2.1.1.

Optical trap formation

Schematic of the experimental setup is shown in Fig. 1. The optical trap was formed on an inverted microscope (Ti-Eclipse, Nikon Inc., Melville, New York) using a $100\times$ oil immersion objective lens with 1.49 N.A. and $120\mu \mathrm{m}$ working distance (Apo TIRF, Nikon). A solid-state $\mathrm{Nd}:{\mathrm{YVO}}_4$ laser (Prisma 1064-V, Coherent, Santa Clara, CA) generating a laser beam with a 1064 nm wavelength in ${\mathrm{TEM}}_{00}$ mode was used for optical trapping. The laser beam was expanded by a set of plano-convex lenses to slightly overfill the back aperture of the microscope objective. The output power of the trapping beam was measured at the back aperture of the objective lens by a power meter (PD300, Ophir Optronics, North Andover, MA). We used beads with two different diameters for trapping. The smaller beads were sulfate-modified fluorescent polystyrene beads $4.2\pm 0.21$ ($\text{mean}\pm \text{standard deviation}$) $\mu \mathrm{m}$ in diameter (F-8858, Molecular probes, Eugene, OR). These beads had an excitation spectrum between 480 and 590 nm with peak fluorescence emission at 605 nm when photo-excited at 580 nm. We refer to these particles as red beads. In addition to the $4\text{-}\mu \mathrm{m}$-diameter red beads, we used larger polystyrene beads, approximately twice in diameter ($8\pm 0.7\mu \mathrm{m}$; 2106E, Phosphorex, Hopkinton, MA) while maintaining a comparable position tracking signal-to-noise ratio (SNR) between the experiments. These beads had an excitation spectrum between 400 and 490 nm with peak fluorescence emission at 500 nm when photo-excited at 460 nm. We refer to these particles as green beads. Both particles were chosen within the same volume-scale to limit the uncertainties related to the volume-dependent differences in how particles in different volume-scales interact with the highly focused laser in optical traps. Beads were suspended in deionized (DI) water during optical trapping experiments.

Fig. 1

Schematic of the experimental setup. (1) $\mathrm{Nd}:{\mathrm{YVO}}_4$ laser. (2) Beam expander. (3) piezoelectric translation (PZT) stage. (4) PZT controller. (5) $100\times$ microscope objective. (6) Dichroic mirror. (7) TRITC/FITC filter set. (8) Arc lamp. (9) Condenser. (10) Steering mirror. (11) Focusing optics. (12) Quadrant photodetector. (13) Analog to digital converter. (14) CCD camera. (15) Computer.

The trapping laser beam (1064 nm) can give rise to two-photon absorption phenomena in red fluorescent beads over all trapping laser powers. To investigate the effects of two-photon absorption on the measured stiffness, we repeated the trap stiffness calibrations using a control bead that had material properties and drag coefficient similar to those of the red fluorescent beads, but without significant two-photon absorption over the trapping laser powers. Specifically, we used $4.18\pm 0.4\text{-}\mu \mathrm{m}$-diameter dragon green fluorescent beads (Bangs Laboratories Inc., Fishers, IN) with peak absorption and emission at 480 and 520 nm, respectively. We repeated the trap stiffness calibrations using the viscous drag force method over the low (50 and 136 mW), medium (674 mW), and high (1.38 W) laser powers (delivered at the specimen plane) in samples containing both $4\mu \mathrm{m}$ beads, with results presented in Sec. 3.2.1. We used a dichroic beam splitter with $>90\%$ transmittance between 470 and 650 nm, and $>98\%$ reflectance at 1064 nm (680dcspxr-laser, Chroma Technology Corp, Bellow Falls, VT) to stir the laser beam toward the objective. The optical setup allowed for simultaneous trapping and fluorescent imaging of the trapped beads.

The transmissions of the microscope objective at 1050 and 1100 nm are reported by the manufacturer as 60% and 57%, respectively. We assumed a 60% transmission for the objective lens at 1064 nm, which is in agreement with other reports^77–79 that measured the transmittance of high N.A. microscope objectives of the same brand and similar optical properties at 1064 nm using the dual objective method.^79–81 We increased the laser output power in eight steps from 85 mW to a maximum power of 2.294 W, measured at the back pupil of the objective. The calculated trapping laser powers delivered to the specimen plane ranged between 50 mW and 1.38 W.

