## Abstract

Conventional optical tweezers suffer from several complications when applying axial forces to surface-tethered molecules. Aberrations from the refractive-index mismatch between an oil-immersion objective’s medium and the aqueous trapping environment both shift the trap centre and degrade the trapping strength with focal depth. Furthermore, interference effects from back-scattered light make it difficult to use back-focal-plane interferometry for high-bandwidth position detection. Holographic optical tweezers were employed to correct for aberrations to achieve a constant axial stiffness and modulate artifacts from backscattered light. Once the aberrations are corrected for, the trap height can be precisely determined from either the back-scattered light or Brenner’s formula.

© 2015 Optical Society of America

## 1. Introduction

Optical tweezers take advantage of the intense gradient force felt by small dielectric particles near a tightly focused laser beam [1]. While useful for organizing or rearranging assemblies of micro- or nanoscopic objects [2], single trapped particles can also be used as probes for precisely measuring and applying force. Since the range of forces accessible to optical tweezers (from tens of femtonewtons to hundreds of piconewtons) is of the same order as many biologically relevant forces, optical tweezers have served as a powerful tool for exploring mechanical effects in cell and molecular biology [3–5].

A common experimental geometry involves tethering a molecule between an optically trapped bead and a glass coverslip. Forces are applied on the molecule by moving the laser focus relative to the coverslip. High numerical aperture (N.A. > 1.2), oil-immersion objectives are typically used to provide a tight trap. However, the refractive index mismatch between the immersion oil and the aqueous trapping medium introduces aberrations that result in both a downstream shift to the trap centre and an axial stiffness that decreases as the trap is moved deeper into the sample. For this reason, conventional optical tweezers restrict motion of the trap to the plane parallel to the coverslip surface where the trap stiffness remains constant. This geometry works well for long surface-tethered molecules, but as the tether length decreases, motion in the lateral and axial planes becomes increasingly coupled making measurements on short molecules (< 1*μ*m) very difficult. While molecular linkers can be introduced, this only introduces more thermal noise, which scales with the length of the tether [6].

There has been much progress in the last decade on modifying and calibrating optical tweezers for purely axial measurements. For a discussion of these advances as well as the novel utility of such a geometry in single-molecule biophysics, we refer the reader to our recent review article [7]. In the present manuscript, we present a simple approach to axial manipulation by employing holographic optical tweezers (HOT) to correct for aberrations and provide a constant axial trap stiffness. While the height dependence of the trap strength can, alternatively, be corrected by employing a water-immersion objective [8], high-N.A. oil-immersion objectives are the preferred choice for many experiments as they can provide a tighter trap and are not prone to evaporation, which may limit the duration of a measurement. Furthermore, combining force-spectroscopy with single-molecule fluorescence measurements such as TIRF, explicitly requires an oil-immersion objective. For these reasons, much effort has gone into mitigating the effects of aberrations in optical trapping and micromanipulation, to create stable traps deep within a liquid medium [9, 10].

We show that by correcting for first order spherical aberrations, we are able to maintain a constant trap stiffness at depths of up to 6 *μ*m. We then show, by measuring the axial drag on a trapped microsphere near the coverslip surface, that we can precisely determine the axial position of the trap. This is in excellent agreement with an alternative method that makes use of intensity oscillations from light backscattered between the microsphere and the coverslip. While the backscattered light is useful for locating the trap, it also gives rise to a detector sensitivity that varies with axial position, which complicates an axial force measurement. We tried to reduce the oscillatory behaviour of the detector sensitivity by introducing a variable beam block at the centre of the incident laser light. While we could reduce the amplitude of the oscillations, we were unable to attenuate them sufficiently without severely degrading the trap stiffness. Finally, we show that this axial holographic optical trap is capable of single-molecule force measurements.

## 2. Experimental method

#### 2.1. Optical setup

The HOT setup (Fig. 1) consists of a hologram generated via a phase-only spatial light modulator (SLM) placed within the Fourier plane of an oil-immersion objective (Olympus PlanApo 100×, 1.4 N.A.). A half wave plate (HWP), placed in front of a polarizing beam splitter (PBS), allows for manual tuning of the intensity from a 1064 nm Nd:YAG laser (4W, Coherent BL-106C). The phase imparted to the wavefront by the SLM (Hamamatsu X10468) separates the light into diffraction orders propagating at various angles from the optical axis. An iris placed in an intermediate Fourier plane allows only the first order beam to pass through while the zeroth (unmodulated light) and unwanted higher orders are blocked. The remaining optics were all aligned with the first order beam. The beam is magnified to slightly overfill the objective and the SLM plane is 4f imaged onto the back focal plane (BFP) of the objective. After passing through the sample, the light is collected with a second objective (Olympus LUCPLanFL 40×, 0.75 N.A.) and its BFP is imaged onto a position sensitive diode (PSD) (FirstSensor DL100-7-PCBA3). Two dichroic mirrors (DM) allow visible light to illuminate the sample and image it onto a CCD camera (PixeLINK PL-B771U).

