## Abstract

The accurate calibration of the force constant of the probe in atomic force microscopy and optical tweezers applications is extremely important for force spectroscopy. The commonly used silicon detectors exhibit a complex transfer function for wavelengths >850 nm, which limits the detection bandwidth leading to serious errors in the force constant determination. We show that this low-pass effect can be compensated for using the frequency response of the detector. This is applicable for calibrations in both atomic force microscopy and optical tweezers. For optical tweezers an additional correction method is discussed based on fitting an expression in which the low-pass characteristics are already accounted for.

©2006 Optical Society of America

## 1. Introduction

In addition to high resolution imaging applications the atomic force microscope (AFM) has proven its usefulness in force spectroscopy in the (sub)nN force range where the microscope is used to measure mechanical properties of biomolecules and polymers and the strength of interand intra-molecular bonds [1, 2, 3, 4, 5]. In the (sub)pN force range a widely used force spectroscopy microscope are the optical tweezers (OT) used for measuring the mechanical properties and unfolding pathways of single molecules and for determining the forces involved in molecular interactions [6, 7, 8, 9, 10].

The appropriate calibration of the force constant of the probe in these microscopes is important for force spectroscopy. The probe, a cantilever in AFM and a micron-sized bead in OT, is subjected to thermal collisions of the surrounding medium that can be considered as a ‘white-noise’ thermal force, resulting in the so-called Brownian motion of the probe. The power spectral density (*S _{x}*) of the deflection signal therefore reflects the frequency response of the probe. Based on the equipartition theorem $\frac{1}{2}{k}_{\mathit{pr}}\u3008{x}^{2}\u3009=\frac{1}{2}{k}_{b}T$ [11], such that the force constant

*k*of the probe can be derived by integrating the power spectral density yielding 〈

_{pr}*x*

^{2}〉, which is the variance of the probe position.

*T*is the absolute temperature and

*k*is Boltzmanns constant.

_{b}An important issue for both applications is that silicon-based position detectors, the type of detector that is often required, can behave as a low-pass filter at frequencies >5 kHz for near-infrared light (>800 nm). Themagnitude of this effect is a function of wavelength, incident light power and the bias voltage [12, 13, 14, 15]. Due to the low-pass effect the integral of *S _{x}* (yielding 〈

*x*

^{2}〉) will be underestimated resulting in an overestimated force constant. The LED-wobbler developed recently allowed us to determine the frequency response of a position detector up to 600 kHz with high accuracy [15]. This frequency response was used to compensate

*S*. For optical tweezers an additional correction method is discussed based on fitting an expression in which the low-pass characteristics are already accounted for.

_{x}## 2. Results

The frequency response of the Brownian motion of a cantilever (MSCT-AUHW, Cantilever B, Veeco, New York) was measured in air in a custom-built AFM [16] with a 785 nm laser diode. The same cantilever was transferred to a hybrid AFM-confocal optical microscope [17] that uses a 1050 nm laser diode to prevent induced photo bleaching during simultaneous fluorescence (at 800 nm) and topography measurements. Both setups include the same type of detector (UDT, Spot9-DMI). Calibration of the detector by determining the deflection sensitivity (nm/Volt)is carried out in a range of 10–100 Hz. The power spectral densities for AFM shown in this letter were corrected for the background noise that was determined by switching off the laser diode.

In Fig. 1 we show the power spectral density function (*S _{x}*) for the same cantilever as acquired in both setups (blue and red curve). The

*S*at 1050 nm corrected for the frequency response of the detector is also given (green curve). The detector response (gain versus frequency) obtained with the LED-wobbler is depicted by the black curve.

_{x}According to the equipartition theorem [11] each mode of vibration of the cantilever equals an energy $\frac{1}{2}{k}_{b}T$ and therefore the first mode of vibration equals $\frac{1}{2}{k}_{pr}\u3008{x}^{2}\u3009$. For a high Q-factor 〈*x ^{2}*〉 can be approximated by integrating around the first resonance peak. Integrating the spectra (Fig. 1) from 6–21 kHz we found a stiffness of (3.8±0.2)·10

^{-2}N/m for the probe acquired at a wavelength of 785 nm. At a wavelength of 1050 nm the calculated stiffness was (11.1±0.6)·10

^{-2}N/m. That means that due to the low-pass effect the stiffness is overestimated by a factor of 3. After correcting Sx for the response of the detector the stiffness was found to be (3.5±0.2)·10

^{-2}N/m, within 10% of the value at 785 nm.

In contrast to AFM, in OT applications the power spectral density *S _{x}* looks different (see Fig. 2, black circles) due to the fact that in this case the damping forces are dominant. The same method of integrating

*S*can be applied, but in this case it is also possible to fit the spectrum with a Lorentzian function. From the cut-off frequency found here the trap stiffness can be deduced, even when the detector is not calibrated. The Lorentzian is given by

_{x}where *D* is the diffusion constant, *f* the frequency and *f _{c}* the cut-off frequency. The trap stiffness is calculated according to

*k*=2

_{tr}*πγ0 fc*with

*γ0*Stokes friction coefficient for a sphere [18]. In the Lorentzian curve-fitting procedure hydrodynamic and aliasing corrections were included [19].

We used three different position detectors to acquire spectra for a trapped 2.67 *µm* polystyrene bead in an optical tweezers setup (Bangs Labs, Fishers, IN) with a 2.5 W Nd:YAG

laser (Coherent, Compass 1064–2500MN, Santa Clara, CA) operating at a wavelength of 1064 nm [9]. In contrast to AFM, the light power on the detector in OT applications can differ for various trapped beads and therefore the low-pass effect. In transmission-based OT the variation in intensity is approximately 10–20% and for a reflection-based setup it can be a factor of 2 [20]. To investigate the effect of light power, spectra were recorded at powers of 500 and 150 *µW*. Reduction of the light power was achieved by the use of a neutral density filter in front of the detector through which the optical trap was unaffected. The detectors used were the DL100-7-KER (Pacific Silicon Sensor), the DLS10 (UDT) and the SPOT-9DMI (UDT). Fig. 2 shows the power spectral density of a trapped bead acquired with the SPOT9-DMI (black circles).