We measured the temperature of the DI water using a thermistor connected to a Vernier LabQuest device (Vernier Software & Technology, Beaverton, OR). The $\text{mean}\pm \text{standard deviation}$ value of water temperature was $24.98\pm 0.23°\mathrm{C}$ (water dynamic viscosity $\eta =0.8925·{10}^{-3}\mathrm{Pa}·\mathrm{s}$ at $T=24.98°\mathrm{C}$). The laser-induced temperature increase, which occurs within the laser focus, has a significant effect on the measured trap stiffness through its effect on the viscosity of the medium and the thermal motion of the trapped particle.^78,82 This effect becomes more pronounced at higher trapping laser powers and higher trapping depths when using an oil immersion microscope objective.⁸³ We accounted for these effects by assuming a temperature increase of $8\mathrm{K}/\mathrm{W}$ at 1064 nm in DI water as our optical trapping medium.⁷⁸ Therefore, the expected range of the DI water temperature in our experiments during the applied range of trapping powers was $\approx 24.98$ to 35.99°C. Furthermore, all experiments were conducted within 30 s of the initial trapping of the bead to avoid or minimize excessive temperature buildup at laser focus.

2.1.2.

Fluorescence and bright field imaging and position detection

Both white light and fluorescence modes can be used for particle tracking using the QPD in the imaging configuration. Under white light, the contrast between the shadow of the optically trapped particle and the bright background is used for particle tracking. In fluorescence, the emission from a fluorescent probe against the dark background is used to track the optically trapped fluorescent particle. For bright field imaging of the particles, a 100 W tungsten lamp (TI-DH Diascopic Illumination Pillar, Nikon) was used to illuminate the samples. For fluorescence imaging of the particles, an arc lamp (LUDL Electronic products, Hawthorne, NY) was used to optically excite the fluorescent beads.

We used a TRITC-B-NTE filter set (Semrock, Rochester, NY) to optically excite the red fluorescent beads in the 530 to 550 nm band and collect the fluorescence emission in the 570 to 620 nm range. A FITC-3540B-NTE filter set (Semrock), which allows the transmission of an excitation band in the range of 446 to 500 nm and an emission band of 513 to 725 nm, was used during the optical excitation of the green fluorescent beads and collection of fluorescence emission. For comparative purposes, we employed two commonly used position sensing methods: first, a nonvideo position detection using a QPD (QP50-6SD, Pacific Silicon Sensor, Westlake Village, CA) and second, video particle tracking using a CCD camera (C9100-13, Hamamatsu Corp., Bridgewater, NJ).

Using the QPD allowed for high-bandwidth position recordings with sampling frequencies in the kilohertz range. The QPD was aligned such that the image positions along the primary $x$, $y$, and $z$ axes of the lab were, respectively, proportional to differential voltages $V_x=(V_1+V_4)-(V_2+V_3)$, $V_y=(V_3+V_4)-(V_1+V_2)$, and the sum voltage $V_{\mathrm{S}}=V_1+V_2+V_3+V_4$ (where $V_{1-4}$ are diode voltages). Differential voltages were digitized by an analog-to-digital converter (BNC-2110 and NI PCI-6133, National Instrument, Austin, TX) and stored in a computer using LabVIEW software (version 8.2, National Instruments) for further analysis. Additionally, bead positions were determined using a CCD camera. For this purpose, images of the fluorescent bead were acquired at 179 frames per second (fps) by HCImage software (version 2.0, Hamamatsu) and analyzed in MATLAB® (The MathWorks, Natick, MA) for particle tracking.

2.2.1.

QPD-based position sensing

To accurately control the position of the optical trapping chamber and apply viscous drag forces of known magnitude on trapped particles, we used a piezoelectric translation (PZT) stage (P-527.C3, Physik Instrumente, Waldbronn, Germany) with a transverse resolution of $<10\mathrm{nm}$. Output voltage of the QPD was calibrated against displacement using the PZT as follows. The position of a trapped bead in the field of view was initially taken as the reference point to center the QPD. Subsequently, in the absence of the optical trap, a bead was attached to the cover slip and brought to the exact transverse position of the trapped bead using the PZT. The fluorescent image of the bead (or the shadow of the bead when white light was used for particle tracking) was centered on the QPD by moving the QPD using a three-dimensional translational stage.

Next, the bead was displaced multiple times in steps of known increments (0.01 to $1\mu \mathrm{m}$) using the PZT while QPD’s differential voltages were recorded. Recording of larger bead displacements (up to $1\mu \mathrm{m}$) in $x$ and $y$ directions showed that the QPD response within this range was linear. Finally, a linear fit to the displacement-voltage data points provided a voltage-displacement calibration equation for the QPD. The differential voltages $V_x$ and $V_y$ were normalized by $V_{\mathrm{S}}$ to create normalized output-displacement calibration graphs in $x$ and $y$ directions. The slope of the linear fit to the voltage-displacement graphs was used to convert QPD outputs to bead displacement during stiffness calibrations.