The trap can be steered in all three dimensions within the sample, (*x*, *y*, *z*), by directly changing the hologram on the SLM, controlled with custom LabView and MATLAB software. This was achieved by displaying a superposition of a phase ramp and a Fresnel lens onto the SLM:

*x′*,

*y′*) are the SLM coordinates,

*f*is the focal length of the objective, and

*λ*is the laser wavelength. The phase ramp acts like a prism to deflect the laser light in

*x*and

*y*while a Fresnel lens effectively focuses and defocuses the light shifting the trap along

*z*. While more accurate and efficient algorithms exist for generating complex holograms [11], this simple approach to steering the beam is adequate for a single trap. However, due to the iris, which filters out unwanted orders of diffraction (and ghost traps), the eventual clipping of the beam limits both the lateral and axial range that the trap can be translated. Moreover, the axial step size was limited by the bit depth of the SLM. The minimum axial step size was found to be ∼ 20 nm, below which there was negligible difference between the displayed holograms.

#### 2.2. Optical trap calibration

Optical tweezers often employ back-focal-plane-interferometry (BFPI), which images the light intensity at the back focal plane of the microscope condenser, to rapidly track the bead’s position [12]. Due to the Gouy phase shift between the scattered and unscattered light, a photodiode placed downstream from the trap will provide a voltage signal *V*(*t*) proportional to the trapped bead’s relative axial displacement from the trap centre *δz̃*(*t*). Knowledge of the detector sensitivity *β* is required to convert this voltage into a distance *δz̃*(*t*) = *βV*(*t*). Both the trap stiffness *κ* and the sensitivity *β* can be acquired by measuring the power spectrum of an untethered, trapped bead over the relevant detection frequencies *f*. Over this frequency range, the power spectrum has the form of a Lorentzian [13]:

*D*(measured in volts) and the corner frequency

_{V}*f*taken as fitting variables. The trap stiffness and detector sensitivity are determined by the following relations, respectively: and where

_{c}*k*= 4.1pN · nm and

_{B}T*γ*is the hydrodynamic drag coefficient. For a spherical particle of radius

*R*, the axial drag on the particle, close to the coverslip surface, can be approximated by Brenner’s formula

*h*is the trap height, defined as the distance between the trapped bead’s centre and the coverslip, and

*γ*

_{0}= 6

*πηR*is the drag in an infinite medium of viscosity

*η*[14]. Because of the nonlinear form of Eq. 5, when trapping close to the coverslip surface (roughly,

*h*∼ 3

*R*), this calibration is acutely sensitive to inaccuracies in the measured trap height.

#### 2.3. Measuring the trap height

Due to both aberrations and competition between the gradient and scattering forces, the location of the trap centre does not coincide with the laser focus. We’ll refer to the nominal focal depth as the axial position within the sample at which the laser focus would be if not for the refractive index mismatch between the immersion oil and the aqueous trapping medium. When the nominal focal depth is changed by a known displacement relative to the coverslip *δz*, either with a translation stage or by directly moving the laser focus with the SLM, the height *h* of the trap above the coverslip will be shifted by some function of *z*. Under the simplifying assumption that *h* = *L*_{1}*z* +*R*, the scaling factor *L*_{1}, within our linear approximation, is referred to as the effective focal shift [15]. Note, we have defined *z* = 0 to be the point where the bead touches the coverslip (i.e., *h* = *R*). Likewise, the effective focal shift is defined similar to [15] as the shift in the centre of the optical trap, which distinguishes it from the standard definition as the shift of the optical focus. To determine the focal shift, and thus the trap height *h*, we can repeatedly extract the drag from the corner frequency, Eq. (3), and fit with Eq. (5). However, this requires us to assume that the stiffness *κ* remains constant, which, due to aberrations, is typically not valid.

A more accurate measure of the trap height utilizes back-scattered light from the bead, which is reflected from the coverslip and modulates the total intensity incident on the photodiode (see Fig. 2) [15]. This results in an oscillating intensity pattern at varying axial positions. These oscillations can be shown to display a sinusoidal variation in intensity as a function of the trap height *h* with constant spatial frequency:

*n*is the refractive-index of the trapping medium (

*n*∼ 1.33 for H

_{2}O) and

*λ*is the wavelength of the detection laser, which, in our case, is also the 1064 nm trap laser. After scanning the axial position of an untethered, optically trapped bead, the measured intensity can be fit as suggested in [16]: where

*P*(

*z*) is a polynomial of second or third order in

*z*and

*L*

_{1}is the focal shift, taken as a fitting parameter along with the constants

*A*,

*B*and

*ζ*. Again, under a linear approximation

*h*=

*L*

_{1}

*z*+

*R*. Other trap parameters, such as the diffusivity

*D*, the sensitivity

_{V}*β*and the standard deviation of the intensity

*σ*, exhibit similar oscillatory behaviour and can be fit with the same functional form as Eq. (7) to confirm the measured focal shift. As we will show, once we’ve corrected for aberrations to maintain a constant trap stiffness, both the backscattering and the axial drag approach agree yielding the same value for the measured trap height above the coverslip surface.

