Calibration of the optical torque wrench

doi:10.1364/OE.20.003787

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Calibration of the optical torque wrench

Francesco Pedaci, Zhuangxiong Huang, Maarten van Oene, and Nynke H. Dekker »View Author Affiliations

Optics Express, Vol. 20, Issue 4, pp. 3787-3802 (2012)
http://dx.doi.org/10.1364/OE.20.003787

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▸ Article Outline
– Abstract
1. Introduction
2. The optical torque wrench
3. Similarities and differences in the calibration of OT and OTW
4. Approaches for angular calibration
5. Discussion
– References and links

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Abstract

The optical torque wrench is a laser trapping technique that expands the capability of standard optical tweezers to torque manipulation and measurement, using the laser linear polarization to orient tailored microscopic birefringent particles. The ability to measure torque of the order of k_BT (∼4 pN nm) is especially important in the study of biophysical systems at the molecular and cellular level. Quantitative torque measurements rely on an accurate calibration of the instrument. Here we describe and implement a set of calibration approaches for the optical torque wrench, including methods that have direct analogs in linear optical tweezers as well as introducing others that are specifically developed for the angular variables. We compare the different methods, analyze their differences, and make recommendations regarding their implementations.

1. Introduction

The ability of optical tweezers (OT) to manipulate microscopic particles held in the focus of a laser has paved the way for many important lines of research in fields ranging from physics to biology. In single-molecule biological physics in particular, OT is one of the techniques that have ushered in a revolution in the way biological systems can be explored. In such studies, a microscopic particle is used as a force transducer and sensor through which the optical force can be transferred to a single biopolymer (e.g. DNA, RNA or protein). This allows one to monitor e.g. the mechanical response of a single biopolymer to linear tension (force spectroscopy), or the action of enzymes that employ the tethered molecule as substrate [1

F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics 5, 318–321 (2011). [CrossRef] [PubMed]

–3

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23, 247 (1994). [CrossRef] [PubMed]

In addition to force, torque also is a physical parameter of biological relevance, as shown by its role in diverse cellular processes such as DNA replication [4

B. E. Funnell, T. A. Baker, and A. Kornberg, “In vitro assembly of a prepriming complex at the origin of the escherichia coli chromosome,” J. Biol. Chem. 262, 10327–10334 (1987). [PubMed]

], transcription [5

L. F. Liu and J. C. Wang, “Supercoiling of the DNA template during transcription,” Proc. Natl. Acad. Sci. U.S.A. 84, 7024–7027 (1987). [CrossRef] [PubMed]

], ATP synthesis [6

M. Yoshida, E. Muneyuki, and T. Hisabori, “ATP synthase, a marvellous rotary engine of the cell,” Nat. Rev. Mol. Cell Biol. 2, 669–677 (2001). [CrossRef] [PubMed]

, 7

S. Saroussi and N. Nelson, “The little we know on the structure and machinery of V-ATPase,” J. Exp. Biol. 212, 1604–1610 (2009). [CrossRef] [PubMed]

], and bacterial propulsion [8

Y. Sowa and R. M. Berry, “Bacterial flagellar motor,” Q. Rev. Biophys. 41, 103–132 (2008). [CrossRef] [PubMed]

]. However, investigating the role of torque has proved more challenging, in part due to a lack of straightforward single-molecule manipulation techniques. Fortunately, recent efforts have focused on the development of techniques that allow the control and measurement of both force and torque on microscopic actuators [9

J. Lipfert, J. W. J. Kerssemakers, T. Jager, and N. H. Dekker, “Magnetic torque tweezers: measuring torsional stiffness in DNA and RecA-DNA filaments,” Nat. Methods 7, 977–980 (2010). [CrossRef] [PubMed]

–12

A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman, and S. X. Sun, “Magnetic tweezers measurement of single molecule torque,” Nano Lett. 9, 1720–1725 (2009). [CrossRef] [PubMed]

The optical torque wrench (OTW) is one such technique and distinguishes itself by its all-optical manipulation and detection of trapped microscopic birefringent particles. The OTW can easily control and detect linear and angular displacements of trapped particles with fast μs temporal resolution, allowing one to implement fast feedback loops to actively clamp the value of force or torque [10

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

,13

A. LaPorta and M. D. Wang, “Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92, 190801 (2004).

]. Recent studies have continued to expand our understanding of OTW physics by focusing on the system’s angular dynamics [14

J. Inman, S. Forth, and M. Wang, “Passive torque wrench and angular position detection using a single-beam optical trap,” Opt. Lett. 35, 2949–2951 (2010). [CrossRef] [PubMed]

, 15

F. Pedaci, Z. Huang, M. v. Oene, S. Barland, and N. H. Dekker, “Excitable particle in an optical torque wrench,” Nat. Phys. 7, 259–264 (2011). [CrossRef]

], but already the OTW has found applications in the study of torque-induced structural transitions of single DNA molecules [16

S. Forth, C. Deufel, M. Y. Sheinin, B. Daniels, J. P. Sethna, and M. D. Wang, “Abrupt buckling transition observed during the plectoneme formation of individual DNA molecules,” Phys. Rev. Lett. 100, 148301 (2008). [CrossRef] [PubMed]

–18

S. Forth, C. Deufel, S. S. Patel, and M. D. Wang, “Direct measurements of torque during Holliday junction migration,” Biophys. J. 101, L05–L07 (2011). [CrossRef]

]. Together, these advances demonstrate that the OTW has the potential to apply and dynamically measure torque at the scale of k_BT (∼ 4 pN nm), opening up new applications in the study of biophysical systems.

A key component of any trapping technique, linear or angular, is the ability to properly calibrate the instrument to perform absolute quantitative measurements. Diverse methods for the calibration of linear OT can be found in the literature [19

K. Visscher and S. M. Block, “Versatile optical traps with feedback control,” Method Enzymol. 298, 460–489 (1998). [CrossRef]

–23

C. Deufel and M. D. Wang, “Detection of forces and displacements along the axial direction in an optical trap,” Biophys. J. 90, 657–667 (2006). [CrossRef]

], while calibration of the OTW has been less fully explored [16

, 24

B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. LaPorta, and S. M. Block, “An optical apparatus for rotation and trapping,” Method Enzymol. 475, 377–404 (2010). [CrossRef]

]. Here we describe several different procedures to calibrate the angular trap of an OTW, detail their experimental implementations, and compare their outcomes. As we will show, several methods have their direct analogs in the calibration of linear traps, while others are more tailored to the particularities of angular variables.

This work is organized as follows: in section 2 we provide an overview of OTW theory (sec. 2.1) and describe our experimental system in detail (sec. 2.2); in section 3 we compare the calibration principles of OT to those of OTW; in section 4 we detail five methods for calibrating the OTW; and in section 5 we discuss their outcomes.

2. The optical torque wrench

In this section, we will describe the theoretical framework of the optical torque wrench and its experimental implementation, which together provide the necessary background for the calibration methods described subsequently.

2.1. Theoretical overview

In an OTW, torque is optically applied to a birefringent particle through spin momentum change in the linearly polarized laser field [10

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

, 13

A. LaPorta and M. D. Wang, “Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92, 190801 (2004).

]. Through its interaction with the birefringent material, the macroscopic polarization p⃗ induced in the medium is in general not parallel to the laser polarization E⃗ (Fig. 1a). Therefore the product τ⃗ = p⃗ × E⃗ is non-zero and a finite torque τ is applied to the particle. The net transferred torque is measured at the trap output as an imbalance between the intensities of the left- and right-circular components of the light that propagated inside the birefringent particle.

Fig. 1 Experimental configuration. a) Schematic depicting of the torque generation inside a birefringent crystal that has a larger susceptibility along its extraordinary axis χ_e than along its ordinary axis χ_o. b) SEM image of a nano-fabricated birefringent quartz cylinder used in the OTW. c) Diagram of the optical setup. OI: optical isolator, AOM: acousto-optic modulator, EOM: electro-optic modulator, NPBS: 50% non-polarizing beam splitter, λ/4: quarter wave-plate, PBS: polarizing beam splitter, PD: photo-detector, PSD: position sensitive detector, OBJ: 1.2NA microscope objective. The optical trap is surrounded by red dashed lines, the torque reference system is labeled and surrounded by grey dots, and the polarization state controlled by the EOM is indicated by red arrows inside the black dashed squares.