The two-photon fluorescence of the red fluorescent beads increases with the trapping laser power. Moreover, it has been shown that the axial position of the laser beam changes with the laser power as a result of laser-induced thermal expansions in the microscope objective,⁸⁴ affecting the position calibration. For the case of particles with strong two-photon absorption/emission, the laser power related differences in the displacement calibration coefficients become more prominent under white light and at higher trapping laser powers as the image of the trapped particle tends to fade away with the increased fluorescence. Therefore, it is important to obtain a separate displacement-calibration curve for the QPD at each trapping laser power.

The choice between using white light and fluorescence for particle tracking is based on the specific conditions of the intended experiments. Moreover, the quality and maximum resolution of the particle tracking scheme is related to the positional SNR, which in turn depends on the characteristics of optically trapped probe and the detector’s spectral sensitivity. For example, our group has used 800 and 1064 nm optical tweezer setups to conduct biophysical studies of cell membranes by extracting membrane tethers from cells using optically trapped particles.^28–30 The shadow cast by the membrane tether introduces large errors in position sensing under white light. To overcome this obstacle while maintaining the SNR, we use fluorescent probes with an emission band in a region where our QPD has a high spectral sensitivity ($\sim 600\mathrm{nm}$). In fact, the position tracking SNR for the red fluorescent beads increases by $\sim 8\%$ in our system when switching from the white light mode to the fluorescence. However, the emission band of the dragon green fluorescent particles used in Sec. 3.2.1 coincides with an area of poor spectral sensitivity of our QPD, resulting in low SNR for tracking these particles in fluorescence mode. Therefore, the QPD position detection in these experiments was conducted in bright field.

2.2.2.

Video-based position sensing

Images acquired by the CCD camera were spatially calibrated by imaging a microruler with $10\mu \mathrm{m}$ spacing. Although the pixel resolution of our CCD imaging system is diffraction limited, there are a number of algorithms for spherical objects^85,86 that can estimate the centroid of the particle to subpixel accuracy, thereby increasing the spatial resolutions of video particle tracking to nanometer range.

We used the radial symmetry method developed by Parthasarathy⁸⁶ to track the position of the optically trapped particles to subpixel resolution. The results of the video-based position sensing are ultimately dependent on the accuracy and pixel noise levels associated with the particle tracking algorithm used. To demonstrate this matter, we adopted the interactive data language algorithm for the centroid method developed by Crocker and Grier⁸⁷ for MATLAB® to track the position of the beads in the same sets of CCD images used in the radial symmetry method. An example of the stiffness values calculated using the two particle tracking algorithms is provided in the Results section. Since the image exiting the microscope objective was slightly diverging, the pixel calibration was sensitive to the position of microscope objective. Therefore, we performed a calibration against an in-focus microruler by averaging the pixel size while the height of the stage (and, therefore, that of the objective) was changed in a range of $30\mu \mathrm{m}$ (covering the actual positions of microscope objective during stiffness calibrations).

2.3.1.

Active stiffness calibration based on viscous drag force method

Forces applied on a microsphere in the proximity of the trapping chamber’s bottom can be calculated using Eq. (1). Holding the trapped bead at a constant height above the cover slip, we moved the PZT stage at controlled velocities to induce known viscous forces on the optically trapped microsphere based on Eq. (1). Displacement of the bead from the center of the trap was recorded as a differential voltage by the QPD. These voltages were converted to displacements using the voltage-displacement equations. The resulting data for displacement-force graph were linearly fitted for small displacements from the center of the trap ($<1\mu \mathrm{m}$) using $f=-k·x$, where the stiffness of the trap ($k$) is extracted from the slope of the fit.

2.3.2.

Passive stiffness calibration method based on power spectral density method

The Brownian motion of a particle in an optical trap can be described by the Langevin equation. For particles in fluids with low Reynolds numbers, the power spectrum of the Brownian motion is a Lorentzian function.

The position sensing method used in this approach should be able to acquire data at frequencies that are considerably higher than the optical trap’s roll-off frequency. Position samplings at 10 to 50 kHz are commonly reported in the literature. Although high-speed cameras along with algorithms that account for the aliasing and blur artifacts intrinsic to camera position sensing have been implemented in studying the PSD of an optical trap,^45,88,89 the use of such apparatus in the absence of multiple or drifting optical traps is not common. Instead, high-bandwidth nonvideo position sensing is commonly employed for stiffness calibration using PSD at a higher accuracy and a lower equipment cost. Our video position sensing system cannot be used to study the full spectrum of the PSD because its acquisition frequency (179 Hz) is well below the roll-off frequency of our optical trap. For stiffness calibration using the PSD method, we used the QPD position data recorded at 50 kHz. A MATLAB® application developed by Tolic-Nørrelykke et al.⁴² was used to accurately extract the roll-off frequency, with results discussed in Sec. 3.

