Calibrating oscillation response of a piezo-stage using optical tweezers

Jin-Hua Zhou; Di Li; Xin-Yao Hu; Min-Cheng Zhong; Zi-Qiang Wang; Lei Gong; Wei-Wei Liu; Yin-Mei Li

doi:10.1364/OE.23.024108

1. Introduction

Piezo-stages (PZTs) are capable to precisely position samples to target locations in nanometer level. They are widely used in atomic force microscope (AFM), scanning tunneling microscope and optical tweezers. In many applications, the precision of subsequent measurements relying on PZT movement are deeply affected by the feedback of piezoelectric actuators. The hysteresis of the actuator will degenerate the response of movement amplitude [1]. Therefore, accurate modeling of the friction between the stage and motor is important to optimize the stage [2].Due to nonlinear relations between the input and the output, a feedback controller with a feed forward compensator can improve tracking performance [3]. However, the timescale of feedback loop limits the response time of the stage. Generally, the commercial stage should be calibrated precisely in advance. Furthermore, the actual movements of sample are related not only with intrinsic characteristics of PZT but also with many other factors, such as mechanical coupling between stage and chamber.

In optical tweezers, the PZT was widely used for positioning, calibration [4, 5 ], force clamp [6, 7 ], stretching cell membrane [8] and DNA [9]. The movement response of a trapped bead in an oscillating flow induced by PZT can be used to measure photodiode’s sensitivity, bead’s size or viscosity of solution [10–13 ].Since PZT is versatile in optical tweezers, the positioning accuracy of PZT and its dynamic response are very important for accurate calibration and precisely quantifying force. Before measurements, the PZT can be calibrated by image analysis of grating’s movement [14]. However, the actual response of the oscillating flow will be affected by dynamics of PZT, coupling of the stage to transfer a chamber motion and laminar flow in a chamber. Therefore, how to quantify those response in situ is vital for precise calibrations in optical tweezers.

Although the bead adhered to coverslip can be used to track the PZT movement, tracking the time lapse images is hardly used at vibrational frequency >10 Hz since a normal camera has low frame rate. Furthermore, the coupling transfer motion for trapped bead in solution is most important for calibrations of viscosity, temperature and sensitivity of detector. When a chamber is driven by a PZT, a trapped bead has subtle responses of a fluid with varied flow velocities. Then, the trapped bead can be regarded as a sensor to detect actual amplitudes of this transfer motion. In this manuscript, we present that power spectral density can be used to calibrate the in situ oscillation response of PZT in optical tweezers. In Section 2, we will briefly introduce how to calibrate actual amplitude of fluid in a chamber and our optical tweezers. In Section 3, we will demonstrate the validity of calibration methods and what factors dominate the actual amplitude. The last section will give our conclusions about these oscillation responses.

2. Experimental methods and setup

2.1 Sinusoidal oscillations for calibrating coupling ratio

When a PZT drives the chamber with a sinusoidal movement of $A (t) = A_{P Z T} \sin (2 π f_{d} t)$ at a given nominal amplitude A_PZT and frequency f_d, the fluid in this channel will move with $V (t) = 2 π f_{d} α A_{P Z T} \cos (2 π f_{d} t)$ , where α is a coupling ratio of actual amplitude of the fluid to nominal amplitude of PZT. If the PZT is accurate and full coupling motion, α = 1. For a trapped bead in a stationary optical trap, its response movements can be expressed as [15]

S (t) = A_{b e a d} \cos (2 π f_{d} t + φ),

where A_bead is the oscillation amplitude of bead and φ is a lag phase of bead’s response to fluid. The drag force

3 π η d V (t)

induced by the fluid will keep balance with optical force κS(t), where η is dynamic viscosity, d is bead’s diameter, and κ is trap stiffness. If the bead is small, the lag phase can be omitted in an overdamped motion, and the coupling ratio can be approximately expressed as

α = κ A_{b e a d} / (12 π^{2} η d f_{d} A_{P Z T}) .

