1. Introduction

With optical tweezers, picoNewton-range forces can be exerted and measured, which has found widespread application in a large variety of fields [1]. To enable quantitative force measurements, the trap stiffness κ of an optical trap needs to be calibrated, i.e., the linear relation – valid over a certain distance – between the force F exerted on a particle in the trap and the displacement Δx of this particle with respect to the trap center has to be determined (κ = –F/Δx). Unlike position detection with a photodiode, camera imaging enables the tracking of many particles simultaneously. However, as shown in detail by Wong and Halvorsen [2], motion blur due to the finite acquisition time of a camera affects the detected particle position and thus the apparent stiffness κ, especially for stiff traps. The authors describe the effect of blur on the measured variance <x ²>, which is proportional to 1/κ, and give an experimentally verified correction function. The use of <x ²> as a measure of trap stiffness, however, is very sensitive to drift and noise, as these will increase <x ²> thereby underestimating κ. By contrast, when using power spectral analysis for trap calibration, the contributions of drift and noise can be readily identified and omitted before analysis. Power spectral analysis is especially useful for our holographic optical tweezers instrument, in which the addressing of the spatial light modulator pixels introduces apparent trap position modulations at specific frequencies [3,4].

In power spectral analysis [5], the power spectrum is fit with a Lorentzian, yielding the corner frequency f_c, which is proportional to the trap stiffness (κ = 2π γ f_c, with γ the viscous drag coefficient). Although the bandwidth of a high-speed camera (several kHz) is much smaller than that of a photodiode, it does allow for trap calibration using power spectral analysis, as demonstrated by Otto et al. [6]. However, they used relatively low trap stiffnesses, with f_c less than a quarter of the Nyquist frequency f_Nyq, and did not account for the effects of blur. In Ref [2], a formula is presented for a blur-corrected fit to the power spectrum that also accounts for aliasing, but although the authors investigated the quality of the fit (for f_c~0.25 × f_Nyq), they did not experimentally validate the obtained corner frequency f_c.

In this work, we investigate the use of high-speed camera power spectral analysis for the calibration of very stiff optical traps with f_c > f_Nyq. To do so, we fit the power spectra taking blur, aliasing and position detection error into account, to obtain f_c for a trapped particle at different laser powers. We compare the results to those from power spectra obtained simultaneously using a position sensitive photodiode (PSD) and find good agreement. To demonstrate the use of this technique to calibrate traps that display spatial modulations and drift, we apply it to calibrate multiple holographic optical traps (HOTs) simultaneously, finding the expected linear relation with laser power. Our work demonstrates, for the first time, the viability of using camera power spectral analysis for calibration of very stiff optical traps, with f_c up to at least 3 × f_Nyq.

2. The effect of blur, aliasing, and detection error on measured power spectra

The double-sided power spectrum S_xx of the x position of a particle trapped in a harmonic potential well as a function of the frequency f is described by a Lorentzian [5]:

 

Fig. 1 Theoretical power spectra with f_Nyq = 1250Hz and W = 0.4ms for f_c = 0.2 × f_Nyq (a), 0.8 × f_Nyq (b) and 2 × f_Nyq (c), showing the effects of aliasing and blur. Plotted: pure Lorentzian (Eq. (1); black solid), aliased Lorentzian (black dashed), blurred (Eq. (2); solid red) and blurred aliased spectrum (Eq. (3); dash-dotted red). In blue, the blurred aliased spectra are plotted for the same f_Nyq and f_c, but for W = 0.1ms.

Download Full Size | PDF

For position detection with a finite acquisition time W, e.g. camera imaging, the blur due to this integration time affects the acquired spectrum and has to be taken into account [2]:

In effect, as a result of blur the power spectrum is increasingly underestimated for higher frequencies (thin red lines in Fig. 1).

A limited data acquisition bandwidth gives rise to aliasing of a power spectrum S. With the Nyquist frequency equal to half the sampling frequency, f_Nyq = 0.5 × f_s, we find:

Due to aliasing (dashed lines in Fig. 1), the power spectrum is overestimated. As can be seen, the effect of aliasing is more pronounced for higher f_c with respect to f_Nyq. For the blurred spectrum, however, the effect of aliasing is limited for f_s = 1/W (red dash-dotted lines), but increases for 1/W > f_s (solid blue lines: 1/W = 8 × f_Nyq). The relation among the parameters f_Nyq, f_c and W affects the extent to which an acquired power spectrum differs from a pure Lorentzian. As can be seen from Fig. 1, the greater f_c and 1/W are compared to f_Nyq, the larger the deviation of the spectrum from Eq. (1), suggesting that it becomes more important to use the adjusted Lorentzian (Eq. (2)) to obtain the correct f_c.