2.3.3.

Passive stiffness calibration based on equipartition theorem method

Equipartition theorem assumes $0.5K_{B}T$ of thermal energy for each degree of freedom, where $K_B$ is the Boltzmann constant ($\approx 1.38\times {10}^{-23}\mathrm{J}{\mathrm{K}}^{-1}$) and $T$ (K) is the absolute temperature. On the other hand, energy associated with thermal fluctuations of a particle in an optical trap with stiffness $k_x$ in $x$ direction equals $0.5k_x{\langle x^2\rangle }^{-1}$, where $\langle x^2\rangle$ is the position variance of the trapped particle in the $x$ direction. By calculating the position variance of the trapped bead, we can measure the trap stiffness along the $x$ axis ($k_x$) as

Similarly, we can measure the trap stiffness in the $y$ direction ($k_y$) using the particle position variance along the $y$ axis. This method of determining the trap stiffness requires a calibrated position sensing device, but does not require calculating the drag coefficient of the trapped bead or knowing the viscosity of the trapping medium. We recorded the position of the bead in the trap using the CCD camera, with stiffness calibration results presented in Sec. 3.

2.3.4.

Passive stiffness calibration based on Boltzmann statistics method

In the classical Boltzmann statistics approach, the trapped particle’s position histogram is assumed to have a normal distribution resulting from a Gaussian trapping laser beam. In thermal equilibrium, Boltzmann statistics describes the probability density $p(x)$ of the particle position as a function of the optical trap’s potential $E(x)$.⁵⁸

The contribution of $dx$ is incorporated in $C$, the normalization factor, and represented in the second term in Eq. (6). This term is an energy offset and is neglected by assuming zero potential at the center of the optical trap. To determine the potential energy, the logarithm of the function fitted to the calculated distribution of particle positions is taken and multiplied by $-K_{B}T$. The resulting distribution is fitted by a quadratic equation $E(x)=(k_x/2)x^2+c$, where $k_x$ is the trap stiffness along the $x$ axis and $c$ is an energy offset assumed to be zero. The trap stiffness along $y$ axis is measured in the same way as that along the $x$ axis.

By assuming a normal distribution for the trapped particle’s position histogram and fitting a parabola to the logarithm of the normal distribution, the results calculated by the Boltzmann statistics method become identical to those by the equipartition theorem. In theory, this approach can enhance the reconstruction of the optical trap potential by relying on the wings of the position histogram. However, this advantage is absent in practice because of the relatively small counts in the wings of the histogram (especially at higher trapping laser powers), rendering them as sources of additional uncertainty. Moreover, the spatial mode of the trapping laser may be perturbed by other modes at high laser output powers, resulting in a non-Gaussian energy distribution across the diameter of the optical trap.

Therefore, we modified the Boltzmann statistics approach by fitting the trapped particle’s position histogram using a kernel function with a smoothing bandwidth of $\sim 1\mathrm{nm}$ instead of a Gaussian function. We then numerically took the logarithm of the function fitted to the calculated distribution of particle positions and multiplied it by $-K_{B}T$. Instead of fitting the resulting potential distribution function with a parabola, we numerically took the position derivative of the energy function. The trap stiffness was measured from the slope of a linear fit to the resulting displacement-force distribution over a region equivalent to one standard deviation away from the center of the trap where a linear distribution was generally observed. We present the stiffness calibration results of the modified Boltzmann statistics approach in Sec. 3. Calibrating an optical trap using the Boltzmann method does not require knowing the trapped particle’s shape, drag coefficient, or medium’s viscosity. We applied the Boltzmann statistics calibration method to the same position data sets recorded for the equipartition theorem calibration method using the CCD camera.

3.1.1.

QPD-based position sensing

Bead movements with steps as small as 10 nm were resolvable by the QPD. Voltage-displacement curves showed a linear QPD response for bead displacements $<1\mu \mathrm{m}$ from the center of the optical trap.

Normalizing the differential voltage of the $x$ and $y$ channels by $V_{\mathrm{S}}$ did not improve the linear fits to the voltage-displacement data, mainly due to the absence of large axial displacements. While normalization decreases the susceptibility of linear fits to ambient optical noise, it also makes the slope of the fit dependent on background brightness. Therefore, in controlled experimental environments like ours with negligible changes in the background signal and the trapped particle’s axial position, using the single differential voltages $V_x$ and $V_y$ (as opposed to normalized QPD outputs) is preferred.