The torque transferred to a positive (negative) uniaxial birefringent particle is related to the angle x = θ_cyl – θ_pol between the linear laser polarization θ_pol and the extraordinary (ordinary) axis θ_cyl according to

τ = τ_{o} sin (2 x) .

(1)

where τ_o is the maximum optical torque, which depends on laser intensity, particle size, and birefringence [10

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

]. For x ≪ 1, linearization of the optical torque τ ≃ 2τ_ox can be used to define the angular stiffness κ = 2τ_o.

The over-damped angular equation of motion for the birefringent particle in an OTW can therefore be written as

- γ (\dot{x} + {\dot{θ}}_{pol}) - τ_{o} sin (2 x) + η (t) = 0

(2)

where γ is the angular drag coefficient and η(t) is the Langevin force, a Gaussian-distributed white noise term obeying 〈η(t)η(t′)〉 = 2k_BTγδ(t – t′).

The dynamics of the OTW are best understood by rewriting the equation of motion to make explicit the potential landscape experienced by the particle. When the laser polarization rotates at constant frequency ω (i.e. θ_pol = ωt), the equation of motion can be written in terms of a total potential U(x) = V(x) – Fx, where V(x) is a periodic potential tilted by an external force F, as

γ \dot{x} = - U^{'} (x) + η (t)

(3)

which is equivalent to Adler’s equation [25

R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE 34, 351–357 (1946). [CrossRef]

]. In the OTW, the periodic angular potential takes the form of

V (x) = - \frac{τ_{o}}{2} cos (2 x)

, with V (x+π) =V(x), and the tilting force is F = −γω. Hence, the result of the polarization rotation is to tilt the periodic potential.

The dynamics of the OTW can be separated into two distinct dynamic regimes, separated by a saddle-node bifurcation occurring at ω = ω_c = τ_o/γ. The first regime, in which |ω| < ω_c, is characterized by the existence of a potential barrier separating two successive stable states. At these low frequencies, the drag torque γω can be balanced by the optical torque τ(x), for a particular angle x, and the particle rotates in phase with the driving polarization. When |ω| ≲ ω_c, the presence of thermal noise can allow the system to escape the barrier between two potential minima, resulting in the appearance of regular spikes in the torque signal, a characteristic feature of the excitability of the system [15

F. Pedaci, Z. Huang, M. v. Oene, S. Barland, and N. H. Dekker, “Excitable particle in an optical torque wrench,” Nat. Phys. 7, 259–264 (2011). [CrossRef]

]. Physically, this corresponds to the extraordinary axis of the particle transiently slipping out of the polarization direction during rotation. In the second regime in which |ω| > ω_c, the potential barrier disappears, the stable and unstable points merge, and a limit cycle is created giving rise to a deterministic periodic torque signal with period given by

T_{o} = \frac{π}{\sqrt{ω^{2} - ω_{c}^{2}}} .

(4)

In presence of noise and for ω sufficiently greater than ω_c, the period of the torque signal becomes a statistical variable T_s distributed around its mean value T_o, hence 〈T_s〉 ≈ T_o.

The physics of the OTW can be experimentally verified (Fig. 2). In Fig. 2a, we demonstrate the excellent agreement between the theoretically predicted (Eq. (1)) torque τ (solid lines) and the experimentally measured values (red and blue data-points), plotted over a full period of the angle between the extraordinary axis and the laser polarization. In Fig. 2b, we plot the experimentally measured mean value of the torque transferred to the trapped birefringent cylinder (Fig. 1b) at different polarization frequencies (for experimental methods, see sec. 2.2). The transition found at |ω| = ω_c illustrates the passage between the two dynamical regimes discussed above. For |ω| < ω_c the torque is constant in time and its value increases linearly with the frequency to balance the drag torque. Beyond the critical frequency, the drag torque γω overcomes the maximum optical torque τ_o, and the axis of the particle escapes the direction of the rotating polarization, reducing the average value of τ_m.

Fig. 2 Torque on a birefringent cylinder. a) Calibrated optical torque as a function of the angle x between the polarization and the extraordinary axis for two laser intensities (blue points: 100 mW, red points: 50 mW, power measured at the objective input). The traces are reconstructed from torque traces recorded at ω > ω_c. Calibration was performed following the method described in sec. 4.2.1. The black lines are sinusoidal fits (Eq. (1)) of the experimental points. b) Mean value of the measured torque as a function of the polarization rotation frequency. Within the region |ω| < ω_c ≃ 37(×2π) Hz the torque is constant in time (τ_m = β_τγω), while for |ω| > ω_c the torque becomes periodic, with mean period given by Eq. (4). Negative frequencies indicate opposite rotation direction. Note that the cylinder used here has a larger volume and lower ω_c than the one used in the subsequent figures and table.

2.2. Experimental configuration

In the OTW, we employ cylindrically-shaped microscopic quartz (positive uniaxial) particles [26

C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods 4, 223–225 (2007). [CrossRef] [PubMed]

] fabricated by electron-beam lithography [27

Z. Huang, F. Pedaci, M. Wiggin, M. v. Oene, and N. H. Dekker, “Electron beam fabrication of micron-scale birefringent quartz particles for use in optical trapping,” ACS Nano 5, 1418–1427 (2011). [CrossRef] [PubMed]

]. The particles have a tapered profile, with the dimensions indicated in Fig. 1b, and their extraordinary axis is perpendicular to their geometrical axis. Inside the optical trap, the cylinder aligns itself parallel to the direction of laser propagation to minimize the scattering force. This, together with the three-dimensional linear trap, constrains five of the particle’s degrees of freedom. The remaining degree of freedom (rotation of the cylinder around its axis) is controlled by the linear polarization of the laser, which defines the stable orientation for the extraordinary axis at x = 0 (mod π) (Fig. 2a).

To control the orientation of the laser polarization, we use a fast electro optic modulator (EOM) with its axis oriented at 45° with respect to the laser polarization (Fig. 1c) in combination with a quarter wave-plate (aligned at 45° with respect to the EOM axis) [24

B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. LaPorta, and S. M. Block, “An optical apparatus for rotation and trapping,” Method Enzymol. 475, 377–404 (2010). [CrossRef]

]. The resulting polarization inside the optical trap is linear (ellipticity ≲ 5%), and its angle is linearly proportional to the voltage applied to the EOM. The torque transferred to the trapped particle is deduced from the difference of the intensity signals of the two circular components of the laser light at the output of the trap.

By applying a repetitive sawtooth signal to the EOM voltage (corresponding to a rotation between 0 and π), we can quasi-continuously rotate the polarization inside the trap. We observe that rotating the laser polarization in an empty trap generates a torque signal that is modulated at the frequency of the EOM voltage, instead of the expected constant (zero) value. This can be due to imperfections of the EOM such as a small misalignment of the internal crystals. We overcome this problem by recording a torque reference signal equivalent to the torque transferred in an empty trap (Fig. 1c, gray dotted box labeled Reference), and defining this as the zero-torque level. This reduces the spurious torque modulation to few percent of τ_o when a particle is trapped and rotated.

The optical trap is formed at the matched focus of a pair of identical microscope objectives (1.2 NA, water immersion). The intensity of the 1064 nm trapping laser is intentionally limited to 100 mW at the trap input. A calibrated position sensitive detector (PSD), placed in a plane conjugate to the back focal plane of the condenser, is used to monitor the displacement of the trapped particle in x,y,z. We employ an acousto-optic modulator to actively control the laser intensity, and we utilize two independent imaging systems to monitor both the laser’s transverse profile at the back focal plane of the condenser as well as the surroundings of the trapped particle inside the flow cell. An FPGA card runs the feedback loops that can be used to actively control the laser intensity and the torque amplitude, while a DAQ card acquires the PSD signals. The maximum overall temporal resolution is 5 μs. The flow cell is prepared with two glass slides (thickness 170 μm each) spaced by one parafilm layer, and buffer exchange is possible through inlet and outlet holes in the top glass slide.

3. Similarities and differences in the calibration of OT and OTW

In OT, quantitative force measurements can be accomplished by obtaining a complete set of three calibration parameters. First, the sensitivity of the detection system (in units of V/m) must be known in order to convert the measured voltage signal, proportional to the displacement of the bead inside the trap, from Volts to meters. Second, once the displacement is known in meters, the force on the trapped bead is calculated as the product between the trap stiffness (in units of pN/nm) and the displacement. From the analysis of the Brownian fluctuations of the particle inside the trap, one can extract the sensitivity and trap stiffness provided that the particle drag coefficient (in units of pN s/nm) is known. Thus in total three independent measurements are required.