Any particle tracking technique has an inherent position detection error. Assuming that this error is independent of f, an error with variance ε ² will introduce a noise level of ε ²/2f_Nyq to the power spectrum. Taking the effects of blur, aliasing and tracking error into account, the measured double-sided power spectral density is described by:

3. Experimental setup and fitting of measured power spectra

In our holographic optical tweezers setup (Fig. 2(a) ), described in detail in [3], we use a high-speed CMOS camera (PCO, 1200 hs, 1280 × 1024 pixel²) to image polystyrene spheres (Spherotech) trapped in water. Particles were trapped in the center of our sample cell (~40 μm thick), which consisted of two cover slips closed off with candle wax. Due to our current illumination, we limited the acquisition rate to 2500 frames/s. Image correlation was used for particle position detection [3]. Laser powers indicated in the text were measured in between mirror M1 and lens L2 (see Fig. 2(a)); we estimate ~50% of this power is delivered to the trapping chamber.

 

Fig. 2 (a) Simplified schematic of the setup. (b) Camera power spectrum (average over 15 spectra; f_Nyq = 1250Hz) for a stuck particle. The median noise power from 400 to 600Hz is taken as the noise level: 7.2 × 10⁻⁴ nm²/Hz in x (shown) and 14.5 × 10⁻⁴ nm²/Hz in y (not shown).

Download Full Size | PDF

In the work presented here we use power spectral analysis of position data acquired with a PSD and with a camera. Before fitting the power spectra to obtain f_c, distinctive noise peaks are eliminated. For the camera data, we average over 5 to 15 power spectra and fit to the measured frequency range (f = 0, f_Nyq = 1250 Hz). For the position sensitive photodiode (f_Nyq = 16384 Hz) we average over >100 spectra. We fit only up to f = 5 kHz, due to the decreased sensitivity of our silicon-based PSD (OSI Optoelectronics, DL-10) above ~5 kHz for 1064 nm light [7].

Although Eq. (4) in principle holds for either of the two detection techniques, its application varies due to the differences between the techniques regarding data acquisition parameters such as bandwidth and integration time.

For photodiode position detection, the bandwidth is typically large, with f_Nyq>10 kHz. As a consequence, the noise level due to the detection error, ε ²/2f_Nyq, is usually not significant, even for high trap stiffness. In addition, the detection integration time is on the order of (sub)μs, making the effect of blur negligible. Because of the effective low-pass filtering by our photodiode, we do not include aliasing and therefore fit our single-sided PSD power spectra, measured in V²/Hz, with:

For camera position detection, the conditions are different. The pixel integration time W is on the order of ms and blur has to be taken into account. In addition, with a bandwidth of typically no more than several kHz, both the noise level and aliasing cannot a priori be neglected. Thus, we use f_c, ε ² and/or γ as free fitting parameters (see below) to fit the camera power spectra with the single-sided version of Eq. (4). Even though γ can be calculated, using it as a fitting parameter allows us to account for possible errors in the estimate of particle radius and medium viscosity. The conversion from pixels to μm is constant (and known) for all of our measurements, so no free conversion parameter like β is necessary. We stop the summation in Eq. (4) at n = ± 3, which we found to be sufficient to account for the effect of blur.

For f_c>>f_Nyq (Fig. 1(c)), the power spectrum shows so little distinct curvature that f_c cannot easily be obtained from the shape of the spectrum. In addition, for higher trap stiffness, the motion of the particle in the trap decreases and the apparent motion due to the detection error ε ² becomes increasingly more significant. Therefore, at these higher trap stiffnesses, f_c can only be determined when both γ and ε ² are known. The viscous drag coefficient γ can be obtained from a power spectrum for which f_c<<f_Nyq (Fig. 1(a)). In this regime, the relative contribution of ε ² is very limited. The plateau region of the power spectrum at low frequencies and the Brownian motion at high frequencies are clearly visible and fitting will yield both f_c and γ. Alternatively, γ can be obtained by fitting to only the high-frequency region, since for f>4 × f_c, the tail of the power spectrum is practically independent of f_c. It is clear that the range of f_c over which this calibration method gives valid results will largely be determined by the accuracy to which γ and ε ² are known.

The accuracy of f_c can be determined by least-squares analysis of power spectra in which data points are statistically independent and normally distributed, criteria that are relatively easy to meet for PSD-derived data [5]. For camera measurements, however, the much lower f_Nyq means that measurements need to be made over orders of magnitude longer time to achieve a comparable number of data points. Instead, we average our camera data (25,000 to 75,000 points for each laser power) over 5 to 15 spectra to reduce noise, but do not apply additional blocking.

