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Performance of optical traps with geometric aberrations

Yael Roichman, Alex Waldron; Emily Gardel and David G. Grier
Department of Physics and Center for Soft Matter Research, New York University, New York, NY 10003


Date: November 12, 2005


Abstract:

We assess the influence of geometric aberrations on optical traps' in-plane performance by studying the dynamics of trapped colloidal spheres in deliberately distorted holographic optical tweezers. The traps' lateral stiffness turns out to be insensitive to moderate amounts of coma, astigmatism and spherical aberration. Moreover, holographic aberration correction enables us to compensate inherent shortcomings in the optical train, thereby adaptively improving its performance. We also demonstrate the effects of geometric aberrations on the intensity profiles of optical vortices, whose readily measured deformations suggest a method for rapidly estimating and correcting geometric aberrations in holographic trapping systems.

An optical tweezer exploits forces exerted by intensity gradients in a single strongly focused beam of light to trap mesoscopic objects in three dimensions (1). The ability to noninvasively exert and monitor forces in microscopic systems has led to exciting new insights in a wide range of fields, including the functioning of biological molecular motors and the statistical mechanics of thermally driven systems. Increasingly, these applications require optical traps with highly optimized and precisely characterized properties. The three-dimensional potential energy well associated with an optical tweezer is determined by the intensity distribution near the focal point, and thus on the detailed structure of the light's wavefronts. The sharpest focus, which yields the strongest intensity gradients and thus the best traps, requires a converging beam with spherical wavefronts, and is said to be diffraction limited. Deviations from this ideal structure are dubbed aberrations and degrade the trap's performance.

It has bean shown both experimentally (6,2,3,5,4) and theoretically (7,18) that optical tweezers' axial trapping efficiency is sensitive to the presence of spherical aberration. Less has been published about spherical aberration's influence on lateral trapping efficiency (25,8), and less still about the influence of other geometric aberrations. This is a growing concern as optical trapping systems become increasingly sophisticated and are combined with complementary optical techniques.

Here we report measurements of lateral trap stiffness under the influence of spherical aberration, coma and astigmatism, using computer-generated holograms to introduce controlled aberrations and digital particle tracking (10,9) to characterize the traps' performance. Not only does this approach provide a means to measure aberrations' influence on optical trapping efficiency, it also provides the basis for real-time adaptive optimization. Holographic optical tweezers (HOT) (11,12,10,16,13,14,15) are especially suited for such studies because their use of computer-generated holograms to structure the wavefronts of light affords a quick and flexible way to introduce aberrations without otherwise modifying the optical train. The degraded beam is subsequently used to create an array of traps whose performance can be assessed in parallel.

Figure 1: Schematic implementation of dynamic holographic optical tweezers with imposed aberrations. A computer-generated hologram, $ \varphi(\vec {\rho})$ , encoding a $ 4 \times 4$ pattern of optical tweezers is distorted by the phase function $ \varphi_1(\vec {\rho})$ encoding a geometric aberration, in this case coma, before being imposed on a Gaussian laser beam with a spatial light modulator (SLM). Particles trapped in the array are imaged with a CCD camera.
\begin{figure}\centering
\includegraphics[width=0.9\columnwidth]{schematic}
\end{figure}

Our HOT implementation, shown schematically in Fig. 1, uses light from a frequency-doubled Nd:YVO$ _4$ laser (Coherent Verdi) expanded by a Galilean telescope to fill the 25 mm diagonal face of a reflective liquid crystal spatial light modulator (SLM). (Hamamatsu X8267-16 PPM). The SLM can impose phase shifts, $ \varphi(\vec{\rho}) \in [0,2\pi]$ , at each position $ \vec{\rho}$ in a $ 768 \times 768$ array. The phase-modified beam is relayed by a second telescope and a wavelength-selective polarizing beam splitter to the input pupil of a 100$ \times$ , NA 1.4, Plan Apo oil immersion objective mounted in a Nikon TE-2000U inverted optical microscope, which focuses the light into optical traps. Trapped samples are imaged onto a CCD camera using the microscope's standard optical train, with the imaging illumination passing through the beam splitter.