The three corresponding angular parameters that need to be quantified during the calibration of an OTW are the torque sensitivity β_τ (in units of V/pN nm), the angular stiffness κ (in units of pN nm/rad), or equivalently, the maximum torque τ_o (in units of pN nm), and the angular drag coefficient γ (in units of pN nm s).

We can nonetheless underline several differences between the calibration of OT and OTW:

The quantity directly measured in an OTW is torque, from which the angle of the particle can be determined, while in OT the displacement is measured and the force is determined from it.
The sensitivity in OT varies outside the linear response region of the detector. In the OTW, by contrast, the measured torque signal τ_m (in Volts) is always linearly proportional to the optical torque τ applied to the birefringent particle via the sensitivity β_τ, according to

$τ_{m} = β_{τ} τ .$
(5)
In OT, a single stable point for the position of the particle is defined and it is common (even though not necessary [28
W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, “Passive all-optical force clamp for high-resolution laser trapping,” Phys. Rev. Lett. 95, 208102 (2005). [CrossRef] [PubMed]
]) to employ only the region of the optical potential where the stiffness is constant and the linear approximation between force and displacement holds. In the OTW, the optical potential is periodic. This makes the anharmonic region of the angular potential readily accessible, because even when the particle explores angles far from the stable solution, it never escapes the 3D trap. In the following, to characterize the sinusoidal optical torque (Fig. 2a) we will use the maximum available torque τ_o.
In both OT and OTW, when the medium viscosity is known, one has to consider how the drag coefficients depend on the particle geometry. For OT experiments, fully spherical dielectric beads with precisely known radii are readily available, hence for calibration purposes one can safely assume the theoretical value for the linear drag given by the Stokes relation (corrected for the proximity of a surface as necessary). Calibrations with other particle geometries are also possible [29
O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, and A. C. Ferrari, “FemtoNewton force sensing with optically trapped nanotubes,” Nano Lett. 8, 3211–3216 (2008). [CrossRef] [PubMed]
, 30
P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, and C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett. 11, 2375–2381 (2011). [CrossRef] [PubMed]
]. This has the advantage of reducing the number of independent calibration measurements from three to two. By contrast, for the OTW one typically nanofabricates the birefringent particles [26
C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods 4, 223–225 (2007). [CrossRef] [PubMed]
, 27
Z. Huang, F. Pedaci, M. Wiggin, M. v. Oene, and N. H. Dekker, “Electron beam fabrication of micron-scale birefringent quartz particles for use in optical trapping,” ACS Nano 5, 1418–1427 (2011). [CrossRef] [PubMed]
]. We nano-fabricated quartz cylinders with ∼ 10% volume variation [27
Z. Huang, F. Pedaci, M. Wiggin, M. v. Oene, and N. H. Dekker, “Electron beam fabrication of micron-scale birefringent quartz particles for use in optical trapping,” ACS Nano 5, 1418–1427 (2011). [CrossRef] [PubMed]
] that have a lightly tapered form; due to this variability in particle size and the fact that an exact analytical expression for the drag coefficient of tapered cylinders is lacking, we develop methods that allow us to directly measure the particle drag in question. Interestingly, our experimental results find quite good agreement between the values of the measured and the theoretical drag, provided that the latter is computed for a perfect cylinder with a volume equivalent to that of the cylinders employed in the experiments.

4. Approaches for angular calibration

We now discuss different approaches to perform the three independent measurements needed for OTW calibration. The approaches have in common that they provide independent measurements of the angular thermal fluctuations, of the angular response of the system to controlled modulations of the direction of the laser polarization, and of the characteristic time-scale of the system. The interplay between these three measured quantities then serves to fix the three calibration parameters. The main experimental control parameter is the frequency ω at which the polarization is rotated, also referred to as the polarization rotation frequency (which may be zero or even negative). For convenience, we classify the different calibration methods by the number of polarization rotation frequencies employed.

4.1. Calibration approach involving measurement over the full range of frequencies: fitting the standard deviation of the torque signal

In this first approach, we rotate the laser polarization over a wide range of frequencies to probe the system’s complete dynamic response while recording the torque transferred to the trapped cylinder. For each polarization rotation frequency ω, we determine the standard deviation δτ_m (in Volts) of the measured optical torque signal τ_m (Fig. 3, blue squares). Similar to the data shown in Fig. 2a, an abrupt transition in the particles response at ω = ω_c is evident.

Fig. 3 Calibration approach involving measurement over the full range of frequencies. We plot the standard deviation of the measured torque (in Volts) as a function of the polarization rotation frequency. The quantities needed for calibration are indicated by arrows. The red line is a fit of the data to Eqs. (6) and (7), yielding ω_c = 429 rad/s, δτ_m(0) = 5.7 mV, and

\sqrt{2} δ τ_{m} (\infty) = V_{o} = 64.9 mV

(see text). The top diagram uses blue circles to schematically indicate the polarization frequencies generated by the EOM in this method.

We can derive an analytical expression for δτ_m (Appendix I), for which the correct form depends on whether ω is above or below the critical frequency ω_c

δ τ_{m} = β_{τ} \sqrt{2 τ_{o} k_{B} T} {[1 - {(ω / ω_{c})}^{2}]}^{\frac{1}{4}} for ω < ω_{c}

(6)

= β_{τ} τ_{o} {[\frac{\sqrt{{(ω / ω_{c})}^{2} - 1}}{(ω / ω_{c}) + \sqrt{{(ω / ω_{c})}^{2} - 1}}]}^{\frac{1}{2}} for ω > ω_{c}

(7)

Fitting the experimentally determined standard deviation of the torque to Eq. (6) and Eq. (7), as shown by the red line in Fig. 3, we obtain the complete set of calibration parameters β_τ, τ_o and γ = τ_o/ω_c. Alternatively, the fit can be used to find the three quantities δτ_m(ω = 0), δτ_m(ω = ∞) and ω_c, from which the calibration parameters can be subsequently deduced according to:

τ_{o} = 4 k_{B} T δ τ_{m}^{2} (\infty) / δ τ_{m}^{2} (0)

(8)

γ = τ_{o} / ω_{c}

(9)

β_{τ} = \sqrt{2} δ τ_{m} (\infty) / τ_{o}

(10)

4.2. Calibration approaches involving separate measurements at two frequencies

In this second approach, we perform separate measurements at two different polarization rotation frequencies. Within this approach, there are two variants that we describe in turn.

4.2.1. Power spectrum analysis at ω = 0 followed by fast polarization rotation at ω > ω_c

In this method [13

A. LaPorta and M. D. Wang, “Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett. 92, 190801 (2004).

], one first considers the power spectral density of the measured torque signal at ω = 0, which is a measurement that yields two independent quantities (Fig. 4a). To see this, we note that at fixed laser polarization the power spectral density of the measured torque signal is described by a Lorentzian (provided that linearization around the particle’s stable point is possible, i.e. τ_o ≫ k_BT). This Lorentzian,

P_{o} (τ_{m}, f) = A_{o} / (f^{2} + f_{c}^{2})

, is parameterized by an amplitude

A_{o} = 4 β_{τ}^{2} τ_{o}^{2} k_{B} T / (π^{2} γ)

and a corner frequency f_c = τ_o/(πγ) = ω_c/π. Fitting the experimental spectrum to this function therefore yields two independent variables A_o and f_c (Fig. 4a top).

Fig. 4 Calibration approaches involving separate measurements at two frequencies. a) Power spectrum analysis at ω = 0 followed by fast rotation at ω > ω_c. From top to bottom: a schematic of the EOM frequencies used (blue circles), the power spectrum at ω = 0 (where the red points result from binning the experimental points (blue) into bins of variable size and the green line is a fit of the red points to a Lorentzian), and a segment of the torque trace acquired at ω/2π = 300Hz. For this dataset, the measured variables (indicated by arrows) are f_c = 152 Hz, A_o = 3.1E-3 V²Hz, and V_o = 67 mV. b) Measurement of the torque variance, period, and amplitude. From top to bottom: a schematic of the EOM frequencies used (blue circles), the probability distribution of the torque readout at ω = 0, and a segment of the torque trace acquired at ω/2π = 300Hz. For this data set, the measured variables (indicated by arrows) are δτ_m = 5.5 mV, 〈T_s〉 = 3.8 ms, and V_o = 66 mV.

