1. Introduction

Detailed knowledge of the force exerted by a single-beam optical trap on a microsphere [1] and its variation with position [2] provide the theoretical basis for the utilization of optical tweezers as “picotensiometers” [3]. The optical trap acts as an elastic spring, since the force is proportional to the displacement of the sample from its equilibrium position (within a small range; only some 100-200 nm in many practical cases [4–6]). Measuring the position of the sample can thus eventually be used to calculate the force; to be useful, however, it is necessary for this to be carried out with nanometer and millisecond accuracy as well as being integrated into an experimental device that minimizes the different sources of instability (laser, microscope, etc.) [4].

Almost simultaneously, three procedures compatible with these requirements were devised in the early 1990s. Finer et al. [5], relying on previous work by Kamimura and Kamiya [7], utilized a method consisting of imaging the sample on a quadrant photodetector (QPD), which replaced the video camera, and using the trapping laser for illumination. Although the instrumental error was notably reduced (from ~10 nm [4] to ~1 nm [7]) and the acquisition rate increased to 4 kHz [5], the method requires repeated and delicate alignment.

This problem does not occur in non-imaging methods. Svoboda et al. [8] advanced by adapting the uniaxial differential laser microinterferometer devised by Denk and Webb [9] to be used simultaneously as an optical trap. Their approach was based on the determination of the polarization changes of two overlapping light beams when intercepted by the trapped microsphere. This method, together with the reduction in Brownian noise caused by laser trapping, enabled, for example, the measurement of the processive motion of kinesin at the molecular scale (8 nm) [8] and later of other mechanical properties (maximum force, force–velocity curve, etc.) [6].

Nonetheless, it was the procedure devised by Ghislain and Webb that was to have the greatest impact on the subsequent evolution of the measurement methods. Possibly inspired by the operation of scanning probe microscopes [10], this method measured the deflection of the trapping beam when it traversed the sample [11]. An intensity detector in a non-imaging plane generated a signal that was a function of the overlap between its active area and the deflected light cone, thus enabling precise three-dimensional tracking of the sample without requiring the polarizing optics of the Svoboda et al. approach.

However, using the deflection of the trapping beam to measure the motion of the sample connected the measurements of positions and of momenta. The deflection of the light cone used by Ghislain and Webb naturally contains information on the change in the momentum of the photons, as S. Smith et al. [12] noticed shortly thereafter. If this light is captured with a lens that fulfills the Abbe sine condition, the spatial position of the light distribution after the collecting lens is not merely proportional to the displacement of the sample, but it directly and immediately indicates the force exerted by the trap [13–15].

Measuring forces directly by means of the analysis of the angular distribution of scattered light has many advantageous features [15]. Unfortunately, the light that goes through the sample has to be captured in full, at least nominally, which is notoriously difficult with an optical trap based on a large-aperture beam [13–15]. Smith et al. solved this problem by using two weakly focused counter-propagating beams, but, because of the simplicity and flexibility of single-beam optical traps compared to this much more complex geometry, this direct force method has not generally been adopted. Instead, the indirect route via the harmonic approximation has been the method of choice; specifically, determining positions through back-focal-plane interferometry (BFPI), a method finally proposed by Visscher et al. [16] in 1996 and theoretically explained by Gittes and Schmidt two years later [17]. The result is a setup equivalent to that of Smith et al., but for single-beam traps where the light scattered by the sample is only collected in part. In this case, asymmetries in the far-field distribution of the radiated intensity can only be related to the transverse displacements of the sample, and not directly to trapping forces.

Despite the ideas of Ghislain and Webb being a common ingredient in the genesis of the method of Smith et al. and BFPI, the degree to which the latter incorporates the clear advantages of the method based upon momentum conservation has not been sufficiently studied. The relationship between the two methods has been theoretically highlighted by Gittes and Schmidt in their first-order interference model [17], and exploited specifically for counter-propagating traps by Smith himself [15]. However, the difficulty in correctly measuring the light momentum with large-aperture beams has impeded similar results for gradient traps. This eventually led the two methods to develop in different directions. To the best of our knowledge, only in two unrelated experiments have particular aspects of BFPI been identified as seeming to imply direct measurements of the force in optical tweezers [18,19]. Bartlett and Henderson [18], studying the functional dependence of the elastic constant on different experimental variables, found a linear relationship between stiffness and detector sensitivity (equivalent to our Eq. (6) and Fig. 3, below). More recently, by means of an experimental setup consisting of two traps of different stiffness simultaneously trapping the same sample, Jahnel et al. [19] observed that the range over which their sensor output was proportional to the force is larger than the linear (with position) range of the trap itself.

Following the work of Smith et al., we recently reported the conditions under which the momentum changes of a large-aperture beam can be measured accurately [20], which opens the door to experimentally tackling this question. Here we explore the connection between BFPI and the measurement of the light momentum for single-beam gradient traps. We start by indicating the possibility that there is an extraordinary range of validity for the force measurements and show the existence of a relationship between the calibration constants β and κ. We then proceed to prove that the product of the two factors does not depend on the trapping conditions and, more specifically, that this product corresponds to the calibration factor that converts the detector signals into momentum changes, according to Ref [15]. More importantly, we demonstrate that these observations are not necessarily restricted to fully optimized instruments but are valid more generally, depending only on the proportion of light collected.

We point out that this has clear practical consequences so that the link between BFPI and the method of Smith et al. must always be kept in mind when measuring the force that optical tweezers exert on a trapped particle.

2. Setup

The instrument used here has been briefly described in Ref [20]. and is presented in more detail in the Appendix. It has the same structure as a conventional BFPI system [16]. A high-numerical-aperture lens collects the light deflected by the sample and a photodetector placed in a plane conjugate to its back focal plane (BFP) provides a measurement of the optical force in volts. Some specific requirements ensure the proper measurement of the light momentum. In particular, we use a position sensitive detector (PSD), an aplanatic, long-focal-length condenser lens with a numerical aperture (NA) of 1.4 and the sample is kept close (< 30 µm) to the upper surface of the microchamber. Experiments were first carried out with this optimized setup as the effects we want to show are more clearly displayed under these favorable conditions. Then we discuss more widely valid results obtained under more typical BFPI conditions.