To assess the holographic traps' performance under the influence of geometric aberrations, we studied the dynamics of silica beads 1.53  $ \unit{\mu m}$ in diameter (Duke Scientific Lot 5238) dispersed in water and trapped in a $ 4 \times 4$ array of optical tweezers. We used standard methods of digital video microscopy (9) to track the trapped particles' motions with $ \Delta r = 10~\unit{nm}$ spatial resolution at $ \Delta t = 1/60~\unit{s}$ intervals. The resulting trajectories were analyzed with statistically optimal methods (10) to extract the traps' stiffness along and perpendicular to the video scan lines while simultaneously measuring the spheres' hydrodynamic radii.

The five primary geometric aberrations in a beam of light can be characterized with Zernike's orthogonal polynomials (17):

$\displaystyle \varphi_1(\vec{\rho})$ $\displaystyle = \frac{a_1}{\sqrt{2}} \, ( 6 x^4 - 6 x^2 + 1 )$ spherical aberration    
$\displaystyle \varphi_2(\vec{\rho})$ $\displaystyle = a_2 \, ( 3 x^3 - 2 x ) \, \cos(\theta - \theta_2)$ coma    
$\displaystyle \varphi_3(\vec{\rho})$ $\displaystyle = a_3 \, x^2 \, \left[ 2 \cos^2(\theta-\theta_3) - 1 \right]$ astigmatism    
$\displaystyle \varphi_4(\vec{\rho})$ $\displaystyle = \frac{a_4}{\sqrt{2}} \, (2 x^2 - 1)$ curvature of field    
$\displaystyle \varphi_5(\vec{\rho})$ $\displaystyle = a_5 \, x \,\cos(\theta-\theta_5)$ distortion (1)

where $ x = \rho/R$ is the radial coordinate in units of the input pupil's radius, $ R$ , $ \theta$ is the polar angle around the optical axis, and where the coefficients $ a_i$ are measured in units of the wavelength of light. The three angles, $ \theta_2$ , $ \theta_3$ and $ \theta_5$ , establish the orientation of coma, astigmatism and distortion, respectively. All five primary geometric aberrations can be introduced independently by adding their phase representations $ \varphi_i(\vec{\rho})$ to the hologram $ \varphi(\vec {\rho})$ encoding the $ 4 \times 4$ array of traps, as shown in Fig. 1.

Figure 2 shows the effect of coma on optical traps' structure and performance. Coma is typically introduced by failure to center lenses along the optical axis. This is a concern, therefore, for optical trapping systems that translate their traps by displacing one or more lenses. The nine axial cross-sections through holographically projected traps in Fig. 2(a) demonstrate that coma ruins the beam's radial symmetry at large distances ( $ z = 10~\ensuremath{\unit{\mu m}}\xspace $ ) from the focus, but does not have an obvious effect on the intensity distribution in the plane of best focus. Even so, a typical trajectory in an aberratd trap such as the inset to Fig. 2(b) shows a slight diagonal elongation along that is not evident for acomatic traps.

Figure 2: Effect of coma. a) Intensity profiles of severely comatic optical tweezers ($ a_2 = 20$ ) at several heights relative to the focal plane. b) Average trap stiffness in the $ \hat{x}$ and $ \hat{y}$ directions of 16 holographic optical traps as a function of coma. The inset shows a single particle's trajectory in a comatic optical tweezer at $ a_2 = 20$ . Coma clearly distorts the intensity profile of optical vortices with topological charge $ \ell = 80$ : in c) $ a_2 = -20$ , d) $ a_2 = 0$ and e) $ a_2 = 20$ .
\begin{figure}\centering
\includegraphics[width=0.9\columnwidth]{comagraph4}
\end{figure}

To quantify the traps' performance, we model each as a harmonic potential well

$\displaystyle u(\vec{r}) = \frac{1}{2} \, \left(k_x x^2 + k_y y^2\right),$ (2)

characterized by distinct stiffnesses $ k_x$ and $ k_y$ along and transverse to the camera's scan direction, respectively. A measured trajectory $ \vec{r}(t) = (x(t),y(t))$ consisting of $ N$ time intervals $ \Delta t$ then yields the associated trap's spring constants $ k_i$ and the sphere's viscous drag coefficient $ \gamma_i$ as (10)