The QPD displacement calibration coefficients for the 4 and $8\mu \mathrm{m}$ green fluorescent beads, respectively, changed by 8% and 4% under the fluorescence mode over the range of trapping laser powers in this study. These coefficient changes increase to 19% and 6% for the 4 and $8\mu \mathrm{m}$ green fluorescent beads under the white light, respectively. For the case of the $4\mu \mathrm{m}$ red fluorescent beads, the QPD displacement calibration coefficients vary over the trapping laser powers by 22% in fluorescence mode and 37% in the bright field mode. While we did not use the normalized QPD outputs, it should be noted that in all cases, normalization of the differential QPD voltages by $V_{\mathrm{S}}$ decreased the dependency of the displacement calibration coefficient on the trapping laser power in an arbitrary manner but did not eliminate it.

3.1.2.

Video-based position sensing

Pixel calibration resulted in $158\mathrm{nm}/\text{pixel}$ in both $x$ and $y$ directions. The bead’s centroid position was estimated with subpixel accuracy using the particle tracking algorithms. Data recorded from a stationary $4\mu \mathrm{m}$ bead attached to a cover slip showed a displacement noise of $\sim 0.03\text{pixels}$ for the radial symmetry and 0.06 pixels for the centroid methods corresponding to 4.7 and 9.5 nm, respectively. This value was measured as 0.042 pixels for the $8\mu \mathrm{m}$ beads using the radial symmetry method, corresponding to 6.6 nm.

3.2.1.

Active stiffness calibration method based on viscous drag force

In Fig. 2, we present the optical trap stiffness for the 4 and $8\mu \mathrm{m}$ beads as a function of laser power delivered to the specimen plane using the active stiffness calibration method based on application of viscous drag force. Each point in the graph is extracted from the slope of a linear fit to the displacement-force graph obtained at the corresponding laser power (Fig. 2, inset). The stiffness values of our optical trap for the $4\mu \mathrm{m}$ beads ranged from $63.7\pm 2.9\mathrm{pN}·\mu {\mathrm{m}}^{-1}$ at 50 mW to $771.1\pm 69.9\mathrm{pN}·\mu {\mathrm{m}}^{-1}$ at 1.38 W, while the stiffness values for the $8\mu \mathrm{m}$ beads ranged from $51.6\pm 8.5\mathrm{pN}·\mu {\mathrm{m}}^{-1}$ at 50 mW to $733.8\pm 78.8\mathrm{pN}·\mu {\mathrm{m}}^{-1}$ at 1.38 W. As expected, a linear increase in optical trap stiffness was observed for both beads as the laser power increased. If not accounted for, the laser-induced temperature increase results in an overestimation of optical trap stiffness, which progressively increases with trapping power.

Fig. 2

Trap stiffness as a function of laser power at the specimen plane, using the active calibration method based on application of viscous drag force. The $k_{Ø4(\mathrm{Visc.})}$ data points (represented as larger filled circles) and $k_{Ø8(\mathrm{Visc.})}$ data points (represented as smaller unfilled circles) correspond to the estimated stiffness values of $4\mu \mathrm{m}$ red and $8\mu \mathrm{m}$ green fluorescent beads, respectively. The inset shows a typical displacement-force graph for the $4\mu \mathrm{m}$ red beads at 0.05 W. The $Ø4$ data points (shown as unfilled triangles) represent bead displacements from the center of the trap resulting from calculated viscous drag forces. The error bars to the mean estimated stiffness values using the red and green fluorescent beads are represented by thick vertical lines with short horizontal caps and thin vertical lines with long horizontal caps, respectively.

The average standard deviation from the mean stiffness values measured for the $4\mu \mathrm{m}$ bead is 7.6%. The average standard deviation increases to 12.7% for the $8\mu \mathrm{m}$ bead. These values are comparable with the uncertainties in the particle sizes (5 and 8.75% for the 4 and $8\mu \mathrm{m}$ beads, respectively). The large Pearson’s $r$ value (0.995) of the linear fit to the displacement-force graphs show that the optical trap behaves as a linear spring for bead displacements $<1\mu \mathrm{m}$ (Fig. 2, inset). Thus, optical trap stiffness calibration based on viscous drag force does not suffer the theoretical or numerical approximations in modeling the optical trap behavior associated with the other calibration approaches used in this study.