3. Results and discussion

In BFPI, sample displacement is measured by means of in situ calibration that converts the electric photodetector signal (volts) into a length (micrometers). Figures 1(a) and 1(b) show, respectively, a typical V-to-µm curve for an 8.06-µm polymethacrylate microsphere and the intensity at the detector plane (see Methods). Beads were left to settle on the coverslip and then the signals were recorded as one of the adhered particles was moved across the laser beam (NA = 1.2) with a piezoelectric stage. Although the particle motion generates a single-valued signal [21], S_x(x₀), even for large displacements (as much as x₀ ~4 µm in this case), only a small range of positions around the trap centre (x₀ = 0) is generally used. In that region of the curve, position is proportional to the detector signal (x₀ = β_x·S_x) so changes in voltage are easily translated into sample displacements once β is known.

 

Fig. 1 Position and force detection capabilities of the BFPI instrument. (a) Position signal and (b) fraction of total intensity. (c) Comparison between the position reading from the detector and the actual piezo displacement. Two different V-to-µm conversions are shown: one obtained with the correct β (orange dots) and the other by multiplying by the β corresponding to a different bead size, (a 1.16-µm particle; hollow squares). The dark shaded area indicates the region where positions are correctly measured with the first calibration factor. (d) The orange dots in (c) are multiplied by the trap stiffness to indicate the force. The light shaded area shows the region where this matches the theoretical force–displacement curve. The hollow squares are obtained by multiplying the corresponding curve in (c) (that with a mismatched β) by the 1.16-µm-bead stiffness. The product of the two mismatched factors gives a puzzlingly accurate force curve. Finally, the dashed vertical lines indicate the bead limits. (e) Error between theory and experiment in (d). The recorded data match the force curve for values up to 2.8 µm within a 6% error, comparable to the uncertainty of the absolute calibration of the instrument (Table 1). (f) Variation of trap stiffness as a function of bead position. (g) Theoretical and experimental force curves for a 0.61-µm bead corresponding to a measured laser power of 17.5 ± 0.9 mW.

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From the different possibilities [22], we chose power spectrum analysis of the thermal motion of the trapped sample to obtain the constant of proportionality, β. In this analysis, comparison between one of the free parameters in the theoretical expression and the value of the diffusion constant of the sample, D = k_BT/γ, gives the value of β; so position can be calibrated if the medium viscosity, the particle size and the sample temperature are all known. We used the value from the calibration (β_x = 39 µm/V) to convert the detector signal in Fig. 1(a) into displacements in real units (µm), and thus explore the position detection capabilities of our instrument. We observed that the position was correctly measured only for displacements smaller than 2 µm (Fig. 1(c)). Beyond that range, changes in the angular distribution of the light scattered by the sample do not correlate linearly with sample positions, so that correct measurements are not possible with this method.

The region in which linearity is valid, although dependent on the physical properties of the object [17] as well as on the NA of the collecting lens [23], is smaller than the harmonic region of the trap (as we discuss below), which in turn typically covers a small range of forces [2]. The range can be increased by normalizing the detector signal, which in addition provides a measurement of β that is insensitive to laser power. Further improvements along these lines were proposed by Svoboda and Block [6] and Lang et al. [21], who applied a polynomial fit to the curve S_x(x₀), and by Perrone et al. [24], who took advantage of the cross-talk between the x and the sum channel to achieve a ten-fold increase in the working range. However, even when the position is measured in this more general fashion, it still remains very sensitive to the particular experimental conditions.

Motion of the probe is indirectly determined from changes in the angular distribution of the light scattered by the sample [17], which in turn are due to the difference between the refractive indices of the object and of the surrounding medium, so any variation in either the sample or the laser properties may require new calibration. The parameter β critically depends on the size of the sample so that, for instance, the value for the 8.06-µm microsphere in Fig. 1 (β_x = 39 µm/V) is ten times that for a 1.16-µm microsphere (β_x = 4 µm/V).

It is therefore evident that when we multiply the 8.06-µm curve in Fig. 1(a) by the calibration factor for the 1.16-µm particle, the instrument cannot provide an accurate measurement (Fig. 1(c)). Typically the system needs to be recalibrated before every experiment, which is a serious drawback for several potential uses of BFPI.

If we go a step further and calibrate the trap stiffness, κ, we can use the position measurement to calculate the optical force. In our experiment, the value of κ was also determined from a fit of the power spectrum data. The force for the different positions of the sample (Fig. 1(d)) was obtained by multiplying the curve in Fig. 1(c) by the constant κ. The range over which our data matched the theoretical curve, to within the 6% error associated with the absolute calibration of the instrument (see Table 1), was found to extend to 0.7 times the particle radius (Fig. 1(e)), notably beyond the region where positions are accurately measured in (c), and also further than the linear regime of the trap, where κ is defined. The theoretical curve was obtained with a T-matrix simulation [25] using an estimated laser power at the sample plane of 11.4 ± 0.6 mW, which was derived previously from measurements of the transmittance of the objective (see below).

An extended force detection region has similarly been observed by Jahnel et al. for a 2.01-µm bead [19]. Using an experimental design similar to that in Ref [26], they compared the force exerted by a stiff trap calibrated using thermal analysis with the force exerted by a second, less powerful trap on a single trapped microsphere. Although unaware of the reasons, they point out that the force could be measured with this second beam beyond the linear regime. Smith et al. [15] found an analogous result with a counter-propagating beam system.