$\displaystyle k_i$ $\displaystyle = \frac{1}{c_{i,0}} + A_i \, \Delta r^2$ (3)
$\displaystyle \gamma_i$ $\displaystyle = - \frac{1}{c_{i,0}} \, \ln\left(\frac{c_{i,1}}{c_{i,0}}\right) + \left(A_i \, \ln \left(\frac{c_{i,1}}{c_{i,0}}\right) + B_i\right) \, \Delta r^2$ (4)

where

$\displaystyle A_i$ $\displaystyle = \frac{c_{i,0}^4 - 2 c_{i,0}^2 c_{i,1}^2 - 3 c_{i,1}^4 + 2 c_{i,0} c_{i,1}^2 c_{i,2}}{ (c_{i,0}^3 - c_{i,0} c_{i,1}^2)^2},$ (5)
$\displaystyle B_i$ $\displaystyle = \frac{c_{i,0}^2 - c_{i,1}^2 + c_{i,0} c_{i,2}}{c_{i,0}^2(c_{i,0}^2 - c_{i,1}^2)}$ (6)

and where

$\displaystyle c_{i,n} = \frac{1}{N-n}\sum_{j=0}^{N-1-n} r_i(j\Delta t) \, r_i((j+n)\Delta t)$ (7)

is the autocorrelation at lag $ n \Delta t$ of the trajectory's $ i$ -th component. Error estimates for $ k_i$ and $ \gamma_i$ also follow from the methods of Ref. (10). Equations (3) through (7) yield accurate results provided the sampling interval $ \Delta t$ is comparable to the viscous relaxation time $ \tau_i = \gamma_i/k_i$ . With the unaberrated traps powered by $ 6.25~\unit{mW}$ per trap, we obtain $ k_{0x} = 0.50 \pm 0.07~\unit{pN/\ensuremath{\unit{\mu m}}\xspace }$ and $ k_{0y} = 0.44 \pm 0.11~\unit{pN/\ensuremath{\unit{\mu m}}\xspace }$ and $ \gamma_0 = \gamma_x = \gamma_y = 1.51 \pm 0.03~\unit{pN \, s/\ensuremath{\unit{\mu m}}\xspace }$ . The variances in these results include the measured (10) $ \pm 10\%$ variation in the holographically projected traps' intensities and the rated $ 10\%$ polydispersity in the spheres' radii. This yields $ \tau = 0.93 \pm 0.06~\unit{ms} > \Delta t$ . In all cases, the thermally driven excursions at temperature $ T = 300\unit{^\circ K}$ exceed the variance due to measurement error, $ 2 k_B T/k_i > \Delta r^2$ , so that the approximations (10) used in deriving Eqs. (3) and (4) are satisfied. With these conditions satisfied, $ N = 1800$ video frames obtained over $ 60~\unit{s}$ suffice to measure the traps' stiffnesses to within a relative error of $ 8\%$ . As a final consistency check, we ensure that the spheres' viscous drag coefficients measured along different direction agree even if the traps themselves are anisotropic.

To best compile results from the entire array of traps, we normalize results for each trap by their values measured without imposed aberrations and then average over the array. The trapping light is linearly polarized, with its plane of polarization aligned with the $ \hat{y}$ direction to within $ 2^\circ$ . The direction of polarization could influence the traps' performance in our high-NA optical train. The potential wells' measured curvatures along the $ \hat{x}$ and $ \hat{y}$ directions, however, indistinguishable in the absence of applied aberrations. This is consistent with recent advances in the theory of optical trapping, which suggest that polarization effects in high-NA optical trapping systems only are important for smaller objects (18).

The results in Fig. 2(b) demonstrate that coma measurably degrades the traps' lateral stiffness despite its comparatively modest influence on the individual traps' intensity distributions. Coma's influence can be made more apparent by employing the SLM also as a mode converter. For example, imposing the phase profile $ \varphi_\ell(\vec{\rho}) = \ell \theta$ transforms a conventional point-like optical tweezer into a ring-like optical vortex (20,21,19) whose radius is determined (22,23) by the integer winding number $ \ell$ . In the absence of aberrations, a well-focused optical vortex appears as a uniformly bright ring of light. By distorting the helical phase profile that defines an optical vortex, geometric aberrations distort its shape and redistribute its intensity. The $ \cos\theta$ structure of coma, for instance, displaces an optical vortex's center away from the optical axis, while the two-fold modulation of astigmatism extends it into an elliptical shape (24). These effects are accentuated by these distortion's radial dependence. The comatic optical vortices in Fig. 2(c) and 2(e) therefore are not just displaced relative to the optical axis, but also are elongated. The example with no imposed distortions in Fig. 2(d) reflects the inherent aberrations in our optical train. The distortions in such images can serve as a metric for estimating the magnitude, $ a_j$ , and orientation, $ \theta_j$ of intrinsic geometric aberrations. Once measured, these can be removed by inverting the signs of the coefficients in Eqs. (1) and adding the resulting phase modulation to any trap-forming holograms.