To determine the third independent variable, a subsequent measurement is performed at ω > ω_c, the regime where the torque experienced by the particle is periodic as a function of time. A typical trace showing the amplitude of the torque signal V_o (defined as half of the peak-to-peak value in Volts) is shown in Fig. 4a bottom. In practice, the value of V_o can vary along the torque trace due to the small spurious modulation discussed in sec.2.2 (see also Discussion below). Therefore, to systematically determine an accurate average value of V_o from the entire recorded torque signal, we measure the standard deviation δτ_m and invert Eq. (7) to obtain V_o = β_ττ_o.

From the measurement of the three independent variables A_o, f_c, and V_o, the calibration parameters are deduced according to:

γ = 4 k_{B} T V_{o}^{2} / (π^{2} A_{o})

(11)

τ_{o} = π γ f_{c}

(12)

β_{τ} = V_{o} / τ_{o}

(13)

4.2.2. Calibration by measurement of the torque variance, period and amplitude

A second method that probes the response of the birefringent particle at two polarization rotation frequencies follows a similar approach. Again, we start by examining the regime in which the polarization direction is fixed (ω = 0), but now we measure the standard deviation of the torque signal

δ τ_{m} (ω = 0) = β_{τ} \sqrt{2 k_{B} T τ_{o}}

(Eq. (6)). This yields the first independent variable (Fig. 4b top).

In a subsequent step, we consider the regime in which the polarization rotation frequency is set to ω > ω_c. From the resulting periodic torque trace in the temporal domain (Fig. 4b bottom), we can extract the torque amplitude V_o = β_ττ_o as before. In addition we extract the mean period 〈T_s〉 of the oscillating torque experienced by the particle. Using Eq. (4), we can extract the value of ω_c = τ_o/γ from 〈T_s〉.

Hence, from the measurement of the three independent variables V_o, 〈T_s〉 and δτ_m(0), the calibration parameters are then given by

τ_{o} = 2 k_{B} T {[V_{o} / δ τ_{m} (0)]}^{2}

(14)

γ = τ_{o} {[ω^{2} - {(π / 〈 T_{s} 〉)}^{2}]}^{- \frac{1}{2}}

(15)

β_{τ} = V_{o} / τ_{o}

(16)

4.3. Calibration approaches using measurements at a single frequency

Lastly, we discuss calibration approaches that yield all three calibration parameters from a measurement performed at a single polarization rotation frequency. As we will show, the two variants presented rest on quite different theoretical approaches.

4.3.1. Sinusoidal modulation of the laser polarization direction

The first method in this category closely follows a strategy that has been used successfully for the calibration of linear traps [31

S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum. 77, 103101 (2006). [CrossRef]

]: we insert a small sinusoidal modulation of amplitude A and frequency f_mod into the voltage driving the EOM, which produces a polarization that oscillates about θ_pol at a fixed frequency: θ_pol = A sin(2πf_modt). Provided that A is sufficiently small, the power spectral density P(τ_m, f) of the measured torque τ_m experienced by the particle can be described as the sum of two components: a Lorentzian with cutoff frequency

f_{c} = \frac{τ_{o}}{π γ}

as in sec. 4.2.1 and, superimposed on this, a peak centered at f = f_mod. Mathematically, this is expressed as

P (τ_{m}, f) = 2 τ_{o}^{2} β_{τ}^{2} [\frac{2 k_{B} T}{γ π^{2} (f^{2} + f_{c}^{2})} + \frac{A^{2}}{(1 + f_{c}^{2} / f_{\mod}^{2})} δ (f - f_{\mod})]

(17)

Under these conditions, three independent variables can be measured from the experimental spectrum (Fig. 5a), to directly yield the three calibration parameters. First, a fit of the back-ground power spectral density

P_{o} = (τ_{m}, f) = A_{o} / (f^{2} + f_{c}^{2})

to a Lorentzian (performed excluding the peak at f_mod) provides the plateau value A_o and the cutoff frequency f_c as in sec. 4.2.1. Second, the peak power A_m at the modulation frequency, given by A_m = P(τ_m, f_mod) – P_o(τ_m, f_mod), is measured.

Fig. 5 Calibration approaches using measurements at a single frequency. a) Sinusoidal modulation of the laser polarization direction. Top: schematic of the EOM frequency used (the blue circle indicates the frequency of the sinusoidal modulation). Bottom: power spectrum of the measured torque signal including the contribution from the imposed modulation of the direction of the laser polarization (f_mod = 300Hz, A = 0.018 rad, Δf = 2 Hz); red points result from binning the experimental points (blue) into bins of variable size and the green line fits the red points to a Lorentzian. From the data shown, we obtain f_c = 150 Hz, A_o = 3.2E-3 V²Hz, and A_m = 2.18E-6 V²/Hz. b) Analysis of the diffusion in a tilted potential landscape. From top to bottom: schematic of the EOM frequency used (blue circle), a segment of the torque trace recorded at ω > ω_c, and a histogram of the measured torque period T_s. From these data, we obtain 〈T_s〉 = 3.8 ms, δT_S = 0.16 ms, and

δ τ_{m} = V_{o} / \sqrt{2} = 45.3

mV.

From the measurement of A_o, f_c, and A_m one then obtains the calibration parameters according to

γ = \frac{2 k_{B} T A_{m}}{π^{2} A^{2} A_{o}} (1 + f_{c}^{2} / f_{\mod}^{2}) Δ f

(18)

τ_{o} = π γ f_{c}

(19)

β_{τ} = \sqrt{\frac{A_{m} Δ f}{2 τ_{o}^{2} A^{2}} (1 + f_{c}^{2} / f_{\mod}^{2})}

(20)

where Δf = 1/t_msr and t_msr is the measurement time, chosen as a multiple of the period of the applied modulation [31

4.3.2. Analysis of the diffusion in a tilted potential landscape

A second single-frequency approach for calibration relies on detailed analysis of the diffusion of a particle in the periodic optical potential tilted by the polarization rotation (Eq. (3)).

The diffusion of a particle in a periodic potential tilted by an external force has been the object of numerous analytical and numerical studies as a result of its interesting physics and recurrence in scientific phenomena [32

P. Reimann, C. V. den Broeck, H. Linke, P. Hanggi, J. M. Rubi, and A. Pérez-Madrid, “Giant acceleration of free diffusion by use of tilted periodic potentials,” Phys. Rev. Lett. 87, 010602 (2001). [CrossRef] [PubMed]

]. In such a system, which is at steady-state but not at the thermodynamic equilibrium, the effective diffusion coefficient D_eff depends on the tilt of the potential and differs from that provided by the Einstein relation D_o = k_BT/γ [33

K. S. Asakia and S. A. Mari, “Diffusion coefficient and mobility of a brownian particle in a tilted periodic potential,” J. Phys. Soc. Jpn. 74, 2226–2232 (2005). [CrossRef]

In the case of the optical periodic potential U(x) of the OTW, tilted by the rotation of the polarization as described by Eq. (3), the theoretical results can be effectively used for calibration. In Appendix II we show that the effective diffusion coefficient for the OTW, when ω > ω_c, can be expressed as [32

]

D_{eff} = \frac{π^{2} δ T_{s}^{2}}{2 {〈 T_{s} 〉}^{3}} = \frac{k_{B} T}{γ} f (r)

(21)

where

δ T_{s}^{2}

is the variance of the torque period and f (r) is a function of r =ω_c/ω (see Appendix II).