This method is a simple way to calibrate the coupling ratios for small bead with calibrated stiffness when the hydrodynamic correction dependent on oscillation frequency is ignored.

In contrast, the power spectral density with hydrodynamic correction [10, 11, 16 ] is another way to precisely calibrate the coupling ratio. For a trapped bead at a depth h from coverslip, the power spectral density of its displacements can be expressed as $P (f, k_{h}) = P_{t h e r m} (f, k_{h}) + P_{r e s p o n s e} (f, k_{h})$ , where the ratio is $k_{h} = d / h$ . In one-sided power spectral density [10, 16 ], the component of thermal motion is

P_{t h e r m} (f, k_{h}) = \frac{D_{0} Re (γ (f, k_{h}) / γ_{0})}{π^{2} {{[f_{c, 0} + f Im (γ (f, k_{h}) / γ_{0}) - f^{2} / f_{m, 0}]}^{2} + {[f Re (γ (f, k_{h}) / γ_{0})]}^{2}}},

and the component of response to fluid flow is

P_{r e s p o n s e} (f, k_{h}) = \frac{{(A f_{d} | γ (f, k_{h}) / γ_{0} |)}^{2} δ (f - f_{d})}{2 {{[f_{c, 0} + f Im (γ (f, k_{h}) / γ_{0}) - f^{2} / f_{m, 0}]}^{2} + {[f Re (γ (f, k_{h}) / γ_{0})]}^{2}}},

where

D_{0} = k_{B} T / γ_{0}

, k_B is the Boltzmann constant,

γ_{0} = 3 π d η

at a temperature T,

f_{c, 0} = κ / (2 π γ_{0})

,

f_{m, 0} = γ_{0} / (2 π m)

, and m is bead’s mass. The friction coefficient ratio

γ (f, k_{h}) / γ_{0}

in frequency domain includes a real part

Re (γ (f, k_{h}) / γ_{0}) = 1 + \sqrt{\frac{f}{f_{v}}} - \frac{3 k_{h}}{32} + \frac{3 k_{h}}{8} \exp (- \frac{4}{k_{h}} \sqrt{\frac{f}{f_{v}}}) \cos (\frac{4}{k_{h}} \sqrt{\frac{f}{f_{v}}})

and an imaginary part

Im (γ (f, k_{h}) / γ_{0}) = - \sqrt{\frac{f}{f_{v}}} + \frac{3 k_{h}}{8} \exp (- \frac{4}{k_{h}} \sqrt{\frac{f}{f_{v}}}) \sin (\frac{4}{k_{h}} \sqrt{\frac{f}{f_{v}}}),

where

f_{v} = 4 η / (ρ π d^{2})

, ρ is the density of bead. Experimentally,

P (f) = β^{2} P^{D} (f)

, where

P^{D} (f)

is the power spectral density of signals from a photo-detector. According to Eq. (3), diffusion coefficient

D_{0}^{D}

in volt and corner frequency

f_{c, 0}

can be fitted. The detector sensitivity will be determined by

β = D_{0} / D_{0}^{D}

for known d, η and T. Then, the coupling ratio for oscillation amplitude

α = β \frac{\sqrt{2 w_{e x} [{(f_{c, 0} / f_{d} + Im (γ (f_{d}, k_{h}) / γ_{0}) - f_{d} / f_{m, 0})}^{2} + {Re}^{2} (γ (f_{d}, k_{h}) / γ_{0})]}}{A_{P Z T} | γ (f_{d}, k_{h}) / γ_{0} |},

where the spike height

w_{e x} = [P^{D} (f_{d}) - P_{t h e r m}^{D} (f_{d})] Δ f

at the specific oscillative frequency f_d.