4. Experimental results and discussion

First, we obtained single-sided camera power spectra for a stuck 2.10-µm-diameter particle, from which we determined the position detection noise levels ε_x ²/f_Nyq = 7.2 × 10⁻⁴ nm²/Hz (ε_x ² = 0.90 nm²; Fig. 2(b)) and ε_y ²/f_Nyq = 14.5 × 10⁻⁴ nm²/Hz (ε_y ² = 1.8 nm²). Large peaks in the power spectra, small or absent in power spectra of trapped particles, are from mechanical noise coupled in through the sample surface. As ε ² depends on the illumination and on W, the images were taken under the same conditions as used in the following experiment (W = 0.4 ms).

Subsequently, we obtained power spectra for a 2.10 µm particle in a single trap (SLM off) using the PSD (for laser powers 3.3, 6.1, 12.7, 27, 52, 53 (2 ×), 92, 105, 216 and 416 mW) and fit these with Eq. (5) with f_c and γ/β as independent fitting parameters for each spectrum. The normalized PSD power spectra in x and their fits (red lines) for three laser powers are shown in Fig. 3(a) .

 

Fig. 3 (a and b) Measured power spectra in x with fits (red lines) for three laser powers for (a) PSD (fit: Eq. (5)) and (b) camera (Eq. (4)). In the power spectra shown in (b), before removal of noise, mechanical noise peaks can be seen. In (c) a camera spectrum in y for 105 mW is shown, fit with Eq. (5) (blue line; f_c = 583 Hz) and with Eq. (4) (red; f_c = 774 Hz).

Download Full Size | PDF

Simultaneously, we acquired particle images with our camera, and from these, determined power spectra. The high-frequency (f>4 × f_c) region of a power spectrum depends very little on f_c. Therefore, to obtain γ, we first fit this region of spectra at low laser power (6.1 mW) with the error fixed at ε_x ² = 0.90 nm² and ε_y ² = 1.8 nm². (The parameter f_c was fixed at an initially estimated value obtained from fitting to the whole frequency range.) We found γ_x = 1.93 × 10⁻⁸ kg/s and γ_y = 1.96 × 10⁻⁸ kg/s, which are close to the expected value at 20 °C of 1.98 × 10⁻⁸ kg/s. Next, we fixed γ at these measured values and fit spectra at all laser powers with f_c as the only free fitting parameter. Figure 3(b) shows three camera power spectra and their fits. In Fig. 3(c), a camera power spectrum for 105 mW and its fit (red) are shown, together with a pure Lorentzian fit (Eq. (5); blue), with f_c and γ as free fitting parameters. Both curves seem to fit the data well, yet the corner frequencies found differ greatly (f_c = 774 and 583 Hz, respectively), demonstrating the significant differences in obtained corner frequencies that result from using Eq. (4) vs. Equation (5) to fit power spectra of stiffer traps.

Figure 4 shows the values obtained for f_c from camera images and from PSD data for all measured laser powers. We find agreement between PSD and camera results (within 10%), up to f_c = 2.9 × f_Nyq (at 416 mW). At this high trap stiffness (κ_x = 432 pN/µm), the height of the plateau is 23.7 × 10⁻⁴ nm²/Hz. Therefore, 30% of the particle’s apparent motion is due to the detection error ε ², making it quite remarkable that an accurate f_c can be obtained. Also plotted are the results from pure Lorentzian fits to the camera spectra. As can be seen, for higher laser power, f_c is increasingly underestimated when using Eq. (5) rather than Eq. (4) to fit power spectra from the camera.

 

Fig. 4 (a) Fit results for f_c as a function of laser power in x (solid symbols) and y (open), for camera and PSD. Also shown are the results for pure Lorentzian fits to the camera data (Eq. (1)). (b) Expanded view of the low laser power region. Lines are linear fits forced through zero to camera results up to 53 mW (x: solid; y: dashed).

Download Full Size | PDF

In this study, we have not accounted for possible effects of heating and have assumed the viscosity and hence γ to be constant for all laser powers. However, due to the absorption of laser light by the solvent and resultant heating, the viscosity is expected to decrease and therefore f_c to increase with laser power [8,9]. As a result, even though κ is independent of temperature, f_c is expected not to be [9]. Based on results given in Ref [8], we estimate the temperature increase for our highest laser power to be ΔT<3K, resulting in a decrease in γ of <7%. Because γ/β is a free fitting parameter, able to capture temperature changes, our fits should correctly return f_c, which we expect to increase from heating effects by <7%. For the camera data, we expect to observe a smaller increase in fit values of f_c with increasing power, because the use of a fixed γ obtained at low laser power results in underestimating the true f_c. The lines in Fig. 4 are linear fits to the x and y camera results at low laser power (up to 53 mW) for which we expect the possible effect of heating to be minimal. We see that for laser powers >100mW, f_c lies above this linear fit for both x and y. Further investigation is needed to confirm whether this is indeed a result of a decrease in γ due to heating.