For the force to be measured correctly in the extended region, the stiffening of the trap for large displacements of the sample, also observed in Ref [27], has to be compensated by changes in β along the curve in such a way that its product with the trap stiffness remains constant. The derivative of the trace in Fig. 1(d) varies by a factor 3 when the trap is moved from the centre of the bead to the edge (Fig. 1(f)), so the calibration of the detection instrument must change by the same amount. These results indicate that the product κ·β is more universal than each parameter separately. The two calibration factors, κ and β, are local magnitudes, defined in the vicinity of a certain position (typically the trap centre), but their product can be used at all sample positions. In our case, a single constant value describes almost the entire force curve. As we discuss below, the range over which the detector readings provide an accurate measurement of the force is connected solely to the amount of light collected, so theory and experiment start diverging at large forces because the recorded intensity decreases (due to the larger angles through which light gets deflected), as indicated in Fig. 1(b). This is not observed, by contrast, when we repeat the experiment for a 0.61-µm bead (Fig. 1(g)). For Rayleigh scatterers, the measuring error (small in our case) is independent of the sample position (see discussion on the fraction of light collected below).

More importantly, when we use the trap stiffness corresponding to the 1.16-µm particle to obtain the second force curve for the larger bead in Fig. 1(d), the result is essentially correct (solid line vs. hollow squares). That is, although the position calibration parameter, β, was not interchangeable between the different samples, the calibration of the detector signal into force measurements seems to be independent of the sample properties, i.e. κ_1µm·β_1µm = κ_8µm·β_8µm. This therefore suggests that β is such that its product with the trap stiffness becomes constant not only along the curve but also regardless of the sample.

In order to show the extent to which the product of the two calibration factors remains fixed, we systematically compared trap stiffness, κ, and position sensitivity, 1/β, for different samples and trapping conditions. We obtained the values from the power spectra of the Brownian motion of trapped particles (Fig. 2(a) , see also Methods). The Lorentzian fitting to the experimental data was corrected to include different effects [28]: aliasing, the detector transparency in the infrared and the frequency dependence of the drag coefficient (Figs. 2(b) and 2(c)). The fitting software [29] was also modified to include the A/D converter quantization noise and to eliminate high-frequency noise from the laser when necessary (Fig. 2(d)).

 

Fig. 2 Power spectrum calibration. (a) Typical two-sided power spectrum of the Brownian motion of a trapped bead (grey dots) and fitting to a corrected Lorentzian curve (orange line). More details about the data recording and analysis are given in the Methods section. The linear dependence of both stiffness, κ, and sensitivity, 1/β, on the laser power (inset) is evidence of correct measurement of the two parameters. (b) Different effects were taken into account to obtain correct measurements of the two constants κ and β. (c) From the fitting of the power spectrum data to a corrected Lorentzian curve, we obtained a mean value for the 3dB-frequency used to characterize the frequency response of the photodetector as a first-order filter, at λ = 1064 nm (f_3dB = 6830 ± 170 Hz; mean + SD; n = 60). We checked this result by fitting a simple Lorentzian function to the power spectrum of the laser alone, obtaining a value of 6.7 kHz. (d) In some cases, the digitization error from the analogue-to-digital converter of our detection system showed up in the spectra. We took this into account in the fitting. The dashed line indicates the noise level in this experiment.

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The experiment shows that the sensitivity, 1/β, is proportional to κ regardless of the properties of the sample or the trapping laser (Fig. 3 ). We trapped beads of five different sizes (0.61 µm, 1.16 µm, 2.19 µm, 3.06 µm and 8.06 µm), made of three different materials (n = 1.48, n = 1.57 and n = 1.68), with both water-immersion and oil-immersion objectives (NA = 1.2 and NA = 1.3, respectively) under different laser powers, from 50 mW to 150 mW. The 30 points lie perfectly along a straight line with only 4% dispersion, despite each factor varying by up to 1500%. Clearly, there is a parameter associated with the instrument which is a constant and results from multiplying κ and β.

 

Fig. 3 Relationship between κ and β. (a) Sensitivity is plotted against stiffness for every pair of parameters; the experiment was repeated for 30 different sets of experimental conditions. The linear fit shows the proportionality of the two constants. The variety of experimental conditions is highlighted in (b-k).

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The existence of such a hidden parameter in single-beam traps has often been overlooked since a constant relationship between κ and β does not typically appear in BFPI measurements. More often, variation by a factor of two or more can be observed for different experimental conditions [30] (to reproduce our results the conditions explained in the Appendix have to be met). To the best of our knowledge, only Barlett and Henderson [18] have reported an experimental correlation between κ and β similar to ours. However, their data show a larger dispersion for a smaller range of stiffness (probably because they use a QPD, see the Appendix for a discussion) and were obtained mainly by modifying the refractive index of the sample.

This is the first time that this clear-cut, characteristic relationship between the two calibration parameters, regardless of the trapping conditions, and its connection with an extraordinary force-measuring range has been reported. The results suggest that BFPI is better interpreted as a method for measuring forces than for measuring positions. This can be explained on the basis of a relationship between BFPI and the determination of light momentum. The following theoretical development follows a prior exposition in Ref [15].

In the presence of a particle, the beam in an optical trap undergoes a change in its momentum structure (Fig. 4 ). This is naturally reflected in the angular spectrum of the time-independent part of the field, which is a solution of the Helmholtz equation [31,32]:

 

Fig. 4 Schematic representation of the telecentric system used for position detection (not to scale). After interacting with the sample, the focused laser beam is scattered in all directions, but mostly concentrated in the forward direction. When the sample remains at the centre of the trap, the light pattern at the BFP of the condenser lens is symmetric, as it is for the incident beam, and the detector signal is zero. The distribution of light in this plane changes when the position, x₀, of the trapped sample relative to the incident beam varies. The refraction of the beam at the water–glass interface at the exit surface of the microchamber allows a large fraction of all the scattered light to be captured (Appendix and Ref [20].). The initial structure of the beam, observed at the front focal plane of the objective, and the changes due to the sample are shown.

