We report a novel method for characterizing the stiffness of white light supercontinuum tweezers, in which the nonlinear photonic crystal fiber used for supercontinuum generation is also utilized as an effective confocal pinhole to track the motion of a trapped bead and as a scan head to realize rapid scanning of the optical trap. By measuring the phase of the bead’s motion in following the trap, a lateral stiffness value of about 7.9 μN/m was obtained with supercontinumm power of about 75mW. Our technique can potentially allow for trap stiffness calibration along an arbitrary direction in three dimensions.
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Optical tweezers have found numerous applications in biological science, materials science, and a variety of other fields in the past two decades due to their unique ability to trap and manipulate microscopic objects without contact [1–3]. Different types of optical sources such as single-wavelength laser or white light supercontinuum (SC) , and continuous-wave or pulsed laser sources  have been explored for optical trapping. SC tweezers, in particular, are of interest in that broadband spectroscopic measurement (e.g., broadband optical scattering  and multiplex coherent anti-Stokes Raman spectroscopy [7,8]) can be performed in situ to characterize a trapped object. In order to quantitatively understand the trapping performance of the SC tweezers, it is important to calibrate their trap stiffness. There are many techniques developed for calibrating narrow-band laser tweezers [9–12], which may also be adapted to calibrate SC tweezers. For instance, SC trap stiffness can be obtained from the position histogram of a trapped bead undergoing Brownian motion . Alternatively, one of the authors has developed an active approach, in which an acousto-optic modulator (AOM) was utilized to oscillate an optical trap at a range of frequencies . Trap stiffness can be obtained from the phase information of the motion of a trapped bead and direct positional measurement is not needed. However, an AOM has limited bandwidth and is not suitable for modulating SC. More importantly, nearly all existing methods can often only characterize the trap stiffness along either lateral or axial direction without significant change of the measurement setup. Here we extend the method in Ref. . and demonstrate a technique to measure SC trap stiffness by exploiting the very photonic crystal fiber (PCF) for SC generation as both an effective confocal pinhole and a scan head. We present the lateral stiffness calibration of an SC trap. By scanning the PCF in three dimensions, our method can have the potential to calibrate the stiffness of a complex optical trap along any arbitrary direction in three dimensions.
Let us consider the motion of a bead in a periodically oscillating trap. It is modeled as one-dimensional movement in a Newtonian fluid. By neglecting the inertial term, which is small under our experimental condition, we have the following equation of motion.Eq. (1) is given byEq. (2), i.e., .
3. Experiment setup and calibrations
The schematic diagram of our experimental system is shown in Fig. 1 . Briefly, femtosecond pulses from a mode-locked laser (KMLabs) were coupled into a nonlinear PCF (Crystal Fibre NL-2.0-770, length: ~40cm) to generate SC. A typical spectrum of the generated SC is shown in Fig. 2(a) , which covers a wavelength range from about 550 nm to 1200 nm. The output end of the PCF was cleaved at an angle in order to minimize reflection and was mounted on a home-modified speaker. During our experiments, the speaker and hence the PCF can oscillate when driven by a sinusoidal signal generated from a function generator. The SC was then collimated by an objective. It was subsequently steered upward by a mirror and tightly focused by an oil-immersion objective (Nikon Plan Apo 100 x 1.4 NA) to form an inverted SC trap. An imaging setup was also built for real-time monitoring of the trapping process. It is worthwhile to point out that the PCF plays a unique triple role here. Firstly, it is the nonlinear medium for generating SC. Secondly, it enables rapid scanning of the SC beam and hence the optical trap. Finally, it serves as an effective confocal pinhole to detect the position of a trapped bead relative to the center of the trap. Note that only the backscattered light from the trapped bead, which was located near the focal point of the objective, can be efficiently coupled back to the PCF and therefore detected. This confocal effect also allows for effective suppression of undesired background reflection and gives rise to an improved signal-to-noise ratio. To confirm this effect, we placed a mirror near the focal plane of the objective lens, mechanically scanned it along the axial direction at a step of 100 nm, and measured the power of the reflected light after it was re-coupled into the PCF and separated from the incoming femtosecond pump beam with a beam-splitter. The measured axial response curve is presented in Fig. 2(b), in which the horizontal axis shows the axial position of the mirror while the vertical axis represents the normalized power of the measured signal. The full width at half maximum (FWHM) of the response curve is about 2.6 μm, a relative large value likely as a result of the chromatic aberration of our system due to the use of a broadband supercontinuum.