It is clear that including detection error in the fits is critical for fitting power spectra of high-stiffness traps. When no camera data for a stuck particle under similar conditions are available, an alternative estimate of ε ² can be found by fitting a spectrum whose shape is predominantly determined by the corner frequency f_c, but to which the detection error ε ² nonetheless has a significant contribution. As mentioned before, γ can be found from the high-frequency region of a power spectrum at relatively low laser power. Now, by fitting two power spectra iteratively, one at low and one at high laser power, we can obtain both ε ² and γ. Because the power spectra at the highest laser power show very little curvature (416 mW, see Fig. 3(b)), making it hard to independently find f_c, we used the spectra for 6.1 and 216 mW and found ε_x ² = 0.83 nm² (γ_x = 1.93 × 10⁻⁸ kg/s) and ε_y ² = 1.0 nm² (γ_x = 1.95 × 10⁻⁸ kg/s), which are within a factor of two of the results for the stuck particle.

Finally, we trapped two particles (2.10 and 3.17 μm in diameter) in two holographic traps separated by 8.5 μm and obtained camera power spectra for laser powers 63 (2.10-μm particle only), 122, 244, 353 (3 ×), 520, 613, 711, 806, and 921 mW (Fig. 5(a) ). A particle’s position can be determined with a PSD only when a single particle is trapped, so no PSD power spectra were measured in this set of experiments. As we did not obtain power spectra for stuck particles under the illumination conditions used here, we employed the strategy put forward in the previous paragraph, i.e., iteratively fitting two power spectra to determine ε ² and γ and subsequently fitting all power spectra with f_c as the only free fitting parameter. For the 2.10-μm-diameter particle we used the power spectra for 63 and 806 mW and found ε_x ² = 0.65 nm² (γ_x = 2.07 × 10⁻⁸ kg/s) and ε_y ² = 1.1 nm² (γ_y = 1.98 × 10⁻⁸ kg/s). For the 3.17-μm-diameter particle we used the power spectra for 122 and 921 mW, which gave ε_x ² = 0.73 nm² (γ_x = 3.18 × 10⁻⁸ kg/s) and ε_y ² = 2.3 nm² (γ_y = 2.96 × 10⁻⁸ kg/s). These values for γ compare well to the expected values of γ (γ = 2.99 × 10⁻⁸ kg/s for a 3.17-μm particle at 20 °C). In Fig. 5(b), f_c is plotted as a function of the laser power for x and y, showing a linear relation for both particles. We see no apparent effect of heating.

 

Fig. 5 (a) Power spectrum of camera y positions for a 2.10-μm-diameter particle in a HOT trap (711 mW). Deleted noise peak at 300 Hz is due to SLM pixel addressing [4]. Inset: camera image. Scale bar is 3 μm. (b) f_c in x and y as a function of laser power for both 2.10 μm and 3.17 μm particles. Lines are linear fits forced through zero (x: solid; y: dashed).

Download Full Size | PDF

5. Conclusions

The use of a high-speed camera for position detection combined with power spectral analysis enables simultaneous trap stiffness calibration for multiple particles, while low-frequency noise and noise peaks can be identified and eliminated. To acquire the stiffness κ for a trapped particle at different laser powers, we obtained the corner frequency f_c by fitting our camera power spectra with an adjusted Lorentzian (Eq. (4)) that accounts for motion blur, aliasing, and position detection error ε ², as given in [2]. We compared our results from camera images with results from the PSD and found good agreement, demonstrating that with known γ and ε ² (which can be obtained independently or from a pair of power spectra measured at high and low laser powers), f_c can be determined for any camera power spectrum up to at least f_c = 3 × f_Nyq. In addition, we showed that fitting a pure rather than an adjusted Lorentzian would increasingly underestimate f_c for higher laser power.

We applied our method to simultaneously calibrate multiple holographic optical traps that displayed spatial modulations due to SLM pixel addressing and found the expected linear relation of f_c with laser power. The ability to calibrate multiple very stiff HOT traps simultaneously will find its application in the use of such traps for quantitative force measurements [10].