_{I}#### 2.4. Correcting aberrations

While both the imaging optics, as well as the SLM, can lead to distortion of the optical trap, the dominant source of aberration arises from a phase shift to the trap light introduced by the index mismatch between the oil/glass (*n _{g}* ∼ 1.52) and the aqueous buffer (

*n*∼ 1.33). This phase shift can be expressed as a sum of Zernicke polynomials

_{b}*ρ*is the radial coordinate normalized to the objective’s entrance pupil radius [17]. Each term in the series represents a different order of aberration (see Table 1) and can be corrected for by subtraction of the corresponding Zernicke polynomial from Eq. (1). The first few terms reflect an overall phase shift to the wavefront (

*n*= 0, Piston), a translation of the wavefront along the optical axis (

*n*= 2, Defocus), followed by increasing orders of spherical aberration (

*n*= 4, 6). Note, here we consider only radially symmetric distortions, so are neglecting effects such as coma or astigmatism.

## 3. Results

#### 3.1. Constant axial stiffness

We find that correcting for first order spherical aberrations alone is sufficient to achieve a constant axial trap stiffness. At each focal depth *z*, this correction is obtained by subtracting the *n* = 4 term of Eq. (8) from Eq. (1) and displaying the resulting phase pattern on the SLM:

*A*(found empirically to be

*A*∼ 0.7). Figure 3 shows the axial stiffness for a 1

*μm*in diameter bead over a range of 3

*μ*m before and after applying this correction. We found we are able to keep a constant stiffness up to ∼ 6

*μ*m into the sample until significant clipping of the beam begins to degrade the trap. However, for surface tethered assays, this is more than enough range.

#### 3.2. Effective focal shift and trap height

With a constant axial trap stiffness, the standard deviation in the intensity of an untethered bead *σ _{I}* simply behaves like a decaying sinusoidal as a function of the trap height

*h*. This oscillatory behaviour is an artifact of the detection and can be removed by scaling it with the sensitivity

*β*, which provides a measure of the trap width (a slowly varying function at worst). However, left in units of Volts,

*σ*provides a robust visualization of the oscillations arising from back-reflected light. A fit to this sinusoidal variation with Eq. (7) yields the focal shift, from which we may deduce the trap height

_{I}*h*above the coverslip surface (see Fig. 4). We can extend the range over which we model the focal shift by including an additional quadratic term such that

*h*=

*L*

_{1}

*z*+

*L*

_{2}

*z*

^{2}+

*R*[16]. Figure 4 shows a comparison between fitting the results with and without the quadratic term. We find that below a focal depth of ∼ 2.5

*μ*m there is negligible difference. Therefore, a linear shift is a good approximation for the regime we are interested in.

As mentioned, another way to determine the position of the optical trap is to measure the axial dependence of the drag arising from surface effects near the coverslip. This approach is typically only an approximation, but for a constant axial trap stiffness, the method becomes exact, so should precisely yield the trap height. To confirm this, the effective focal shift was determined from fits to both the intensity and the normalized axial drag *γ/γ*_{0} at increasing bead-coverslip separations. To calculate *γ* at each height, the corner frequencies were used with Eq. (3) and a constant stiffness was assumed throughout. With aberration corrections applied, the relationship is simply
$\gamma /{\gamma}_{0}={f}_{c}^{\mathit{max}}/{f}_{c}$, where
${f}_{c}^{\mathit{max}}$ is taken far from the surface. When the aberrations are present, however,
${f}_{c}^{\mathit{max}}$ peaks and then begins to decrease as the assumption of a constant trap strength begins to fail. In this case, we had to estimate the true asymptotic stiffness. Effectively, this amounted to modifying the previous relation to
$\gamma /{\gamma}_{0}={f}_{c}^{\mathit{max}}/\left({f}_{c}\mathrm{\Gamma}(h)\right)$, where Γ(*h*) is simply the denominator of Brenner’s law (Eq. (5)) and *h* is obtained under the paraxial approximation for the effective focal shift (i.e., *h* = 0.79*z* + *R*) [15].

