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We demonstrate a lock-in particle tracking scheme in optical tweezers based on stroboscopic modulation of an illuminating optical field. This scheme is found to evade low frequency noise sources while otherwise producing an equivalent position measurement to continuous measurement. This was demonstrated to yield up to 20 dB of noise suppression at both low frequencies (< 1 kHz), where low frequency electronic noise was significant, and around 630 kHz where laser relaxation oscillations introduced laser noise. The setup is simple, and compatible with any trapping optics.
© 2013 Optical Society of America
1. Introduction
In optical tweezers, small particles are trapped by the electric field gradient at the focus of a tightly focused laser beam [1]. After the light interacts with the particle, it encodes information about the particle position [2], which is typically extracted with a quadrant detector at the back-focal plane of a condenser lens [3]. This allows particle tracking with sub-nanometer sensitivity [4] as forces ranging from subpiconewton [5,6] to nanonewton [7] are controllably applied. Such experiments have uncovered important phenomena in biophysics, including both the dynamics and magnitude of the forces applied by biological motors [6, 8, 9], the stretching and folding properties of DNA and RNA [9–11], the dynamics of virus-host coupling [12], the strain on an enzyme during catalysis [9], and the rheological properties of cellular cytoplasm [13–16]. Furthermore, optical tweezers provide a significant tool for studying the fundamental physics of Brownian motion [17, 18] and quantum optomechanics [19, 20].
While shot-noise establishes the fundamental sensitivity limit for optical tweezers based measurements [2, 21, 22], real experiments are generally limited by technical noise sources such as laser noise, electronic noise in the detector, or drifts of mirrors in the experiment. These technical sources of error can be a significant hindrance to precision measurement, so much effort has gone into reducing them [4, 5, 12, 21]. Recently, an optical lock-in particle tracking scheme was developed which allowed evasion of low-frequency technical noise without needing to remove the noise sources from the experiment. This was applied in a dark-field particle tracking setup in conjunction with quantum correlated light to break the shot-noise limit in particle tracking sensitivity [13]. In principle, this optical lock-in particle tracking scheme offers near immunity to low frequency laser noise and electronic noise, which could make it a highly practical method for a wide range of experiments. In the initial demonstration, however, the noise suppression attained through use of the optical lock-in tracking was not characterized [13]. Furthermore, since that experiment relied on dark-field illumination, the experimental setup was incompatible with standard optical tweezers equipment. Here we demonstrate optical lock-in particle tracking in a bright-field optical tweezers setup, and characterize the noise suppression attained. It is shown that lock-in based particle tracking allows evasion of low frequency electronic noise and laser intensity noise, and achieves equivalent sensitivity where the dominant noise source is fundamental shot-noise. The reduction in laser noise and electronic noise yields up to 20 dB of noise suppression below 1 kHz, where low frequency electronic noise is significant, and over 20 dB of noise suppression around 600 kHz where the laser crystal relaxation oscillations introduce an intensity noise feature.
2. Basic concept
The lock-in based particle tracking measurement demonstrated here is qualitatively similar to a continuous position tracking experiment. In optical tweezers based measurements, scattering particles are illuminated and the spatial distribution of the resulting scattered field is measured to infer particle position [2, 3]. Any modulation on the incident illumination is carried onto the scattered field, shifting some of the optical power from the laser carrier frequency into modulation side-bands. Once the scattered field is measured, the optical modulation translates into a modulation on the electrical signal, with the particle position information centered about the modulation frequency (see Fig. 1). The particle position can be recovered by demodulating this signal.
Fig. 1 An illumination optical field is modulated by its interaction with the particle (red). In order to measure this, it is mixed with another bright local oscillator field (dark blue). However, the local oscillator also has some low-frequency noise present. If the illumination field frequency matches the local oscillator frequency, as shown on the left, then the low-frequency noise competes with the low-frequency particle motion signal. However, if it is in amplitude modulated side-bands, as shown on the right, then the low-frequency particle motion can be isolated from the low-frequency noise, thereby improving the measurement sensitivity.