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Every plane wave that makes up the field at the sample plane, Aⁱ(θ, φ)e^iks·r, experiences a change in its direction of propagation (Fig. 4). Due to this, the laser transfers part of its momentum to the sample thus producing a net force on it. The modification of the individual momentum of each plane wave, from $\hslash k{\widehat{s}}^i$ to $\hslash k{\widehat{s}}^s$ ($\widehat{s}$ being the ray vector with components (sinθ cosφ, sinθ sinφ, cosθ), and the superscript s indicating the scattered field), gives rise to an asymmetric distribution of the scattered light. If we picture the beam as a whole, the mean deflection angle of the radiant intensity, I_Ω(θ,φ), which is related to the square of the Fourier transform of e^s(r), gives, for small excursions from the centre of the trap, a measurement of the sample displacement, x₀:

After the interaction with the object, the beam is collected by a condenser lens. A high-NA system can be used to increase the position sensitivity [23]. Displacements of the sample are easily tracked at the BFP of this collecting lens since, according to Eq. (1), it contains the Fourier transform information (that is, the new amplitudes $A_0^s$) and therefore, the irradiance at this plane, I(x’,y’), is the projection of the scattered intensity, I_Ω(θ,φ). The intensity can only be projected without distortion, for the large solid angles used in these experiments, if the lens is aplanatic, that is if plane waves are stigmatically imaged at its BFP according to the Abbe sine condition [35] (x’ = f’nsinθ). If that is the case, the position of the light distribution centroid, 〈x’〉, is proportional to the sample displacement. Using Eq. (2), we obtain:

The connection between the measurements of positions and momenta can finally be summarized in the following identity:

We next supply experimental evidence for the validity of Eq. (8). In a new experiment, we calibrated the instrument using the two different routes: through the product of κ and β, and from separate determinations of R_D, ψ and f’. We then compared the results. The latter approach required some extra measurements.

The force exerted by the laser, F_x = -(n/c) P〈sinθ〉, is described by only its energy rate and the change in the propagation direction; magnitudes which can easily be measured by a position-sensitive photodetector located at the BFP of a lens, where angles are converted into positions. The problem then is reduced to the calibration of the conversion from angles and laser power at the sample plane, to the detector position and intensity readings, respectively. In practice, this corresponds to measuring the total focal length of the instrument, f’, and the efficiency, ψ.

A Ronchi ruling experiment, analogous to that described elsewhere [20], gave us a measurement of the total focal length of f’ = 2.62 ± 0.08 mm (3% error). A paraxial calculation based on the distance between the condenser and relay lenses and on their focal lengths (Fig. 5(a) ), as a first approximation, and a computer simulation with Zemax as an additional verification (Fig. 5(b), black line) provided very similar results (f’ = 2.6 mm and f’ = 2.64 mm respectively). However, in the experiment, the main source of error was the difference between the axial positions of an auxiliary CCD camera and the PSD. The latter was positioned as shown in Fig. 5(c). We determined, both through experiment and simulation, that a 1-mm error translated into ~6% uncertainty in the effective value of f’ (Fig. 5(b), orange line). To minimize this, an alternative method that involved obtaining the focal length from the photodetector itself was devised (Fig. 5(d)). The same Ronchi ruling that was used earlier was again employed to generate discrete plane waves with known angles of propagation. An iris was placed at the BFP of the condenser lens to select a single diffraction order and the voltage from the detector was recorded as the iris was slid across the plane. This method allowed us, furthermore, to establish that no significant discrepancies in detector size, R_D, existed in the x and y directions. According to the manufacturer, its half-size was R_D = 4.5 mm, the value we utilized in our calculations.

 

Fig. 5 Determination of f’. (a) Layout of our BFPI instrument. H and H’ indicate the principal planes of both the condenser and the relay lens. (b) The relay lens (or the PSD) position affects the effective focal length of the system. A change of 1 mm in the relay lens position leads to a variation of 6% in f’, which translates into a similar error in α_detector (δα_detector /α_detector = δf’/f’). This was observed in the calibration experiments where we found a change from 100 to 109 pN/V. We also found that a further reduction of the distance between the lens and the detector (~10 mm) eventually led to a 100% difference in α_detector. In contrast, such changes in the position of the optical elements did not have any impact on the calibration of the instrument efficiency, ψ. (c) In order to establish the correct position of the relay lens, the photodetector signal was recorded as an empty trap was holographically moved in steps of 10 µm across the field of view between two extreme points separated by 100 µm. Taking advantage of the Fourier transform relation between the sample plane and the BFP of the condenser and its shift property (inset), the proper axial position of the relay lens was identified as the one for which the variation in the voltage was minimum. (d) An alternative Ronchi ruling experiment with the photodetector was used both to measure the focal length of the instrument and to determine the contribution of the asymmetries of the PSD responsivity along its two independent axes. Plane waves with known transverse momentum were generated and sequentially projected onto the PSD; they were selected by means of an iris located at the BFP of the condenser lens. The sequence of points was first along the x-axis, then the y-axis and at 45° between the two. The normalized signal for each plane wave, S_r/S_sum = r/R_D, where r is the position of the focused wave on the detector and R_D is the detector radius, was plotted against its transverse momentum, that is, n·sinθ. The fitting was used to determine the quotient f’/R_D. No significant differences were observed between the results for the three directions (~1%).

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The other parameter required to determine α_detector is the efficiency, ψ (Fig. 6 ). This relates laser power at the sample plane with the PSD detector intensity output in volts, so it is a measurement of the optical transmittance of the detection apparatus as well as of the responsivity of the PSD. We employed the dual-objective method [38,39] to calibrate the amount of light transmitted by our trapping lens (Nikon, NA = 1.2, water-immersion) and therefore that reaching the sample. An iris was placed in a plane conjugate to the entrance pupil of the objective and a second identical objective was aligned with the optical axis of the first making their focal planes coincident. Then, as the iris diameter was increased, the light transmitted through the system was measured with a power meter, and the transmission was calculated as T = (P_out/P_in)^1/2. The same experiment was repeated for a different objective (Nikon, NA = 1.3, oil-immersion). The results were 55% and 62%, respectively; the latter in agreement with Ref [38].