First we demonstrate the influence of applying no corrections by fitting directly the Lorentzian function, given by Eq. 1 to the acquired spectra. The resulting Lorentzian fit is shown in Fig. 2 (green curve). The fit results for the uncorrected *S _{x}* for all three detectors at the two light powers are given in the second column of Table 1. If a detector that reveals a low-pass effect starting at lower frequencies was used, a lower value for the trap stiffness was found.

As for AFM applications *S _{x}* can be corrected for the frequency response of the detector as measured with the LED wobbler (at 1070 nm and the same light powers). After correction

*S*is curve-fitted again with the Lorentzian function (Eq. 1). For all spectra the fit-range was 110 Hz–10 kHz. The resulting optical trap stiffness values are given in the third column of Table 1 showing that the trap stiffness values deduced with the detectors are in good agreement with a variation of <2%.

_{x}For OT a second method is available based on fitting an expression included in the Lorentzian function to account for the low-pass effect. Actually this expression for the low-pass effect, given by Eq. 2, is derived by Berg-Sørensen *et al*. (2003) to account for the low-pass effect and to circumvent the requirement to determine the total light power. Before discussing this correction method we validate the expression by fitting it to the frequency response of the detector as measured with the LED wobbler. The expression is given by

In this equation *α* describes the fraction of light absorbed in the depletion layer, which is a function of the effective thickness of the depletion layer and therefore a function of wavelength, bias voltage and light power. Since the fraction of light absorbed outside the depletion layer has to diffuse towards the depletion layer it is low-pass filtered with a characteristic frequency *f _{diode}*. Depending on the detector

*f*is in the order of 5 kHz. In Fig. 3 we show frequency responses of the DL100-7-KER, the DLS10 and the SPOT-9DMI for light powers of 50 and 500

_{diode}*µW*. Additionally the fit based on Eq. 2 for each frequency response is shown. The model describes the effect quite well for lower frequencies. Above a certain frequency the model deviates significantly from the measured frequency response. The fit range was increased stepwise until the model did differ more than 5% from the value at the maximum frequency of the selected range (45 kHz for the DL100-7-KER, 15 kHz for the DLS10 and 10 kHz for the SPOT9DMI). For earlier results reported in the literature [12, 14] the frequency range was below these critical frequencies; consequently, this effect has not been observed before. For frequencies much higher than the characteristic frequency

*f*the second term in Eq. 2 approaches zero and the term,

_{diode}*α*becomes dominant. Thus Eq. 2 will always be able to fit a low-pass filtered system up to a certain frequency, as long as the slope of the attenuation part is ≤6 dB/octave.

^{2}Now we use the Lorentzian function with the expression for the low-pass effect included, to fit to the power spectral densities for the 2.67 *µm* trapped bead trapped in the OT setup discussed previously. For this ‘new’ Lorentzian there are four fit parameters, *D* and *f _{c}* describing the dynamics of the trapped bead and a and

*f*describing the detector response. Again hydrodynamic and aliasing corrections are included. For all spectra the fit range was 110 Hz–10 kHz. In Fig. 2 (red curve) the result is shown for this correction method. The resulting key fit parameters values are given in the last column of Table. 1. The trap stiffness values found for each detector are in good agreement. The trap stiffness found for the DL100-7-KER in the case no correction method was used is remarkably close to (<5%) the values found for the two corrections methods, indicating the small low-pass effect of this detector. For the other two detectors the trap stiffness is strongly underestimated in case of no corrections. In separate experiments the trap stiffness values found with the two correction methods were compared with another calibration technique for OT (sinusoidal driving force [18]) where the trap stiffness did not differ more than <5% from this value.

_{diode}Finally we investigated the overall effect of the light power on the deduced trap stiffness using the first correction method discussed for both AFM and OT. Instead of correcting the spectra acquired at 500 *µW* with the frequency response of the detector at the appropriate power we used the detector response determined at 50 *µW*. For the DL100-7-KER we found a stiffness of 590 *pN/µm* instead of 577 *pN/µm* using the appropriate correction light power. For the DLS10 we found 654 *pN/µm* instead of 595 *pN/µm* and for the SPOT-9DMI 584 *pN/µm* instead of 591 *pN/µm*. Although the difference in the power for the correction curves was a factor of 10, we found for both the DL100-7-KER and the SPOT9DMI an acceptable change of <3%. For the DLS10 a change of 10% was found.

## 3. Conclusion

Due to the low-pass effect of silicon position detectors large errors are found in the force constant determination in AFM and OT applications. However we have shown that the power spectral density can be corrected for the frequency response of the position detector as determined with the LED wobbler. Applying the equipartition theorem for AFM or curve-fitting a Lorentzian in OT the correct force constant can be determined. For variations of the light power within a factor of ten we found a variation of <3% for the force constant for the DL100-7-KER and the SPOT9-DMI. This makes acquisition of the frequency response of these two detectors for various powers redundant.

For OT an additional correction method is available by including a model for the detector low-pass effect into the Lorentzian curve-fit. However we have shown that this model partially describes the low-pass effect of the detector and is thus only applicable up to a maximum frequency range (~15 kHz). Knowledge of the frequency response of the detector beforehand is thus required.

## Acknowledgements

This work is supported by the MESA+ institute for nanotechnology.

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