We may ask how the expected sensitivity of such a measurement compares to a usual continuous measurement. When optical fields are measured, the resulting photocurrent at time t is given by
where NE is the electronic noise, G is the detector gain, E is the total electric field at the detector at the coordinates X and Y, and U(X, Y) represents the spatial gain of the detector; for instance, if the photocurrent from two halves of a split detector are subtracted from one another, this is represented as U(X, Y) = sign(X), while a bulk detector has U(X, Y) = 1. Here we assume that the fields present are a scattered field Es which depends on particle position, and a local oscillator ELO with which the scattered field interferes, such that E = ELO + Es. In most optical tweezers experiments, the local oscillator is simply given by the component of the trapping field which has not scattered from the particle. For lock-in experiments, the fields are separated to allow the particle illumination to be modulated independently of the local oscillator. The scattered field is assumed to be much smaller than the local oscillator (|Es| ≪ |ELO|) as is typically the case, such that the measured photocurrent is given by The explicit time dependence of the scattered field may be separated from the spatial mode shape as Es = (As(t) + ξ(t))ψs(X, Y), where As(t) and ξ(t) are respectively the real expectation value of the field amplitude from which the particle tracking signal originates, and its fluctuations which contribute noise such as shot noise. ψs(X, Y) is the complex spatial mode-shape of the scattered field; this does not have explicit time dependence, but is modified as the particle moves. It is this spatial modification which is ultimately monitored to retrieve a particle tracking signal. To find the dependence of the scattered field on a small particle displacement x(t), it can be expanded to first order as Substituting this expression into Eq.(2), the component of the photocurrent which gives a linear particle tracking signal can be seen to be where for brevity we define a gain . The position sensitivity is optimized when this gain is maximized, as occurs when both the phase and shape of the local oscillator field are optimized to perfectly interfere with the scattered field component [13, 22]. Substituting this into Eq.(2) we can represent the measured photocurrent as where all the terms in the integrand which did not contribute to the tracking signal are included as optical noise Nopt. For a continuous measurement, the expectation value of the scattered field amplitude As(t) should be constant. Alternatively, we can perform lock-in measurement if we modulate the scattered field amplitude at frequency ω such that , where the modulated amplitude has an RMS value of Ās. Provided the modulation frequency is much faster than the mechanical motion, the position can then be extracted by demodulation; Thus, the effect of the lock-in is to shift the low frequency noise to high frequencies, and generate a second harmonic term proportional to x while leaving the signal term unchanged. The second harmonic term and the low frequency noise can then be removed via a low-pass filter, such that only the noise originally near the modulation frequency enters the measurement. Wherever low-frequency noise is dominant, lock-in measurement allows suppression of the noise floor. This does not influence white noise sources such as shot-noise, since these are equally present at low frequencies and around the modulation frequency. Thus, the fundamental shot-noise limit on position sensitivity is not influenced by a choice between continuous or lock-in measurement. The two schemes have equivalent shot-noise limits to sensitivity when the lock-in scattered amplitude Ās matches the amplitude As of a the continuous measurment, or equivalently, when the same number of scattered photons in modulation side-bands are collected for the lock-in measurement as are collected for a continuous measurement.3. Experiment
The optical lock-in particle tracking experiment shown in Fig. 2 was built, and the sensitivity attainable with continuous and lock-in measurements characterized. A particle is trapped in water between two Nikon Plan Fluorite objectives with numerical aperture (NA) of 1.3. The trapping field and local oscillator are produced by a low noise [23] Innolight Prometheus Nd:YAG laser at 1064 nm. The trapping power is 30 mW, and this was amplitude modulated at 2.5 MHz to allow lock-in measurement. This modulation frequency is sufficiently high that the resulting modulation of the trap strength should not measurably disturb the particle motion. The backscatter from this modulated field is combined with a 3 mW local oscillator field which also propagates through the trap. The resulting interference is then measured on a split detector to determine particle position [3]. The split detector follows the design used in Ref. [20], where the left and right halves of the laser beam are separated with a mirror, and then measured with a balanced New Focus 1817 detector. This approach extends the scheme first developed for optical lock-in particle tracking [13] to a more typical optical trapping setup. There the scattered field was produced from dark-field illumination which was incident from the side, whereas here the trapping field provides the illumination. Side-illumination is only possible if there is room for a free-space probe field to reach the trap center, which requires use of long working distance objectives (in Ref. [13], 6 mm) which are not typically used for optical trapping.
