser power is set to 5 mW. (a) Translational trajectory of the nanorod. (b) Probability distribution of the nanorod position along x. The orange line is a fit to the data using a Gaussian distribution function. (c) Power spectral density (PSD) of the simulated x displacement. The orange line is an averaged PSD obtained using Welch’s method to reduce the noise [41]. (d) and (e) x-displacement standard deviation σx (d) and PSD corner frequency fc (e) of the trapped nanorod as the translational effective Brownian temperature Tt varies. The triangles in the plots are calculated from theoretical formulas while the red circles represent the results obtained by fitting the calculated probability distribution and the PSD curve. Error bars in (d) and (e) correspond to 95% confidence intervals obtained from the fits.

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B. Rotational Fluctuations

We next investigate how the temperatures affect the rotational motion of a nanorod. As mentioned previously, the nanorod will feel different optical driving torques when it moves within the trap, i.e., Mopt,i=Mopt(xi,yi), because of the Gaussian distribution of the laser intensity. The nanorod thus has continuously varying average rotation frequency favg,i=Mopt(xi,yi)/(2πγr). The probability distribution histogram of favg [Fig. 4(a)] is consistent with the nanoparticle position probability distribution shown in Fig. 3(b). We computed the rotation frequency and its fluctuation by simulating the experimental measurement, that is, we calculated the intensity of the light scattered from the nanorod after passing through a linear polarizer, Isca,iP=σsca,CPI(xi,yi)cos2φi [19]. A typical plot of the calculated IscaP is shown in Fig. 4(b), where both Tt and Tr have been set to room temperature by assuming that the particle is not absorbing any energy. The spikes in the curve indicate the fluctuations in the nanorod position and orientation. We then computed the ACF and PSD of IscaP. Curve fitting then gives the nanorod rotation dynamics, favg and τ0, as in the experiments.

 figure: Fig. 4.

Fig. 4. Simulation results of the rotational movement of a rotating Brownian gold nanorod in 2D optical trap. (a) Probability distribution of the nanorod rotational frequency. (b) Calculated intensity of the scattered light by the nanorod recorded after a polarizer IscaP. The intensity variation indicates the rotational movement of the nanorod. (c) and (d) ACFs (c) and PSDs (d) of the calculated IscaP. The autocorrelation decay and PSD peak broadening result from the rotational fluctuation caused by nanorod Brownian motion. The orange lines in the ACF plots represent fits to the data using the theoretically derived correlation function. The rotation frequency and autocorrelation decay time are obtained from the fitting. The orange lines in the PSD plots are averaged PSDs using Welch’s method to reduce the noise [41].

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We next tried to determine the contributions of the nanorod translational and rotational Brownian motion to its overall rotational fluctuation. First, to isolate the translational contribution, the thermal noise term representing the nanorod rotational fluctuation in Eq. (3) is omitted (Section 3 of Supplement 1). The ACF of the calculated IscaP exhibits a well-defined oscillation [Fig. 4(c)] in accordance with the distinct sharp peak that appears in the PSD trace [Fig. 4(d)]. However, the translation dynamics by itself obviously leads to a decaying ACF amplitude and a PSD peak broadening, verifying the fluctuation in the rotation frequency. Note here that the PSD peak exhibits an asymmetry due to the asymmetric rotation frequency distribution [Fig. 4(a)]. Second, in order to instead isolate the contribution from the rotational fluctuation, we assume that the nanorod rotates at a fixed position, namely the laser focus (Section 3 of Supplement 1). This also results in ACF decay and PSD peak broadening, as expected [Figs. 4(c) and 4(d)]. However, note that the peak is now symmetric since the rotational speed fluctuation follows the Maxwell–Boltzmann velocity distribution, and that a corner frequency is not physically defined since the translational motion is excluded from the simulation. Finally, when both translation and rotation are considered, a broader peak can be observed in the PSD and the oscillating amplitude in the ACF decays faster. The extracted rotation frequencies favg and decay times τ0 extracted for the three cases (i.e., translations only, rotations only, or both taken into account) are indicated in Fig. 4(c) and found to differ quite significantly. One sees that ft+rft<fr, which is expected since Mopt(0,0)>Mopt is always valid. More importantly, we found that 1/τ0t+r1/τ0t+1/τ0r. This interesting result holds for a variety of different simulation parameter settings and shows that the translational and rotational contributions to the total ACF decay constant contributes “in parallel”. The slight deviation between the left and right sides of the equation comes from the fact that we used the autocorrelation fitting formula to describe the ACFs even though the rotation frequency distribution is not perfectly symmetric. Similarly, the PSD peak widths Γt, Γr, and Γt+r are related: Γt+rΓt+Γr. These results thus indicate that the contributions from translational and rotational Brownian motion to the rotation fluctuations of a trapped particle can, in principle, be clearly separated. Finally, just like in the experiment, we can also obtain Tt and Tr from the simulated particle trajectory data using Eq. (1) and compare these values with the actual input temperatures to Eq. (3). This is further described in Section 4 of Supplement 1. The comparison indicates differences on the order of a few percent or less for the settings used. Specifically, using fr+t and τ0r+t from the experiment as an approximation to fr and τ0r when calculating Tr leads to a slight underestimation and overestimation, respectively, of the actual effective rotational temperature.

4. CONCLUSION

To summarize, we have shown that by tracking the translational and rotational movements of a spinning gold nanorod optically trapped in 2D, we are able to estimate the translational and rotational effective Brownian temperatures Tt and Tr of the nanorod. The two temperatures differ very significantly due to the optothermal heating of the plasmonic nanoparticle, and they basically agree with average medium temperatures sampled at a distance (0.1μm) away from and in very close proximity (10nm) to the particle surface, respectively. In the preceding analysis, we have assumed that the dependence of medium viscosity on temperature and the detailed properties of the particle (size, shape, and optical cross-sections) are known. Further, using stochastic simulations, we have shown that translational motion can significantly influence the determination of the rotational Brownian temperature. This effect needs to be considered in all measurements of hot Brownian dynamics of trapped nanoparticles.

Funding

Knut och Alice Wallenbergs Stiftelse.

Acknowledgment

We thank S. Nader and S. Reihani for valuable discussions.

 

See Supplement 1 for supporting content.

