Optical potential energy landscapes created using acousto-optical deflectors are characterized via solvent-driven colloidal particles. The full potential energy of both single optical traps and complex landscapes composed of multiple overlapping traps are determined using a simple force balance argument. The potential of a single trap is shown to be well described by a Gaussian trap with stiffness found to be consistent with those obtained by a thermal equilibrium method. We also obtain directly the depth of the well, which (as with stiffness) varies with laser power. Finally, various complex systems ranging from double-well potentials to random landscapes are generated from individually controlled optical traps. Predictions of these landscapes as a sum of single Gaussian wells are shown to be a good description of experimental results, offering the potential for fully controlled design of optical landscapes, constructed from single optical traps.
© 2012 OSA
Undulating potential energy landscapes appear in a vast range of physical systems, from the surfaces in atomic crystals  to Josephson junctions in superconductors . Modeling such landscapes has become an important challenge in science and engineering as many devices and electronic systems become smaller and their features become more difficult to probe directly . Model potential energy landscapes have been created with a wide range of techniques varying from directly structuring surfaces using lithographic techniques [4–6] to selectively adsorbing nanoparticles to surfaces .
The discovery of optical trapping of small particles using tightly focused laser light by Ashkin  opened the field of optical manipulation, offering the possibility of generating potential energy landscapes using light. Optical landscapes have proved very versatile, addressing problems ranging from particle sorting [9–12], phase behaviour  and Kramer’s hopping [14–17] to nanotribology [18, 19]. For all of these applications a detailed knowledge of the potential energy landscape is crucial. In laser based particle sorting techniques for instance, the efficiency depends sensitively on the deflection angle, whose optimum may be predicted from the details of the landscapes [9–12].
Optical potential energy landscapes, U(x), are generally characterized as a function of position, x, by inverting the probability distribution, P(x), of Brownian particles in the landscape using Boltzmann’s factor P(x) ∝ exp [−βU(x)] [13, 15–17, 20, 21], where β = (kBT)−1 with kB Boltzmann’s constant and T the absolute temperature. However, with thermal fluctuations being the motive force this approach is typically limited to optical potentials with a depth, U0, of a few kBT. Also, characterizing single optical traps by this method leads to the restoring force, Fopt = −kx, of a harmonic optical potential [20, 21], where k is the trapping stiffness and x the displacement of a particle from the center of the trap. This linear regime is generally sufficient in the limit of small fluctuations, but breaks down for larger displacements [22–24]. This is particularly important for non-equilibrium optical trapping applications such as active micro-rheology [25,26]; during the dragging of a trapped particle though a viscoelastic medium the particle is likely to sample the non-linear regions of an optical trap.
To characterize the full potential energy of single optical traps and deeper potential energy landscapes with wells of tens of kBT, driving forces up to typically a pico-Newton are required. Piezoelectric stages [22, 23] and a dual-beam optical trap  have been used to drive single trapped particles away from the trap center, and a stiffening of the optical trap has been observed. In the case of extended potential energy landscapes Arzola et. al. reported a study of a particle on an inclined plane, passing an optical washboard potential created by interference fringes . This approach uses various particles on the order of 10 μm in diameter, such that gravity is a sufficient driving force for particles to escape individual traps. Further studies report particle sorting using the solvent driven flow of colloidal particles over optical landscapes created using interference patterns , spatial light modulators (SLMs) [9, 12, 28, 29] and acousto-optical deflectors (AODs) .
Here, we combine acousto-optically generated ‘one dimensional’ (1D) optical potential energy landscapes with colloidal particles driven using a well-defined solvent flow to quantitatively characterize the full optical potential energy landscape. The particles are driven along an axis through the centers of the 2D circularly symmetric optical traps, resulting in a 1D potential measurement. The solvent flow gives a driving force up to tens of pico-Newtons, which, based on a simple force balance [12, 27], allows us to simultaneously measure the trapping stiffness, k, and the depth, U0, of the traps constituting the optical landscape. Compared to other driving methods based on piezoelectric stages [22, 23] or dual optical trap arrangements , our method is cheap and fast, and in addition flowing multiple single particles through the landscape allows us to gain good statistics.
