1、Calibration of position dataRaw position signals recorded with quadrant photodiode detector are in units of volts, which have to be converted into units of meters before the local drift method can be used to analyze the force field. To do this, three calibration factors along x, y and z axes need to
2、 be determined. In the described experiment, the following calibration method was applied: First, position autocorrelation function was calculated and fitted with an exponential to get the position autocorrelation time which is equal to assuming a Brownian particle moving in a /kharmonic potential 1
3、. Here k is the stiffness of the trap and is the Stokes drag 6acoefficient for a sphere with radius a in a fluid with viscosity . With a known a and , the stiffness k can be readily calculated from . Fig. S1(A) shows the exponential fit to the position autocorrelation data along the y axis. Second,
4、a histogram of raw position signals is fitted with a Gaussian function as shown in Fig. S1(B), which gives the width of the distribution volt. Using the equipartition theorem, the width of the distribution is related to the thermal energy as , where S is the calibration factor with a unit of m/V, kB
5、 is the Boltzmann constant kS2voltBTand T is the temperature 2. Using this method, we obtained calibration factors Sx=2.9610-8 m/V, Sy=3.5410-8 m/V, and Sz=1.0010-7 m/V from the raw position data. Although our calibration method assumes a harmonic potential and we have just demonstrated that the for
6、ce field of optical tweezers has a nonconservative component, we still think that this assumption introduces a minimal error in the trap calibration for the following reasons: 1) If we assume the nonconservative force changed in some way the position histogram or the autocorrelation function, it wou
7、ld result in an error in the calibration factor and thus in the stiffness of the trap. However, an error in the position calibration factor cannot introduce a nonconservative force. It will just affect the precision of measured force constant. 2) The most probable position of the particle is on the
8、optical axis where the scattering force is largest but has the smallest rate of change with the lateral distance from the optical axis. Because the nonconservative force is a result of the change of the scattering force with the lateral distance, the particles motion is least influenced by the nonco
9、nservative force when closest to the optical axis. In our calibration method, position data around the trapping center are most significant in determining the fitting result because of small statistical errors. Thus the error introduced in our calibration by the assumption of a harmonic potential is
10、 minimal and does not affect our conclusions. Of course, one may think of independent calibration methods without assuming a harmonic potential. For example, one can immobilize a particle with polymer gel in three dimensions and then scan it through the focus spot in order to calibrate the detector
11、response. However, such a calibration method cannot reproduce the experimental conditions and other errors are introduced for instance due to the refractive index difference between the polymer gels and the medium used in the experiment. In summary, the calibration method applied is not without shor
12、tcomings but it does not affect the conclusions of the paper. Fig. S1. Calibration procedure. (A) Position autocorrelation function and (B) position histogram of raw position signals along the y-axis. The signals were sampled at 50 kHz for 10 seconds in all three dimensions simultaneously. The autoc
13、orrelation function (red circles in A) was fitted with an exponential function (blue curve in A), which gave a position autocorrelation time = 0.487 ms. A Gaussian function (blue curve in B) fitted to the position histogram (red bars in B) gave the f(x)D01exp(2)width of the distribution of = 0.982 V
14、. Experimental force profiles for small particle displacement After calibration, the average position in each dimension x, y, and z is subtracted from calibrated position data, which brings the origin of the coordinate system to the trapping center. We then apply the local drift method to the data a
15、nd obtain the 3D force field of the optical trap. The forces along each axis (calculated as the average of data falling into a column 10 nm10 nm centered on each axis) are displayed in Fig. S2 to show that the force increases linearly away from the axes for small displacement. Since these force prof
16、iles only contain the data measured on the axes, without averaging over the whole trapping volume, the force constants in the figure are slightly larger than the overall stiffnesses of the trap, which are kx=6.2x10-6 N/m, ky=3.5x10-6 N/m and kz=0.9x10-6 N/m. These values are in good agreement with p
17、revious work 3.Fig. S2. Force profiles and linear fits along each axis for the experimental force field shown in Figure 2 in the main text. Each data point represents the value in one volume element 10x10x10 nm3 centered at this point. Brownian dynamics simulation using a Gaussian beam model for the