To carry out a calibration measurement, we fix ω > ω_c and record the resulting periodic torque signal (Fig. 5b top). From this single time-trace, we directly measure the average value of the torque period 〈T_s〉 and its variance

δ T_{s}^{2}

in the temporal domain. The distribution of the period T_s is shown (Fig. 5b (bottom)). The value of ω_c can then be determined from Eq. (4) using

ω_{c} = τ_{o} / γ = \sqrt{ω^{2} - {(π / 〈 T_{s} 〉)}^{2}}

, allowing us to calculate r and f (r). The drag coefficient is then found from Eq. (21) as

γ = \frac{2 k_{B} T {〈 T_{s} 〉}^{3}}{π^{2} δ T_{s}^{2}} f (r)

(22)

and the maximum optical torque is then easily calculated from γ and ω_c using

τ_{o} = γ ω_{c}

(23)

Finally, the standard deviation of the same torque signal δτ_m is measured (Fig. 5), in order to yield the angular sensitivity β_τ (Eq. (7))

β_{τ} = \frac{δ τ_{m}}{τ_{o}} \sqrt{\frac{(ω / ω_{c}) + \sqrt{{(ω / ω_{c})}^{2} - 1}}{\sqrt{{(ω / ω_{c})}^{2} - 1}}}

(24)

5. Discussion

Using the experimental configuration described in sec. 2.2, we have experimentally implemented the five different calibration methods for the OTW described in the previous section on the same trapped birefringent cylinder. The resulting values of the calibration parameters (γ, β_τ, τ_o) are summarized in Table 1. Overall the numbers obtained with the different approaches agree well with one another. The errors shown in the table, indicated by the notation

\pm_{b}^{a}

, reflect the precision obtained repeating the same measurement (a) and the single measurement error that results from error propagation of experimental uncertainties in the analytical expressions (b).

Table 1 Experimental results of the different calibration methods obtained with the same trapped birefringent cylinder. For every method, the notation

m \pm_{b}^{a}

indicates the mean value m, obtained in N successive measurements, the standard deviation a of the N measurements, and the error propagated from the uncertainties of the parameters measured in the method and calculated from the analytical expression of m. Here

β_{τ}^{'} = β_{τ}^{- 1}

. References to sections, figures and equations of this work for each method and parameter are provided.

Method	Drag γ		Sensitivity β′_τ		Max. Torque τ_o
Method	(pN nms)	Eq.	(pN nm/mV)	Eq.	(pN nm)	Eq.
Fit of the standard deviation of the torque (N = 2, sec. 4.1, Fig. 3)	$2.4 \pm_{0.2}^{0.1}$	(9)	$15.9 \pm_{0.9}^{0.3}$	(10)	$1041 \pm_{63}^{7}$	(8)
Spectrum at ω = 0 and fast rotation at ω > ω_c (N = 6, sec. 4.2.1 Fig. 4a)	$2.1 \pm_{0.3}^{0.2}$	(11)	$15.7 \pm_{2.9}^{0.6}$	(13)	$1008 \pm_{189}^{37}$	(12)
Torque variance, period and amplitude (N = 6, sec. 4.2.2, Fig. 4b)	$2.1 \pm_{0.1}^{0.1}$	(15)	$16.7 \pm_{0.1}^{0.6}$	(16)	$1100 \pm_{25}^{41}$	(14)
Sinusoidal modulation of the polarization direction (N = 2, sec. 4.3.1, Fig. 5a)	$2.2 \pm_{0.3}^{0.1}$	(18)	$16.0 \pm_{2.9}^{0.3}$	(20)	$1044 \pm_{129}^{22}$	(19)
Diffusion in a tilted potential landscape (N = 6, sec. 4.3.2, Fig. 5b)	$2.4 \pm_{0.1}^{0.1}$	(22)	$15.9 \pm_{1.4}^{0.6}$	(24)	$1046 \pm_{47}^{40}$	(23)

For an absolute and independent check of the measurements, we can compare the experimentally determined values for the particle drag to the theoretically determined value for an appropriately-shaped cylinder (Fig. 1b). For a cylinder in water of length 1.8 μm and diameter 0.6 μm, a theoretical value of γ = 2.3 pN nm s is expected for the drag coefficient [34

M. M. Tirado and J. Garciadelatorre, “Rotational-dynamics of rigid, symmetric top macromolecules; application to circular-cylinders,” J. Chem. Phys. 73, 198–1993 (1980). [CrossRef]

], which is well within the range of experimentally measured values for γ, and therefore validates our results.

When using these different methods, it is important to minimize potential sources of systematic errors. For example, there is a first source of error that primarily affects the methods that measure torque fluctuations during rotation of the laser polarization (sec. 4.1, 4.2, 4.3.2). This is due to the fact that it is not straightforward to obtain a torque signal that is fully free of spurious modulation (at the frequency of the EOM) in torque when ω > 0, despite the improvements offered through the use of a reference signal as discussed in sec. 2.2. We find that the precise alignment of the axes of the two quarter-waveplates surrounding the optical trap is the critical parameter in achieving a flat torque signal when the polarization is rotated at a finite frequency below ω_c. Once such alignment is optimized, the residual amount of spurious torque modulation depends also on the trapped particles, probably reflecting small differences in their geometries and scattering. The data presented in this work correspond to a particle for which the spurious modulation could be reduced below the thermal noise. For other particles this was not always possible, and as a consequence the precision with which the parameters were obtained by the different methods was significantly lower (with deviations from the mean value up to ±20%).

If the torque modulation is appreciable, the first method introduced, relying on the use of multiple frequencies (sec. 4.1) and measurement of the standard deviation of the torque, suffers from a systematic error particularly strongly. This is due to the fact that for the part of the measurement that is conducted at 0 < ω < ω_c, the spurious modulation can be relatively large compared to the predicted value of the standard deviation (see Fig. 3), artificially increasing the value of δτ_m; this effect is less pronounced when ω > ω_c.

A second source of error is the accuracy of the fit of the experimental power spectrum to the theoretical Lorentzian curve [21

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

], affecting the two methods described in ”Power spectrum analysis at ω = 0 followed by fast rotation at ω > ω_c” (sec.4.2.1) and ”Sinusoidal modulation of the laser polarization direction” (sec.4.3.1). This is reflected in the relatively large propagated uncertainty of these two methods. Variations in the fitting parameters critically depend on the weight given to different frequency regions, which can be controlled by binning the data with a variable bin size (as done in Fig. 4a and Fig. 5a to increase the weight of the Lorentzian plateau), and on the value of the maximum frequency present in the fitted spectrum, which delimits the region to which an f⁻²-dependence is fitted.

Which type of calibration method is the be most useful to the experimentalist? Based on the above considerations of sources of error, it seems likely that the methods that rely on the analysis of torque fluctuations for 0 < ω < ω_c (sec. 4.1) and the methods that rely on spectral analysis (sec. 4.2.1 and sec. 4.3.1) should be the most prone to inaccuracies (see Table). To reduce systematic errors, one would preferably acquire and analyze torque traces in the time domain using either a fixed (ω = 0) or rapidly rotating (ω ≫ ω_c) polarization (methods of sec. 4.2.2 and sec. 4.3.2). On this basis, we recommend these two methods. For the practical implementation in an experimental setup, the rapidity with which calibration can be performed is also an important consideration. Amongst the methods presented here, the one based on the use of multiple polarization rotation frequencies (sec. 4.1) is surely the most time consuming; conversely, the single-frequency methods of sec. 4.3.1 and 4.3.2, which only require acquisition at a single polarization rotation frequency, are the most rapid. When high throughput is required, these two methods are preferred. Taking both speed and accuracy into account, we conclude that the single-frequency method of sec. 4.3.2 will be typically the most suitable. This method also has the advantage of allowing one to dynamically measure possible variations of the drag coefficient within few polarization cycles, which can be useful in micro-rheology measurements.

In conclusion, we have described and performed various methods of OTW calibration, some with direct OT analog and others developed specifically for the angular variables. Overall, the different methods lead to close results, which also agree with the theoretical prediction for the particle drag coefficient. However, the absolute values of the variables measured by the instrument should be expected to depend on the details of calibration method chosen. We hope this work can contribute to a wider use of the OTW, a versatile technique with many potential applications in physical and biophysical studies.

Appendices

Appendix I Derivation of the standard deviation of the torque

We derive here the analytical expressions (Eq. (6) and Eq. (7)) for the standard deviation of the torque (sec. 4.1). We recall that τ_m = β_ττ (Eq. (5)), where τ_m is the measured torque signal in Volts, β_τ is the sensitivity, τ = τ_o sin(2x) is the optical torque (Eq. (1)), and x is the angle between the laser polarization and the extraordinary axis of the birefringent cylinder. When the laser polarization rotates at a constant frequency ω, the noise-free equation of motion can be written as

\dot{x} = - \frac{τ_{o}}{γ} sin (2 x) - ω

(25)

(see Eq. (3)). We treat the two cases ω < ω_c and ω > ω_c separately.