We find that the magnitude of the trapping stiffness is linearly proportional to the laser power, I0, consistent with thermal equilibrium methods, although these methods are limited to relatively weak traps . The depth of the optical trap, which is not accessible using thermal equilibrium methods, is also found to be proportional to I0[12, 16]. Finally we extend our measurements to more complex, periodic and aperiodic potentials, and show that we can fully control, engineer and predict the exact nature of optical potential energy landscapes.
The paper is organized as follows: in section 2 the theoretical basis of this work is laid out, followed by a description of the experimental methods in section 3. Section 4 discusses the results from experiments on single traps (section 4.1) and landscapes consisting of multiple traps (section 4.2). We make our conclusions in section 5.
2. Dynamics of particles driven over optical traps
Particles driven by solvent flowing at velocity, vflow, at low Reynolds numbers (Re) and high Péclet numbers (Pe), experience a driving force Fflow = ξvflow, where ξ = 6πηR is the friction coefficient of a spherical particle of radius R in a solvent of viscosity η. Re is a ratio of inertial to viscous forces, so particles in this low Re system are governed by the viscosity of the solvent, while Pe is a ratio of advection and diffusion in a fluid, so particles in this high Pe system are governed by the overall motion of the fluid. The presence of an optical potential energy landscape introduces an additional force, Fopt (x), which depends on the position coordinate, x, and causes perturbations to the velocity of the particle. A particle’s velocity, v(x), under the influence of both of these effects is described by,32]. The optical landscape produced through the combination of N single optical traps may be described by a sum of the potentials of the individual traps, j: Eq. (1) to obtain the instantaneous velocity of a particle traveling across the potential energy landscape: 30, 33].
3. Experimental methods
3.1. Colloidal model system and flow
The flow is generated by a micro-fluidics pump (Harvard PHD 2000 with milled glass syringes), connected via micro-fluidics tubing to a quartz glass sample cell (Hellma) with a 200 μm vertical cross section (see Fig. 1(c)). The cell is cleaned between fills using 2% Hellmanex solution, rinsing with 20% EtOHaq, and plasma treatment. The particles used are Dynabeads® M-270 Carboxylic Acid, diameter σ = 2.8μm. These particles are highly monodisperse crosslinked polystyrene spheres with CO2H surface groups, making the particles negatively charged, and causing them to repel the negatively charged glass wall of the sample cell, preventing sticking. Because the particles have a high density, 1.6g cm−3, and consequently small gravitational height, ( , where m* is the buoyant mass of the particle and g is gravitational field strength), the system is considered to be two-dimensional. Hence, in the presence of flow the particles follow a well defined one dimensional trajectory. We use very dilute suspensions of particles and consider a single particle driven through the potential.
3.2. Optical setup
The one-dimensional optical potential energy landscapes are generated by acousto-optically controlled optical laser tweezers . Figure 1 illustrates the optical setup used in this work. A 1064 nm laser beam (Coherent Compass) is expanded by two lenses (T1, T2), reflected by a mirror (M1), and directed through a pair of perpendicular acousto-optical deflectors (AODs, AA Opto-electronics). The AODs produce four output beams, of which one, the (1,1) order, is selected for further use, as it is deflected both horizontally and vertically. The beam is guided through the telescope optics, lenses T3 and T4, and redirected (M2 – M5) into the back aperture of a vertically positioned tweezing objective (Leica, 50x, NA=0.55), which focuses the beam to a waist of ∼ 2μm. The optical traps are controlled via the AODs using Aresis software .
Illumination of the sample is provided by a optical fiber light source (Thorlabs OSL1), also focused through the tweezing objective. The light transmitted through the sample cell is then refocused by the imaging objective (40x, NA=0.50) onto a PixeLINK CMOS camera (PL-B741U). A bandpass filter (BPF) is used to block the laser light, in order to protect the camera. Movies recorded are decomposed into individual frames using ImageJ, which are then analyzed using IDL particle tracking routines adapted from Crocker, Grier and Weeks [35, 36].
3.3. Flow measurements
Flow rate is chosen as the minimum sufficient to enable the particle to escape from the trap. The stability of the flow is crucial, and flow fluctuations have been confirmed to be negligible compared to than those resulting from Brownian motion. Images of typically 1024 pixels × 200 pixels are recorded at 50–80 fps using the video capture mode of PixeLINK capture OEM.