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30. R. L. Fogel’son and E. R. Likhachev, “Temperature dependence of viscosity,” Tech. Phys. 46, 1056–1059 (2001). [CrossRef]  

31. C.-K. Chou, W.-L. Chen, P. T. Fwu, S.-J. Lin, H.-S. Lee, and C.-Y. Dong, “Polarization ellipticity compensation in polarization second-harmonic generation microscopy without specimen rotation,” J. Biomed. Opt. 13, 014005 (2008). [CrossRef]  

32. F. Gittes and C. F. Schmidt, “Interference model for back-focal-plane displacement detection in optical tweezers,” Opt. Lett. 23, 7–9 (1998). [CrossRef]  

33. F. Hajizadeh, S. M. Mousavi, Z. S. Khaksar, and S. N. S. Reihani, “Extended linear detection range for optical tweezers using image-plane detection scheme,” J. Opt. 16, 105706 (2014). [CrossRef]  

34. M. Grießhammer and A. Rohrbach, “5D-Tracking of a nanorod in a focused laser beam—a theoretical concept,” Opt. Express 22, 6114–6132 (2014). [CrossRef]  

35. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004). [CrossRef]  

36. A. Lehmuskero, R. Ogier, T. Gschneidtner, P. Johansson, and M. Käll, “Ultrafast spinning of gold nanoparticles in water using circularly polarized light,” Nano Lett. 13, 3129–3134 (2013). [CrossRef]  

37. G. Volpe and G. Volpe, “Simulation of a Brownian particle in an optical trap,” Am. J. Phys. 81, 224–230 (2013). [CrossRef]  

38. J. K. G. Dhont, An Introduction to Dynamics of Colloids (Elsevier, 1996).

39. S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–89 (1943). [CrossRef]  

40. H. Risken, Fokker-Planck Equation (Springer, 1996).

41. P. D. Welch, “The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. 15, 70–73 (1967). [CrossRef]  

42. I. M. Tolić-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159, 225–240 (2004). [CrossRef]  

References

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  1. M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
    [Crossref]
  2. S. Kheifets, A. Simha, K. Melin, T. C. Li, and M. G. Raizen, “Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,” Science 343, 1493–1496 (2014).
    [Crossref]
  3. H. Turlier, D. A. Fedosov, B. Audoly, T. Auth, N. S. Gov, C. Sykes, J.-F. Joanny, G. Gompper, and T. Betz, “Equilibrium physics breakdown reveals the active nature of red blood cell flickering,” Nat. Phys. 12, 513–519 (2016).
    [Crossref]
  4. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 21–22 (2003).
    [Crossref]
  5. G. Volpe and D. Petrov, “Torque detection using Brownian fluctuations,” Phys. Rev. Lett. 97, 210603 (2006).
    [Crossref]
  6. A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical microrheology using rotating laser-trapped particles,” Phys. Rev. Lett. 92, 198104 (2004).
    [Crossref]
  7. J. Millen, T. Deesuwan, P. Barker, and J. Anders, “Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,” Nat. Nanotechnol. 9, 425–429 (2014).
    [Crossref]
  8. K. Neupane, D. A. Foster, D. R. Dee, H. Yu, F. Wang, and M. T. Woodside, “Direct observation of transition paths during the folding of proteins and nucleic acids,” Science 352, 239–242 (2016).
    [Crossref]
  9. K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5, 491–505 (2008).
    [Crossref]
  10. J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Ann. Rev. Biochem. 77, 205–228 (2008).
    [Crossref]
  11. K. Svoboda and S. M. Block, “Optical trapping of metallic Rayleigh particles,” Opt. Lett. 19, 930–932 (1994).
    [Crossref]
  12. M. Pelton, M. Z. Liu, H. Y. Kim, G. Smith, P. Guyot-Sionnest, and N. F. Scherer, “Optical trapping and alignment of single gold nanorods by using plasmon resonances,” Opt. Lett. 31, 2075–2077 (2006).
    [Crossref]
  13. L. M. Tong, V. D. Miljković, and M. Käll, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Lett. 10, 268–273 (2010).
    [Crossref]
  14. P. V. Ruijgrok, N. R. Verhart, P. Zijlstra, A. L. Tchebotareva, and M. Orrit, “Brownian fluctuations and heating of an optically aligned gold nanorod,” Phys. Rev. Lett. 107, 037401 (2011).
    [Crossref]
  15. A. Ohlinger, A. Deak, A. A. Lutich, and J. Feldmann, “Optically trapped gold nanoparticle enables listening at the microscale,” Phys. Rev. Lett. 108, 018101 (2012).
    [Crossref]
  16. O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013).
    [Crossref]
  17. A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. M. Tong, and M. Käll, “Laser trapping of colloidal metal nanoparticles,” ACS Nano 9, 3453–3469 (2015).
    [Crossref]
  18. F. Hajizadeh and S. N. S. Reihani, “Optimized optical trapping of gold nanoparticles,” Opt. Express 18, 551–559 (2010).
    [Crossref]
  19. L. Shao, Z.-J. Yang, D. Andrén, P. Johansson, and M. Käll, “Gold nanorod rotary motors driven by resonant light scattering,” ACS Nano 9, 12542–12551 (2015).
    [Crossref]
  20. G. Baffou and R. Quidant, “Thermo-plasmonics: using metallic nanostructures as nano-sources of heat,” Laser Photon. Rev. 7, 171–187 (2013).
    [Crossref]
  21. S. E. Lee and L. P. Lee, “Biomolecular plasmonics for quantitative biology and nanomedicine,” Curr. Opin. Biotechnol. 21, 489–497 (2010).
    [Crossref]
  22. L. Osinkina, S. Carretero-Palacios, J. Stehr, A. A. Lutich, F. Jäckel, and J. Feldmann, “Tuning DNA binding kinetics in an optical trap by plasmonic nanoparticle heating,” Nano Lett. 13, 3140–3144 (2013).
    [Crossref]
  23. J. Croissant and J. I. Zink, “Nanovalve-controlled cargo release activated by plasmonic heating,” J. Am. Chem. Soc. 134, 7628–7631 (2012).
    [Crossref]
  24. L. Shao and J. F. Wang, “Functional metal nanocrystals for biomedical applications,” in Handbook of Photonics for Biomedical Engineering, A. H.-P. Ho, D. Kim, and M. G. Somekh, eds. (Springer, 2015), pp. 1–32.
  25. X. H. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc. 128, 2115–2120 (2006).
    [Crossref]
  26. M. Braun and F. Cichos, “Optically controlled thermophoretic trapping of single nano-objects,” ACS Nano 7, 11200–11208 (2013).
    [Crossref]
  27. D. Rings, R. Schachoff, M. Selmke, F. Cichos, and K. Kroy, “Hot Brownian motion,” Phys. Rev. Lett. 105, 090604 (2010).
    [Crossref]
  28. D. Rings, D. Chakraborty, and K. Kroy, “Rotational hot Brownian motion,” New J. Phys. 14, 053012 (2012).
    [Crossref]
  29. G. Falasco, M. V. Gnann, D. Rings, and K. Kroy, “Effective temperatures of hot Brownian motion,” Phys. Rev. E 90, 032131 (2014).
    [Crossref]
  30. R. L. Fogel’son and E. R. Likhachev, “Temperature dependence of viscosity,” Tech. Phys. 46, 1056–1059 (2001).
    [Crossref]
  31. C.-K. Chou, W.-L. Chen, P. T. Fwu, S.-J. Lin, H.-S. Lee, and C.-Y. Dong, “Polarization ellipticity compensation in polarization second-harmonic generation microscopy without specimen rotation,” J. Biomed. Opt. 13, 014005 (2008).
    [Crossref]
  32. F. Gittes and C. F. Schmidt, “Interference model for back-focal-plane displacement detection in optical tweezers,” Opt. Lett. 23, 7–9 (1998).
    [Crossref]
  33. F. Hajizadeh, S. M. Mousavi, Z. S. Khaksar, and S. N. S. Reihani, “Extended linear detection range for optical tweezers using image-plane detection scheme,” J. Opt. 16, 105706 (2014).
    [Crossref]
  34. M. Grießhammer and A. Rohrbach, “5D-Tracking of a nanorod in a focused laser beam—a theoretical concept,” Opt. Express 22, 6114–6132 (2014).
    [Crossref]
  35. K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
    [Crossref]
  36. A. Lehmuskero, R. Ogier, T. Gschneidtner, P. Johansson, and M. Käll, “Ultrafast spinning of gold nanoparticles in water using circularly polarized light,” Nano Lett. 13, 3129–3134 (2013).
    [Crossref]
  37. G. Volpe and G. Volpe, “Simulation of a Brownian particle in an optical trap,” Am. J. Phys. 81, 224–230 (2013).
    [Crossref]
  38. J. K. G. Dhont, An Introduction to Dynamics of Colloids (Elsevier, 1996).
  39. S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–89 (1943).
    [Crossref]
  40. H. Risken, Fokker-Planck Equation (Springer, 1996).
  41. P. D. Welch, “The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. 15, 70–73 (1967).
    [Crossref]
  42. I. M. Tolić-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159, 225–240 (2004).
    [Crossref]