18、 optical trapConventional Brownian dynamics simulations 4 in a known force field were performed to validate the accuracy of the “local drift” method. The theoretical force field of the optical trap was calculated using a paraxial Gaussian beam model for Rayleigh particles 5. Even though this model i
19、s only a zero-order approximation of the focused beam and it does not take into account its vectorial character, it has the advantage of providing analytical formulas for the gradient and scattering forces that can be readily inserted into the Brownian dynamics simulation. Our particles with radius
20、a = 100 nm lie outside the commonly accepted valid range of the Rayleigh approximation where is the radiation wavelength in medium; in our case 20= 1064 / 1.33 nm 800 nm which gives the size limit for Rayleigh approximation equal to 40 nm. However, even for slightly larger particles, Rayleigh approx
21、imation still provides the values of forces that lie to within a few tens of percent from the exact values 5. To be able to compare the simulations and the experimental results, simulation parameters were chosen to mimic those in the experiments (laser wavelength: 1064 nm; laser power at the focus s
22、pot: 25 mW; numerical aperture: 1.2; beam waist: 459 nm; particle diameter: 200 nm; refractive index of particle: 1.57; refractive index of medium: 1.33). The position fluctuation of the particle is simulated such that for each time interval, the particles displacement is the sum of the directional
23、drift caused by the local force and a random diffusion from free Brownian motion: rn1rF ndom isplacentThe directional drift part is directly proportional to the local trapping force . The random rFdisplacement is Gaussian-distributed with a standard deviation of , where D is the rn 2tdiffusion const
24、ant of the particle and s is the time interval between successive t1.6706steps. After the position data were simulated, the local drift method was applied to calculate the force field.Fig. S3 shows the forces recovered from the simulated position data and their comparison with the theory. For direct
25、 comparison, the force profiles along each axis are chosen. The recovered forces agree with high accuracy with the theorectial forces inserted in the Brownians dynamics simulations. The corresponding force constants for each dimension are kx= ky=3.4x10-6 N/m and kz=0.5x10-6 N/m. Although we chose on
26、ly a small portion of the force field for direct comparison, the remaining force field also shows good agreement. Fig. S3(d) shows the force field projected on the x-y plane at the position where Fz=0 on the optical axis. This position is about 180 nm above the focal plane. Fig. S4 shows the compari
27、son of the magnitudes for the curl of the simulated and theoretical force field. The agreement between simulation and theory further proves the accuracy of the local drift method .Fig. S3. Comparison of simulated force field in an optical trap using the local drift method and theory. (a), (b), and (
28、c) show the force profiles along three axes. The blue dots with error bars are forces recovered from the simulated position data using the “local drift” method. The solid red lines are theoretical forces directly calculated using the formulas for the gradient and scattering forces given in 5. (d) sh
29、ows the x and y component of the force field at z=180 nm, where the z component of the force is zero on the optical axis. Fig. S4. Magnitudes of the curl of the force field obtained from simulated data and the curl of the theoretical force field . The curl for the simulation is calculated by averagi
30、ng the magnitudes of the curl in Fig. 3 in the main text around circles centered on the optical axis. The theoretical value is calculated from the direct derivative of the force field from 5. The agreement of the magnitudes confirms that our local drift method is sufficiently precise to determine th
31、e derivative of the force field. Estimation of average energy flowFig. S5. Work done on a trapped particle by the nonconservative force along different paths. The experimental force field shown in Fig. 2 was used. The work for each path is given in units of kBT. We integrated the force along differe
32、nt closed path shown in Fig. S5. The average nonconservative force can be estimated from , where W is the work done along Fnc/sthe path and s is the length of the closed path.1 C.W. Wang and G.E. Uhlenbeck, in Selected Papers on Noise and Stochastic Processed (ed. Wax, N.), 113 (Dover, New York, 195
33、4).2 E.-L. Florin, A. Pralle, E.H.K. Stelzer, J.K.H. Horber, Appl. Phys. A 66, S75 (1998). Photonic force microscope calibration by thermal noise analysis3 A. Rohrbach, Phys. Rev. Lett. 95, 168102 (2005). Stiffness of optical traps: quantitative agreement between experiment and electromagnetic theory4 A.C. Branka and D.M. Heyes, Phys. Rev. E 58, 2611 (1998). Algorithms for Brownian dynamics simulation5 Yasuhiro Harada, Toshimitsu Asakura, Opt. Commu. 124, 529-541 (1996). Radiation forces on a dielectric sphere in the Rayleigh scattering regime