 

Fig. 6 Determination of the efficiency, ψ, of the detection instrument. The infrared laser (λ = 1064 nm), with circular polarization, was focused by the objective lenses: (a) Nikon CFI Plan Apo VC 60xA WI and (b) Nikon CFI Plan Fluor 100xH. The former is an NA = 1.2 water-immersion lens with an entrance pupil diameter r_pupil = 4 mm ( = f’NA, f’ = 3.33 mm); the latter is an NA = 1.3 oil-immersion objective with r_pupil = 2.6 mm (f’ = 2 mm). The laser power at the back aperture of the objective (triangles) was measured as the diameter of an iris, r, located in a conjugate plane before the telescope was changed. The beam waist, w = 5.6 ± 0.2 mm, was calculated by fitting the data to P(r) = P₀(1 - exp(-2r²/w²)), where P₀ is the incident laser power, and it was found to be coincident, to within the error, with the product m·r_beam, where m = 2.22 is the magnification between the laser fiber diameter and the back aperture of the objective in our setup, and r_beam = 2.55 mm is the output laser radius. The power in the sample plane (circles) was then modulated by the transmittance function of the objective (top plots), which was measured using the dual objective method [38]. As pointed out in Ref [39], we found a non-homogeneous radial transmission. The profile, obtained for each value of the pupil radius as the ratio (P_out(r)/P_in(r))^1/2, fitted a function T_offset + T₀exp(-r²/2 σ²) with T_offset = 52.6 and σ = 3.6 mm for the water-immersion lens, and T_offset = 52.9 and σ = 3.3 mm for the oil-immersion objective. The measured transmissions were 55% and 62%, respectively, in good agreement with Ref [38]. The error between these values, corresponding to 〈T²〉^1/2, and the actual transmissions 〈T〉 [39] were 5% and 1.5%. Finally, the detector intensity reading (orange dots) was measured and was used (c) to determine the efficiency, ψ, for both objectives as the ratio S_sum(r)/P_sample(r), with values of 56 V/W and 59 V/W, respectively. We found that these values were independent of the laser power (as expected) but they showed a certain (small) dependence on radial distance. (d) The mean value of the efficiency was obtained from the distribution of ψ for all the data analyzed, 58 ± 3 V/W, where the standard deviation represents a 5% error. The value depends on the filters in front of the PSD, but it can be corrected by their attenuation without a recalibration.

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We followed the same procedure to find the value of ψ. In this case, we changed the second objective for the force measuring apparatus, using the position sensitive detector to record the intensity in volts. The laser power at the sample was deduced from the laser power reading at the entrance pupil of the objective multiplied by the transmission, T, that we just obtained. The ratio between the two measurements gave an efficiency of ψ = 58 ± 3 V/W (5% error).

Finally, we computed the factor α_detector and found that, as expected from Eq. (8), it matched the mean value of α_trap in both x and y directions to within the estimated error (Fig. 7 and Table 1 ). The constants α_trap^x and α_trap^y followed distributions with a standard deviation of 3-4% in both cases (Fig. 7(b)).

 

Fig. 7 Comparison between α_trap and α_detector. (a) Values of α_trap in units of α_detector obtained from the power spectra for different experimental conditions. The bead size (1.16 µm, 3.06 µm and 8.06 µm) its refractive index (1.48 and 1.57) and the laser power were varied. The shaded area indicates the 6% error in α_detector, which was determined from the propagation of errors in f’ (3%) and Ψ (5%). The error bars were also obtained from the propagation of errors. (b) The separate distributions for x and y show that the two components follow Gaussian functions and are centered at different values: 98 ± 3 pN/V (mean ± SD; 3% error) and 94 ± 4 pN/V (mean ± SD; 4% error), respectively. The result for the y-component is 5% smaller than that for the x-component.

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Table 1. Values of α.

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The independence of the product κ·β from experimental conditions such as particle size or refractive index is demonstrated in Fig. 3. However, from the theoretical discussion culminating in Eq. (8) and the results of the last experiment summarized in Table 1, we can state a more general conclusion: the calibration only depends on three properties of the sensor apparatus (its focal length, the detector size and the efficiency), and is hence totally independent of the trapping phenomenon. We find it illustrative in this regard that the 14% difference in κ_x and κ_y observed in some experiments does not automatically translate into an equivalent discrepancy between α_trap^x and α_trap^y. However, although very similar, the results for the two components exhibited a small difference (Fig. 7(b)). We found that α_trap^y = 0.95 α_trap^x. The origin of this discrepancy lies in radial asymmetries of the light patterns projected onto the photodetector. Small losses of light during the measurements translated into slightly different calibration along the two axes.

The experimental proof of the connection between the method of Smith et al. and BFPI for single-beam traps has important consequences for the latter. First, it means that we can achieve a robust and permanent calibration of the apparatus. The value of α_trap obtained inside a homogeneous buffer or for a regular sample should still be valid, for instance, in a more complex environment such as the cytoplasm of a cell (preliminary results in Ref [40].), or for an arbitrarily shaped object. Second, the calibration is not restricted to the harmonic approximation of the trap; the measurement of the force is valid across a larger range, minimizing the power used for a given trapping force, which is of interest for different biological applications [41–44]. We thus close the loop by giving a unified explanation of the apparently unconnected results in Figs. 1 and 3. Now, the two properties follow as simple corollaries from the interpretation of BFPI measurements as changes in light momentum, and therefore, as direct force determinations.