2016 (2)

H. Turlier, D. A. Fedosov, B. Audoly, T. Auth, N. S. Gov, C. Sykes, J.-F. Joanny, G. Gompper, and T. Betz, “Equilibrium physics breakdown reveals the active nature of red blood cell flickering,” Nat. Phys. 12, 513–519 (2016).
[Crossref]

K. Neupane, D. A. Foster, D. R. Dee, H. Yu, F. Wang, and M. T. Woodside, “Direct observation of transition paths during the folding of proteins and nucleic acids,” Science 352, 239–242 (2016).
[Crossref]

2015 (2)

A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. M. Tong, and M. Käll, “Laser trapping of colloidal metal nanoparticles,” ACS Nano 9, 3453–3469 (2015).
[Crossref]

L. Shao, Z.-J. Yang, D. Andrén, P. Johansson, and M. Käll, “Gold nanorod rotary motors driven by resonant light scattering,” ACS Nano 9, 12542–12551 (2015).
[Crossref]

2014 (5)

G. Falasco, M. V. Gnann, D. Rings, and K. Kroy, “Effective temperatures of hot Brownian motion,” Phys. Rev. E 90, 032131 (2014).
[Crossref]

F. Hajizadeh, S. M. Mousavi, Z. S. Khaksar, and S. N. S. Reihani, “Extended linear detection range for optical tweezers using image-plane detection scheme,” J. Opt. 16, 105706 (2014).
[Crossref]

M. Grießhammer and A. Rohrbach, “5D-Tracking of a nanorod in a focused laser beam—a theoretical concept,” Opt. Express 22, 6114–6132 (2014).
[Crossref]

S. Kheifets, A. Simha, K. Melin, T. C. Li, and M. G. Raizen, “Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,” Science 343, 1493–1496 (2014).
[Crossref]

J. Millen, T. Deesuwan, P. Barker, and J. Anders, “Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,” Nat. Nanotechnol. 9, 425–429 (2014).
[Crossref]

2013 (6)

O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013).
[Crossref]

A. Lehmuskero, R. Ogier, T. Gschneidtner, P. Johansson, and M. Käll, “Ultrafast spinning of gold nanoparticles in water using circularly polarized light,” Nano Lett. 13, 3129–3134 (2013).
[Crossref]

G. Volpe and G. Volpe, “Simulation of a Brownian particle in an optical trap,” Am. J. Phys. 81, 224–230 (2013).
[Crossref]

G. Baffou and R. Quidant, “Thermo-plasmonics: using metallic nanostructures as nano-sources of heat,” Laser Photon. Rev. 7, 171–187 (2013).
[Crossref]

L. Osinkina, S. Carretero-Palacios, J. Stehr, A. A. Lutich, F. Jäckel, and J. Feldmann, “Tuning DNA binding kinetics in an optical trap by plasmonic nanoparticle heating,” Nano Lett. 13, 3140–3144 (2013).
[Crossref]

M. Braun and F. Cichos, “Optically controlled thermophoretic trapping of single nano-objects,” ACS Nano 7, 11200–11208 (2013).
[Crossref]

2012 (3)

J. Croissant and J. I. Zink, “Nanovalve-controlled cargo release activated by plasmonic heating,” J. Am. Chem. Soc. 134, 7628–7631 (2012).
[Crossref]

A. Ohlinger, A. Deak, A. A. Lutich, and J. Feldmann, “Optically trapped gold nanoparticle enables listening at the microscale,” Phys. Rev. Lett. 108, 018101 (2012).
[Crossref]