3.4. Equilibrium trap measurements
One of the traditional methods for finding the stiffness constant of an optical trap involves taking a long time series of a Brownian particle held by an optical trap . The images obtained may then be analyzed by fitting a parabolic potential to a plot of probability against position. This method was used to compare to the data obtained from the flow method. To this end, a particle was placed in the optical trap previously used for the flow experiments, and images recorded at 2 fps for up to an hour.
4. Results and discussion
4.1. Single optical traps
Firstly, we drive the colloidal particles at ∼ 15μm s−1 along a linear path, running through the center of a single optical trap. Figure 2a shows the position of the particle as a function of time as obtained from the tracking routines. The early and late parts of the position versus time graph are linear and of the same gradient, indicating the particle traveling at the imposed flow velocity far away from the optical trap. When the particle travels through the optical trap its velocity is affected and a characteristic position-time pattern is observed, indicated in Fig. 2a by the markers A to D. The corresponding velocity versus position graph is shown in Fig. 2b. Initially the particle proceeds at the constant velocity of the solvent flow, until point A, when there is a period of acceleration due to gradient forces of the optical trap. The velocity then rapidly drops to its minimum at C, where the gradient force now opposes the flow forces. Beyond this point, the particles accelerates to reach the flow velocity at D.
Assuming a Gaussian potential for the optical trap (Eq. (2)), we fit the experimental data according to Eq. (4) to obtain the complete optical potential, fully characterised by the trapping stiffness, k, and the depth of the optical trap, U0. The solid curve in Fig. 2b, v(x), shows that this assumption is valid, and that the region of influence of the trap is around 10μm. We also show U(x) in Figs. 2a and 2b for completeness. The velocity profile in Fig. 2b is also a direct measure for the force profile (see Eq. (1)) and can be directly compared to earlier work which reported a stiffening of the potential at larger displacements [22–24]. Our velocity profile appears very similar to the force profile reported in [23, 24], although the uncertainty in our data does not allow us to resolve the expected stiffening of the trap, which would appear between points B and C as a deviation from the derivative of a Gaussian potential.
We next determine the laser power dependence of k, which is presented in Fig. 3a. As expected k increases linearly with laser power consistent with earlier reports, e.g. [21, 37]. Also, the k values from flow measurements match well with those from the thermal equilibrium method , suggesting that our flow technique is indeed an effective, easy and fast way to obtain this information. The depth, U0, of the optical trap, can only be determined from the full optical force field as shown in Fig. 2b[23, 24]. Our measurements show that U0 is linearly proportional to the laser power for depths up to 200 kBT, consistent with theoretical predictions [12, 38]. So far this has only been experimentally demonstrated for potential energy landscapes with maximum energy wells on the order of a few kBT using the thermal equilibrium method [13, 15–17].
4.2. Optical potential energy landscapes
We now extend our measurements to 1D potential energy landscapes generated by multiple overlapping single optical traps. This introduces additional contributions to the optical force exerted on the particle. Completely analogously to the single trap experiments, we use the velocity profile v(x) to reconstruct the potential energy landscape using Eq. (4). We will show that once the trap stiffness kj and well depth U0,j of the single optical traps are known, optical landscapes can be fully engineered and described by a simple sum of single Gaussian potentials as given by Eq. (3) .
First we consider a simple ‘extended’ landscape consisting of two pairs of identical optical traps. Figure 4a shows the velocity profile v(x) of a particle driven at a flow velocity of ∼ 11μm s−1 through four optical traps separated by distances δ12 = δ34 = 3μm and δ23 = 10μm. This arrangement is well described by four Gaussian wells (Eq. (4)), exhibiting additional small peaks in between the two traps constituting the pairs, which clearly illustrates the additivity of the potentials. Fitting may be achieved using the same parameters for each trap or independent parameters: we found the results from both methods to be within 3.5% of each other. The good correlation of the reconstructed potential energy landscape to that predicted by Eq. (3) (Fig. 4a, top panel) shows that the velocity profile in Fig. 4a corresponds to two double-well potentials separated by δ23 + 0.5 (δ12 + δ34) = 13μm.