In the absence of aberration corrections, the calculation of *γ/γ*_{0}, under the assumption of a constant trap stiffness, diverges quickly from the functional form of Brenner’s formula (Eq. (5)). The normalized drag should approach unity as the bead is moved away from the coverslip, but since the trap stiffness is actually progressively decreasing, the experimental measure of *γ/γ*_{0} (shown in Fig. 5(a)) begins to deviate after ∼ 1*μ*m. Therefore, a fit of Brenner’s law to the data is only applicable within this range and tends to overestimate the actual trap height compared to the more accurate method of using the periodic modulations in intensity to measure the trap position (see Fig. 5(b)). However, after correcting for aberrations to provide a constant axial trap stiffness, the normalized drag is now well described by the theory, and the effective focal shift obtained from a fit to the axial drag agrees with that obtained from fitting the intensity modulations (see Fig. 5(b)).

#### 3.3. Modulating the intensity fluctuations with a variable beam block

The oscillations present in the scattered intensity, which are convenient for measuring the trap height, are also, unfortunately, visible in the detector sensitivity *β*. This axial variation in the sensitivity complicates measurement of the bead displacement from the trap centre. We thought that, if we could controllably switch the oscillations on and off, we might be able to further simplify an axial force measurement. Toward this end, we tried to obtain a constant sensitivity *β* by selectively blocking the central rays that contribute most to the backscattered signal.

A customizable beam block was created by turning off a circular region of pixels on the SLM. In this way, the light incident on those pixels is reflected into the zeroth order and subsequently blocked by the iris in the beam path. To centre the laser beam incident on the SLM, the intensity was profiled by turning on a small (20×20) subset of pixels and raster scanning through the SLM, measuring the total intensity in the 1^{st} order deflected beam at each step. The resulting profile was fit with a 2D Gaussian to find the location of the peak intensity on the SLM window. This approach allowed us to centre the beam block to sub-pixel accuracy.

We found that, although the amplitude of the oscillations could be modulated in this way, they could not be completely eliminated without significant degradation to the trapping strength and detection sensitivity (see Fig. 6). This is in agreement with previous results showing that oscillations in the intensity persist deep into the sample [18]. An alternative approach is to simply introduce a separate detection laser at lower power and reduced temporal coherence; however, this would only work for a limited range of separation between the detection and trapping beams’ foci.

#### 3.4. Force-extension measurement on λ-DNA

To illustrate that our axial holographic trap is amenable to single-molecule force measurements, a dsDNA construct (5kbp, from *λ*-DNA) was stretched axially to obtain a force-extension curve and fit with a worm-like chain (WLC) model [19]. The bead’s relative displacement is proportional to the difference of the measured intensity with that of an untethered bead *I*_{0}. Thus, at each step *z* made by updating the SLM, the extension *L* is:

*κ*is: where

*I*

_{0}(

*z*) and

*β*(

*z*) were measured before stretching by axially scanning an untethered bead. Small variations in bead size and coverslip thickness had the effect of shifting the intensity curves by a constant. However, the force extension and calibration curves could be nicely aligned due to the signature oscillatory behavior near the coverslip surface. Likewise,

*β*(

*z*) showed great repeatability between beads. The intensity for both a tethered and an untethered bead are shown in Fig. 7(a). The resulting force-extension curve can then be fit by the commonly accepted interpolation formula for the worm-like chain (WLC) model of dsDNA [19,20]:

*L*is the contour length and

_{c}*ξ*is the persistence length. The corresponding force-extension data with the WLC fit is shown in Fig. 7(b). The persistence length obtained from the fit, 42 nm, agrees with conventional force-extension measurements (i.e., along the lateral direction) in our buffer.

_{p}## 4. Conclusion

We have presented an axial holographic optical tweezers that is capable of applying controlled axial forces on surface-tethered molecules. By correcting only the first order spherical aberrations, we could realize an optical trap that maintained a constant axial trap stiffness up to 6*μ*m into the sample, greatly simplifying axial force measurements. For a trap of constant stiffness, determining the axial position of the trap by fitting Brenner’s formula for the normalized axial drag shows excellent agreement with an alternative approach that exploits oscillations in the intensity arising from backscattered light. We also tried to control the oscillatory intensity signal through the introduction of a variable beam block on the SLM, in an attempt to generate a uniform detector sensitivity *β*. Though the oscillations in the sensitivity could not be eliminated completely, knowledge of the behaviour of *β*(*z*) is sufficient for single-molecule force measurements. Without the need for a piezoelectric stage, or complicated calibration procedures, this simple method of axial trapping with a HOT will open the door to low-noise measurements on submicron length molecules.

## Acknowledgments

R. Pollari and J. N. Milstein acknowledge support from the Natural Sciences and Engineering Research Council of Canada (NSERC) (NSERC, RGPIN 418251-13) and the Canada Foundation for Innovation (CFI, PN 30735). We also thank S. Ishitani-Silva and S. Yehoshua for their helpful assistance and feedback on the manuscript.

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