The viscous drag force calibration method reports stiffness values for the $8\mu \mathrm{m}$ beads that are, on average, 23% less than those of the $4\mu \mathrm{m}$ beads across the laser powers used in this study. The stiffness values are directly extracted from the displacement-force graphs involving controlled external forces and report the smallest standard deviations compared with the other calibration methods used in this study. Therefore, we regard the trap stiffness calculated using the viscous drag force method as the most accurate and reproducible optical trap stiffness values in our experiments.

We repeated the trap stiffness calibrations in samples consisting of the $4.2\pm 0.21\mu \mathrm{m}$ red fluorescent beads and the $4.18\pm 0.4\mu \mathrm{m}$ dragon green fluorescent beads to investigate the effect of two-photon absorption on the measured stiffness. Figure 3 shows the results of the trap stiffness calibrations using the viscous drag force method over the low, mid, and high laser powers for both beads. The $k_{Ø4R(\mathrm{Visc.})}$ and $k_{Ø4G(\mathrm{Visc.})}$ data points correspond to the estimated stiffness values of $4\mu \mathrm{m}$ red and $4\mu \mathrm{m}$ green beads under white light, respectively. The difference between the mean stiffness values calculated using the two beads at each trapping laser power was smaller than the sum of standard deviations to the means and ranged from 10.6% at 50 mW to $<0.01\%$ at 1.38 W. Thus, the effects of two-photon absorption on the optical forces and the measured stiffness using particles in this volume-scale are negligible.

Fig. 3

Trap stiffness as a function of laser power at the specimen plane, using the active calibration method based on application of viscous drag force. The $k_{Ø4R(\mathrm{Visc.})}$ data points (represented as asterisks) and $k_{Ø4G(\mathrm{Visc.})}$ data points (represented as circles) correspond to the estimated stiffness values of $4\mu \mathrm{m}$ red and $4\mu \mathrm{m}$ dragon green beads under white light, respectively. The error bars to the mean estimated stiffness values using the red and dragon green fluorescent beads are represented by thin vertical lines with short horizontal caps and thick vertical lines with long horizontal caps, respectively.

3.2.2.

Passive static calibration based on power spectral density method

To find the characteristic roll-off frequency ($f_0$) of the optical trap, we recorded the position of 10 sequentially trapped beads of either diameter at various laser powers at a sampling rate of 50 kHz. We analyzed the PSD of bead displacements using an application developed by Tolic-Nørrelykke et al.⁴² Data sets recorded for 10 s, and cropped at 5 and 2 s yielded the same roll-off frequencies. The roll-off frequencies for the experiments using the $4\mu \mathrm{m}$ beads were measured in the range of $255.6\pm 66\mathrm{Hz}$ at a trapping laser power of 50 mW to $1422\pm 256\mathrm{Hz}$ at 504 mW. The roll-off frequencies using the $8\mu \mathrm{m}$ beads were measured to range from $195\pm 30.3\mathrm{Hz}$ at 50 mW to $830.1\pm 64.9\mathrm{Hz}$ at 504 mW. Taking the effects of laser-induced temperature increase into consideration, these roll-off frequencies correspond to stiffness values in the range of $60.2\pm 16\mathrm{pN}·\mu {\mathrm{m}}^{-1}$ at 50 mW to $276.6\pm 39.1\mathrm{pN}·\mu {\mathrm{m}}^{-1}$ at 504 mW for the $4\mu \mathrm{m}$ beads, and $40.3\pm 7.9\mathrm{pN}·\mu {\mathrm{m}}^{-1}$ at 50 mW to $161.5\pm 11\mathrm{pN}·\mu {\mathrm{m}}^{-1}$ at 504 mW for the $8\mu \mathrm{m}$ beads.

Figure 4 shows the optical trap stiffness calculated for the 4 and $8\mu \mathrm{m}$ particles using the PSD method. The correlation between the trap stiffness and laser powers is linear up to the trapping laser power of 504 mW. However, at higher trapping powers, the uncertainty in the estimated stiffness values by the PSD calibration method increases, with reduced sensitivity of the stiffness estimates to power increments and deviation from a linear trend. This observation is in agreement with the previous reports that employed this stiffness calibration method.⁷⁸ The maximum laser trapping power range at which the PSD calibration method could be used to estimate reproducible stiffness calibration results was higher than the other two passive calibration methods as shown in the next section, but lower than the active stiffness calibration method. Within this range, the PSD method reports stiffness values for the $8\mu \mathrm{m}$ bead that are, on average, 42% smaller than those reported for the $4\mu \mathrm{m}$ bead. The average standard deviations from the mean stiffness values calculated for the 4 and $8\mu \mathrm{m}$ beads using the PSD method were 18 and 12%, respectively.