2.2 Setup for experiments

Our optical tweezers [8] were built based on an inverted microscope (IX71, Olympus, Japan), as shown in Fig. 1 . The trapping laser is a fiber laser at wavelength 1064nm with maximum output 10W (AFL-1064-40-R-CL, Amonics Limited, Hong Kong). Passing through a polarizing beamsplitter (PBS), the p-polarization beam can be regulated by an attenuator A (Glan-Taylor prism, LGP-4A10, Qufu Normal University). The attenuated beam is expanded by lenses L1 and L2, and coupled into a microscope by a dichroic mirror (DM1). After passing through lens L3 and tube lens, the beam size is 6.5 mm in the back focal plane of an objective (UPLSAPO60XW, NA 1.2, water immersion, Olympus). The highly focused beam can trap particles in three dimensions. The scattering light of a trapped bead is collected by a condenser. The beam profile of scattering light in the back focal plane of condenser is imaged by lenses L4 and L5 on a position sensing detector (PSD, DL100-7PCBA3, Pacific Silicon SensorInc., USA). The voltage signals from the PSD can be recorded by a custom-designed software acquisition software (LabView, National Instruments, USA) from an acquisition card (PCI-6251, National Instruments). The positions of trapped bead can be tracked with PSD according to an integration of position [17]. A chamber is held on a slide holder (P-545.SH3, PI, Germany) and driven by combined stages, which are composed of a three-dimensional piezo-stage (PZT,P-545.3R7,200 × 200 × 200μm, PI) with 1 nm resolution and a mechanical stage (M545, 25 × 25 mm, PI).The PZT is fixed on the mechanical stage. Then, the travel of chamber on PZT is over a large range with high resolution. The movements of confined beads can be also recorded by a CCD camera (CoolSNAP-HQ2, photometrics, USA).

Fig. 1 Schematic of optical tweezers. A, attenuator; L1- L3, lens; DM1- DM2, dichroic mirror; PSD, position sensing detector. The PSD was located in a conjugate plane of the back focal plane of condenser. Stages consist of a PZT and a mechanical stage.

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In our experiments, a flow chamber consists of a glass slide and a coverslip. There is a designed pattern parafilm between the slide and coverslip. When the parafilm was heated beyond 60 °C, it melted to stick to both the slide and coverslip. Thereafter, the parafilm was placed in room temperature, and the intermediary parafilm concreted to seal the chamber with a flow channel of ~22 mm × 5 mm × 160 μm. Our sample was a diluted suspend of polystyrene beads with a diameter of d = 2μm (1.998 µm ± 0.022 µm, 4202A, Duke Scientific). When the chamber was filled with sample, both ends of the channel were sealed with vaselinum. Our experiments were conducted at room temperature ~24°C

3. Results and discussions

For the flow channel, length of ~22 mm was along x-direction and width of ~5 mm was along y-direction.

3.1 Power spectral density for calibration

To avoid the laser-induced heating in focal region [18], the trapping beam was attenuated to 33 mW in water. Ignoring the heating, the viscosity η = 0.911 × 10⁻³ Ns/m² for water at room temperature. When PZT drove the chamber with 10-Hz sinusoidal oscillations with A_PZT = 0.5 μm at a depth of h = 30 μm, the movements of trapped bead were recorded by PSD. In one measurement, the signals were sampled for 40 seconds at 10 kHz. The response oscillations within one second are shown in Fig. 2(a) . The 40-points smoothed data (black lines) can be fitted (red curve) with $S (t) = A_{P S D} \sin (2 π f_{d} t + φ) + B$ according to Eq. (1), where A_PSD is the amplitude of bead’s movements from PSD, B is the baseline. For this 40-seconds data, the fitted amplitude is A_PSD = 5.563 × 10⁻³ V.

Fig. 2 Oscillation response of trapped bead (a) and its power spectral density (b). The PZT oscillated at 10 Hz. The one-second original data (a, grey lines) are smoothed by 40 points (black lines) and fitted (red curve) with a sinusoidal function. The power spectral density (b) of 40-seconds data is fitted in a red curve according to Eq. (3) by blocking the 10-Hz spike.