These results were clearly observed with our instrument since it was intentionally built to fulfill specific conditions, which are typically not met in other setups. The distinctive element, provided that a PSD is used (instead of a QPD, see Appendix), is the high-NA condenser that captures and analyzes most of the scattered light. Deviation from the optimal configuration hides, to a varying degree, the properties that we have discussed. That may explain why our observations have few precedents; however, the preceding results should still be generally recognizable since the method is intrinsically the same.

In practice, the effect of a loss of information regarding momentum translates simply into a larger dispersion of the product κ.β and a reduction of the range within which absolute calibration of the instrument is maintained. In a typical BFPI system, the calibration factor may diverge from α_detector, as the measured intensity, $S_{sum}^m$, and the measured position of the centroid of the captured light distribution, $x{\text{'}}^m$, differ from their original values, S_sum and x’, (Fig. 8(a) ):

 

Fig. 8 Effect of light losses on force measurements. (a) Sketch of the measurement process for a collecting lens with a small NA. (b) As for the results in Fig. 1, we show experimental curves of force for an 8.1-µm bead for different NAs of the condenser lens. The vertical dashed line indicates the limit where the results with NA~1.1 overlap with the correct force curve to within a 6% error. The shaded area corresponds to the harmonic region of the trap, where the force can be described by the orange (and blue) line (-κx₀) to within a 6% error. The deviation from the linear approximation starts at 1.8 µm for NA = 1.1, although exact force measurements can be obtained at up to 2.8 µm. The horizontal dashed lines indicate again a 6% error. The range where measurements are correct for the reduced NA correlates with the amount of light collected. (c) An excessive reduction in the amount of light captured can make the system lose the robustness in the force calibration even for small displacements of the sample, as shown in (b) for NA~1. This may eventually restrict the use of the instrument to position detection only. This plot shows the relationship between stiffness and detector sensitivity for two different bead sizes and several laser powers for small NA values of the condenser (we chose 0.65 for this example, since it is a typical value [53]). The data obtained at different laser powers are still correlated, but show two different slopes for the two beads. There is no single calibration constant, α_trap, that characterizes the instrument, so recalibration for different experimental conditions would be necessary in this case.

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The size of the sample is important in this regard. The angular distribution of scattered light is different for different sizes of trapped particle, so the loss of momentum information varies between samples (Fig. 9(a) ). For example, for dipoles (particles of only a few hundred nanometers) or large particles (several microns) the most relevant information is concentrated within a reduced-angle cone [15]. So, for both large and small particles, the effect of reducing the NA of the capturing lens is less significant. With small beads, the beam is scattered in the form of a spherical wave that carries zero net transverse momentum (Fig. 9(b)). In addition, all the momentum change arises due to interference with the unscattered light, so the information remains limited to the cone defined by the NA of the trapping objective, NA_obj, regardless of the sample position. In this case, the decrease of the condenser NA is only noticeable for values smaller than NA_obj. In contrast, large samples behave roughly as converging lenses; the incident beam is focused, thus reducing its NA (Figs. 9(c) and 9(d)). The force calibration may still be valid for wide displacements of the sample, even if the NA of the collecting system is further reduced.

 

Fig. 9 Fraction of light collected. (a) The value of the force at which the experimental data deviates from the exact force curve in Fig. 8(b) for the low-NA condenser, depends on the sample, and more particularly, on its size. This is observed in a Mie scattering simulation of the fraction of forward-scattered light for different sizes of a polystyrene bead. The beam waist was 0.4 µm. (b) A faint scattering disk is the only evidence of the presence of a trapped sample for Rayleigh scatterers (arrows). (c, d) For large microspheres, the deflected beam (NA~1.1) remains inside a cone of NA = 1.2 (dashed circle) for a large range of displacements.

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Equation (9) and Figs. 8 and 9 clearly indicate that force measurements bypass the Hookean approximation of the trap and are ultimately conditioned solely by the capacity to collect a significant fraction of the light, which can be achieved with relative ease. In the range where this fraction is close to 100% (where α_trap = α_detector) the measurement displays the properties that we have discussed here.

4. Methods

Analysis of position and force measurement capabilities

For the analysis of the position and force detection capabilities of our BFPI instrument, we recorded the detector signal as an 8.06-µm polymethacrylate microsphere attached to a coverslip was moved across the beam, in a direction perpendicular to the trap axis (Fig. 1). The sample was moved with a piezoelectric stage (Piezosystem Jena, TRITOR 102 SG) in steps of 8.4 nm.

It should be noted that this procedure entails some difficulties and is often ruled out as a method to determine the voltage-to-position calibration factor of the instrument. For example, the scattered field can be affected by the presence of the glass surface [45], which can interfere with the position signal. However, we did not find differences between the results obtained with beads stuck to coverslips and those with particles embedded in an agarose gel for any of the sizes used in the experiments.

A second typical problem is the imprecise three-dimensional positioning of the particle relative to the trap. We addressed this issue by using the light pattern from a trapped particle at the BFP of the condenser as a visual reference. Since such patterns provide a very sensitive measurement of the location of the sample, the adhered bead could be positioned precisely, both transversally and axially.

In the comparison of the experimental results with the T-matrix simulations, we used the beam waist at the focus, ω₀, as an adjustable parameter. This is, in general, a magnitude that is difficult to measure. The knife-edge scanning method [46], for example, can introduce errors as large as 13% (50 nm) [47] or more [46]. Other approaches, such as the analysis of the image of the focused laser spot reflected from a coverslip can reach a 20% error [48]. In several papers, the value has been estimated [17] or simulated [34]. Given the uncertainties, we used ω₀ as a free parameter to match the theoretical curves to the experimental data in a similar way to [49]. The results for the 8-µm and the 0.6-µm beads, 340 nm and 410 nm, respectively, are in agreement with values that we obtained from images of the focused beam spots and similar to those reported by others. The difference between the two values may arise from changes in the correction collar of the water-immersion objective, whose position was not fixed.