Fig. 4

Trap stiffness as a function of laser power at the specimen plane, using the passive stiffness calibration method based on the power spectral density. The $k_{Ø4(\mathrm{PSD})}$ data points (represented as larger filled squares) and $k_{Ø8(\mathrm{PSD})}$ data points (represented as smaller unfilled squares) correspond to the estimated stiffness values of $4\mu \mathrm{m}$ red and $8\mu \mathrm{m}$ green beads, respectively. The standard deviation from the mean measured stiffness values using the red and green fluorescent beads at each trapping laser power are represented by thick vertical lines with short horizontal caps and thin vertical lines with long horizontal caps, respectively.

3.2.3.

Passive stiffness calibration based on equipartition theorem

Using the variance of bead displacements as observed by the camera, the optical trap stiffness values for the $4\mu \mathrm{m}$ beads were estimated in the range of $36.8\pm 7.5\mathrm{pN}·\mu {\mathrm{m}}^{-1}$ at 50 mW of trapping laser power in the focal plane to $192\pm 59.6\mathrm{pN}·\mu {\mathrm{m}}^{-1}$ at 355 mW. The stiffness values for the $8\mu \mathrm{m}$ beads were estimated in the range of $38.9\pm 18\mathrm{pN}·\mu {\mathrm{m}}^{-1}$ at 50 mW to $196.5\pm 64.6\mathrm{pN}·\mu {\mathrm{m}}^{-1}$ at 355 mW [Fig. 5(a)]. The trap stiffness calculated by the equipartition theorem is in good agreement with the stiffness measured using the other methods at lower trapping laser powers. While video particle tracking at 179 fps was able to measure the optical trap stiffness using the equipartition theorem at trapping laser powers of up to 355 mW and observe a linear increase in the measured trap stiffness versus laser power, the method lost sensitivity to the increments in the trapping laser power beyond 355 mW, and no consistent results were reported at higher laser powers [Fig. 5(b)].

Fig. 5

Trap stiffness as a function of laser power at the specimen plane for 4 and $8\mu \mathrm{m}$ beads, using passive stiffness calibration methods based on the Boltzmann statistics and the equipartition theorem. The $k_{Ø4(m.\mathrm{Bolt.})}$ data points (represented as circles) and $k_{Ø4(\mathrm{Equi.})}$ data points (represented as triangles) correspond to the estimated stiffness values for the $4\mu \mathrm{m}$ beads using the modified Boltzmann statistics and the equipartition theorem, respectively. The $k_{Ø8(m.\mathrm{Bolt.})}$ data points (represented as unfilled columns) and $k_{Ø8(\mathrm{Equi.})}$ data points (represented as shaded columns) correspond to the estimated stiffness values for the $8\mu \mathrm{m}$ beads using the modified Boltzmann statistics and the equipartition theorem methods, respectively. Particles in the aforementioned data sets were tracked using the radial symmetry method. The $k_{Ø4*(m.\mathrm{Bolt.})}$ data points (represented as diamonds) correspond to the estimated stiffness values for the $4\mu \mathrm{m}$ beads using the same particles images and calibration methods as in $k_{Ø4(m.\mathrm{Bolt.})}$, but with the centroid method used to track the optically trapped particles. The standard deviations from the mean measured stiffness values at each trapping laser power are shown as error bars. The error bars to the estimated stiffness values $k_{Ø4(m.\mathrm{Bolt.})}$, $k_{Ø4(\mathrm{Equi.})}$, and $k_{Ø4*(m.\mathrm{Bolt.})}$ are represented by solid horizontal lines, capped vertical lines, and dashed horizontal lines, respectively. The error bars to the estimated stiffness values $k_{Ø8(m.\mathrm{Bolt.})}$ and $k_{Ø8(\mathrm{Equi.})}$ are represented by dotted horizontal lines and a dash-dot spline, respectively. (a) Stiffness values $k_{Ø4(m.\mathrm{Bolt.})}$, $k_{Ø4(\mathrm{Equi.})}$, $k_{Ø8(m.\mathrm{Bolt.})}$, and $k_{Ø8(\mathrm{Equi.})}$ at lower trapping laser powers (0.05 to 0.55 W). (b) Stiffness values $k_{Ø4(m.\mathrm{Bolt.})}$, $k_{Ø4(\mathrm{Equi.})}$, $k_{Ø8(m.\mathrm{Bolt.})}$, and $k_{Ø8(\mathrm{Equi.})}$ over an extended range of trapping laser powers (0.05 to 1.5 W). Neither of the approaches produced reliable stiffness results at trapping laser powers $>0.5\mathrm{W}$. (c) Dependence of the measured trap stiffness on the particle tracking algorithm.