Next we use multiple overlapping optical traps to produce periodic 1D potential energy landscapes. Figure 4b shows the result of driving a particle through four identical optical traps with a spacing of δ = 3μm. The excellent fit to Eq. (4) further corroborates the applicability of the superposition of Gaussian traps (Eq. (4)). The corresponding potential energy landscape is again in good agreement with Eq. (3) (Fig. 4b, top panel). The sinusoidal character of the landscape is clearly observed in the velocity profile, and is made evident by the fit in Fig. 4b. The period and amplitude of the sinusoidal landscape may now be tuned via the spacing δ and the laser power, which in turn controls k and U0. Note that the mean velocity of the sinusoidal fit is slightly lower than the fit based on a sum of Gaussian potentials, reflecting the fact that the particles are slightly closer to the wall due to the increased radiation pressure from overlapping traps, and consequently feel a slightly higher friction coefficient .
Finally, we generate two types of random potential energy landscapes consisting of overlapping optical traps with randomly generated positions and/or intensities (Fig. 5). In Fig. 5a we show a landscape of 6 identical but randomly spaced traps, and in Fig. 5b a landscape of 4 traps with random spacing, k and U0. Again the good fits and close correlation of theoretical and subsequently generated experimental potentials confirm that in principle once simple optical traps are fully characterized, any sort of optical landscape can be generated in a fully predictable manner. Experimental characterization in 2D, particularly of anisotropic traps, would however pose more of a challenge, because the particle can avoid steep increases in the potential.
We have generated 1D optical potential energy landscapes using optical traps controlled by acousto-optical deflectors. Colloidal particles are driven through the optical landscapes by a well defined solvent flow. The full optical potential energy of both single optical traps and complex landscapes composed of multiple overlapping traps is determined by a simple force balance argument. Our flow based method is simple and straightforward to implement, allowing for fast and reliable characterizations of optical landscapes.
We find that the potential of a single optical trap is Gaussian, although noise in the experimental data may obscure previously reported stiffening of the optical trap. The magnitude of the trap stiffness and its linear dependence on laser power is found to be consistent with those found from thermal equilibrium methods. We also directly obtain the depth of the optical trap, which is shown to be linear with the laser power. Subsequently, various optical landscapes ranging from double-well potentials to random landscapes are generated from individually controlled optical traps. The corresponding energy landscapes as obtained from the velocity profiles are well described by a sum of single Gaussian optical traps. This in principle allows the fully controlled engineering of any sort of optical potential energy landscape, and we believe that our technique will be of interest for applications in micro-fluidic devices.
We thank the group of Clemens Bechinger (University of Stuttgart, Germany) for the design of the optical trapping setup. Thomas Skinner, Arran Curran and Samantha Ivell are thanked for useful discussions and critically reading the manuscript. This work was funded by the John Fell Fund and EPSRC.
References and links
1. E. Gnecco, R. Bennewitz, T. Gyalog, Ch. Loppacher, M. Bammerlin, E. Meyer, and H.-J. Güntherodt, “Velocity Dependence of Atomic Friction,” Phys. Rev. Lett. 84, 1172–1175 (2000). [CrossRef] [PubMed]
2. K. E. Kürten and C. Krattenthaler, “Multistability and multi 2π-kinks in the Frenkel-Kontorova model: an application to arrays of Josephson Junctions,” Int. J. Mod. Phys. B 21, 2324–2334 (2007). [CrossRef]
3. V. K. Vlasko-Vlasov, A. E. Koshelev, U. Welp, W. Kwok, A. Rydh, G. W. Crabtree, and K. Kadowaki, “Magneto-optical imaging of Josephson vortices in layered superconductors,” in Magneto-Optical Imaging, T. H. Johansed and D. V. Shantsev, eds. (Springer, 2004), pp. 39–46. [CrossRef]