D. Rings, D. Chakraborty, and K. Kroy, “Rotational hot Brownian motion,” New J. Phys. 14, 053012 (2012).
[Crossref]

2011 (1)

P. V. Ruijgrok, N. R. Verhart, P. Zijlstra, A. L. Tchebotareva, and M. Orrit, “Brownian fluctuations and heating of an optically aligned gold nanorod,” Phys. Rev. Lett. 107, 037401 (2011).
[Crossref]

2010 (4)

L. M. Tong, V. D. Miljković, and M. Käll, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Lett. 10, 268–273 (2010).
[Crossref]

D. Rings, R. Schachoff, M. Selmke, F. Cichos, and K. Kroy, “Hot Brownian motion,” Phys. Rev. Lett. 105, 090604 (2010).
[Crossref]

S. E. Lee and L. P. Lee, “Biomolecular plasmonics for quantitative biology and nanomedicine,” Curr. Opin. Biotechnol. 21, 489–497 (2010).
[Crossref]

F. Hajizadeh and S. N. S. Reihani, “Optimized optical trapping of gold nanoparticles,” Opt. Express 18, 551–559 (2010).
[Crossref]

2008 (3)

C.-K. Chou, W.-L. Chen, P. T. Fwu, S.-J. Lin, H.-S. Lee, and C.-Y. Dong, “Polarization ellipticity compensation in polarization second-harmonic generation microscopy without specimen rotation,” J. Biomed. Opt. 13, 014005 (2008).
[Crossref]

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5, 491–505 (2008).
[Crossref]

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Ann. Rev. Biochem. 77, 205–228 (2008).
[Crossref]

2006 (3)

M. Pelton, M. Z. Liu, H. Y. Kim, G. Smith, P. Guyot-Sionnest, and N. F. Scherer, “Optical trapping and alignment of single gold nanorods by using plasmon resonances,” Opt. Lett. 31, 2075–2077 (2006).
[Crossref]

G. Volpe and D. Petrov, “Torque detection using Brownian fluctuations,” Phys. Rev. Lett. 97, 210603 (2006).
[Crossref]

X. H. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc. 128, 2115–2120 (2006).
[Crossref]

2004 (3)

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[Crossref]

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical microrheology using rotating laser-trapped particles,” Phys. Rev. Lett. 92, 198104 (2004).
[Crossref]

I. M. Tolić-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159, 225–240 (2004).
[Crossref]

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 21–22 (2003).
[Crossref]

2001 (1)

R. L. Fogel’son and E. R. Likhachev, “Temperature dependence of viscosity,” Tech. Phys. 46, 1056–1059 (2001).
[Crossref]

1998 (2)

F. Gittes and C. F. Schmidt, “Interference model for back-focal-plane displacement detection in optical tweezers,” Opt. Lett. 23, 7–9 (1998).
[Crossref]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[Crossref]

1994 (1)

1967 (1)

P. D. Welch, “The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. 15, 70–73 (1967).
[Crossref]

1943 (1)

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–89 (1943).
[Crossref]

Anders, J.

J. Millen, T. Deesuwan, P. Barker, and J. Anders, “Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,” Nat. Nanotechnol. 9, 425–429 (2014).
[Crossref]

Andrén, D.

L. Shao, Z.-J. Yang, D. Andrén, P. Johansson, and M. Käll, “Gold nanorod rotary motors driven by resonant light scattering,” ACS Nano 9, 12542–12551 (2015).
[Crossref]

Audoly, B.

H. Turlier, D. A. Fedosov, B. Audoly, T. Auth, N. S. Gov, C. Sykes, J.-F. Joanny, G. Gompper, and T. Betz, “Equilibrium physics breakdown reveals the active nature of red blood cell flickering,” Nat. Phys. 12, 513–519 (2016).
[Crossref]

Auth, T.

H. Turlier, D. A. Fedosov, B. Audoly, T. Auth, N. S. Gov, C. Sykes, J.-F. Joanny, G. Gompper, and T. Betz, “Equilibrium physics breakdown reveals the active nature of red blood cell flickering,” Nat. Phys. 12, 513–519 (2016).
[Crossref]

Baffou, G.

G. Baffou and R. Quidant, “Thermo-plasmonics: using metallic nanostructures as nano-sources of heat,” Laser Photon. Rev. 7, 171–187 (2013).
[Crossref]

Barker, P.

J. Millen, T. Deesuwan, P. Barker, and J. Anders, “Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,” Nat. Nanotechnol. 9, 425–429 (2014).
[Crossref]

Berg-Sørensen, K.

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[Crossref]

I. M. Tolić-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159, 225–240 (2004).
[Crossref]

Betz, T.

H. Turlier, D. A. Fedosov, B. Audoly, T. Auth, N. S. Gov, C. Sykes, J.-F. Joanny, G. Gompper, and T. Betz, “Equilibrium physics breakdown reveals the active nature of red blood cell flickering,” Nat. Phys. 12, 513–519 (2016).
[Crossref]

Bishop, A. I.

A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical microrheology using rotating laser-trapped particles,” Phys. Rev. Lett. 92, 198104 (2004).
[Crossref]

Block, S. M.

Braun, M.

M. Braun and F. Cichos, “Optically controlled thermophoretic trapping of single nano-objects,” ACS Nano 7, 11200–11208 (2013).
[Crossref]

Bustamante, C.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Ann. Rev. Biochem. 77, 205–228 (2008).
[Crossref]

Carretero-Palacios, S.

L. Osinkina, S. Carretero-Palacios, J. Stehr, A. A. Lutich, F. Jäckel, and J. Feldmann, “Tuning DNA binding kinetics in an optical trap by plasmonic nanoparticle heating,” Nano Lett. 13, 3140–3144 (2013).
[Crossref]

Chakraborty, D.

D. Rings, D. Chakraborty, and K. Kroy, “Rotational hot Brownian motion,” New J. Phys. 14, 053012 (2012).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar, “Stochastic problems in physics and astronomy,” Rev. Mod. Phys. 15, 1–89 (1943).
[Crossref]

Chemla, Y. R.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Ann. Rev. Biochem. 77, 205–228 (2008).
[Crossref]

Chen, W.-L.

C.-K. Chou, W.-L. Chen, P. T. Fwu, S.-J. Lin, H.-S. Lee, and C.-Y. Dong, “Polarization ellipticity compensation in polarization second-harmonic generation microscopy without specimen rotation,” J. Biomed. Opt. 13, 014005 (2008).
[Crossref]

Chou, C.-K.