For ω < ω_c = τ_o/γ, the cylinder rotates in phase with the polarization. At equilibrium, when x = x_eq, the mean torque is given by

〈 τ 〉 = - τ_{o} sin (2 x_{eq}) = γ ω

(26)

from which we deduce the equilibrium position

x_{eq} = \frac{1}{2} arcsin (\frac{ω}{ω_{c}})

. We can express the variance of x from the equipartition theorem as δx² = k_BT/κ, where the angular stiffness κ can be written as κ = −(∂τ/∂x)_eq = 2τ_ocos(2x_eq). This allows us to write the variance of the torque (see Eq. (28)) as

δ τ^{2} = {(\frac{\partial τ}{\partial x})}^{2} δ x^{2} = 2 k_{B} T τ_{o} cos (2 x_{eq})

(27)

= 2 k_{B} T τ_{o} \sqrt{1 - {(ω / ω_{c})}^{2}}

(28)

Equation (6) is obtained from Eq. (28) by multiplying by the sensitivity β_τ.

For ω > ω_c, the cylinder does not rotate in phase with the driving polarization, and the noise-free solution is a periodic torque trace with period T_o given by Eq. (4). In this case, the mean value of the torque is given by

\begin{array}{l} 〈 τ 〉 = \frac{1}{T_{o}} \int_{0}^{T_{o}} τ d t = \frac{1}{T_{o}} \int_{0}^{T_{o}} γ (\dot{x} + ω) d t \\ = \frac{1}{T_{o}} γ [x (T_{o}) - x (0)] + γ ω = - \frac{1}{T_{o}} γ π + γ ω \\ = γ (ω - \sqrt{ω^{2} - ω_{c}^{2}}) \end{array}

(29)

The expression of 〈τ〉 is then used to calculate the torque variance δτ² according to

δ τ^{2} = 〈 τ^{2} 〉 - {〈 τ 〉}^{2} = \frac{1}{T_{τ}} \int_{0}^{T_{τ}} τ γ (d x / d t + ω) d t - {〈 τ 〉}^{2} = γ ω 〈 τ 〉 - {〈 τ 〉}^{2}

(30)

= τ_{0}^{2} \frac{\sqrt{{(ω / ω_{c})}^{2} - 1}}{(ω / ω_{c}) + \sqrt{{(ω / ω_{c})}^{2} - 1}}

(31)

Equation (7) is obtained from Eq. (31) by multiplying by the sensitivity β_τ.

Appendix II Diffusion in a tilted periodic potential

We discuss here the theoretical results used in the calibration method based on analysis of the diffusion of a particle in a tilted periodic potential (sec. 4.3.2). Specifically, we consider the diffusion in a total potential U(x) = V(x)–Fx formed by a periodic potential V(x), tilted by an external force F. In the case of the OTW, the motion of the over-damped particle is described by Eq. (3).

When the polarization rotation frequency is set to ω > ω_c, the system is far from the thermodynamic equilibrium as there is a non-zero probability flux resulting from the disappearance of the energy barrier between successive minima. In this case, the Einstein relation D_o – μk_BT = 0 (expressed in terms of the mobility μ = 1/γ) is not valid. Rather, the following approximate expression can be derived for the effective values of the diffusion coefficient D_eff and the effective mobility μ_eff, provided γω ≫ k_BT [33

K. S. Asakia and S. A. Mari, “Diffusion coefficient and mobility of a brownian particle in a tilted periodic potential,” J. Phys. Soc. Jpn. 74, 2226–2232 (2005). [CrossRef]

D_{eff} - k_{B} T μ_{eff} \approx \frac{k_{B} T}{γ} \frac{\sqrt{ω^{2} - ω_{c}^{2}}}{ω} [\frac{2 〈 τ^{2} 〉}{γ^{2} ω^{2}} + \frac{5 〈 τ^{3} 〉}{γ^{3} ω^{3}}]

(32)

The effective mobility is in general defined by μ_eff = dv/dF [33

K. S. Asakia and S. A. Mari, “Diffusion coefficient and mobility of a brownian particle in a tilted periodic potential,” J. Phys. Soc. Jpn. 74, 2226–2232 (2005). [CrossRef]

], where v is the speed of the particle under the action of the force F. In our case, recalling that for ω > ω_c the torque is a periodic function of time with period 〈T_s〉 (Eq. (4)) and F = −γω (Eq. (3)), the effective mobility can be expressed as

μ_{eff} = - \frac{1}{γ} \frac{d}{d ω} 〈 v (ω) 〉 = \frac{1}{γ} \frac{d}{d ω} \frac{π}{〈 T_{s} 〉} \approx \frac{ω}{γ \sqrt{ω^{2} - ω_{c}^{2}}}

(33)

Next, we use the expressions for 〈τ²〉 (see Eq. (30)), and derive

〈 τ^{3} 〉 = - (γ / 2) τ_{o}^{2} \sqrt{ω^{2} - ω_{c}^{2}} + γ^{3} ω^{2} (ω - \sqrt{ω^{2} - ω_{c}^{2}})

. Substituting these expressions together with with Eq. (33) into Eq. (32), we find that the effective diffusion coefficient can be written as

D_{eff} = \frac{k_{B} T}{γ} f (r)

(34)

where

r = \frac{ω_{c}}{ω}

and where f(r) is given by

f (r) \approx \frac{1}{\sqrt{1 - r^{2}}} + \frac{2 \sqrt{1 - r^{2}}}{1 + \sqrt{1 - r^{2}}} [r^{2} + \frac{5}{4} r^{4} (1 + \frac{1}{1 + \sqrt{1 - r^{2}}})] .

(35)

Acknowledgments

We thank J. Lipfert and M. Versteegh for fruitful discussions, J. van der Does, D. de Roos and J. Beekman for help with instrumentation and infrastructure, and TU Delft, FOM (Dutch Foundation for Research on Matter), and the European Science Foundation for financial support.

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OCIS Codes
(200.4880) Optics in computing : Optomechanics
(350.4855) Other areas of optics : Optical tweezers or optical manipulation

ToC Category:
Optical Trapping and Manipulation

History
Original Manuscript: September 30, 2011
Revised Manuscript: December 15, 2011
Manuscript Accepted: December 15, 2011
Published: February 1, 2012

Virtual Issues
Vol. 7, Iss. 4 Virtual Journal for Biomedical Optics

Citation
Francesco Pedaci, Zhuangxiong Huang, Maarten van Oene, and Nynke H. Dekker, "Calibration of the optical torque wrench," Opt. Express 20, 3787-3802 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-4-3787