In order to devise an optimal scheme for the driving voltage, we characterized the oscillation amplitude of the speaker. An imaging system was set up to image the PCF output end onto a CCD camera. A series of frames were first acquired from the CCD camera. By summing them up, a time-integrated image representing the trajectory of the fiber tip was obtained. Next, a threshold was properly chosen to binarize the image. The leftmost, rightmost, uppermost, and lowermost “on” pixels were then identified and the horizontal and vertical dimensions of the trajectory were estimated. Figure 2(c) shows the result in a range of frequencies from 2 Hz to 595 Hz. At each frequency, we adjusted the voltage output from the function generator so that the dimension of the PCF trajectory was close to that at other frequencies. Note that the speaker has a resonant frequency at around 120 Hz, which leads to a much larger amplitude compared with other off resonant frequencies. Since the function generator has a minimum output voltage of 50 mV, the amplitude near the resonance frequency was not equalized. The oscillation amplitude of the fiber tip can be estimated by subtracting the dimension of the trajectory by a pre-calibrated number of pixels corresponding to the PCF fiber size. It is estimated to be about 15 μm near the resonance frequency and less than 1 μm at the off-resonance frequencies.
We then proceeded to calibrate the phase response of the speaker system in relation to the driving signal. To this end, we inserted a mirror to redirect the SC beam towards a spatial filter comprising an objective lens (10X, NA = 0.25, focal length: 16.5 mm) and a pinhole (nominal diameter: 20μm) placed at its focal plane. The transmitted light was modulated when the speaker was driven by a sinusoidal signal. The second harmonic phase of the detected signal was measured by a lock-in amplifier. Figure 2(d) shows the measured phase at different driving frequencies. Essentially, the measurement calibrates the system phase of the oscillating SC trap.
4. Trap stiffness measurement
To characterize the lateral stiffness of the SC trap, we scanned the trap transversely by applying a sinusoidal driving signal to the speaker. The average power of the SC beam measured before the oil-immersion objective was around 75 mW. A sample cell was placed near the focal point of the objective. It contained 2-μm-diameter silica beads suspended in water and sandwiched between a pair of microscope cover glass (with the edges sealed by vacuum grease). A single silica bead was then identified and trapped. The backscattered light was collected by the same objective and coupled back into the PCF. It was subsequently separated from the incoming femtosecond beam by a pellicle beamsplitter (Thorlabs BP108, 92% transmission 8% reflection), and detected by a photo-detector (Thorlabs PDA36A). Finally, the detected signal was sent to a lock-in amplifier to measure its second harmonic phase in relation to the sinusoidal driving signal. The time constant of the lock-in amplifier was set to 1 s and a low-pass filter parameter of 24 dB/octave was chosen. The frequency of the driving signal was varied from 2 Hz to 597 Hz with a step of 7 Hz. At each frequency, four groups of phase data were sequentially collected with an interval of five seconds between them. A total of 100 data samples were thus obtained and averaged. Additionally, after changing to a new frequency a 40-second delay was applied to ensure that the lock-in amplifier converged to a stable phase output. It should be noted that the strategy of long measurement time was employed to aggressively ensure the accuracy of the phase measurement [c. f. the error bar in Fig. 3(a) ], which can be shortened by reducing the waiting time and adjusting the parameters (e.g., time constant) of the lock-in amplifier without significantly compromising the signal-to-noise ratio of the measurement. The phase lag of the bead relative to the trap can then be obtained by subtracting the calibrated system phase from the measured second harmonic phase of the backscattered signal and dividing the result by two. Its cotangent value is plotted in Fig. 3(b) as a function of the driving angular frequency. The data can be fitted with a linear curve with a slope of 2.34 ms, from which the trapping stiffness is calculated [c.f. Eq. (2)]. The obtained transverse trapping stiffness is estimated to be about 7.9 μN/m. Note that the measurement data also exhibits notable deviation from an ideal linear relationship. Several factors may have likely contributed to the noise in the measurement, which include the spectrum and power instability of the supercontinuum over a long measurement time, and the deviation of the speaker vibrational mode from an ideal one-dimensional harmonic oscillation.
We have demonstrated a novel method for determining the stiffness of an SC trap. The nonlinear photonic crystal fiber in our technique serves not only as a nonlinear medium for generating SC but also as an effective confocal pinhole for tracking the motion of a trapped bead and as a scanning head for realizing high-speed sinusoidal oscillation of the SC trap. Our method can be further extended to measure the axial stiffness, and potentially allow for characterization of the stiffness of an optical trap in three dimensions. Although our current work is based on SC tweezers, the same technique may also be adopted for traditional optical tweezers using narrow-band laser sources.
This work is supported by the National Science Foundation (Award# ECCS0547475). Z. Zhang would like to thank the China Scholarship Council for the financial support (CSC File No. 2008625013).
References and links
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