C.-K. Chou, W.-L. Chen, P. T. Fwu, S.-J. Lin, H.-S. Lee, and C.-Y. Dong, “Polarization ellipticity compensation in polarization second-harmonic generation microscopy without specimen rotation,” J. Biomed. Opt. 13, 014005 (2008).
[Crossref]

Cichos, F.

M. Braun and F. Cichos, “Optically controlled thermophoretic trapping of single nano-objects,” ACS Nano 7, 11200–11208 (2013).
[Crossref]

D. Rings, R. Schachoff, M. Selmke, F. Cichos, and K. Kroy, “Hot Brownian motion,” Phys. Rev. Lett. 105, 090604 (2010).
[Crossref]

Croissant, J.

J. Croissant and J. I. Zink, “Nanovalve-controlled cargo release activated by plasmonic heating,” J. Am. Chem. Soc. 134, 7628–7631 (2012).
[Crossref]

Deak, A.

A. Ohlinger, A. Deak, A. A. Lutich, and J. Feldmann, “Optically trapped gold nanoparticle enables listening at the microscale,” Phys. Rev. Lett. 108, 018101 (2012).
[Crossref]

Dee, D. R.

K. Neupane, D. A. Foster, D. R. Dee, H. Yu, F. Wang, and M. T. Woodside, “Direct observation of transition paths during the folding of proteins and nucleic acids,” Science 352, 239–242 (2016).
[Crossref]

Deesuwan, T.

J. Millen, T. Deesuwan, P. Barker, and J. Anders, “Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere,” Nat. Nanotechnol. 9, 425–429 (2014).
[Crossref]

Dhont, J. K. G.

J. K. G. Dhont, An Introduction to Dynamics of Colloids (Elsevier, 1996).

Dong, C.-Y.

C.-K. Chou, W.-L. Chen, P. T. Fwu, S.-J. Lin, H.-S. Lee, and C.-Y. Dong, “Polarization ellipticity compensation in polarization second-harmonic generation microscopy without specimen rotation,” J. Biomed. Opt. 13, 014005 (2008).
[Crossref]

El-Sayed, I. H.

X. H. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc. 128, 2115–2120 (2006).
[Crossref]

El-Sayed, M. A.

X. H. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc. 128, 2115–2120 (2006).
[Crossref]

Falasco, G.

G. Falasco, M. V. Gnann, D. Rings, and K. Kroy, “Effective temperatures of hot Brownian motion,” Phys. Rev. E 90, 032131 (2014).
[Crossref]

Fedosov, D. A.

H. Turlier, D. A. Fedosov, B. Audoly, T. Auth, N. S. Gov, C. Sykes, J.-F. Joanny, G. Gompper, and T. Betz, “Equilibrium physics breakdown reveals the active nature of red blood cell flickering,” Nat. Phys. 12, 513–519 (2016).
[Crossref]

Feldmann, J.

L. Osinkina, S. Carretero-Palacios, J. Stehr, A. A. Lutich, F. Jäckel, and J. Feldmann, “Tuning DNA binding kinetics in an optical trap by plasmonic nanoparticle heating,” Nano Lett. 13, 3140–3144 (2013).
[Crossref]

A. Ohlinger, A. Deak, A. A. Lutich, and J. Feldmann, “Optically trapped gold nanoparticle enables listening at the microscale,” Phys. Rev. Lett. 108, 018101 (2012).
[Crossref]

Ferrari, A. C.

O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013).
[Crossref]

Flyvbjerg, H.

K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum. 75, 594–612 (2004).
[Crossref]

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R. L. Fogel’son and E. R. Likhachev, “Temperature dependence of viscosity,” Tech. Phys. 46, 1056–1059 (2001).
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K. Neupane, D. A. Foster, D. R. Dee, H. Yu, F. Wang, and M. T. Woodside, “Direct observation of transition paths during the folding of proteins and nucleic acids,” Science 352, 239–242 (2016).
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M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
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C.-K. Chou, W.-L. Chen, P. T. Fwu, S.-J. Lin, H.-S. Lee, and C.-Y. Dong, “Polarization ellipticity compensation in polarization second-harmonic generation microscopy without specimen rotation,” J. Biomed. Opt. 13, 014005 (2008).
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O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013).
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Hajizadeh, F.

F. Hajizadeh, S. M. Mousavi, Z. S. Khaksar, and S. N. S. Reihani, “Extended linear detection range for optical tweezers using image-plane detection scheme,” J. Opt. 16, 105706 (2014).
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A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical microrheology using rotating laser-trapped particles,” Phys. Rev. Lett. 92, 198104 (2004).
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X. H. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc. 128, 2115–2120 (2006).
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L. Osinkina, S. Carretero-Palacios, J. Stehr, A. A. Lutich, F. Jäckel, and J. Feldmann, “Tuning DNA binding kinetics in an optical trap by plasmonic nanoparticle heating,” Nano Lett. 13, 3140–3144 (2013).
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H. Turlier, D. A. Fedosov, B. Audoly, T. Auth, N. S. Gov, C. Sykes, J.-F. Joanny, G. Gompper, and T. Betz, “Equilibrium physics breakdown reveals the active nature of red blood cell flickering,” Nat. Phys. 12, 513–519 (2016).
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L. Shao, Z.-J. Yang, D. Andrén, P. Johansson, and M. Käll, “Gold nanorod rotary motors driven by resonant light scattering,” ACS Nano 9, 12542–12551 (2015).
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A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. M. Tong, and M. Käll, “Laser trapping of colloidal metal nanoparticles,” ACS Nano 9, 3453–3469 (2015).
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F. Hajizadeh, S. M. Mousavi, Z. S. Khaksar, and S. N. S. Reihani, “Extended linear detection range for optical tweezers using image-plane detection scheme,” J. Opt. 16, 105706 (2014).
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G. Falasco, M. V. Gnann, D. Rings, and K. Kroy, “Effective temperatures of hot Brownian motion,” Phys. Rev. E 90, 032131 (2014).
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C.-K. Chou, W.-L. Chen, P. T. Fwu, S.-J. Lin, H.-S. Lee, and C.-Y. Dong, “Polarization ellipticity compensation in polarization second-harmonic generation microscopy without specimen rotation,” J. Biomed. Opt. 13, 014005 (2008).
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S. E. Lee and L. P. Lee, “Biomolecular plasmonics for quantitative biology and nanomedicine,” Curr. Opin. Biotechnol. 21, 489–497 (2010).
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S. E. Lee and L. P. Lee, “Biomolecular plasmonics for quantitative biology and nanomedicine,” Curr. Opin. Biotechnol. 21, 489–497 (2010).
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A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. M. Tong, and M. Käll, “Laser trapping of colloidal metal nanoparticles,” ACS Nano 9, 3453–3469 (2015).
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S. Kheifets, A. Simha, K. Melin, T. C. Li, and M. G. Raizen, “Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,” Science 343, 1493–1496 (2014).
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R. L. Fogel’son and E. R. Likhachev, “Temperature dependence of viscosity,” Tech. Phys. 46, 1056–1059 (2001).
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C.-K. Chou, W.-L. Chen, P. T. Fwu, S.-J. Lin, H.-S. Lee, and C.-Y. Dong, “Polarization ellipticity compensation in polarization second-harmonic generation microscopy without specimen rotation,” J. Biomed. Opt. 13, 014005 (2008).
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L. Osinkina, S. Carretero-Palacios, J. Stehr, A. A. Lutich, F. Jäckel, and J. Feldmann, “Tuning DNA binding kinetics in an optical trap by plasmonic nanoparticle heating,” Nano Lett. 13, 3140–3144 (2013).
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O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013).
[Crossref]