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References

F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics5, 318–321 (2011). [CrossRef] [PubMed]
K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods5, 491–505 (2008). [CrossRef] [PubMed]
K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct.23, 247 (1994). [CrossRef] [PubMed]
B. E. Funnell, T. A. Baker, and A. Kornberg, “In vitro assembly of a prepriming complex at the origin of the escherichia coli chromosome,” J. Biol. Chem.262, 10327–10334 (1987). [PubMed]
L. F. Liu and J. C. Wang, “Supercoiling of the DNA template during transcription,” Proc. Natl. Acad. Sci. U.S.A.84, 7024–7027 (1987). [CrossRef] [PubMed]
M. Yoshida, E. Muneyuki, and T. Hisabori, “ATP synthase, a marvellous rotary engine of the cell,” Nat. Rev. Mol. Cell Biol.2, 669–677 (2001). [CrossRef] [PubMed]
S. Saroussi and N. Nelson, “The little we know on the structure and machinery of V-ATPase,” J. Exp. Biol.212, 1604–1610 (2009). [CrossRef] [PubMed]
Y. Sowa and R. M. Berry, “Bacterial flagellar motor,” Q. Rev. Biophys.41, 103–132 (2008). [CrossRef] [PubMed]
J. Lipfert, J. W. J. Kerssemakers, T. Jager, and N. H. Dekker, “Magnetic torque tweezers: measuring torsional stiffness in DNA and RecA-DNA filaments,” Nat. Methods7, 977–980 (2010). [CrossRef] [PubMed]
M. E. J. Friese, T. A. Nieminem, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature394, 348–350 (1998). [CrossRef]
M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics5, 343–348 (2011). [CrossRef]
A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman, and S. X. Sun, “Magnetic tweezers measurement of single molecule torque,” Nano Lett.9, 1720–1725 (2009). [CrossRef] [PubMed]
A. LaPorta and M. D. Wang, “Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett.92, 190801 (2004).
J. Inman, S. Forth, and M. Wang, “Passive torque wrench and angular position detection using a single-beam optical trap,” Opt. Lett.35, 2949–2951 (2010). [CrossRef] [PubMed]
F. Pedaci, Z. Huang, M. v. Oene, S. Barland, and N. H. Dekker, “Excitable particle in an optical torque wrench,” Nat. Phys.7, 259–264 (2011). [CrossRef]
S. Forth, C. Deufel, M. Y. Sheinin, B. Daniels, J. P. Sethna, and M. D. Wang, “Abrupt buckling transition observed during the plectoneme formation of individual DNA molecules,” Phys. Rev. Lett.100, 148301 (2008). [CrossRef] [PubMed]
B. C. Daniels, S. Forth, M. Y. Sheinin, M. D. Wang, and J. P. Sethna, “Discontinuities at the DNA supercoiling transition,” Phys. Rev. E80, 040901 (2009). [CrossRef]
S. Forth, C. Deufel, S. S. Patel, and M. D. Wang, “Direct measurements of torque during Holliday junction migration,” Biophys. J.101, L05–L07 (2011). [CrossRef]
K. Visscher and S. M. Block, “Versatile optical traps with feedback control,” Method Enzymol.298, 460–489 (1998). [CrossRef]
M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum.73, 1687 (2002). [CrossRef]
K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum.75, 594–612 (2004). [CrossRef]
K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum.75, 2787–2809 (2004). [CrossRef]
C. Deufel and M. D. Wang, “Detection of forces and displacements along the axial direction in an optical trap,” Biophys. J.90, 657–667 (2006). [CrossRef]
B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. LaPorta, and S. M. Block, “An optical apparatus for rotation and trapping,” Method Enzymol.475, 377–404 (2010). [CrossRef]
R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE34, 351–357 (1946). [CrossRef]
C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods4, 223–225 (2007). [CrossRef] [PubMed]
Z. Huang, F. Pedaci, M. Wiggin, M. v. Oene, and N. H. Dekker, “Electron beam fabrication of micron-scale birefringent quartz particles for use in optical trapping,” ACS Nano5, 1418–1427 (2011). [CrossRef] [PubMed]
W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, “Passive all-optical force clamp for high-resolution laser trapping,” Phys. Rev. Lett.95, 208102 (2005). [CrossRef] [PubMed]
O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, and A. C. Ferrari, “FemtoNewton force sensing with optically trapped nanotubes,” Nano Lett.8, 3211–3216 (2008). [CrossRef] [PubMed]
P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, and C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett.11, 2375–2381 (2011). [CrossRef] [PubMed]
S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum.77, 103101 (2006). [CrossRef]
P. Reimann, C. V. den Broeck, H. Linke, P. Hanggi, J. M. Rubi, and A. Pérez-Madrid, “Giant acceleration of free diffusion by use of tilted periodic potentials,” Phys. Rev. Lett.87, 010602 (2001). [CrossRef] [PubMed]
K. S. Asakia and S. A. Mari, “Diffusion coefficient and mobility of a brownian particle in a tilted periodic potential,” J. Phys. Soc. Jpn.74, 2226–2232 (2005). [CrossRef]
M. M. Tirado and J. Garciadelatorre, “Rotational-dynamics of rigid, symmetric top macromolecules; application to circular-cylinders,” J. Chem. Phys.73, 198–1993 (1980). [CrossRef]

Abbondanzieri, E. A.
Adler, R.
Andreasson, J. O. L.
Asakia, K. S.
Baker, T. A.
Ballerini, R.
Barland, S.
Berg-Sørensen, K.
Berry, R. M.
Block, S. M.
Bonaccorso, F.
Bowman, G. D.
Bowman, R.
Capitanio, M.
Celedon, A.
Daniels, B.
Daniels, B. C.
Dejgosha, S.
Dekker, N. H.
den Broeck, C. V.
Deufel, C.
Dewan, R.
Dunlap, D.
Fazal, F. M.
Ferrari, A. C.
Finzi, L.
Flyvbjerg, H.
Forth, S.
Friese, M. E. J.
Funnell, B. E.
Gao, Q.
Garciadelatorre, J.
Giuntini, M.
Greenleaf, W. J.
Gucciardi, P. G.
Gutierrez-Medina, B.
Hanggi, P.
Heckenberg, N. R.
Hisabori, T.
Howard, J.
Huang, Z.
Inman, J.
Jagadish, C.
Jager, T.
Jones, P. H.
Jülicher, F.
Kerssemakers, J. W. J.
Kornberg, A.
LaPorta, A.
Linke, H.
Lipfert, J.
Liu, L. F.
Maragò, O. M.
Mari, S. A.
Muneyuki, E.
Nagy, A.
Nelson, N.
Neuman, K. C.
Nieminem, T. A.
Nodelman, I. M.
Oene, M. v.
Padgett, M.
Paiman, S.
Patel, S. S.
Pavone, F. S.
Pedaci, F.
Pérez-Madrid, A.
Reece, P. J.
Reimann, P.
Romano, G.
Rozhin, A. G.
Rubi, J. M.
Rubinsztein-Dunlop, H.
Saroussi, S.
Scardaci, V.
Schäffer, E.
Searson, P.
Sethna, J. P.
Sheinin, M. Y.
Simmons, C. R.
Sowa, Y.
Sun, S. X.
Svoboda, K.
Tan, H. H.
Tirado, M. M.
Toe, W. J.
Tolic-Nørrelykke, S. F.
Visscher, K.
Wang, F.
Wang, J. C.
Wang, M.
Wang, M. D.
Wiggin, M.
Wildt, B.
Wirtz, D.
Woodside, M. T.
Yoshida, M.

ACS Nano

Z. Huang, F. Pedaci, M. Wiggin, M. v. Oene, and N. H. Dekker, “Electron beam fabrication of micron-scale birefringent quartz particles for use in optical trapping,” ACS Nano5, 1418–1427 (2011). [CrossRef] [PubMed]

Annu. Rev. Biophys. Biomol. Struct.

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct.23, 247 (1994). [CrossRef] [PubMed]

Biophys. J.

C. Deufel and M. D. Wang, “Detection of forces and displacements along the axial direction in an optical trap,” Biophys. J.90, 657–667 (2006). [CrossRef]
S. Forth, C. Deufel, S. S. Patel, and M. D. Wang, “Direct measurements of torque during Holliday junction migration,” Biophys. J.101, L05–L07 (2011). [CrossRef]

J. Biol. Chem.

B. E. Funnell, T. A. Baker, and A. Kornberg, “In vitro assembly of a prepriming complex at the origin of the escherichia coli chromosome,” J. Biol. Chem.262, 10327–10334 (1987). [PubMed]

J. Chem. Phys.

M. M. Tirado and J. Garciadelatorre, “Rotational-dynamics of rigid, symmetric top macromolecules; application to circular-cylinders,” J. Chem. Phys.73, 198–1993 (1980). [CrossRef]

J. Exp. Biol.

S. Saroussi and N. Nelson, “The little we know on the structure and machinery of V-ATPase,” J. Exp. Biol.212, 1604–1610 (2009). [CrossRef] [PubMed]

J. Phys. Soc. Jpn.

K. S. Asakia and S. A. Mari, “Diffusion coefficient and mobility of a brownian particle in a tilted periodic potential,” J. Phys. Soc. Jpn.74, 2226–2232 (2005). [CrossRef]

Method Enzymol.

K. Visscher and S. M. Block, “Versatile optical traps with feedback control,” Method Enzymol.298, 460–489 (1998). [CrossRef]
B. Gutierrez-Medina, J. O. L. Andreasson, W. J. Greenleaf, A. LaPorta, and S. M. Block, “An optical apparatus for rotation and trapping,” Method Enzymol.475, 377–404 (2010). [CrossRef]

Nano Lett.