Fig. 2 Layout of the optical lock-in tracking scheme used here. PBS: polarizing beam-splitter, DM: dichroic mirror, λ/4: quarter waveplate. Particles are trapped with a 30 mW trapping field which has been amplitude modulated at 2.5 MHz in a fiber Mach-Zehnder modulator (dark red). Back-scattered light from this trapping field is combined with a 3 mW local oscillator field (light red). Since the back-scattered light passes through the quarter waveplate twice, its polarization is rotated and it passes straight through the polarizing beamsplitter. The interference between the scattered field and the local oscillator is measured on a split detector to track the particles. A separate green field is used to image the particles in the trap.
The amplitude modulation on the trapping field is chosen to leave approximately equal power in the central laser frequency and the first modulation side-band. This allows the continuous and pulsed measurements to occur simultaneously with a single detector, and with equivalent recording conditions. Some non-linearity in the modulator resulted in a number of higher harmonics being generated, which were suppressed in the data acquisition with analog electronic filters.
Using this setup, the Brownian motion of a 0.5 μm polystyrene bead was simultaneously measured both continuously and from side-bands around the 2.5 MHz modulation, with spectra shown in Figs. 3(a) and 3(b) respectively. The background noise was characterized by performing equivalent measurements in the absence of a trapped bead. As expected, the lock-in measurement is very similar to the continuous measurement, but with a reduction in the included electronic and laser noise. The reduction in included noise (shown in Fig. 3(c)) causes the measurement imprecision to improve at the frequencies where laser and electronic noise are dominant. Below 1 kHz, the noise floor was dominated by 1/f electronic noise, with some additional noise spikes. In this regime, the imprecision is improved by an average of 8 dB, with peaks suppressed by up to 20 dB. A prominent laser noise peak was also present at 550–710 kHz due to the laser crystal relaxation oscillations, and this was also suppressed by 20 dB in the lock-in results. At higher frequencies the measurements are both shot-noise limited, and their noise floors converge. These results verify that the lock-in measurement is equivalent to a continuous measurement, except that it evades low frequency technical noise.
Fig. 3 Particle tracking spectra are shown from simultaneous continuous (a) and lock-in (b) measurements. The gold trace shows the noise floor present in the absence of a trapped particle which corresponds to the measurement imprecision, and the blue shows the measured signal with a 0.5 μm polystyrene bead held in the trap. For clarity, the traces are smoothed with a logarithmic width filter, and this broadens the sharp electrical noise features. The prominent spectral peak around 630 kHz in the continuous data is laser noise which arises from the laser diode relaxation oscillations, and this noise feature is barely visible in the lock-in results. Subplot (c) shows the factor by which the noise floor has been lowered for the lock-in results.
While the particle motion was only measured along a single axis in this experiment, lock-in particle tracking is fully compatible with the 3D tracking performed with quadrant detectors, provided the quadrant has sufficient bandwidth to capture the particle motion around the modulation side-bands. It should be noted that while lock-in particle tracking evades laser noise and electronic noise, this can only improve the sensitivity to the particle position relative to the optical fields. As with all other particle tracking experiments, the measurement remains sensitive to mirror drifts or air currents outside the trap which cause the trap center to drift, and conventional methods are needed to stabilize these noise sources.
We have demonstrated that a lock-in measurement scheme provides a simple and robust technique to reduce technical noise in an optical tweezers setup. This can yield a substantial improvement in sensitivity which could be practical for many optical tweezers applications.
Acknowledgments
This work was supported by the Australian Research Council Discovery Project Contract No. DP0985078.
References and links
1. A. Ashkin and J. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987) [CrossRef] [PubMed] .
2. M. A. Taylor, J. Knittel, and W. P. Bowen, “Fundamental constraints on particle tracking with optical tweezers,” New J. Phys. 15, 023018 (2013) [CrossRef] .
3. F. Gittes and C. F. Schmidt, “Interference model for back-focal-plane displacement detection in optical tweezers,” Opt. Lett. 23, 7–9 (1998) [CrossRef] .
4. I. Chavez, R. Huang, K. Henderson, E.-L. Florin, and M. G. Raizen, “Development of a fast position-sensitive laser beam detector,” Rev. Sci. Instrum. 79, 105104 (2008) [CrossRef] [PubMed] .