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The corresponding power spectral density is shown as black lines in Fig. 2(b). The signals are divided into 5 groups. The 5-groups mean of power spectral density can be fitted within 3000 Hz in a red curve according to Eq. (3) by blocking the spike. In this oscillation, β is 5494.8 nm/V according to fitted $D_{0}^{D}$ , fitted f_c,0 is 136.2 Hz, and corresponding stiffness is 14.7 pN/μm. Due to k_h = 1/15, the hydrodynamic interaction between particle and coverslip can be ignored. According to Eq. (2), α = 0.833 for without hydrodynamic correction, which is the same with the coupling ratio from power spectral density with Eqs. (3-7) . Generally, the oscillation amplitude of fluid is the same as PZT. However, our results from two ways show the actual amplitude is smaller than nominal amplitude at this frequency.

3.2 Coupling ratios depending on oscillation frequencies of PZT

From previous calibrations of 10-Hz oscillations, the actual amplitude of fluid is smaller than the nominal amplitude of PZT. Are coupling ratios related with oscillation frequency or transfer motion between chamber and stage? We performed same measurements for fixed chamber and unfixed chamber. In the case of fixed chamber, the chamber was stuck to slide holder on the PZT by double-stick tape and also clamped by two standard pressers. In the case of unfixed chamber, the chamber was only placed on the slide holder without double-stick tape and clamping. In these two cases, the PZT oscillated with 0.5-μm amplitude at 2.5 Hz, 5Hz, 10 Hz, 20 Hz, 25 Hz and 50 Hz.

In the oscillating fluid, response movements of trapped bead can be determined by Eq. (1). The coupling ratio can be calibrated according to Eq. (2) without hydrodynamic correction and ignoring inertia. To obtain more precise results, coupling ratios in this section are calibrated by power spectral density according to Eq. (7) with a dynamical correction, which is required for small depths (in next Section).

When the trapped bead is at a depth of 30 μm, the coupling ratios in two cases are shown in Fig. 3 . In x-direction (Fig. 3(a)), coupling ratios (α_x) for unfixed chamber are very close to the case of fixed chamber. For oscillations at low frequency (< 10 Hz), α_x decreases slowly with frequency increasing. At 10 Hz, α_x is 0.842 ± 0.002 (mean ± std) for fixed chamber and 0.837 ± 0.012 for unfixed chamber. It is indicated that the error is up to ~16% if we take nominal amplitude as actual amplitude. When oscillation is at high frequency (>10 Hz), α_x decreases rapidly with frequency increasing. At 50 Hz, α_x is 0.534 ± 0.003 for fixed chamber and 0.508 ± 0.002 for unfixed chamber. If we ignore this coupling ratio, the actual amplitude will be overestimated up to ~2 times. This error will be transferred to subsequent calibrations based on precise amplitude of movement.

Fig. 3 Dependency of coupling ratio on oscillation frequency for a chamber fixed or unfixed on the PZT. According to Eq. (7) with hydrodynamic correction, α was calibrated in x-direction (a) and in y-direction (b).

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In y-direction (Fig. 3(b)), coupling ratios (α_y) in two cases are very close at each oscillation frequency. With frequency increasing, α_y decreases rapidly. When oscillations are at 2.5 Hz and 5 Hz, α_y> 0.95. These response amplitudes are very close to oscillation amplitudes of PZT. For oscillations at 10 Hz and 50 Hz, however, the coupling ratios in two cases are ~0.87 and ~0.31, respectively. Ignoring the coupling ratio, the actual amplitude will be overestimated up to ~3 times at this oscillation frequency.

In the cases of fixed and unfixed chambers, we also measured coupling ratios using oscillation with amplitudes of 1 μm and 1.5 μm at previous frequency. Though some oscillations have the deviations beyond the linear response region of detector, those calibrated ratios are still close to the results in the case of 0.5-μm oscillation at each frequencies. Those results show that actual amplitude depending on oscillation frequency is not related with fixed status of chamber due to enough friction to transfer motion even for unfixed chamber.