O. M. Maragò, P. H. Jones, F. Bonaccorso, V. Scardaci, P. G. Gucciardi, A. G. Rozhin, and A. C. Ferrari, “FemtoNewton force sensing with optically trapped nanotubes,” Nano Lett.8, 3211–3216 (2008). [CrossRef] [PubMed]
P. J. Reece, W. J. Toe, F. Wang, S. Paiman, Q. Gao, H. H. Tan, and C. Jagadish, “Characterization of semiconductor nanowires using optical tweezers,” Nano Lett.11, 2375–2381 (2011). [CrossRef] [PubMed]
A. Celedon, I. M. Nodelman, B. Wildt, R. Dewan, P. Searson, D. Wirtz, G. D. Bowman, and S. X. Sun, “Magnetic tweezers measurement of single molecule torque,” Nano Lett.9, 1720–1725 (2009). [CrossRef] [PubMed]

Nat. Methods

J. Lipfert, J. W. J. Kerssemakers, T. Jager, and N. H. Dekker, “Magnetic torque tweezers: measuring torsional stiffness in DNA and RecA-DNA filaments,” Nat. Methods7, 977–980 (2010). [CrossRef] [PubMed]
K. C. Neuman and A. Nagy, “Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy,” Nat. Methods5, 491–505 (2008). [CrossRef] [PubMed]
C. Deufel, S. Forth, C. R. Simmons, S. Dejgosha, and M. D. Wang, “Nanofabricated quartz cylinders for angular trapping: DNA supercoiling torque detection,” Nat. Methods4, 223–225 (2007). [CrossRef] [PubMed]

Nat. Photonics

F. M. Fazal and S. M. Block, “Optical tweezers study life under tension,” Nat. Photonics5, 318–321 (2011). [CrossRef] [PubMed]
M. Padgett and R. Bowman, “Tweezers with a twist,” Nat. Photonics5, 343–348 (2011). [CrossRef]

Nat. Phys.

F. Pedaci, Z. Huang, M. v. Oene, S. Barland, and N. H. Dekker, “Excitable particle in an optical torque wrench,” Nat. Phys.7, 259–264 (2011). [CrossRef]

Nat. Rev. Mol. Cell Biol.

M. Yoshida, E. Muneyuki, and T. Hisabori, “ATP synthase, a marvellous rotary engine of the cell,” Nat. Rev. Mol. Cell Biol.2, 669–677 (2001). [CrossRef] [PubMed]

Nature

M. E. J. Friese, T. A. Nieminem, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical alignment and spinning of laser-trapped microscopic particles,” Nature394, 348–350 (1998). [CrossRef]

Opt. Lett.

J. Inman, S. Forth, and M. Wang, “Passive torque wrench and angular position detection using a single-beam optical trap,” Opt. Lett.35, 2949–2951 (2010). [CrossRef] [PubMed]

Phys. Rev. E

B. C. Daniels, S. Forth, M. Y. Sheinin, M. D. Wang, and J. P. Sethna, “Discontinuities at the DNA supercoiling transition,” Phys. Rev. E80, 040901 (2009). [CrossRef]

Phys. Rev. Lett.

S. Forth, C. Deufel, M. Y. Sheinin, B. Daniels, J. P. Sethna, and M. D. Wang, “Abrupt buckling transition observed during the plectoneme formation of individual DNA molecules,” Phys. Rev. Lett.100, 148301 (2008). [CrossRef] [PubMed]
A. LaPorta and M. D. Wang, “Optical torque wrench: angular trapping, rotation, and torque detection of quartz microparticles,” Phys. Rev. Lett.92, 190801 (2004).
P. Reimann, C. V. den Broeck, H. Linke, P. Hanggi, J. M. Rubi, and A. Pérez-Madrid, “Giant acceleration of free diffusion by use of tilted periodic potentials,” Phys. Rev. Lett.87, 010602 (2001). [CrossRef] [PubMed]
W. J. Greenleaf, M. T. Woodside, E. A. Abbondanzieri, and S. M. Block, “Passive all-optical force clamp for high-resolution laser trapping,” Phys. Rev. Lett.95, 208102 (2005). [CrossRef] [PubMed]

Proc. IRE

R. Adler, “A study of locking phenomena in oscillators,” Proc. IRE34, 351–357 (1946). [CrossRef]

Proc. Natl. Acad. Sci. U.S.A.

L. F. Liu and J. C. Wang, “Supercoiling of the DNA template during transcription,” Proc. Natl. Acad. Sci. U.S.A.84, 7024–7027 (1987). [CrossRef] [PubMed]

Q. Rev. Biophys.

Y. Sowa and R. M. Berry, “Bacterial flagellar motor,” Q. Rev. Biophys.41, 103–132 (2008). [CrossRef] [PubMed]

Rev. Sci. Instrum.

M. Capitanio, G. Romano, R. Ballerini, M. Giuntini, F. S. Pavone, D. Dunlap, and L. Finzi, “Calibration of optical tweezers with differential interference contrast signals,” Rev. Sci. Instrum.73, 1687 (2002). [CrossRef]
K. Berg-Sørensen and H. Flyvbjerg, “Power spectrum analysis for optical tweezers,” Rev. Sci. Instrum.75, 594–612 (2004). [CrossRef]
K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum.75, 2787–2809 (2004). [CrossRef]
S. F. Tolić-Nørrelykke, E. Schäffer, J. Howard, F. S. Pavone, F. Jülicher, and H. Flyvbjerg, “Calibration of optical tweezers with positional detection in the back focal plane,” Rev. Sci. Instrum.77, 103101 (2006). [CrossRef]

2011, Fazal, Nat. Photonics
2011, Padgett, Nat. Photonics
2011, Pedaci, Nat. Phys.
2011, Forth, Biophys. J.
2011, Huang, ACS Nano
2011, Reece, Nano Lett.
2010, Gutierrez-Medina, Method Enzymol.
2010, Inman, Opt. Lett.
2010, Lipfert, Nat. Methods
2009, Saroussi, J. Exp. Biol.
2009, Celedon, Nano Lett.
2009, Daniels, Phys. Rev. E
2008, Maragò, Nano Lett.
2008, Forth, Phys. Rev. Lett.
2008, Sowa, Q. Rev. Biophys.
2008, Neuman, Nat. Methods
2007, Deufel, Nat. Methods
2006, Deufel, Biophys. J.
2006, Tolic-Nørrelykke, Rev. Sci. Instrum.
2005, Greenleaf, Phys. Rev. Lett.
2005, Asakia, J. Phys. Soc. Jpn.
2004, Berg-Sørensen, Rev. Sci. Instrum.
2004, Neuman, Rev. Sci. Instrum.
2002, Capitanio, Rev. Sci. Instrum.
2001, Reimann, Phys. Rev. Lett.
2001, Yoshida, Nat. Rev. Mol. Cell Biol.
1998, Visscher, Method Enzymol.
1998, Friese, Nature
1994, Svoboda, Annu. Rev. Biophys. Biomol. Struct.
1987, Funnell, J. Biol. Chem.
1987, Liu, Proc. Natl. Acad. Sci. U.S.A.
1980, Tirado, J. Chem. Phys.
1946, Adler, Proc. IRE
1908, LaPorta, Phys. Rev. Lett.

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Calibration of the optical torque wrench

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Abstract

1. Introduction

2. The optical torque wrench

2.1. Theoretical overview

2.2. Experimental configuration

3. Similarities and differences in the calibration of OT and OTW

4. Approaches for angular calibration

4.1. Calibration approach involving measurement over the full range of frequencies: fitting the standard deviation of the torque signal

4.2. Calibration approaches involving separate measurements at two frequencies

4.2.1. Power spectrum analysis at ω = 0 followed by fast polarization rotation at ω > ωc

4.2.2. Calibration by measurement of the torque variance, period and amplitude

4.3. Calibration approaches using measurements at a single frequency

4.3.1. Sinusoidal modulation of the laser polarization direction

4.3.2. Analysis of the diffusion in a tilted potential landscape

5. Discussion

Appendices

Appendix I Derivation of the standard deviation of the torque

Appendix II Diffusion in a tilted periodic potential

Acknowledgments

References and links

References

ACS Nano

Annu. Rev. Biophys. Biomol. Struct.

Biophys. J.

J. Biol. Chem.

J. Chem. Phys.

J. Exp. Biol.

J. Phys. Soc. Jpn.

Method Enzymol.

Nano Lett.

Nat. Methods

Nat. Photonics

Nat. Phys.

Nat. Rev. Mol. Cell Biol.

Nature

Opt. Lett.

Phys. Rev. E

Phys. Rev. Lett.

Proc. IRE

Proc. Natl. Acad. Sci. U.S.A.

Q. Rev. Biophys.

Rev. Sci. Instrum.

Cited By

Figures

4.2.1. Power spectrum analysis at ω = 0 followed by fast polarization rotation at ω > ω_c