4. B. Michel, A. Bernard, A. Bietsch, E. Delamarche, M. Geissler, D. Juncker, H. Kind, J.-P. Renault, H. Rothuizen, H. Schmid, P. Schmidt-Winkel, R. Stutz, and H. Wolf, “Printing meets lithography: Soft approaches to high-resolution patterning,” IBM J. Res. & Dev. 45, 697–719 (2001). [CrossRef]
6. G. Zhang and D. Wang, “Colloidal Lithography - The Art of Nanochemical Patterning,” Chem. Asian J. 4, 236–245 (2009). [CrossRef]
7. A. J. O’Reilly, C. Francis, and N. J. Quitoriano, “Gold nanoparticle deposition on Si by destabilising gold colloid with HF,” J. Colloid Interf. Sci. 370, 46–50 (2012). [CrossRef]
8. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970). [CrossRef]
11. G. Milne, D. Rhodes, M. MacDonald, and K. Dholakia, “Fractionation of polydisperse colloid with acousto-optical generated potential energy landscapes,” Opt. Lett. 32, 1144–1146 (2007). [CrossRef] [PubMed]
12. K. Xiao and D.G. Grier, “Sorting colloidal particles into multiple channels with optical forces: Prismatic optical fractionation,” Phys. Rev. E 82, 051407 (2010). [CrossRef]
14. D. Babic, C. Schmitt, I. Poberaj, and C. Bechinger, “Stochastic resonance in colloidal systems,” Europhys. Lett. 67, 158–164 (2004). [CrossRef]
15. C. Schmitt, B. Dybiec, P. Hänggi, and C. Bechinger, “Stochastic resonance vs. resonant activation,” Europhys. Lett. 74, 937–943 (2006). [CrossRef]
16. R. D. L. Hanes, C. Dalle-Ferier, M. Schmiedeberg, M. C. Jenkins, and S.U. Egelhaaf, “Colloids in one dimensional random energy landscapes,” Soft Matter 8, 2714–2723 (2012). [CrossRef]
17. A. Curran, M. P. Lee, R. Di Leonardo, J. M. Cooper, and M. J. Padgett, “Partial synchronization of stochastic oscillators through hydrodynamic coupleing,” Phys. Rev. Lett. 108, 240601 (2012). [CrossRef] [PubMed]
18. T. Bohlein, J. Mikhael, and C. Bechinger, “Observation of kinks and antikinks in colloidal monolayers driven across ordered surfaces,” Nat. Materials 11, 126–130 (2012). [CrossRef]
19. T. Bohlein and C. Bechinger, “Experimental observation of directional locking and dynamical ordering of colloidal monolayers driven across quasiperiodic substrates,” Phys. Rev. Lett. 109, 058301 (2012). [CrossRef] [PubMed]
20. K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787 (2004). [CrossRef]
21. J. Dobnikar, M. Brunner, H. H. von Grünberg, and C. Bechinger, “Three-body interactions in colloidal systems,” Phys. Rev. E 69, 031402 (2004). [CrossRef]
25. R. R. Brau, J. M. Ferrer, H. Lee, C. E. Castro, B. K. Tam, P. B. Tarsa, P. Matsudaira, M. C. Boyce, R. D. Kamm, and M. J. Lang, “Passive and active microrheology with optical tweezers,” J. Opt. A: Pure Appl. Opt. 9, S103–S112 (2007). [CrossRef]
26. M. Tassieri, G. M. Gibson, R. M. L. Evans, A. M. Yao, R. Warren, M. J. Padgett, and J. M. Cooper, “Measuring storage and loss moduli using optical tweezers: Broadband microrheology,” Phys. Rev. E 81, 026308 (2012). [CrossRef]
28. K. Ladavac, K. Kasza, and D. G. Grier, “Sorting mesoscopic objects with periodic potential landscapes: Optical fractionation,” Phys. Rev. E 70, 010901 (2004). [CrossRef]
30. J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Kluwer Academic Publishers, Dordrecht, The Netherlands, 1983).
31. A. V. Straube, A. A. Louis, J. Baumgartl, C. Bechinger, and R. P. A. Dullens, “Pattern formation in colloidal explosions,”Europhys. Lett. 94, 48008 (2011). [CrossRef]
32. T. Tlusty, A. Meller, and R. Bar-Ziv, “Optical Gradient Forces of Strongly Localized Fields,” Phys. Rev. Lett. 81, 1738–1741 (1998). [CrossRef]
33. J. Leach, H. Mushfique, S. Keen, R. Di Leonardo, G. Ruocco, J. M. Cooper, and M. J. Padgett, “Comparison of Faxns correction for a microsphere translating or rotating near a surface,” Phys. Rev. E 79, 026301 (2009). [CrossRef]
34. Aresis d.o.o., Aresis beam steering controller and Tweez software (2007).
35. J. C. Crocker and D. G. Grier, “Methods of Digital Video Microscopy for Colloidal Studies,” J. Colloid Interface Sci. 179, 298–310 (1996). [CrossRef]
36. E. Weeks, Particle tracking using IDL, http://www.physics.emory.edu/weeks/idl/.
38. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. 124, 529–541 (1996). [CrossRef]