5. J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77, 205–228 (2008) [CrossRef] .
6. J. T. Finer, R. M. Simmons, and J. A. Spudich, “Single myosin molecule mechanics: piconewton forces and nanometre steps,” Nature 368, 113–119 (1994) [CrossRef] [PubMed] .
7. A. Jannasch, A. F. Demirörs, P. D. J. van Oostrum, A. van Blaaderen, and E. Schäffer, “Nanonewton optical force trap employing anti-reflection coated, high-refractive-index titania microspheres,” Nature Photon. 6, 469–473 (2012) [CrossRef] .
8. K. Svoboda, C. F. Schmidt, B. J. Schnapp, and S. M. Block, 1993 “Direct observation of kinesin stepping by optical trapping interferometry,” Nature 365, 721–727 (1993) [CrossRef] [PubMed] .
9. C. Bustamante, Y. R. Chemla, N. R. Forde, and D. Izhaky, “Mechanical processes in biochemistry,” Annu. Rev. Biochem. 73, 705–748 (2004) [CrossRef] .
10. C. Bustamante, “Unfolding single RNA molecules: bridging the gap between equilibrium and non-equilibrium statistical thermodynamics,” Q. Rev. Biophys. 38, 291–301 (2005) [CrossRef] .
11. W. J. Greenleaf and S. M. Block, “Single-molecule, motion-based DNA sequencing using RNA polymerase,” Science 313, 801 (2006) [CrossRef] [PubMed] .
12. P. Kukura, H. Ewers, C. Müller, A. Renn, A. Helenius, and V. Sandoghdar, “High-speed nanoscopic tracking of the position and orientation of a single virus,” Nat. Methods 6, 923–929 (2009) [CrossRef] [PubMed] .
13. M. A. Taylor, J. Janousek, V. Daria, J. Knittel, B. Hage, H.-A. Bachor, and W. P. Bowen, “Biological measurement beyond the quantum limit,” Nature Photon. (2013) [CrossRef] .
14. S. Yamada, D. Wirtz, and S. C. Kuo, “Mechanics of living cells measured by laser tracking microrheology,” Biophys. J. 78, 1736–1747 (2000) [CrossRef] [PubMed] .
15. E. N. Senning and A. H. Marcus, “Actin polymerization driven mitochondrial transport in mating S. cerevisiae,” Proc. Natl. Acad. Sci. USA 107, 721–725 (2010) [CrossRef] [PubMed] .
16. I. M. Tolić-Nørrelykke, E.-L. Munteanu, G. Thon, L. Oddershede, and K. Berg-Sørensen, “Anomalous diffusion in living yeast cells,” Phys. Rev. Lett. 93, 078102 (2004) [CrossRef] .
17. T. Franosch, M. Grimm, M. Belushkin, F. M. Mor, G. Foffi, L. Forró, and S. Jeney, “Resonances arising from hydrodynamic memory in Brownian motion,” Nature 478, 85–88 (2011) [CrossRef] [PubMed] .
18. R. Huang, I. Chavez I, K. M. Taute, B. Lukić, S. Jeney, M. G. Raizen, and E.-L. Florin, “Direct observation of the full transition from ballistic to diffusive Brownian motion in a liquid,” Nature Phys. 7, 576–580 (2011) [CrossRef] .
19. D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, and P. Zoller, “Cavity opto-mechanics using an optically levitated nanosphere,” Proc. Natl. Acad. Sci. USA 107, 1005–1010 (2010) [CrossRef] [PubMed] .
20. T. Li, S. Kheifets, and M. G. Raizen, “Millikelvin cooling of an optically trapped microsphere in vacuum,” Nature Phys. 7, 527–530 (2011) [CrossRef] .
21. M. A. Taylor, J. Knittel, M. T. L. Hsu, H.-A. Bachor, and W. P. Bowen, “Sagnac interferometer-enhanced particle tracking in optical tweezers,” J. Opt. 13, 044014 (2011) [CrossRef] .
22. J. W. Tay, M. T. L. Hsu, and W. P. Bowen, “Quantum limited particle sensing in optical tweezers,” Phys. Rev. A 80063806 (2009) [CrossRef] .
23. P. Kwee and B. Willke, “Automatic laser beam characterization of monolithic Nd:YAG nonplanar ring lasers,” Appl. Opt. 476022–6032 (2008) [CrossRef] [PubMed]