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S. Kheifets, A. Simha, K. Melin, T. C. Li, and M. G. Raizen, “Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,” Science 343, 1493–1496 (2014).
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L. M. Tong, V. D. Miljković, and M. Käll, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Lett. 10, 268–273 (2010).
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F. Hajizadeh, S. M. Mousavi, Z. S. Khaksar, and S. N. S. Reihani, “Extended linear detection range for optical tweezers using image-plane detection scheme,” J. Opt. 16, 105706 (2014).
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K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5, 491–505 (2008).
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K. Neupane, D. A. Foster, D. R. Dee, H. Yu, F. Wang, and M. T. Woodside, “Direct observation of transition paths during the folding of proteins and nucleic acids,” Science 352, 239–242 (2016).
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A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical microrheology using rotating laser-trapped particles,” Phys. Rev. Lett. 92, 198104 (2004).
[Crossref]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[Crossref]

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A. Lehmuskero, R. Ogier, T. Gschneidtner, P. Johansson, and M. Käll, “Ultrafast spinning of gold nanoparticles in water using circularly polarized light,” Nano Lett. 13, 3129–3134 (2013).
[Crossref]

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A. Ohlinger, A. Deak, A. A. Lutich, and J. Feldmann, “Optically trapped gold nanoparticle enables listening at the microscale,” Phys. Rev. Lett. 108, 018101 (2012).
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P. V. Ruijgrok, N. R. Verhart, P. Zijlstra, A. L. Tchebotareva, and M. Orrit, “Brownian fluctuations and heating of an optically aligned gold nanorod,” Phys. Rev. Lett. 107, 037401 (2011).
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L. Osinkina, S. Carretero-Palacios, J. Stehr, A. A. Lutich, F. Jäckel, and J. Feldmann, “Tuning DNA binding kinetics in an optical trap by plasmonic nanoparticle heating,” Nano Lett. 13, 3140–3144 (2013).
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Petrov, D.

G. Volpe and D. Petrov, “Torque detection using Brownian fluctuations,” Phys. Rev. Lett. 97, 210603 (2006).
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X. H. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc. 128, 2115–2120 (2006).
[Crossref]

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G. Baffou and R. Quidant, “Thermo-plasmonics: using metallic nanostructures as nano-sources of heat,” Laser Photon. Rev. 7, 171–187 (2013).
[Crossref]

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S. Kheifets, A. Simha, K. Melin, T. C. Li, and M. G. Raizen, “Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,” Science 343, 1493–1496 (2014).
[Crossref]

Reihani, S. N. S.

F. Hajizadeh, S. M. Mousavi, Z. S. Khaksar, and S. N. S. Reihani, “Extended linear detection range for optical tweezers using image-plane detection scheme,” J. Opt. 16, 105706 (2014).
[Crossref]

F. Hajizadeh and S. N. S. Reihani, “Optimized optical trapping of gold nanoparticles,” Opt. Express 18, 551–559 (2010).
[Crossref]

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G. Falasco, M. V. Gnann, D. Rings, and K. Kroy, “Effective temperatures of hot Brownian motion,” Phys. Rev. E 90, 032131 (2014).
[Crossref]

D. Rings, D. Chakraborty, and K. Kroy, “Rotational hot Brownian motion,” New J. Phys. 14, 053012 (2012).
[Crossref]

D. Rings, R. Schachoff, M. Selmke, F. Cichos, and K. Kroy, “Hot Brownian motion,” Phys. Rev. Lett. 105, 090604 (2010).
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A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. M. Tong, and M. Käll, “Laser trapping of colloidal metal nanoparticles,” ACS Nano 9, 3453–3469 (2015).
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A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical microrheology using rotating laser-trapped particles,” Phys. Rev. Lett. 92, 198104 (2004).
[Crossref]

M. E. J. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature 394, 348–350 (1998).
[Crossref]

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P. V. Ruijgrok, N. R. Verhart, P. Zijlstra, A. L. Tchebotareva, and M. Orrit, “Brownian fluctuations and heating of an optically aligned gold nanorod,” Phys. Rev. Lett. 107, 037401 (2011).
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Schmidt, C. F.

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D. Rings, R. Schachoff, M. Selmke, F. Cichos, and K. Kroy, “Hot Brownian motion,” Phys. Rev. Lett. 105, 090604 (2010).
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L. Shao, Z.-J. Yang, D. Andrén, P. Johansson, and M. Käll, “Gold nanorod rotary motors driven by resonant light scattering,” ACS Nano 9, 12542–12551 (2015).
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L. Shao and J. F. Wang, “Functional metal nanocrystals for biomedical applications,” in Handbook of Photonics for Biomedical Engineering, A. H.-P. Ho, D. Kim, and M. G. Somekh, eds. (Springer, 2015), pp. 1–32.

Simha, A.