3.3 Coupling ratios versus trap depths

Since an incompressible solution was driven by chamber movement induced by PZT, the response movement for solution forms an oscillation flow relative to trapped bead. According to boundary conditions, the flow velocity response to stage’s movement depends on depth from coverslip [10].

Does this laminar flow in chamber affect the dependency of coupling ratio on frequency? Using a trapped bead with 0.5-μm amplitude of oscillations at depths of 10μm, 20μm and 30μm, the coupling ratios at previous frequencies are shown in Fig. 4 . The dependencies in x-direction (Fig. 4(a)) are similar at three depths. These results show the velocity of fluid does not depend on depths. Since those experiments in three cases were performed with different chambers, the differences are probably related with chamber sealed with vaselinum in this direction. Surprising to us, the ratios in y-direction (Fig. 4(b)) at three depths are very close at each frequency for three chambers. These results show that the actual amplitudes in y-direction are not related with chambers. In Fig. 4, the coupling ratio depending on oscillation frequency does not originate from the laminar flow at different depths.

Fig. 4 Coupling ratios at depths of 10μm, 20μm and 30μm. a, in x-direction; b, in y-direction.

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3.4 Calibrating coupling ratios with light scattering

For oscillations at high frequency, the actual amplitude is far smaller than nominal amplitude. Previous results indicate that the coupling ratio does not depend on laminar flow and transfer motion between chamber and stage is enough. Does the dependency of coupling ratio on oscillation frequency originate from our combined stages? In previous calibrations, a trapped bead was regarded as a sensor to probe the relative motion of solution. However, the deformability of chamber will distort the actual response of PZT. To remove the disturbance of deformability, here, we used two aligned beads fixed on the glass slide of chamber as sensors to calibrate coupling ratio.

With scanning two beads successively, the intensity of light scattering from beads can be recorded by PSD. To obtain small probes and align them, we diluted polystyrene beads with diameter of 500 nm (491nm ± 4 nm, 3495A, Duke Scientific) in phosphate buffered saline. Two beads were trapped to contact the slide for 1 second, then, they were adhered to slide soon. We can align two beads along x- or y-directions. When one of two beads is scanned through the focused beam, a peak of scattering intensity will appear. Figure 5 shows the scattering intensity of two beads with sinusoidal oscillations at 25 Hz in x-direction. While the slide moved at a nominal amplitude A_PZT, the actual amplitude is A, as shown in the inset figure. The oscillation can be described as $S (t) = A - A \cos (2 π f_{d} t)$ at start point in the left. Therefore, the peak 1 relative to start point appears at a time interval t ₁, which is half of time interval of two adjacent peaks from bead 1. Likewise, the interval t ₂ for bead 2 also can be determined. The distance S ₁₂ between two beads can be measured by tracking particle centroid from images. Then, the actual amplitude meets $A = S_{12} / [\cos (2 π f_{d} t_{1}) - \cos (2 π f_{d} t_{2})]$ , and the coupling ratio can determined by $α = A / A_{P Z T}$ .

Fig. 5 Light scattering of scanning two beads adhered to glass slide. While the slide moves with sinusoidal oscillations at 25 Hz motions in x-direction, peak 1 or peak 2 (indicated by arrowhead) of the scattering intensity appears at when bead 1 or bead 2 (inset figure) passes through the laser beam. A is the actual amplitude. The original data (black lines) were sampled at 50 kHz and smoothed by 50 points (red lines).

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According to particle tracking, S ₁₂ is 963 nm for aligning in x-direction and 1009 nm for aligning in y-direction, respectively. In the two cases, A_PZT = 3 μm. The coupling ratios calibrated with light scattering of scanning two aligned beads are shown in Fig. 6 . Coupling ratios in x-direction (red circles) decrease with oscillation frequency increasing. In y-direction, the ratios (blue squares) rapidly decrease with oscillation frequency increasing. In two cases at high frequencies, the actual amplitudes are much less than nominal amplitudes. At oscillation frequencies increasing, the coupling ratios in y-direction are much less than in x-direction. Since beads were fixed on the slide and the slide was also fixed on the PZT, these coupling ratios can be regarded as dynamic responses of the PZT ensemble, which includes actuator, sensors for feedback and interior mechanical coupling.