Optical trap stiffness calibration methods based on measuring the Brownian displacements of an optically trapped particle are highly dependent on accurate position sensing and easily disturbed by the presence of displacement noise. The accuracy of the stiffness values reported by this statistical calibration approach can be improved at higher trapping laser powers by enhancing the position detection’s spatial and temporal resolution. Estimated optical trap stiffness by the equipartition theorem method has a direct correlation with the absolute temperature of the trapping medium [Eq. (4)]. The estimated 2.83 K temperature increase at 355 mW attributes to $<1\%$ underestimation of stiffness if the laser-induced temperature increase is neglected. The equipartition theorem calibration method reports stiffness values for the $8\mu \mathrm{m}$ bead that are, on average, 6% smaller than those reported for the $4\mu \mathrm{m}$ bead in its effective range of 50 to 355 mW. The average standard deviations from the mean stiffness values calculated using the equipartition theorem over this range are 26% and 34% for the 4 and $8\mu \mathrm{m}$ beads, respectively. Therefore, we regard the equipartition theorem as a less accurate method in calibrating optical trap stiffness as compared with the PSD and viscous drag force methods.

3.2.4.

Passive stiffness calibration based on Boltzmann statistics method

In Fig. 5, optical trap stiffness values, estimated by the modified Boltzmann statistics calibration method using the CCD camera as position sensing means, are presented as a function of trapping laser powers in the focal plane. Estimated optical trap stiffness values for the $4\mu \mathrm{m}$ beads ranged from $35\pm 6.1\mathrm{pN}·\mu {\mathrm{m}}^{-1}$ at 50 mW to $198.3\pm 60.7\mathrm{pN}·\mu {\mathrm{m}}^{-1}$ at 355 mW. The stiffness values for the $8\mu \mathrm{m}$ beads ranged from $37.1\pm 12.9\mathrm{pN}·\mu {\mathrm{m}}^{-1}$ at 50 mW to $193.8\pm 53.2\mathrm{pN}·\mu {\mathrm{m}}^{-1}$ at 355 mW. The average standard deviations from the mean stiffness values calculated using the modified Boltzmann statistics method over this range are 29% and 33% for the 4 and $8\mu \mathrm{m}$ beads, respectively. The trap stiffness measurements by the equipartition theorem and the modified Boltzmann statistics calibration methods for the 4 and $8\mu \mathrm{m}$ beads over the trapping laser power range of 50 to 504 mW are presented in Fig. 5(a) for comparison. Both calibration methods are capable of observing the size-dependent differences in optical trap stiffness within their effective range. However, the average difference of 6% between the measured stiffness for the 4 and $8\mu \mathrm{m}$ beads using the equipartition theorem increases to 13.6% when using the modified Boltzmann statistics method.

The average standard deviation from the mean calculated stiffness values does not change in a meaningful way between the two methods. This suggests that the decreased dependency of the stiffness calculations on the wings of the particle’s position distribution in the modified Boltzmann statistics method does not introduce significant uncertainties in the calculated results. In turn, the discrepancies between the two methods become apparent as the trapping laser powers increase, resulting in fewer data points in the far bins of the trapped particles’ position histogram. The classical Boltzmann statistics approach becomes limited in performance at 355 mW, while the modified Boltzmann statistics method allows for the measurements to be extended into a higher laser power of 504 mW for the $4\mu \mathrm{m}$ beads, measuring a stiffness of $245.7\mathrm{pN}·\mu {\mathrm{m}}^{-1}$.

While the modified Boltzmann statistics also shows some improvement over the equipartition theorem in calculating the mean stiffness value for the larger particle at 504 mW, its performance could not be reliably extended into higher powers for the $8\mu \mathrm{m}$ bead mainly because of the higher pixel noise in tracking the larger particles. The performance of any calibration method relying on video-based position sensing method depends on the accuracy of the subpixel particle tracking algorithm used. Figure 5(c) provides an example of how the modified Boltzmann statistics approach is limited in performance using the same bead image sets as tracked by the centroid particle tracking algorithm, a tracking algorithm that provides less accuracy and larger pixel noise for subpixel particle tracking.⁸⁶

The optical trap stiffness values estimated using the Boltzmann statistics method is directly related to the absolute temperature of the trapping medium [Eq. (6)]. Similar to the equipartition theorem method, in our system, the trap stiffness is underestimated by 1.4% at the trapping power of 504 mW using the Boltzmann statistics method if laser-induced temperature increase at the focus is not accounted for.