S. Kheifets, A. Simha, K. Melin, T. C. Li, and M. G. Raizen, “Observation of Brownian motion in liquids at short times: instantaneous velocity and memory loss,” Science 343, 1493–1496 (2014).
[Crossref]

Smith, G.

Smith, S. B.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Ann. Rev. Biochem. 77, 205–228 (2008).
[Crossref]

Stehr, J.

L. Osinkina, S. Carretero-Palacios, J. Stehr, A. A. Lutich, F. Jäckel, and J. Feldmann, “Tuning DNA binding kinetics in an optical trap by plasmonic nanoparticle heating,” Nano Lett. 13, 3140–3144 (2013).
[Crossref]

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Sykes, C.

H. Turlier, D. A. Fedosov, B. Audoly, T. Auth, N. S. Gov, C. Sykes, J.-F. Joanny, G. Gompper, and T. Betz, “Equilibrium physics breakdown reveals the active nature of red blood cell flickering,” Nat. Phys. 12, 513–519 (2016).
[Crossref]

Tchebotareva, A. L.

P. V. Ruijgrok, N. R. Verhart, P. Zijlstra, A. L. Tchebotareva, and M. Orrit, “Brownian fluctuations and heating of an optically aligned gold nanorod,” Phys. Rev. Lett. 107, 037401 (2011).
[Crossref]

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I. M. Tolić-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159, 225–240 (2004).
[Crossref]

Tong, L. M.

A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. M. Tong, and M. Käll, “Laser trapping of colloidal metal nanoparticles,” ACS Nano 9, 3453–3469 (2015).
[Crossref]

L. M. Tong, V. D. Miljković, and M. Käll, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Lett. 10, 268–273 (2010).
[Crossref]

Turlier, H.

H. Turlier, D. A. Fedosov, B. Audoly, T. Auth, N. S. Gov, C. Sykes, J.-F. Joanny, G. Gompper, and T. Betz, “Equilibrium physics breakdown reveals the active nature of red blood cell flickering,” Nat. Phys. 12, 513–519 (2016).
[Crossref]

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P. V. Ruijgrok, N. R. Verhart, P. Zijlstra, A. L. Tchebotareva, and M. Orrit, “Brownian fluctuations and heating of an optically aligned gold nanorod,” Phys. Rev. Lett. 107, 037401 (2011).
[Crossref]

Volpe, G.

O. M. Maragò, P. H. Jones, P. G. Gucciardi, G. Volpe, and A. C. Ferrari, “Optical trapping and manipulation of nanostructures,” Nat. Nanotechnol. 8, 807–819 (2013).
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G. Volpe and G. Volpe, “Simulation of a Brownian particle in an optical trap,” Am. J. Phys. 81, 224–230 (2013).
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G. Volpe and G. Volpe, “Simulation of a Brownian particle in an optical trap,” Am. J. Phys. 81, 224–230 (2013).
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G. Volpe and D. Petrov, “Torque detection using Brownian fluctuations,” Phys. Rev. Lett. 97, 210603 (2006).
[Crossref]

Wang, F.

K. Neupane, D. A. Foster, D. R. Dee, H. Yu, F. Wang, and M. T. Woodside, “Direct observation of transition paths during the folding of proteins and nucleic acids,” Science 352, 239–242 (2016).
[Crossref]

Wang, J. F.

L. Shao and J. F. Wang, “Functional metal nanocrystals for biomedical applications,” in Handbook of Photonics for Biomedical Engineering, A. H.-P. Ho, D. Kim, and M. G. Somekh, eds. (Springer, 2015), pp. 1–32.

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P. D. Welch, “The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. 15, 70–73 (1967).
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Woodside, M. T.

K. Neupane, D. A. Foster, D. R. Dee, H. Yu, F. Wang, and M. T. Woodside, “Direct observation of transition paths during the folding of proteins and nucleic acids,” Science 352, 239–242 (2016).
[Crossref]

Yang, Z.-J.

L. Shao, Z.-J. Yang, D. Andrén, P. Johansson, and M. Käll, “Gold nanorod rotary motors driven by resonant light scattering,” ACS Nano 9, 12542–12551 (2015).
[Crossref]

Yu, H.

K. Neupane, D. A. Foster, D. R. Dee, H. Yu, F. Wang, and M. T. Woodside, “Direct observation of transition paths during the folding of proteins and nucleic acids,” Science 352, 239–242 (2016).
[Crossref]

Zijlstra, P.

P. V. Ruijgrok, N. R. Verhart, P. Zijlstra, A. L. Tchebotareva, and M. Orrit, “Brownian fluctuations and heating of an optically aligned gold nanorod,” Phys. Rev. Lett. 107, 037401 (2011).
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J. Croissant and J. I. Zink, “Nanovalve-controlled cargo release activated by plasmonic heating,” J. Am. Chem. Soc. 134, 7628–7631 (2012).
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ACS Nano (3)

A. Lehmuskero, P. Johansson, H. Rubinsztein-Dunlop, L. M. Tong, and M. Käll, “Laser trapping of colloidal metal nanoparticles,” ACS Nano 9, 3453–3469 (2015).
[Crossref]

L. Shao, Z.-J. Yang, D. Andrén, P. Johansson, and M. Käll, “Gold nanorod rotary motors driven by resonant light scattering,” ACS Nano 9, 12542–12551 (2015).
[Crossref]

M. Braun and F. Cichos, “Optically controlled thermophoretic trapping of single nano-objects,” ACS Nano 7, 11200–11208 (2013).
[Crossref]

Am. J. Phys. (1)

G. Volpe and G. Volpe, “Simulation of a Brownian particle in an optical trap,” Am. J. Phys. 81, 224–230 (2013).
[Crossref]

Ann. Rev. Biochem. (1)

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Ann. Rev. Biochem. 77, 205–228 (2008).
[Crossref]

Comput. Phys. Commun. (1)

I. M. Tolić-Nørrelykke, K. Berg-Sørensen, and H. Flyvbjerg, “MatLab program for precision calibration of optical tweezers,” Comput. Phys. Commun. 159, 225–240 (2004).
[Crossref]

Curr. Opin. Biotechnol. (1)

S. E. Lee and L. P. Lee, “Biomolecular plasmonics for quantitative biology and nanomedicine,” Curr. Opin. Biotechnol. 21, 489–497 (2010).
[Crossref]

IEEE Trans. Audio Electroacoust. (1)

P. D. Welch, “The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms,” IEEE Trans. Audio Electroacoust. 15, 70–73 (1967).
[Crossref]