Fig. 6 Coupling ratios calibrated with optical trap and light scattering of two beads adhered to slide.

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With a comparison of the calibration with optical trap, the ratios calibrated by light scattering in x-direction are relative large at high frequencies in Fig. 6. And the differences from two method enlarge gradually with oscillation frequencies increasing. At 50 Hz, α_x is 0.63 for calibration with light scattering. But for calibration with optical trap, α_x is 0.53 at the same oscillation frequency. These results show that coupling ratios in x-direction probably are related with chamber except for intrinsic responses of PZT. In this direction, the chamber was sealed with soft vaseline to avoid solution evaporation, then, the channel at two ends has deformability for oscillations in x-direction. This deformability could induce previous differences. In y-direction, coupling ratios from two methods are very close at each frequency. In this direction, the chamber was sealed with concreted parafilm with weak deformability.

To further determine whether the vaseline sealing chamber channel affects the coupling ratios or not, we sealed a new chamber with super glue (Ausbond (China) Co. Ltd, Shenzhen) for 10 minutes. In x-direction, the coupling ratios has been calibrated by trapped bead with power spectral density of oscillation. At each oscillation frequency, the coupling ratio is very close to calibration from light scattering of two aligned beads. The maximum difference at each frequency is within 3%. In other words, the actual amplitude is related with the deformability of chamber except for response of PZT.

3.5 Discussions

In some reports [1, 3 ], the actual amplitude of PZT measured by laser interferometer was smaller than nominal amplitude due to nonlinear hysteresis. And there were different performances in x- and y-directions. In our experiments, the coupling ratio at each frequency varies little for a chamber with fixed or unfixed states, so the coupling transfer motion between chamber and PZT is enough. Even for a chamber with solid sealing, coupling ratios in x- and y-directions still depend on oscillation frequency, these dependencies are related with intrinsic characteristics of PZT, which could be related with depolarization of stage after working many years.

In previous investigations, we have employed power spectral density and light scattering to calibrate oscillation response in different cases. In most applications of optical tweezers, the actual response amplitude for trapped bead is our main concern, which can be used to accurately calibrate viscosity, temperature or sensitivity in situ. Due to probing the actual amplitude in solution, the power spectral density is the better choice for calibration.

Due to the coupling ratio in Eqs. (2) and (7) both depending on sensitivity, the accuracy of sensitivity is vital for calibration with power spectral density. To demonstrate the accuracy of this calibration, we have employed the sensitivity β with power spectral density and drag force method. In an optical trap the optical force F and deviation Δx have a linear relationship of F = -κΔx. In ray-optic model [19, 20 ], the stiffness κ can be regarded as a constant when Δx<0.3d for large particles including ellipsoid [21]. For a 2-μm trapped bead in our experiments, the deviation Δx<200 nm can be regarded in the linear response region since axial equilibrium position drifts very little with transverse deviation increasing [22]. According to Stokes’ law [12], the transverse deviation is proportional to fluid velocity induced by PZT.

When the PZT moved with three amplitudes (0.5 μm, 1μm and 1.5 μm) at 2.5 Hz, 5Hz, 10 Hz, 20 Hz, 25 Hz and 50 Hz, the oscillation amplitudes of trapped bead at depth 30μm were fitted as A_PSD in Fig. 2(a), and corresponding β was calibrated by $D_{0} / D_{0}^{D}$ in power spectral density. Using power spectral density, the deviation from trap center can be determined by βA_PSD. To avoid deviations beyond linear response for the detector, the deviations were confined within 140 nm. Figure 7 shows that βA_PSD (red circles) is proportional to A_PSD whatever in x-direction (Fig. 7(a)) or in y-direction (Fig. 7(b)). With a linear fitting (red lines), the sensitivities are 5285.7 nm/V in x-direction and 6289.5 nm/V in y-direction, respectively.