Calibration through spectral analysis of the Brownian motion of the sample

We calculated the trap stiffness, κ, and the sensitivity, 1/β, from the power spectra of the thermal motion of trapped beads (Fig. 2). All the experiments were carried out with the microspheres (Spherotech and Sigma-Aldrich) suspended in deionized water inside a flow chamber (model RC-30WA, Warner Instruments). Five different sizes (0.61 ± 0.01 µm, 1.16 ± 0.04 µm, 2.19 ± 0.05 µm, 3.06 ± 0.08 µm and 8.06 ± 0.10 µm; mean ± SD) and three materials (polymethacrylate, n = 1.48, polystyrene, n = 1.57, and melamine resin, n = 1.68) were selected. The laser power at the sample plane was determined from the transmission factor of the objective, which was previously measured following the dual-objective method [38,39] (Figs. 6(a) and 6(b)). We used two different objective lenses for trapping: water-immersion and oil-immersion (Nikon CFI, PlanApo VC 60xA NA = 1.2 and PlanFluor 100xH NA = 1.3, respectively).

The two calibration constants displayed linear relationships with laser power (Fig. 2(a), inset). Spectra were obtained from 40-s series of 15000 points at f_sample = 15 kHz after blocking (blocks of 350 points). Following the results from [28,29], aliasing, the photodetector transparency at 1064 nm (f_3dB ~6.8 ± 0.2 kHz for our photodetector, similar to previous results [50]; mean ± SD; n = 60; see also Fig. 2(c)), the frequency dependence of the drag coefficient, γ, and the analogue-to-digital converter noise were taken into account for fitting the experimental data to a Lorentzian curve (Figs. 2(b)-2(d)). To determine the two constants κ and β from the fitted parameters, the bead size calibrated by the manufacturer, Faxén’s correction of γ [3], and the nominal value of water viscosity at the operating temperature were considered. Measurements were performed at stable room temperature ( ± 2 K), which was monitored with an electronic thermometer.

Errors in κ, β, and κ·β were determined through the propagation of errors. When, in addition, the final value was obtained as a mean of n different measurements (Table 1), the total error was computed as the standard deviation, SD.

Appendix

In this appendix we give further details of the particular working conditions under which the results in Figs. 1, 3 and 7 were obtained. They are set to ensure the correct measurement of light momentum changes of an optical trap based on a high-NA beam.

Figure 10 shows a ZEMAX simulation of our optimized BFPI optical system, which primarily consists of a high-NA lens (an oil-immersion DIC condenser, Nikon T-CHA, NA=1.4) and an auxiliary lens (a Thorlabs doublet AC254-100-C and an additional singlet LA1986-C). The light scattered by the sample, represented by a set of plane waves propagating within a solid angle of ~2π, is collected by the condenser lens and the relay lens projects the light pattern at the BFP of the condenser onto a photodetector with 1/4 magnification. The NA of the condenser lens, NA_condenser, has to be larger than the refractive index of the suspension medium, n_medium, in order to collect the light propagating almost parallel to the upper surface of the microchamber [20] (inset). In this case: n_medium ~ n_medium sinθ = n_glass sinθ’ < NA_condenser, where θ and θ’ are the convergence angle of the beam before and after refraction at the water–glass interface, respectively. In our case this condition is fulfilled as NA_condenser = 1.4 and n_medium ~ 1.33.

 

Fig. 10 Computer simulation of the setup using ZEMAX. The inset is a magnified view of the sample region showing how the front lens captures plane waves scattered at high angles.

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The second major requirement of our setup is the use of an aplanatic optical system, that is, the system must fulfill the Abbe sine condition. This ensures the proper decomposition of the beam into the transverse momentum (p_r/p₀ = n·sinθ, where p₀ = h/λ) in the detector plane (that is, ensures that the spatial coordinates at the PSD plane are proportional to momentum). Although this is a typical correction in microscopy optics, we checked our system for compliance with this requirement, as illustrated in Fig. 11 . Figure 11(a) plots the results of a ray tracing simulation of the condenser (Nikon T-CHA, oil-immersion, NA=1.4), based on the specifications found in US patent no. 5657166 [51]. Rays travelling at different angles, θ, hit the BFP of the condenser at positions r = f’ n·sinθ (f’ = 10.66 mm) with high fidelity. We observed that, even for far off-axis rays, the geometric image was smaller than the Airy disk (140 µm), thus being, in addition, diffraction limited.

 

Fig. 11 Analysis of the aplanatic lens requirement. (a) ZEMAX Simulation of the condenser lens alone. Light rays travelling at angles specified by their numerical aperture at the x-axis hit the BFP at positions given by coordinate r in the y-axis. (b) The residues are small and can be fitted to a power law. (c) Spherical aberrations are probably behind the observed discrepancies as they would shift rays following a five-order power law, close to that found in (b). (d) Experimental results for the compound system (condenser + relay) and simulation.

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Figure 11(b) plots the small deviations found with respect to the Abbe condition for large angles. The difference between the position r and the expected value, r₀ = f’ n·sinθ, is a power (b = 4.77 ~ 5) of the transverse component of the momentum, and is always smaller than 5%. The spherical aberration, ϕ = (S₁/8)r’⁴ (where 0<r’<1 and S₁ is the Seidel coefficient), which is the main optical aberration of an aplanatic oil-immersion condenser, lies at the heart of this small discrepancy, as shown in Fig. 11(c). In an aberration-free lens, the light propagating in direction θ exits the optical system at position r₀. However, due to the spherical aberration, the actual position in our system is r₀ - ϕ sinθ. This gives rise to the polynomial dependency with sinθ (of order 5) observed in (b). The effect on the detector reading is nevertheless small. A Matlab simulation confirmed that the effect of the spherical aberration depends on the scattered pattern and on the deflection of the beam, but it is typically below 2%. This contrasts with the reported poorer performance of the condenser when used as a trapping lens [52]. Finally, Fig. 11(d) shows simulated and experimental data of the fulfillment of the Abbe sine condition by the total optical system (including the auxiliary lens). We chose a combination of elements for the relay lens, so the momentum structure of the beam was not degraded when projected onto the position sensitive detector. A diffraction grating made holographically in-house with an Agfa holotest plate, type 8E75HD (substrate with n~1.53 at 1064 nm), was used to generate plane waves of known transverse momentum. The position of the diffracted orders in the PSD plane was determined from a CCD image (on the right of Fig. 11(d)). This experiment also allowed us to determine the focal length of the total system (f’ = 2.62 ± 0.08 mm).