J. Am. Chem. Soc. (2)

J. Croissant and J. I. Zink, “Nanovalve-controlled cargo release activated by plasmonic heating,” J. Am. Chem. Soc. 134, 7628–7631 (2012).
[Crossref]

X. H. Huang, I. H. El-Sayed, W. Qian, and M. A. El-Sayed, “Cancer cell imaging and photothermal therapy in the near-infrared region by using gold nanorods,” J. Am. Chem. Soc. 128, 2115–2120 (2006).
[Crossref]

J. Biomed. Opt. (1)

C.-K. Chou, W.-L. Chen, P. T. Fwu, S.-J. Lin, H.-S. Lee, and C.-Y. Dong, “Polarization ellipticity compensation in polarization second-harmonic generation microscopy without specimen rotation,” J. Biomed. Opt. 13, 014005 (2008).
[Crossref]

J. Opt. (1)

F. Hajizadeh, S. M. Mousavi, Z. S. Khaksar, and S. N. S. Reihani, “Extended linear detection range for optical tweezers using image-plane detection scheme,” J. Opt. 16, 105706 (2014).
[Crossref]

Laser Photon. Rev. (1)

G. Baffou and R. Quidant, “Thermo-plasmonics: using metallic nanostructures as nano-sources of heat,” Laser Photon. Rev. 7, 171–187 (2013).
[Crossref]

Nano Lett. (3)

L. Osinkina, S. Carretero-Palacios, J. Stehr, A. A. Lutich, F. Jäckel, and J. Feldmann, “Tuning DNA binding kinetics in an optical trap by plasmonic nanoparticle heating,” Nano Lett. 13, 3140–3144 (2013).
[Crossref]

A. Lehmuskero, R. Ogier, T. Gschneidtner, P. Johansson, and M. Käll, “Ultrafast spinning of gold nanoparticles in water using circularly polarized light,” Nano Lett. 13, 3129–3134 (2013).
[Crossref]

L. M. Tong, V. D. Miljković, and M. Käll, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Lett. 10, 268–273 (2010).
[Crossref]

Nat. Methods (1)

K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods 5, 491–505 (2008).
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Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. Measurement of the rotation dynamics of a 2D trapped gold nanorod. (a) Schematic of the optical tweezers setup. (Inset) Scanning electron microscopy image of the used gold nanorods. (b) Measured QPD voltage dependence on immobilized nanorod displacement from the laser focus at different laser powers. The dashed line shows a fit to the central linear range of the data measured at 11.7 mW. (c) and (d) QPD-measured trajectory of a rotating nanorod in the optical trap (c) and distribution of the rod position (d). The laser power was set at 6.4 mW. (e) PSD of the measured nanorod x displacement. The red smooth curve is the fitting at ffavg to obtain the corner frequency.
Fig. 2.
Fig. 2. Experimentally measured rotation dynamics and estimated Brownian temperatures of a rotating gold nanorod under varying laser powers. (a) PSD of the backscattered laser light intensity collected by the QPD. (b) Autocorrelation plots of scattering intensity after an analyzer collected by the APD. (c) Laser power-dependent rotation frequencies of nanorods measured by APD and QPD. Error bars reveal the standard deviations of the autocorrelation fitting (red) and of a Lorentzian distribution fit to PSD peak (green), respectively. (d) Power spectrum peak widths of signals shown in (a) and ACF decay rates of signals shown in (b). (e) Estimated Tt and Tr from the experimental corner frequency fc and ACF decay time τ0, respectively. Error bars represent 95% confidence intervals. The red solid line indicates the calculated surface temperature of the nanorod while the green solid line indicates the calculated temperature in solution 110 nm away from the nanorod surface. (f) Calculated temperature profile around the gold nanorod when heated by the circularly polarized trapping laser. The yellow and white dashed circles indicate the sizes of the laser beam waist and the movement range of the nanorod center of mass, respectively, with the latter representing two standard deviations of the nanorod displacement.
Fig. 3.
Fig. 3. Simulation of the translational movement of a rotating Brownian gold nanorod in a 2D optical trap. The incident laser power is set to 5 mW. (a) Translational trajectory of the nanorod. (b) Probability distribution of the nanorod position along x. The orange line is a fit to the data using a Gaussian distribution function. (c) Power spectral density (PSD) of the simulated x displacement. The orange line is an averaged PSD obtained using Welch’s method to reduce the noise [41]. (d) and (e) x-displacement standard deviation σx (d) and PSD corner frequency fc (e) of the trapped nanorod as the translational effective Brownian temperature Tt varies. The triangles in the plots are calculated from theoretical formulas while the red circles represent the results obtained by fitting the calculated probability distribution and the PSD curve. Error bars in (d) and (e) correspond to 95% confidence intervals obtained from the fits.
Fig. 4.
Fig. 4. Simulation results of the rotational movement of a rotating Brownian gold nanorod in 2D optical trap. (a) Probability distribution of the nanorod rotational frequency. (b) Calculated intensity of the scattered light by the nanorod recorded after a polarizer IscaP. The intensity variation indicates the rotational movement of the nanorod. (c) and (d) ACFs (c) and PSDs (d) of the calculated IscaP. The autocorrelation decay and PSD peak broadening result from the rotational fluctuation caused by nanorod Brownian motion. The orange lines in the ACF plots represent fits to the data using the theoretically derived correlation function. The rotation frequency and autocorrelation decay time are obtained from the fitting. The orange lines in the PSD plots are averaged PSDs using Welch’s method to reduce the noise [41].

Equations (3)

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{σ=(kBTt/k)1/2fc=k/(2πγt(Tt))favg=Mopt/(2πγr(Tr))τ0=γr(Tr)/(4kBTr),
mx¨(t)=γtx˙(t)+kxx(t)+(2kBTtγt)1/2Wx(t),my¨(t)=γty˙(t)+kyy(t)+(2kBTtγt)1/2Wy(t),Jφ¨(t)=γrφ˙(t)+Mopt+(2kBTrγr)1/2Wφ(t).
xi=xi1(kxi1/γt)Δt+(2DtΔt)1/2wi,x,yi=yi1(kyi1/γt)Δt+(2DtΔt)1/2wi,y,φi=φi1Mopt(xi1,yi1)γr·Δt+(2DrΔt)12wi,φ.

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