Fig. 7 Relation between actual deviations and response amplitudes from PSD for a trapped bead. a, in x-direction; b, in y-direction. Response amplitude from PSD (A_PSD) and CCD camera (A_CCD) were fitted with a sinusoidal function, and sensitivity β was calibrated by power spectral density in each measurement. βA_PSD is the corresponding deviation.

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We also measured the deviations using drag force method [12, 22 ], in which the trapped bead with 2.5-Hz oscillations was recorded with CCD camera and PSD at the same time. The position of trapped bead in an images sequences can be tracked with cross-correlation calculation in nanometer resolution [8, 23 ]. According to Eq. (1), the amplitudes detected by CCD camera and PSD can be fitted as A_CCD and A_PSD, respectively. Sensitivity β can be fitted (black lines) according to $A_{C C D} = β A_{P S D}$ , as shown with black squares in Fig. 7. With drag force method, the sensitivities in x- and y-directions are 5309 nm/V and 6390.3 nm/V, respectively. In each direction, the sensitivity from power spectral density is very close to the result from drag force method. Thus, the sensitivity from power spectral density can be regarded accurately for known bead’s size and viscosity of solution. In other words, the measurement of the coupling ratio depending on sensitivity is accurate.

In our experiments, the coupling ratios in y-direction are more stable than in x-direction due to sealing chamber with concrete parafilm. To precisely calibrate viscosity in situ, the chamber should be sealed solidly. Even so, there will be a large difference for oscillations at >10 Hz according to nominal amplitude. For our PZT with oscillations at 10 Hz, the detector sensitivity will be underestimated up to ~16% according to Eq. (12) in [10], and the measured viscosity will be overestimated up to ~42% according to Eq. (3) in [11]. Thus, the actual amplitude of movement should be calibrated at first. This is vital for calibrating sensitivity and viscosity in optical tweezers.

4. Summary

Generally, PZT is regarded as a precise stage to transfer motion. Its precise movements are the base for many calibrations in optical tweezers, AFM and many other setups. When a stage moves with sinusoidal oscillations, power spectral density of a trapped bead can be used to in situ calibrate the response amplitude. In our experiments, we found that the response amplitude is smaller than corresponding nominal amplitude in x- or y-directions. With oscillation frequency increasing, the coupling ratio decreases rapidly. For our PZT, the coupling ratio in y-direction is much smaller than that in x-directions at oscillation frequency of 50 Hz. We employed light scattering of two aligned beads to confirm that these performances can be affected by sealing states of the chamber and intrinsic characteristics of PZT ensemble. If the actual amplitude is not calibrated, there are large errors for measuring sensitivity or viscosity according to nominal amplitude of PZT. This dependency of actual amplitude on oscillation frequency is vital for calibration. In our applications, the response of PZT at a specific frequency should be calibrated at first. Meanwhile, these methods can also be employed to calibrate other kinds of stage’s oscillations.

Acknowledgments

This work is supported by the National Basic Research Program of China (Grant No. 2011CB910402) and National Natural Science Foundation of China (11374292 and 11302220, 31100555). We thank Dr. Hao-Wei Wang (University of Science and Technology of China) for reading our manuscript, and thank Dr. Lian-Sheng Zhang (Hefei University of Technology) for useful discussions about PZT.

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Calibrating oscillation response of a piezo-stage using optical tweezers

Abstract

1. Introduction

2. Experimental methods and setup

2.1 Sinusoidal oscillations for calibrating coupling ratio

2.2 Setup for experiments

3. Results and discussions

3.1 Power spectral density for calibration

3.2 Coupling ratios depending on oscillation frequencies of PZT

3.3 Coupling ratios versus trap depths

3.4 Calibrating coupling ratios with light scattering

3.5 Discussions

4. Summary

Acknowledgments

References and links

Cited By

Figures (7)

Equations (7)

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