In practice, to capture highly skewed light rays, it is also necessary to work as close to the upper coverslip as possible. Failing to work close to the exit surface reduces the performance of the system in two ways: when the axial distance, z, between the trap and the glass increases, 1) the amount of captured light decreases and 2) the degree of agreement with the Abbe sine condition worsens. These effects can be partly reduced through the use of a long-focal-length collecting lens, as shown in Fig. 12 . In our case, f’ = 10.66 mm. Figure 12(a) shows light patterns at the BFP of the condenser lens (inset), which provide direct information about the solid angle captured (the amount of light collected). We observed, both with a simulation and experimentally, that the effective NA changes as the trap is moved in the axial direction. For our system, the effective NA is 1.3 when the trap is kept within a range of 50 µm from the glass and 1.29 if the distance is greater than this but less than 100 µm. These values correspond to a large fraction of all the scattered light for most samples (Fig. 9). The fit in Fig. 12(a) corresponds to the curve (derived using geometrical optics) z tgθ_max + (wd – z) tgθ’_max = r_pupil, where θ_max is the acceptance angle of the condenser for the cone of light in the sample, θ’_max is the refracted angle at the water–glass interface, wd is the working distance (wd = 1.72 mm) and r_pupil is the pupil radius, which was used as a free parameter, with a final value of 3 mm. Similarly, the condenser still fulfills the Abbe sine condition when the trap is moved deep into the microchamber (Fig. 12(b)), beyond the plane for which the spherical aberration is corrected (i.e. the working distance). The displacement of the laser introduces a certain error but the deviation from the optimal condition is <5%. In contrast, the use of a short-focal-length lens, such as an oil-immersion objective, does not provide the same performance (Figs. 12(c) and 12(d)). A computer simulation of our objective lens (with a much shorter focal length than the condenser, f’ = 2 mm) using ZEMAX (data taken from US patent no. 5805346) shows how the pattern at its back aperture is completely modified when the trap position changes as little as from 3 to 20 µm (Fig. 12(c)). The effective focal length varies by 6%, the linearity for large angles disappears and the effective NA of the lens is drastically reduced from 1.3 to 1.2 (Fig. 12(d)). Such lenses should not be used for this purpose.

 

Fig. 12 Analysis of the requirement of a collecting lens of long focal length. (a) Experimental light patterns at the BFP of the condenser, for two axial positions of the trap, and plot showing the dependence of the effective numerical aperture of the collecting lens with the axial distance. An increasing, but still small, fraction of the light travelling at large angles is lost. (b) Aplanatism of the condenser lens for increasing cover-glass-to-sample distance, the differences are barely noticeable. (c) A computer simulation of an oil-immersion objective with a focal length f’ = 2 mm, used as a substitute condenser and (d) degree of fulfillment of the Abbe sine condition. This lens is more sensitive to axial changes in the sample position because of its shorter focal length.

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Finally, experimental data show that a PSD is a critical requisite. Other kinds of photodiodes, such as QPDs, are often used in BFPI. They do not, however, provide the true position of the centroid of the light pattern at the BFP of the condenser lens, which induces errors in the measurement of the total beam momentum, and as a consequence produces incorrect force values. For example, we think that the relatively high dispersion of the data in Ref [18]. is largely due to their using a QPD, as they did employ a high-NA condenser and the experiment involved small modifications of the sample properties. We have similarly observed a larger data dispersion in the relationship between κ and β when replacing the PSD by a QPD, as shown in Fig. 13 . In the experiment shown, a microsphere was trapped and dragged by the surrounding fluid as the whole microchamber was moved with a piezoelectric stage. The photodetector signal was recorded and plotted against the hydrodynamic force applied on the trapped particle given by Stokes’ equation, F = 6πηav, where η is the medium viscosity at the operating temperature and a and v(t) = 2πfx₀·cos(2πft) correspond to the radius and the velocity of the bead, respectively. The latter was determined from the velocity of the piezoelectric stage, which was driven with a sinusoidal voltage at frequency f << f_c, the corner frequency of the system. We found that small changes in sample properties, for example, a slight increase in the sphere diameter (from 1.16 µm to 3.06 µm), had a clear impact on instrument calibration (Fig. 13(a)). In contrast, such an effect was not observed when the QPD was replaced by a PSD, as shown in Fig. 13(b). Furthermore, we found a clear discrepancy between the value of α computed from the theory in Ref [17], which assumes a QPD at the BFP, and those in Table 1 (α = 42 pN/V). QPDs are more sensitive than PSDs. However, unless needed to measure position alone, their use should be avoided.

 

Fig. 13 QPD vs. PSD. (a) A QPD produces outputs that are sensitive to the size of the sample as opposed to (b) a PSD, whose outputs are proportional to the centroid of the light distribution, and thus faithfully represent the net momentum flux when placed at the BFP.

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Acknowledgments

We are indebted to the company IZASA (Barcelona, Spain) for kindly lending us duplicates of our microscope objectives. We thank Christopher Evans and Prof. E. Martín-Badosa for the careful reading of the manuscript, and J. Mas for fruitful discussions. This work was funded by the Spanish Ministry of Education and Science, under grants FIS2007-65880 and FIS2010-16104, as well as by the Agency for Assessing and Marketing Research Results (AVCRI) of the University of Barcelona. A. Farré is the recipient of a doctoral research grant from the Generalitat de Catalunya.

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