A close examination of the data in Fig. 2(b) suggests that the traps' performance might be improved by deliberately adding a small amount of negative coma. This is consistent with the observed distortion of the optical vortex in Fig. 2(d).

Figure 3: Effect of astigmatism. a) The intensity profile of an astigmatic beam ($ a_3 = 20$ ) at ten axial positions relative to the focal plane. b) Average stiffness in the $ \hat{x}$ and $ \hat{y}$ directions of 16 holographic optical traps as a function of the imposed astigmatism. The inset shows a typical single-particle trajectory at $ a_3 = 20$ . The presence of astigmatism is readily discerned in the structure of optical vortices, as demonstrated for c) $ a_3 = -20$ , d) $ a_3 = 0$ and e) $ a_3 = 20$ .
\begin{figure}\centering
\includegraphics[width=.9\columnwidth]{astigraph3}\end{figure}

Astigmatism, which can result from having lenses canted in the optical train, has a more obvious effect on optical tweezers' structure and function. The cross-sections through an astigmatic trap in Fig. 3(a) demonstrate that the beam focuses along the $ \hat{x}$ and $ \hat{y}$ axes in different planes. The result is that a sphere encounters a substantially different lateral confining potential as it diffuses axially. This gives rise to the cross-shaped trajectory in the inset to Fig. 3(b). Although the derivation of Eq. (3) does not account for this complicated structure, the results obtained for $ k_x$ and $ k_y$ are consistent with astigmatism's degrading lateral trapping performance. Furthermore, because the spheres are displaced downstream by radiation pressure, the traps appear softer along the direction in which the downstream lobe is longer. This changes with the sign of the coefficient $ a_2$ , as shown by the data in Fig. 3(b).

The inherent astigmatism in our system can be estimated by locating the symmetry axis of Fig. 3(b), and is roughly $ a_2 = 5$ . This is consistent also with the images of astigmatic optical vortices in Figs. 3(c), (d) and (e). These clearly show biaxial extension due to astigmatism's two-fold modulation.

Figure 4: Effect of spherical aberration. Average trap stiffness in the $ \hat{x}$ and $ \hat{y}$ directions of 16 holographic optical traps as a function of the imposed spherical aberration.
\begin{figure}\centering
\includegraphics[width=.9\columnwidth]{sphabgraph2}
\end{figure}

The other primary geometric aberrations have more subtle effects on optical traps' efficacy and on optical vortices' structure. Spherical aberration, for example, is characterized by phase terms proportional to $ \rho^4$ and $ \rho^2$ . The quadratic term serves mostly to displace the trap's plane of best focus along the $ \hat{z}$ direction. To emphasize spherical aberration's effect on the optical tweezers' lateral stiffness, we imposed only the quartic term. This enables us to adjust the degree of aberration without shifting the focal plane, which frees our measurement from several possible artifacts including changes in the optical train's inherent spherical aberration (25,8) and changes in the spheres' hydrodynamic coupling to the wall (26). The data in Fig. 4 reveal that even large amounts of this quartic distortion have very little influence on the traps' in-plane performance. This contrasts with spherical aberration's demonstrated degradation of axial trapping stiffness (6,2,3,5,4). Even a well-aligned optical train can suffer from this defect if the sample's index of refraction differs from the design value. In this case, deliberately adding negative spherical aberration can restore the traps' performance (5,25,8), and might account for the slight improvement we observe.

Optimal statistical analysis of optically trapped particles' thermally driven trajectories provides rapid characterization of the traps' performance (10). When combined with direct visualization of the optical train's aberrations through the structure of projected optical vortices, this provides a basis for adaptively assessing and optimizing holographic optical trapping. Optimization takes the form of an estimate for the phase modulation required to correct wavefront distortions that can be added to trap-forming holograms. Although aberrations are known to substantially degrade axial trapping performance, we have demonstrated that their effect on lateral trapping is less pronounced, with astigmatism and coma apparently requiring the most attention.

This work was supported by the National Science Foundation through Grant Number DBI-0233971, through the Research Experiences for Undergraduates program of the NSF, and by the donors of the Petroleum Research Fund of the American Chemical Society.




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David G. Grier 2005-11-12